principles of sonar performance modeling 2010

Principles of Sonar Performance Modeling

Michael A. Ainslie

Principles of SonarPerformance Modeling

Published in association with

PPraxisraxis PPublishingublishingChichester, UK

Dr Michael A. AinslieTNO, Sonar DepartmentThe HagueThe Netherlands

SPRINGER–PRAXIS BOOKS IN GEOPHYSICAL SCIENCESSUBJECT ADVISORY EDITOR: Philippe Blondel, C.Geol., F.G.S., Ph.D., M.Sc., F.I.O.A., Senior Scientist,Department of Physics, University of Bath, Bath, UK

ISBN 978-3-540-87661-8 e-ISBN 978-3-540-87662-5DOI 10.1007/978-3-540-87662-5

Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2010921914

# Springer-Verlag Berlin Heidelberg 2010

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv

PART I FOUNDATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 What is sonar? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Purpose, scope, and intended readership . . . . . . . . . . . . . . . . 41.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Part I: Foundations (Chapters 1–3) . . . . . . . . . . . . . . 61.3.2 Part II: The four pillars (Chapters 4–7) . . . . . . . . . . . 61.3.3 Part III: Towards applications (Chapters 8–11) . . . . . . 71.3.4 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 A brief history of sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.1 Conception and birth of sonar (–1918) . . . . . . . . . . . . 81.4.2 Sonar in its infancy (1918–1939) . . . . . . . . . . . . . . . . 151.4.3 Sonar comes of age (1939–) . . . . . . . . . . . . . . . . . . . 171.4.4 Swords to ploughshares . . . . . . . . . . . . . . . . . . . . . . 22

1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Essential background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1 Essentials of sonar oceanography . . . . . . . . . . . . . . . . . . . . . 27

2.1.1 Acoustical properties of seawater . . . . . . . . . . . . . . . 282.1.2 Acoustical properties of air . . . . . . . . . . . . . . . . . . . 30

2.2 Essentials of underwater acoustics. . . . . . . . . . . . . . . . . . . . . 302.2.1 What is sound? . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.2 Radiation of sound . . . . . . . . . . . . . . . . . . . . . . . . 312.2.3 Scattering of sound . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3 Essentials of sonar signal processing . . . . . . . . . . . . . . . . . . 422.3.1 Temporal filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.2 Spatial filter (beamformer) . . . . . . . . . . . . . . . . . . . . 44

2.4 Essentials of detection theory . . . . . . . . . . . . . . . . . . . . . . . 472.4.1 Gaussian distribution . . . . . . . . . . . . . . . . . . . . . . . 472.4.2 Other distributions . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 The sonar equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1.1 Objectives of sonar performance modeling . . . . . . . . . . 533.1.2 Concepts of ‘‘signal’’ and ‘‘noise’’ . . . . . . . . . . . . . . . 543.1.3 Generic deep-water scenario . . . . . . . . . . . . . . . . . . . 553.1.4 Chapter organization . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Passive sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.2 Definition of standard terms (passive sonar) . . . . . . . . . 583.2.3 Coherent processing: narrowband passive sonar . . . . . . 643.2.4 Incoherent processing: broadband passive sonar . . . . . . 80

3.3 Active sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.3.2 Definition of standard terms (active sonar) . . . . . . . . . 953.3.3 Coherent processing: CW pulseþDoppler filter. . . . . . . 993.3.4 Incoherent processing: CW pulseþ energy detector . . . . 112

3.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

PART II THE FOUR PILLARS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4 Sonar oceanography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.1 Properties of the ocean volume . . . . . . . . . . . . . . . . . . . . . . 126

4.1.1 Terrestrial and universal constants . . . . . . . . . . . . . . . 1264.1.2 Bathymetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.1.3 Factors affecting sound speed and attenuation in pure

seawater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.1.4 Speed of sound in pure seawater . . . . . . . . . . . . . . . . 1394.1.5 Attenuation of sound in pure seawater . . . . . . . . . . . . 146

4.2 Properties of bubbles and marine life . . . . . . . . . . . . . . . . . . 1484.2.1 Properties of air bubbles in water . . . . . . . . . . . . . . . 1484.2.2 Properties of marine life . . . . . . . . . . . . . . . . . . . . . 152

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4.3 Properties of the sea surface . . . . . . . . . . . . . . . . . . . . . . . . 1594.3.1 Effect of wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.3.2 Surface roughness . . . . . . . . . . . . . . . . . . . . . . . . . 1664.3.3 Wind-generated bubbles . . . . . . . . . . . . . . . . . . . . . 169

4.4 Properties of the seabed . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.4.1 Unconsolidated sediments . . . . . . . . . . . . . . . . . . . . 1724.4.2 Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.4.3 Geoacoustic models . . . . . . . . . . . . . . . . . . . . . . . . 183

4.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5 Underwater acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.2 The wave equations for fluid and solid media . . . . . . . . . . . . . 192

5.2.1 Compressional waves in a fluid medium . . . . . . . . . . . 1925.2.2 Compressional waves and shear waves in a solid medium 194

5.3 Reflection of plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 1975.3.1 Reflection from and transmission through a simple fluid–

fluid or fluid–solid boundary . . . . . . . . . . . . . . . . . . 1985.3.2 Reflection from a layered fluid boundary . . . . . . . . . . 2015.3.3 Reflection from a layered solid boundary . . . . . . . . . . 2045.3.4 Reflection from a perfectly reflecting rough surface . . . . 2055.3.5 Reflection from a partially reflecting rough surface . . . . 208

5.4 Scattering of plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 2095.4.1 Scattering cross-sections and the far field . . . . . . . . . . 2095.4.2 Backscattering from solid objects . . . . . . . . . . . . . . . 2105.4.3 Backscattering from fluid objects . . . . . . . . . . . . . . . . 2145.4.4 Scattering from rough boundaries . . . . . . . . . . . . . . . 223

5.5 Dispersion in the presence of impurities . . . . . . . . . . . . . . . . . 2255.5.1 Wood’s model for sediments in dilute suspension . . . . . 2255.5.2 Buckingham’s model for saturated sediments with inter-

granular contact . . . . . . . . . . . . . . . . . . . . . . . . . . 2265.5.3 Effect of bubbles or bladdered fish . . . . . . . . . . . . . . 227

5.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

6 Sonar signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2516.1 Processing gain for passive sonar . . . . . . . . . . . . . . . . . . . . . 252

6.1.1 Beam patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2526.1.2 Directivity index . . . . . . . . . . . . . . . . . . . . . . . . . . 2666.1.3 Array gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2716.1.4 BB application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2786.1.5 Time domain processing . . . . . . . . . . . . . . . . . . . . . 279

6.2 Processing gain for active sonar . . . . . . . . . . . . . . . . . . . . . . 2796.2.1 Signal carrier and envelope . . . . . . . . . . . . . . . . . . . 2806.2.2 Simple envelopes and their spectra . . . . . . . . . . . . . . 282

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6.2.3 Autocorrelation and cross-correlation functions and thematched filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

6.2.4 Ambiguity function . . . . . . . . . . . . . . . . . . . . . . . . 3006.2.5 Matched filter gain for perfect replica . . . . . . . . . . . . 3066.2.6 Matched filter gain for imperfect replica (coherence loss) 3076.2.7 Array gain and total processing gain (active sonar) . . . 308

6.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

7 Statistical detection theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3117.1 Single known pulse in Gaussian noise, coherent processing . . . . 312

7.1.1 False alarm probability for Gaussian-distributed noise . 3127.1.2 Detection probability for signal with random phase . . . 3137.1.3 Detection threshold . . . . . . . . . . . . . . . . . . . . . . . . 3267.1.4 Application to other waveforms . . . . . . . . . . . . . . . . 327

7.2 Multiple known pulses in Gaussian noise, incoherent processing 3277.2.1 False alarm probability for Rayleigh-distributed noise

amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3287.2.2 Detection probability for incoherently processed pulse

train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3297.3 Application to sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

7.3.1 Active sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3447.3.2 Passive sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3447.3.3 Decision strategies and the detection threshold . . . . . . 346

7.4 Multiple looks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3487.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3487.4.2 AND and OR operations . . . . . . . . . . . . . . . . . . . . 3507.4.3 Multiple OR operations . . . . . . . . . . . . . . . . . . . . . 3547.4.4 ‘‘M out of N ’’ operations . . . . . . . . . . . . . . . . . . . . 356

7.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

PART III TOWARDS APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . 359

8 Sources and scatterers of sound . . . . . . . . . . . . . . . . . . . . . . . . . . 3618.1 Reflection and scattering from ocean boundaries . . . . . . . . . . . 361

8.1.1 Reflection from the sea surface . . . . . . . . . . . . . . . . . 3628.1.2 Scattering from the sea surface . . . . . . . . . . . . . . . . . 3698.1.3 Reflection from the seabed . . . . . . . . . . . . . . . . . . . 3758.1.4 Scattering from the seabed . . . . . . . . . . . . . . . . . . . 391

8.2 Target strength, volume backscattering strength, and volumeattenuation coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3998.2.1 Target strength of point-like scatterers . . . . . . . . . . . . 4008.2.2 Volume backscattering strength and attenuation coeffi-

cient of distributed scatterers . . . . . . . . . . . . . . . . . . 4098.2.3 Column strength and wake strength of extended volume

scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

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8.3 Sources of underwater sound . . . . . . . . . . . . . . . . . . . . . . . 4148.3.1 Shipping source spectrum level measurements . . . . . . . 4178.3.2 Distributed sources on the sea surface . . . . . . . . . . . . 4248.3.3 Distributed sources on the seabed (crustacea) . . . . . . . 429

8.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

9 Propagation of underwater sound. . . . . . . . . . . . . . . . . . . . . . . . . . 4399.1 Propagation loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

9.1.1 Effect of the seabed in isovelocity water . . . . . . . . . . . 4409.1.2 Effect of a sound speed profile . . . . . . . . . . . . . . . . . 459

9.2 Noise level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4839.2.1 Deep water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4849.2.2 Shallow water . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4899.2.3 Noise maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

9.3 Signal level (active sonar) . . . . . . . . . . . . . . . . . . . . . . . . . 4919.3.1 The reciprocity principle . . . . . . . . . . . . . . . . . . . . . 4929.3.2 Calculation of echo level . . . . . . . . . . . . . . . . . . . . . 4939.3.3 V-duct propagation (isovelocity case) . . . . . . . . . . . . . 4949.3.4 U-duct propagation (linear profile) . . . . . . . . . . . . . . 494

9.4 Reverberation level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4959.4.1 Isovelocity water . . . . . . . . . . . . . . . . . . . . . . . . . . 4979.4.2 Effect of refraction . . . . . . . . . . . . . . . . . . . . . . . . . 500

9.5 Signal-to-reverberation ratio (active sonar) . . . . . . . . . . . . . . 5089.5.1 V-duct (isovelocity case) . . . . . . . . . . . . . . . . . . . . . 5089.5.2 U-duct (linear profile) . . . . . . . . . . . . . . . . . . . . . . . 509

9.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

10 Transmitter and receiver characteristics. . . . . . . . . . . . . . . . . . . . . . 51310.1 Transmitter characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 514

10.1.1 Of man-made systems . . . . . . . . . . . . . . . . . . . . . . . 51510.1.2 Of marine mammals . . . . . . . . . . . . . . . . . . . . . . . . 542

10.2 Receiver characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54510.2.1 Of man-made sonar . . . . . . . . . . . . . . . . . . . . . . . . 54510.2.2 Of marine mammals, amphibians, human divers, and fish 549

10.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

11 The sonar equations revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57311.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57311.2 Passive sonar with coherent processing: tonal detector . . . . . . . 574

11.2.1 Sonar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 57411.2.2 Source level (SL) . . . . . . . . . . . . . . . . . . . . . . . . . . 57511.2.3 Narrowband propagation loss (PL) . . . . . . . . . . . . . . 57611.2.4 Noise spectrum level (NLf ) . . . . . . . . . . . . . . . . . . . 57811.2.5 Bandwidth (BW) . . . . . . . . . . . . . . . . . . . . . . . . . . 57911.2.6 Array gain (AG) and directivity index (DI) . . . . . . . . . 580

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11.2.7 Detection threshold (DT) . . . . . . . . . . . . . . . . . . . . . 58111.2.8 Worked example . . . . . . . . . . . . . . . . . . . . . . . . . . 583

11.3 Passive sonar with incoherent processing: energy detector . . . . . 59111.3.1 Sonar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 59111.3.2 Source level (SL) . . . . . . . . . . . . . . . . . . . . . . . . . . 59211.3.3 Broadband propagation loss (PL) . . . . . . . . . . . . . . . 59211.3.4 Broadband noise level (NL) . . . . . . . . . . . . . . . . . . . 59311.3.5 Processing gain (PG) . . . . . . . . . . . . . . . . . . . . . . . 59311.3.6 Broadband detection threshold (DT) . . . . . . . . . . . . . 59711.3.7 Worked example . . . . . . . . . . . . . . . . . . . . . . . . . . 599

11.4 Active sonar with coherent processing: matched filter . . . . . . . 60611.4.1 Sonar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 60611.4.2 Echo level (EL), target strength (TS), and equivalent

target strength (TSeq) . . . . . . . . . . . . . . . . . . . . . . . . 60711.4.3 Background level (BL) . . . . . . . . . . . . . . . . . . . . . . . 61011.4.4 Processing gain (PG) . . . . . . . . . . . . . . . . . . . . . . . 61011.4.5 Detection threshold (DT) . . . . . . . . . . . . . . . . . . . . . 61211.4.6 Worked example . . . . . . . . . . . . . . . . . . . . . . . . . . 613

11.5 The future of sonar performance modeling . . . . . . . . . . . . . . 63011.5.1 Advances in signal processing and oceanographic model-

ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63011.5.2 Autonomous platforms . . . . . . . . . . . . . . . . . . . . . . 63111.5.3 Environmental impact of anthropogenic sound . . . . . . . 631

11.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

A Special functions and mathematical operations . . . . . . . . . . . . . . . . . 635A.1 Definitions and basic properties of special functions . . . . . . . . 635

A.1.1 Heaviside step function, sign function, and rectanglefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

A.1.2 Sine cardinal and sinh cardinal functions . . . . . . . . . . 636A.1.3 Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . . 636A.1.4 Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 636A.1.5 Error function, complementary error function, and right-

tail probability function . . . . . . . . . . . . . . . . . . . . . 637A.1.6 Exponential integrals and related functions . . . . . . . . . 639A.1.7 Gamma function and incomplete gamma functions . . . . 640A.1.8 Marcum Q functions . . . . . . . . . . . . . . . . . . . . . . . . 644A.1.9 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 644A.1.10 Bessel and related functions . . . . . . . . . . . . . . . . . . . 645A.1.11 Hypergeometric functions . . . . . . . . . . . . . . . . . . . . 648

A.2 Fourier transforms and related integrals . . . . . . . . . . . . . . . . 649A.2.1 Forward and inverse Fourier transforms . . . . . . . . . . 649A.2.2 Cross-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 650

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A.2.3 Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651A.2.4 Discrete Fourier transform . . . . . . . . . . . . . . . . . . . 651A.2.5 Plancherel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 652

A.3 Stationary phase method for evaluation of integrals . . . . . . . . 652A.3.1 Stationary phase approximation . . . . . . . . . . . . . . . . 652A.3.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

A.4 Solution to quadratic, cubic, and quartic equations . . . . . . . . . 655A.4.1 Quadratic equation . . . . . . . . . . . . . . . . . . . . . . . . 655A.4.2 Cubic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 655A.4.3 Quartic and higher order equations . . . . . . . . . . . . . . 656

A.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656

B Units and nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659B.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659

B.1.1 SI units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659B.1.2 Non-SI units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659B.1.3 Logarithmic units . . . . . . . . . . . . . . . . . . . . . . . . . 659

B.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665B.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665B.2.3 Names of fish and marine mammals . . . . . . . . . . . . . 666

B.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

C Fish and their swimbladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673C.1 Tables of fish and bladder types . . . . . . . . . . . . . . . . . . . . . 673C.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

Contents xi

To Anna

Preface

The science of sonar performance modeling is traditionally separated into a ‘‘wet end’’comprising the disciplines of acoustics and oceanography and a ‘‘dry end’’ of signalprocessing and detection theory. This book is my attempt to bring both aspectstogether to serve as a modern reference for today’s sonar performance modeler,whether for research, design, or analysis, as Urick’s Principles of Underwater Sounddid for sonar engineers of his day. The similarity in the title is no accident.

During the process I made some valuable discoveries that I now share with thereader. The radar literature provides a deep mine of resources, with applicable resultsfrom the theories of wave propagation, signal processing, and (an especially rich vein,largely unexploited in the sonar literature) statistical detection. From oceanographywe learn that each of the world’s oceans has its own unique physical, chemical, andbiological signature, with sometimes profound consequences for sonar.

Marine mammals have evolved a sonar of their own, the remarkable properties ofwhich we are only beginning to unravel, as reported in the increasingly sophisticatedbioacoustics literature. Governments and industry around the world have begun totake seriously the environmental consequences of man’s use, whether deliberate orincidental, of sound in the sea. I have done my best to provide a representativesnapshot of this rapidly developing field.

Some readers will treat this book as a repository of facts, figures, and formulas,while others will seek in it explanations and clarity. It has been my intention to satisfythe needs of both types of reader by including mathematical derivations and workedexamples, supplemented with measurements or estimates of relevant input param-eters. Of all readers I request the patience to overlook the flaws that undoubtedlyremain, despite my best attempts to weed them out.

Michael A. AinslieTNO, The Hague, The Netherlands, March 2010

Foreword

Underwater acoustics is largely a branch of physics, perhaps merging with geophysicsand oceanography, but as soon as one attempts to assess a sonar’s performance underrealistic conditions, a host of other engineering factors come into play. Is the desiredtarget signal louder than all the other natural noise from wind, waves, ship engines,strumming cables? Is it louder than sound scattered from other distant objects? Howdo the standard signal-processing techniques such as beamforming, spectral analysis,and statistical analysis influence the probability of achieving a target detection and theprobability of a false alarm?

The author, Dr. Mike Ainslie, is a physicist with a considerable academicpublication record and many years’ hands-on experience in sonar assessment forthe U.K.’s MOD and for TNO in The Netherlands. Through a firm foundation inphysics, always taking great care over the physical units, Principles of Sonar Per-formance Modeling introduces rigor and clarity into the traditional sonar equationwhile still answering the fundamental engineering questions. As well as dealing withthe more pure disciplines of sound generation, propagation, and reverberation, ittackles sound sources, targets, signal processing, and detection theory for man-madeand biological sonar.

Underlying all this is a desire ‘‘to see the wood for the trees’’. For instance, it isoften the case with propagation that, despite all the complexities of refraction,reflection, diffraction, scattering, and so on, some simple mechanism dominates,and sometimes one can express the entire transmission loss, ambient noise level,or reverberation level by a simple formula. This insight, or even revelation, is animportant bonus and check if one is to have faith in numerical assessment ofcomplicated search scenarios. It can also become a useful shortcut when a particularscenario is to be investigated under many different acoustic, or processing, conditions.Examples of such insights will be found throughout.

The cornerstone is the derivation of the sonar equations—too often presented asindisputable fact—from simple physical principles. The derivation is presented

initially in terms of ratios of simple physical quantities, and converted to decibels onlyat the end. Such an approach provides both clarity and a systematic rationale fordetermining how to evaluate each sonar equation term, and occasionally throws upunexpected new corrections.

The book will provide a useful reference for acousticians, engineers, physicists,mathematicians, sonar designers, and naval sonar operators whether working inresearch labs, the defense industry, or universities.

Chris HarrisonNATO Undersea Research Centre (NURC), Italy, March 2010

xvi Foreword

Acknowledgments

The eight years it has taken me to write this book were spent working at TNO inThe Hague. It has been a pleasure and a privilege to do so. The Sonar Department,despite two changes of name and two changes of leadership in that time, has providedconstant support and understanding for the necessary extra-curricular activities.I wish to thank all at TNO—too many to mention all by name—who helped to makeit possible.

I thank D. A. Abraham, P. Blondel, D. M. F. Chapman, P. H. Dahl,C. A. F. de Jong, P. A. M. de Theije, D. D. Ellis, R. M. Hamson, C. H. Harrison,J. A. Harrison, R. A. Hazelwood, D. V. Holliday, T. G. Leighton, A. J. Robins,S. P. Robinson, C. A.M. vanMoll, K. L.Williams,M. Zampolli, and two anonymousreferees, all of whom reviewed at least one complete chapter and helped to improve thequality of the final product. Any remaining errors that find their way into print areentirely mine and not of the reviewers.

Through his written publications, DavidWeston is an eternal inspiration—I havelost count of the number of times his name is cited. I also benefited from discussionswith Chris Harrison, Chris Morfey, Christ de Jong, Dale Ellis, Frans-Peter Lam,Mario Zampolli, Peter Dahl, and Tim Leighton.

Data or artwork were made available to me by Pascal de Theije (Figure 7.6),Peter Dahl (Figure 8.3), Alvin Robins (Figure 8.5), Vincent van Leijen (Figure 8.13),Peter van Holstein (Figure 8.14), Henry Dol (Figures 9.24 and 9.25), Mathieu Colin(all figures in Chapter 9 making use of either BELLHOP or SCOOTER),Robbert van Vossen (Figures 9.28 and 9.29), Wim Verboom (miscellaneous sealand porpoise audiograms), Garth Mix (thumbnail images of marine mammals),and Paul Wensveen (Figure 11.20).

The computer model INSIGHT (version 1.4.2) was used, with permission ofCORDA Ltd., to illustrate many of the sonar performance calculations. Also usedwere the acoustic propagation models SCOOTER and BELLHOP from the OceanAcoustics Library (http://oalib.hlsresearch.com). Other valuable Internet resources

include FishBase (www.fishbase.org), the Ocean Biogeographic Information System(www.iobis.org), Mathworld (http://mathworld.wolfram.com) and Wikipedia (www.wikipedia.org).

Phillipe Blondel and Clive Horwood were always available when needed foradvice. Neil Shuttlewood is responsible for a professional end-product.

Last but not least, none of this would have been possible without theunquestioning love and support from my wife Pilar and patience of my daughterAnna, whose teenage years are forever tinted with shades of sonar performance.

Michael A. AinslieTNO, The Hague, The Netherlands, March 2010

xviii Acknowledgments

Figures

1.1 Sketch of Beudant’s experiment of ca. 1816 . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Sketch of the Colladon–Sturm experiment of 1826 . . . . . . . . . . . . . . . . . . . 9

1.3 Inventor Reginald Fessenden and physicist Jean Daniel Colladon . . . . . . . . 9

1.4 Physicists Paul Langevin and Robert William Boyle . . . . . . . . . . . . . . . . . . 11

1.5 French statesman and mathematician Paul Painleve . . . . . . . . . . . . . . . . . . 13

1.6 Installation of early U.S. passive-ranging sonar with two towed eels . . . . . . 15

1.7 Sound absorption vs. frequency in seawater . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Attenuation coefficient and audibility vs. frequency in seawater . . . . . . . . . . 30

2.2 Radiation from a point source of power W in free space . . . . . . . . . . . . . . 33

2.3 Radiation from a point source in the presence of a reflecting boundary . . . . 35

2.4 Radiation from a sheet source element of width �r . . . . . . . . . . . . . . . . . . . 38

2.5 Beam patterns for L=� ¼ 5 and steering angles 0, 45 deg. . . . . . . . . . . . . . . 46

2.6 Probability density functions of noise and signal-plus-noise observables . . . 50

3.1 Principles of passive detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Spectral density level of the radiated power at the source and intensity at the

receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Spectral density level of the transmitter source factor and mean square pressure

at the receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 Coherent propagation loss vs. range and target depth . . . . . . . . . . . . . . . . . 66

3.5 Spectral density level of background noise . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6 Spectral density level of signal and noise . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.7 ROC curves for a Rayleigh-distributed signal in Rayleigh noise. . . . . . . . . . 72

3.8 Propagation loss and figure of merit vs. target range . . . . . . . . . . . . . . . . . 76

3.9 Signal level vs. target range, and in-beam noise level . . . . . . . . . . . . . . . . . 77

3.10 Linear signal excess and twice detection probability vs. range for NB passive

sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.11 Signal excess vs. target range and depth . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.12 Spectral density level of the transmitter source factor and mean square pressure

at the receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.13 Propagation loss vs. frequency and target range . . . . . . . . . . . . . . . . . . . . . 83

3.14 Spectral density level of signal and noise . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.15 ROC curves for a BB signal in Rayleigh noise . . . . . . . . . . . . . . . . . . . . . . 86

3.16 Propagation loss and figure of merit vs. range . . . . . . . . . . . . . . . . . . . . . . 91

3.17 Signal spectrum level vs. range, and in-beam noise spectrum level . . . . . . . . 92

3.18 Linear signal excess and twice detection probability vs. range for BB passive

sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.19 Propagation loss vs. range and depth for the BB passive worked example . . 94

3.20 Principles of active detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.21 Propagation loss and figure of merit vs. target range at fixed array depth and vs.

array depth for fixed range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.22 Signal level and in-beam noise level vs. target range at fixed array depth and vs.


3.23 Linear signal excess and twice detection probability for coherent CW active

sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.24 Signal excess vs. target range and array depth . . . . . . . . . . . . . . . . . . . . . . 111

3.25 Signal and (in-beam) background levels vs. target range at fixed array depth and

vs. array depth for fixed range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.26 Total background, background components, and in-beam background level vs.

target range at fixed array depth and vs. array depth for fixed range . . . . . . 119

3.27 Propagation loss and figure of merit vs. target range at fixed array depth and vs.


3.28 Linear signal excess and twice detection probability for incoherent CW active

sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.1 Global bathymetry map derived from satellite measurements of the gravity field 127

4.2 Annual average temperature map at depth 3 km. . . . . . . . . . . . . . . . . . . . . 129

4.3 Geographical variations in surface temperature for northern winter and

northern summer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.4 Temperature profiles for locations in the northwest Pacific Ocean and northeast

Atlantic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.5 Bathymetry map for the northwest Pacific Ocean . . . . . . . . . . . . . . . . . . . . 132

4.6 Bathymetry map for the north Atlantic Ocean . . . . . . . . . . . . . . . . . . . . . . 132

4.7 Annual average salinity map at depth 3 km . . . . . . . . . . . . . . . . . . . . . . . . 133

4.8 Temperature salinity diagram for the World Ocean . . . . . . . . . . . . . . . . . . 134

4.9 Seasonal variations in surface salinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.10 Salinity profiles for locations in the northwest Pacific Ocean and northeast

Atlantic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4.11 Density profiles for locations in the northwest Pacific Ocean and northeast

Atlantic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.12 Global acidity (K) contours at sea surface and at depth 1 km . . . . . . . . . . . 140

4.13 Arctic acidity (K) contours at the sea surface and at depth 1 km . . . . . . . . . 142

4.14 Acidity (K) profiles for major oceans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.15 Sound speed profiles for locations in the northwest Pacific Ocean and northeast

Atlantic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.16 Seawater attenuation coefficient vs. frequency . . . . . . . . . . . . . . . . . . . . . . 149

4.17 Fractional sensitivity of seawater attenuation to temperature, salinity, acidity,

and depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.18 Geographical distribution of herring and Norway pout in the North Sea . . . 160

4.19 Wind speed scaling factors to convert from a 20m reference height to the

standard reference height of 10m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

xx Figures

4.20 Near-surface bubble population density spectra . . . . . . . . . . . . . . . . . . . . . 170

4.21 Compressional and shear speed vs. density of rocks . . . . . . . . . . . . . . . . . . 181

4.22 Compressional and shear speeds vs. density for all rocks and for basalts . . . 183

5.1 Illustration of compressional and shear wave propagation. . . . . . . . . . . . . . 195

5.2 Fluid sediment layer between two uniform half-spaces . . . . . . . . . . . . . . . . 202

5.3 Form function j f ðkaÞj vs. ka for a rigid sphere, a tungsten carbide sphere, and

spheres made of various metals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

5.4 Resonance frequency vs. bubble radius for air bubbles in water. . . . . . . . . . 238

5.5 Resonant bubble radius vs. frequency for air bubbles in water. . . . . . . . . . . 241

6.1 Sinc beam patterns for steering angles 0, 30, 60, and 90 deg . . . . . . . . . . . . 254

6.2 Beam patterns for continuous line array: cosine and Hann shading . . . . . . . 258

6.3 Beam patterns for continuous line array: raised cosine shading . . . . . . . . . . 260

6.4 Hamming family shading patterns and beam patterns. . . . . . . . . . . . . . . . . 262

6.5 Beam pattern of unshaded circular array . . . . . . . . . . . . . . . . . . . . . . . . . . 265

6.6 Directivity index for an unsteered continuous line array vs. normalized array

length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

6.7 Directivity index vs. steering angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

6.8 Shading factor vs. steering angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

6.9 Power spectrum for a Gaussian LFM pulse . . . . . . . . . . . . . . . . . . . . . . . . 288

6.10 Power spectrum for a rectangular LFM pulse . . . . . . . . . . . . . . . . . . . . . . 289

6.11 Generic ambiguity surface for Gaussian CW pulse . . . . . . . . . . . . . . . . . . . 302

6.12 Ambiguity surfaces for Gaussian CW pulses of duration 0.5 s and 2.0 s . . . . 303

6.13 Generic ambiguity surfaces for Gaussian LFM pulse . . . . . . . . . . . . . . . . . 305

7.1 ROC curves for non-fluctuating amplitude signal in Rayleigh noise . . . . . . . 315

7.2 Rayleigh, one-dominant-plus-Rayleigh, Dirac, and Rice probability distribu-

tion functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

7.3 ROC curves for Rayleigh-fading signal in Rayleigh noise . . . . . . . . . . . . . . 319

7.4 ROC curves for Rician fading signal in Rayleigh noise . . . . . . . . . . . . . . . . 321

7.5 Rice probability density functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

7.6 ROC curves for 1DþR signal in Rayleigh noise . . . . . . . . . . . . . . . . . . . . 324

7.7 Graph of xðMÞ vs. M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

7.8 ROC curves (Albersheim approximation) for a non-fluctuating amplitude

signal: variation of detection threshold with M for fixed pfa . . . . . . . . . . . . 332

7.9 ROC curves (Albersheim approximation) for a non-fluctuating amplitude

signal: variation of detection threshold with pfa for fixed M . . . . . . . . . . . . 333

7.10 ROC curves for a non-fluctuating amplitude signal: incoherent addition with

M ¼ 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

7.11 ROC curves for a non-fluctuating amplitude signal: incoherent addition with

M ¼ 1 to M ¼ 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

7.12 ROC curves for a broadband signal: limit of large M . . . . . . . . . . . . . . . . . 338

7.13 Supplementary ROC curves for a broadband non-fluctuating signal. . . . . . . 339

7.14 ROC curves for a Rayleigh fading signal: incoherent addition with M ¼ 30 . 341

7.15 ROC curves for a 1DþR signal: incoherent addition with M ¼ 30 . . . . . . . 343

7.16 Fusion gain vs. pfa for OR operation (fixed pd) . . . . . . . . . . . . . . . . . . . . . 352

7.17 Fusion gain vs. F for OR operation (fixed D) . . . . . . . . . . . . . . . . . . . . . . 353

7.18 ROC curves for a non-fluctuating signal: effect of AND and OR fusion. . . . 354

7.19 ROC curves for a 1DþR signal: effect of AND and OR fusion . . . . . . . . . 355

7.20 ROC curves for a Rayleigh-fading signal: effect of AND and OR fusion . . . 356

8.1 Variation of surface reflection loss with wind speed (1–4 kHz) . . . . . . . . . . . 366

Figures xxi

8.2 Surface reflection loss in nepers calculated vs. angle and frequency . . . . . . . 368

8.3 Surface reflection loss vs. wind speed (30 kHz) . . . . . . . . . . . . . . . . . . . . . . 369

8.4 Seabed reflection loss vs. grazing angle for uniform unconsolidated sediments 376

8.5 Seabed reflection loss vs. angle and frequency–sediment thickness product for a

layered unconsolidated sediment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

8.6 Seabed reflection loss vs. angle for rocks . . . . . . . . . . . . . . . . . . . . . . . . . . 385


sand sediment overlying a granite basement and clay over basalt . . . . . . . . . 387


sand sediment of thickness 10m overlying a granite basement and a clay

sediment of thickness 300m over basalt. . . . . . . . . . . . . . . . . . . . . . . . . . . 390

8.9 Seabed backscattering strength for a medium sand sediment and frequency

1–30 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

8.10 Comparison between predicted and measured seabed backscattering strength

for a fine sand sediment and frequency 35 kHz. . . . . . . . . . . . . . . . . . . . . . 394

8.11 Seabed backscattering strength for a coarse clay sediment. . . . . . . . . . . . . . 395

8.12 Comparison between predicted and measured seabed backscattering strength

for a medium silt sediment and frequency 20 kHz. . . . . . . . . . . . . . . . . . . . 396

8.13 Typical ambient noise spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

8.14 Typical values of sound pressure level and peak pressure level. . . . . . . . . . . 418

8.15 Measured equivalent source spectral density levels: commercial and industrial

shipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

8.16 Estimated third-octave monopole source level: cargo ship Overseas Harriette 423

8.17 Areic dipole source spectrum: wind noise . . . . . . . . . . . . . . . . . . . . . . . . . 426

8.18 Areic dipole source spectrum: rain noise . . . . . . . . . . . . . . . . . . . . . . . . . . 428

8.19 Measured waveform and frequency spectrum of a single shrimp snap . . . . . 430

9.1 Geometry for bottom reflections in deep water . . . . . . . . . . . . . . . . . . . . . 441

9.2 Propagation loss vs. range for reflecting seabed at f ¼ 250Hz . . . . . . . . . . . 442

9.3 Bottom-refracted ray paths travel through the sediment and form a caustic in

the reflected field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

9.4 Propagation loss vs. range for a reflecting and refracting seabed at f ¼ 250Hz 446

9.5 Propagation loss vs. range for a reflecting and refracting seabed: sensitivity to

sediment properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

9.6 Reflection loss vs. angle for sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

9.7 Propagation loss vs. range, and reflection loss vs. angle for sand and mud in

shallow water at frequency 250Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

9.8 Sound speed profile in the northwest Pacific . . . . . . . . . . . . . . . . . . . . . . . 460

9.9 Propagation loss vs. range for northwest Pacific summer and winter at

f ¼ 1500Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

9.10 Propagation loss vs. range and depth for northwest Pacific winter profile: effect

of upward refraction in surface duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

9.11 Depth factor vs. receiver depth in surface duct . . . . . . . . . . . . . . . . . . . . . . 469

9.12 Ray trace illustrating formation of caustics and cusps in surface duct up to a

range of 40 km, for a source depth of 30m, and for the same case as Figure 9.10 470

9.13 Propagation loss vs. frequency and range for a surface duct . . . . . . . . . . . . 473

9.14 Ray trace illustrating the formation of convergence zones at the sea surface 475

9.15 Propagation loss vs. range and depth: effect of downward refraction on Lloyd

mirror interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

xxii Figures

9.16 Propagation loss vs. range for shallow water with a mud bottom for two

different sound speed profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

9.17 Approximation to D�=� for fixed �min . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

9.18 Predicted deep ocean noise spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

9.19 Sensitivity of deep-water ambient noise spectra to rain rate. . . . . . . . . . . . . 486

9.20 Sensitivity of deep-water noise spectra to wind speed . . . . . . . . . . . . . . . . . 487

9.21 Predicted ambient noise spectral density level vs. frequency and depth . . . . . 488

9.22 Effect of the seabed on the ambient noise spectrum in isovelocity water . . . . 489

9.23 Effect of the sound speed profile on the ambient noise spectrum for a clay seabed 490

9.24 Dredger noise map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

9.25 Bathymetry used for Figure 9.24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

9.26 Reverberation for problem RMW11 and frequency 3.5 kHz . . . . . . . . . . . . 499

9.27 Reverberation depth factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

9.28 Reverberation for problem RMW12 and frequency 3.5 kHz . . . . . . . . . . . . 504

9.29 Reverberation for problem RMW12 and frequency 3.5 kHz (close-up) . . . . . 505

9.30 Ray trace illustrating formation of caustics and cusps in a bottom duct, and

propagation loss vs. range and depth at f ¼ 3.5 kHz. . . . . . . . . . . . . . . . . . 506

9.31 SRR depth factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

10.1 Maximum multibeam echo sounder and sidescan sonar source levels vs.

transmitter frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

10.2 Unweighted and Gaussian-weighted cosine pulses from Table 10.16 . . . . . . 530

10.3 Exponentially damped sine and decaying exponential pulses from Table 10.17 532

10.4 Mean square pressure vs. energy fraction. . . . . . . . . . . . . . . . . . . . . . . . . . 535

10.5 Comparison of echolocation pulses made by the harbor porpoise and killer

whale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

10.6 Underwater audiograms for harbor porpoise . . . . . . . . . . . . . . . . . . . . . . . 551

10.7 Underwater audiograms for killer whale . . . . . . . . . . . . . . . . . . . . . . . . . . 552

10.8 Underwater audiograms for harbor seal . . . . . . . . . . . . . . . . . . . . . . . . . . 554

10.9 Underwater audiograms for human divers . . . . . . . . . . . . . . . . . . . . . . . . . 556

10.10 Underwater sound level weighting curves for three groups of cetaceans plus

pinnipeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

11.1 Directivity index DI ¼ 10 log10 GD for an unsteered continuous line array vs.

normalized array length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

11.2 ROC curves for 1DþR amplitude signal in Rayleigh noise . . . . . . . . . . . . 582

11.3 Propagation loss vs. range for NWP winter case . . . . . . . . . . . . . . . . . . . . 584

11.4 In-beam signal and noise levels vs. range for NWP winter and Chapter 3 NBp

worked example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

11.5 Input parameters for northwest Pacific (NWP) problem . . . . . . . . . . . . . . . 588

11.6 Signal excess vs. range and depth for NWP winter . . . . . . . . . . . . . . . . . . . 589

11.7 Signal excess vs. range and depth for NWP winter: close-up of first convergence

zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

11.8 Albersheim’s approximation for the detection threshold . . . . . . . . . . . . . . . 598

11.9 Propagation loss vs. range and depth for SWS and for the Chapter 3 BBp

worked example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

11.10 Signal and noise spectra for SWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

11.11 In-beam signal and noise levels vs. range for SWS and BBp . . . . . . . . . . . . 602

11.12 Signal excess vs. range and depth for SWS . . . . . . . . . . . . . . . . . . . . . . . . 603

11.13 Input parameters for shallow-water sand (SWS). . . . . . . . . . . . . . . . . . . . . 604

11.14 In-beam signal and noise spectra for SWS . . . . . . . . . . . . . . . . . . . . . . . . . 605

Figures xxiii

11.15 Signal excess vs. range and rainfall rate for SWS . . . . . . . . . . . . . . . . . . . . 606

11.16 Geometry for worked example involving killer whale hunting salmon . . . . . 614

11.17 Example measurements of orca pulse shapes and power spectra . . . . . . . . . 615

11.18 Variation in orca source level with distance from target . . . . . . . . . . . . . . . 617

11.19 Propagation loss vs. distance and broadband correction . . . . . . . . . . . . . . . 618

11.20 Orca audiogram and individual hearing threshold measurements . . . . . . . . . 620

11.21 Echo level and noise level vs. distance between orca and salmon: wind speed

2m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

11.22 Echo level and noise level vs. distance between orca and salmon: wind speed 2 to

10m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624

11.23 Background level vs. distance between orca and salmon: wind speed 10m/s . 628

11.24 Array gain vs. distance between orca and salmon: wind speed 10m/s . . . . . . 629

11.25 Signal and background levels vs. distance between orca and salmon: wind speed

10m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630

A.1 The complementary error function erfcðxÞ and three approximations . . . . . . 638

A.2 The gamma function and four approximations. . . . . . . . . . . . . . . . . . . . . . 643

A.3 The modified Bessel function and Levanon’s approximation . . . . . . . . . . . . 647

xxiv Figures

Tables

2.1 Detection truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1 Sonar equation calculation for NB passive example . . . . . . . . . . . . . . . . . . 76

3.2 Sonar equation calculation for BB passive example . . . . . . . . . . . . . . . . . . 90

3.3 Sonar equation calculation for CW active sonar example with Doppler filter 107

3.4 Sonar equation calculation for CW active sonar example with incoherent

energy detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.1 Average salinity and potential temperature by major ocean basin . . . . . . . . 133

4.2 Seawater parameters used for evaluation of attenuation curves plotted in Figure

4.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.3 Mass, length, and aspect ratio of selected sea mammals . . . . . . . . . . . . . . . 154

4.4 Volume and surface area of ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.5 Acoustical properties of fish flesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.6 Acoustical properties of whale tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4.7 Acoustical properties of euphausiids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4.8 Values of zooplankton density and sound speed ratios . . . . . . . . . . . . . . . . 157

4.9 North Sea fish population estimates by species. . . . . . . . . . . . . . . . . . . . . . 158

4.10 WMO Beaufort wind force scale and estimated wind speed. . . . . . . . . . . . . 162

4.11 Comparison of wind speed estimates for Beaufort force 1–11 based on WMO

code 1100 and CMM-IV with those of da Silva . . . . . . . . . . . . . . . . . . . . . 165

4.12 Definition of sea state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.13 Beaufort wind force: relationship between wind speed and wave height . . . . 168

4.14 Sea state: relationship between wave height and wind speed . . . . . . . . . . . . 168

4.15 Sediment type vs. grain diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.16 Definition of sediment grain sizes and qualitative descriptions . . . . . . . . . . . 174

4.17 Default HF geoacoustic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4.18 Default MF geoacoustic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.19 Names of sedimentary rocks resulting from the lithification of different

sediment types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4.20 Geoacoustic parameters for sedimentary and igneous rocks. . . . . . . . . . . . . 183

5.1 Compressional speed, shear speed, and density used to calculate the form

factors for the four metals shown in Figure 5.3 . . . . . . . . . . . . . . . . . . . . . 212

5.2 Backscattering cross-sections of large rigid objects . . . . . . . . . . . . . . . . . . . 213

5.3 Backscattering cross-sections of large fluid objects . . . . . . . . . . . . . . . . . . . 215

5.4 Water and solid grain sediment parameter values needed for Buckingham’s

grain-shearing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

5.5 Values of physical constants used for the evaluation of the bubble resonance

characteristics in Figures 5.4 and 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

6.1 Summary of properties for various taper functions . . . . . . . . . . . . . . . . . . . 264

6.2 Summary of beam properties for selected shading . . . . . . . . . . . . . . . . . . . 265

6.3 Summary of frequency domain properties of simple pulse envelopes . . . . . . 284

6.4 Summary of time domain properties of simple pulse shapes (envelope). . . . . 285

6.5 Summary of time domain properties of simple pulse shapes (phase) . . . . . . . 285

6.6 Summary of amplitude envelopes required to synthesize simple power spectra 291

6.7 Autocorrelation functions for CW and LFM pulses . . . . . . . . . . . . . . . . . . 296

6.8 Derivation of matched filter gain for pulse duration and sample interval . . . 306

6.9 Effect of multipath on matched filter gain . . . . . . . . . . . . . . . . . . . . . . . . . 308

7.1 Comparison table: moments of probability distribution functions . . . . . . . . 320

7.2 DTþ 5 log10 M vs. M and pfa for three different pd values . . . . . . . . . . . . . 334

7.3 Application of the detection theory results of Section 7.1 to active sonar CW

and FM pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

7.4 Equations for the detection probability for different signal amplitude distribu-

tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

7.5 Application of detection theory results to NB and BB passive sonar . . . . . . 346

7.6 Detection threshold for various statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 347

7.7 Detection threshold for a 1DþR amplitude distribution . . . . . . . . . . . . . . 348

7.8 ROC relationships and fusion gain for AND and OR operations for fixed SNR 353

8.1 Sediment properties at top and bottom of the transition layer . . . . . . . . . . . 381

8.2 p and s critical angles for representative rock parameters . . . . . . . . . . . . . . 385

8.3 Parameters for uniform fluid sediment and rock half-space . . . . . . . . . . . . . 388

8.4 Defining parameters for a layered solid medium. . . . . . . . . . . . . . . . . . . . . 389

8.5 Measurements of the Lambert parameter . . . . . . . . . . . . . . . . . . . . . . . . . 397

8.6 Target strength measurements for bladdered fish . . . . . . . . . . . . . . . . . . . . 401

8.7 Target strength measurements for whales . . . . . . . . . . . . . . . . . . . . . . . . . 403

8.8 Target strength measurements for euphausiids and bladder-less fish . . . . . . . 404

8.9 Target strength measurements for jellyfish . . . . . . . . . . . . . . . . . . . . . . . . . 407

8.10 Target strength measurements for siphonophores . . . . . . . . . . . . . . . . . . . . 407

8.11 Second World War measurements of the target strength of man-made objects 408

8.12 Predicted average night-time contribution to VBS, CS, and attenuation due to

pelagic fish in the North Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

8.13 Default advice for VBS for sparse, intermediate, and dense marine life . . . . 411

8.14 Wake strength measurements for various WW2 surface ships . . . . . . . . . . . 414

8.15 Wake strength for various WW2 submarines . . . . . . . . . . . . . . . . . . . . . . . 414

8.16 Third-octave source levels of various commercial and industrial vessels . . . . 421

9.1 Characteristic properties from Chapter 4 of medium sand and mud. . . . . . . 454

9.2 Sound speed profiles for the northwest Pacific location . . . . . . . . . . . . . . . . 461

9.3 Nomenclature used for shipping densities . . . . . . . . . . . . . . . . . . . . . . . . . 484

9.4 Seabed parameters for problems RMW11 and RMW12 . . . . . . . . . . . . . . . 500

xxvi Tables

9.5 Caustic ranges and corresponding two-way travel arrival times for a source at

depth 30m and receiver at depth 50m. . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

10.1 Source level of single-beam echo sounders . . . . . . . . . . . . . . . . . . . . . . . . . 516

10.2 Source level of sidescan sonar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

10.3 Source level of multibeam echo sounders. . . . . . . . . . . . . . . . . . . . . . . . . . 518

10.4 Source level of depth profilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

10.5 Source level of fisheries search sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

10.6 Source level of hull-mounted search sonar . . . . . . . . . . . . . . . . . . . . . . . . . 521

10.7 Source level of helicopter dipping sonar . . . . . . . . . . . . . . . . . . . . . . . . . . 521

10.8 Source level of active towed array sonar . . . . . . . . . . . . . . . . . . . . . . . . . . 522

10.9 Source level of miscellaneous search sonar (including coastguard and risk

mitigation sonar). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

10.10 Source level of low-amplitude acoustic deterrents . . . . . . . . . . . . . . . . . . . . 524

10.11 Source level of high-amplitude acoustic deterrents . . . . . . . . . . . . . . . . . . . 525

10.12 Source level of acoustic communications systems . . . . . . . . . . . . . . . . . . . . 526

10.13 Source level of selected acoustic transponders and alerts . . . . . . . . . . . . . . . 527

10.14 Source level of acoustic cameras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

10.15 Source level of miscellaneous oceanographic sonar . . . . . . . . . . . . . . . . . . . 528

10.16 Relationships between different source level definitions for two symmetrical

wave forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

10.17 Relationships between different source level definitions for two asymmetrical

wave forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

10.18 Relative MSP, averaged over time window during which local average exceeds

specified threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

10.19 Relative MSP, averaged over time window during which pulse energy

accumulates to specified proportion of total. . . . . . . . . . . . . . . . . . . . . . . . 534

10.20 Dipole source level of air guns and air gun arrays . . . . . . . . . . . . . . . . . . . 536

10.21 Zero-to-peak source level of generator–injector air guns . . . . . . . . . . . . . . . 537

10.22 Zero-to-peak source level of seismic survey sources other than air guns . . . . 538

10.23 Summary of peak pressure and pulse energy for three types of explosive . . . 540

10.24 Specific pulse energy and apparent specific SLE for pentolite . . . . . . . . . . . . 541

10.25 Echolocation pulse parameters for selected animals . . . . . . . . . . . . . . . . . . 543

10.26 Maximum peak-to-peak source levels of high-frequency marine mammal clicks 546

10.27 Peak equivalent RMS and peak-to-peak source levels of low-frequency marine

mammal pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

10.28 Hearing thresholds and sensitive frequency bands of selected cetaceans . . . . 553

10.29 MSP and EPWI hearing thresholds in air and water for four pinnipeds plus

human subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

10.30 Hearing thresholds in water for 10 species of fish . . . . . . . . . . . . . . . . . . . . 557

10.31 Parameters of bandpass filter used in M-weighting . . . . . . . . . . . . . . . . . . . 560

10.32 Genera represented by the functional hearing groups . . . . . . . . . . . . . . . . . 561

10.33 Proposed thresholds of M-weighted sound exposure level for permanent and

temporary auditory threshold shift in cetaceans and pinnipeds . . . . . . . . . . 562

10.34 Proposed thresholds of peak pressure for permanent and temporary auditory

threshold shift in cetaceans and pinnipeds . . . . . . . . . . . . . . . . . . . . . . . . . 563

10.35 Outline of the severity scale from Southall et al. (2007). . . . . . . . . . . . . . . . 564

10.36 Spread of sound pressure level values resulting in the specified behavioral

responses in cetaceans and pinnipeds for nonpulses . . . . . . . . . . . . . . . . . . . 564

Tables xxvii

10.37 Spread of sound pressure level values resulting in the specified behavioral

responses in cetaceans and pinnipeds for multiple pulses . . . . . . . . . . . . . . . 565

11.1 List of applications of man-made active and passive underwater acoustic

sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

11.2 Error in DT incurred by assuming 1DþR statistics . . . . . . . . . . . . . . . . . . 582

11.3 Sonar equation calculation for NWP winter . . . . . . . . . . . . . . . . . . . . . . . 587

11.4 Filter gain vs. bandwidth in octaves for a white signal and colored noise . . . 596

11.5 Sonar equation calculation for shallow-water sand . . . . . . . . . . . . . . . . . . . 604

11.6 Active sonar example, limited by hearing threshold . . . . . . . . . . . . . . . . . . 620

11.7 Active sonar example, limited by wind noise . . . . . . . . . . . . . . . . . . . . . . . 624

A.1 Integrals of integer powers of the sine cardinal function . . . . . . . . . . . . . . . 636

A.2 Selected values of the gamma function GðxÞ for 0 < x � 1 . . . . . . . . . . . . . 640

A.3 Examples of Fourier transform pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

B.1 SI prefixes for indices equal to an integer multiple of 3. . . . . . . . . . . . . . . . 660

B.2 SI prefixes for indices equal to an integer between þ3 and �3. . . . . . . . . . . 661

B.3 Frequently encountered non-SI units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662

B.4 List of abbreviations and acronyms, and their meanings . . . . . . . . . . . . . . . 667

C.1 Bladder presence and type key used in Tables C.3, C.4, and C.7 . . . . . . . . . 674

C.2 Reference key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674

C.3 Bladder type by order for ray-finned fishes (Actinopterygii) . . . . . . . . . . . . 675

C.4 Bladder type by family. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676

C.5 ‘‘Catchability’’ key (Yang groups) used in Table C.7 . . . . . . . . . . . . . . . . . 677

C.6 Length key used in Table C.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677

C.7 Fish and their bladders, sorted by scientific name. . . . . . . . . . . . . . . . . . . . 678

xxviii Tables

Part I

Foundations

1

Introduction

Wee represent Small Sounds as Great and Deepe; LikewiseGreat Sounds, Extenuate and Sharpe; Wee make diverse

Tremblings and Warblings of Sounds, which in their Originallare Entire. Wee represent and imitate all Articulate Sounds andLetters, and the Voices and Notes of Beasts and Birds. Wee havecertaine Helps, which sett to the Eare doe further the Hearinggreatly. Wee have also diverse Strange and Artificiall Eccho’s,Reflecting the Voice many times, and as it were Tossing it; And

some that give back the Voice Lowder then it came, someShriller, and Some Deeper; Yea some rendring the Voice,

Differing in the Letters or Articulate Sound, from that theyreceyve. Wee have also meanes to convey Sounds in Trunks and

Pipes in strange Lines, and Distances.

Francis Bacon (1624)

1.1 WHAT IS SONAR?

Sonar can be thought of as a kind of underwater radar, using sound instead of radiowaves to interrogate its surroundings. But what is special about sound in the sea?Radio waves travel unhindered in air, whereas sound energy is absorbed relativelyquickly. In water, the opposite is the case: low absorption and the presence of naturaloceanic waveguides combine to permit propagation of sound over thousands ofkilometers, whereas the sea is opaque to most of the electromagnetic spectrum.

The word sonar is an acronym for sound navigation and ranging. The primarypurpose of sonar is the detection or characterization (estimation of position, velocity,and identity) of submerged, floating, or buried objects. Electronic systems capable of

underwater detection and localization were developed in the 20th century, motivatedinitially by the sinking of RMS Titanic in 1912 and the First World War (WW1), andspurred on later by the Second World War (WW2) and the Cold War. Nevertheless,by comparison with marine fauna, man remains a novice user of underwater sound.Deprived of light in their natural habitat, dolphins have evolved a sophisticated formof sonar over millions of years, without which they would be almost blind. Theytransmit bursts of ultrasound, and sense the world around them by interpreting theechoes. Many fish and other aquatic animals are also capable of both producing andhearing sounds.

1.2 PURPOSE, SCOPE, AND INTENDED READERSHIP

This book is aimed at anyone, novice and experienced practitioner alike, with aninterest in estimating the performance of sonar, or understanding the conditions forwhich a particular existing or hypothetical system is likely to make a successfuldetection. This includes sonar analysts and designers, whether for oceanographicresearch, navigation, or search sonar. It also includes those studying the use of soundby marine mammals and the impact of exposure of these animals to sound. Regard-less of application, the objective of sonar performance modeling is usually to supporta decision-making process. In the case of man-made sonar, the decision is likely toinvolve the optimization of some aspect of the design, procurement, or use of sonar.(What frequency or bandwidth is appropriate? How many sonars are needed tocomplete the task in the time available?) For bio-sonar there is increasing interestin the assessment (and mitigation) of the risk of damage to marine life due toanthropogenic sources of underwater sound. (What level of sound might disrupt adolphin’s ability to locate and capture its prey? How can the risk of hearing damagebe prevented or minimized?)

The nature of the sought object, known as the sonar target, depends on theapplication. Examples include man-made objects of military interest (a mine orsubmarine), shipwrecks (as a navigation hazard or archeological artifact), and fish(the target of interest to a whale or fisherman).

In general, sonar can be grouped into two main categories. These are active sonarand passive sonar, which are distinguished by the presence and absence, respectively,of a sound transmitter as a component of the sonar system.

. An active sonar system comprises a transmitter and a receiver and works on theprinciple of echolocation. If a signal (in this case an echo from the target) isdetected, the position of the target can be estimated from the time delay anddirection of the echo. The echolocation principle is also used by radar, and by thebiological sonar of bats and dolphins.

. A passive sonar includes a receiver but no transmitter. The signal to be detected isthen the sound emitted by the target.

Examples of man-made sonar include

4 Introduction [Ch. 1

. Echo sounder: perhaps the most common of all man-made sonars, an echosounder is a device for measuring water depth by timing the delay of an echofrom the seabed. The strength and character of the echo can also provide anindication of bottom type.

. Fisheries sonar: sonar equipment used by the fisheries industry exploits the sameprinciple as the echo sounder, except that the purpose is to detect fish instead ofthe sea floor.

. Military sonar: modern navies deploy a wide variety of sonar systems, designed todetect and track potential military threats such as surface ships, submarines,mines, or torpedoes. The diverse nature of these threats and of the platformson which the sonar systems are mounted means that military sonars are them-selves diverse, with each specialized system dedicated to a particular task.

. Oceanographic sensor: scientific work aimed at understanding and surveying thesea (acoustical oceanography) makes extensive use of a variety of different kindsof sonar, many of which are variants of the echo sounder.

. Shadow sensor: in exceptional cases, the sonar ‘‘signal’’, instead of being thesound emitted or scattered by the target, might actually be some perturbationto the expected background. For example, the shadow of an object lying on theseabed might be detectable when the object itself is not.

Many readers will be familiar with Urick’s classic Principles of Underwater Sound forEngineers,1 which provided its readers with the tools they needed to carry out sonardesign and assessment studies. These tools come in the form of a set of equationsrelating the predicted signal-to-noise ratio to known parameters such as the radiatedpower of the sonar transmitter, or the size and shape of the target. This set ofequations is known as the ‘‘sonar equations’’. The same basic requirement remainstoday, but the modeling methods have increased in sophistication during the 25 yearsthat have elapsed since Urick’s third and final edition, with a bewildering array ofcomputer models to choose from (Etter, 2003). The present objective is to meet theneeds of the modern user or developer of such models by documenting establishedmethods and relevant research results, using internally consistent definitions andnotation throughout. The discipline of sonar performance modeling is perceivedsometimes as a black art. The purpose of this book is, above all, to demystify thisart by explaining the jargon and deriving the sonar equations from physical princi-ples.

The book’s scope includes underwater sound, the properties of the sea relevantto the generation and propagation of sound, and the processing that occurs afteran acoustic signal has been converted to an electrical one2 and then digitized.The estimation of sonar performance is taken as far as the detection (and falsealarm) probability, but no further than that. While the scope excludes localization,

1.2 Purpose, scope, and intended readership 5]Sec. 1.2

1 See Urick (1967) and two later editions (Urick, 1975, 1983).2 Conversion between electrical and acoustical energy (known as transduction), whether on

transmission or reception, is excluded from the scope. The interested reader is referred to Hunt

(1954) and Stansfield (1991).

classification, and tracking tasks, such as the estimation of position and velocity of asonar target, a satisfactory detection capability is a prerequisite for any of these.

1.3 STRUCTURE

Sonar performance modeling is a multidisciplinary science, requiring knowledge ofsubjects as diverse as mathematics, physics, electrical engineering, chemistry, geology,and biology.3 It is convenient to group the material into four foundation categories(or ‘‘pillars’’), on which the science of sonar performance modeling is built: sonaroceanography, underwater acoustics, sonar signal processing and statistical detectiontheory. The book has three main parts, described below.

1.3.1 Part I: Foundations (Chapters 1–3)

Part I comprises this Introduction and two further chapters, also of an introductorynature. The purpose of Chapter 2 is to describe the essential concepts required for abasic understanding of the sonar equations, which are derived in Chapter 3. Fourgeneric types of sonar are introduced, with a simple worked example provided foreach. The material in Chapters 2 and 3 is intended as a primer, to illustrate theprinciples, and generally preferring simplicity to realism. Advanced readers mightprefer to skip the introductory part and start reading from Chapter 4, consultingChapter 3 only for definitions.

1.3.2 Part II: The four pillars (Chapters 4–7)

Each of the four chapters in Part II is devoted to one of the four pillars. The one onoceanography (Chapter 4) describes the sea as a medium for sound propagation.Relevant properties of the oceans’ contents and boundaries are considered, suchas the geoacoustical properties of sediments and rocks, sea surface waveheightspectra, near-surface bubble density, and the acoustical properties of marine life.

The chapter on acoustics (Chapter 5) provides a theoretical foundation forunderstanding the behavior of sound in the sea, including reflection and scatteringfrom its contents and boundaries. Cumulative propagation effects associated withmultiple boundary reflections are the subject of Chapter 9.

An acoustic signal arriving at a sonar receiver is converted to an electrical signalby a device known as a ‘‘transducer’’. This electrical signal is subjected to a series ofoperations designed to determine the presence or otherwise of a sonar target. Theseoperations are known collectively as signal processing, which is the subject of Chapter6. The purpose of signal processing can be thought of as either to enhance the signalfrom the target or to reduce the background noise. These two points of view are


3 The reader is assumed to have completed a degree-level course in a numerate discipline such

as physics, applied mathematics, or engineering.

entirely equivalent, as in the end what matters is the ratio of signal power to noisepower.

Finally, to be of practical use, the output of the signal processing must beinterpreted by a decision-maker. The chance that a sonar operator correctly (orincorrectly) deduces that a target is present is known as the probability of detection(or false alarm). The quantitative study of detection and false alarm probabilities isknown as statistical detection theory, and this is the subject of Chapter 7.

1.3.3 Part III: Towards applications (Chapters 8–11)

The purpose of the final chapters is to show how to apply the principles from Parts Iand II to more realistic situations. Chapter 8 provides quantitative information aboutthe sources, reflectors, and scatterers of underwater sounds, while Chapter 9 describessound propagation in the sea and its impact on both the signal and background.Chapter 10 describes the characteristics of both man-made and biological sonar,including the sensitivity of marine animals to underwater sound.

Chapter 11 brings together information from all the preceding chapters andapplies it to a set of problems partly based on the worked examples of Chapter 3,introducing more advanced concepts and definitions where necessary. It closes with aspeculative account of possible future development of sonar performance modeling inthe 21st century.

1.3.4 Appendices

In addition to the 11 chapters, there are three appendices. Two of these provideinformation needed for the correct interpretation of the main text, describing specialfunctions and mathematical operations (Appendix A), and units and nomenclature(Appendix B). Finally, Appendix C can be thought of as an extension to Chapter 4. Itcontains information about fish and their swim bladders that will be of use to a readerinterested in the interaction of sound with fish or fish shoals.

1.4 A BRIEF HISTORY OF SONAR

The remainder of this Introduction is devoted to a historical account of the devel-opment of sonar. It is the author’s tribute to the work of Constantin Chilowski,4

Daniel Colladon, Pierre and Jacques Curie, Maurice Ewing, Reginald Fessenden,Harvey Hayes, Paul Langevin, H. Lichte, Leonard Liebermann, J. Marcum, StephenRice, and Albert Beaumont Wood. It owes its existence in no small part to thedetailed accounts of Hunt (1954), Wood (1965), and Hackmann (1984).

The history focuses on developments in France, Britain, and the U.S.A., as theseare the places where the main early advances took place, especially during WW1.Developments in Germany and the U.S.S.R. are mentioned only briefly, partly due to

1.4 A brief history of sonar 7]Sec. 1.4

4 Zhurkovich (2008) transcribes this name as ‘‘K.V. Shilovsky’’.

the difficulty in finding reliable sources for them (in the case of Russian and Sovietacoustics, corrected recently by the publication of the History of Russian UnderwaterAcoustics, edited by Godin and Palmer, 2008).

1.4.1 Conception and birth of sonar (–1918)

1.4.1.1 Discovery and ingenuity

The concept of echo ranging, by which the distance to an object is determined bymeasuring the time delay to an echo from that object, originates from at least as farback as the 17th century. More recent origins of sonar can be traced to two seeminglyunrelated scientific developments in the 19th century, the first being the measurementof the speed of sound in seawater, ca. 1816, by Francois Beudant, in the FrenchMediterranean. Beudant used a crude but effective method (illustrated in Figure 1.1),involving an underwater bell and a swimmer waving a flag. A more precise determina-tion, with improved light–sound synchronization (Figure 1.2), was made in 1826 byColladon (Figure 1.3) and Sturm, in Lake Geneva.5 Both measurements are describedby Colladon and Sturm (1827), and in both cases the values obtained (1,500m/s and


Figure 1.1. Sketch of Beudant’s experiment of ca. 1816 (reprinted fom Girard, 1877).

5 Their purpose was not to measure the speed of sound for its own sake, but to determine the

bulk modulus of water, which can be calculated from the sound speed if its density is known.


Figure 1.2. Sketch of the Colladon–Sturm experiment of 1826 (reprinted fom Girard, 1877).

Figure 1.3. Inventor Reginald Fessenden (left) and physicist Jean Daniel Colladon (right). The

image of Fessenden is reprinted from http://www.ieee.ca/millennium/radio/radio_unsung.html,

last accessed October 22, 2009, RadioScientist.#

1,435m/s) are consistent with modern expectation for the respective measurementconditions.

The second important development is the discovery of piezoelectricity by Pierreand Jacques Curie in 1880. Experiments with certain special dielectric crystals(especially quartz and Rochelle salt) revealed that these materials respond to anapplied pressure by developing a small potential difference. The converse effect,whereby an applied electric field distorts the shape of the crystal, was predictedshortly afterwards by Gabriel Lippmann and confirmed by the Curie brothers in1881.

In the late 1890s and early 1900s, some lightships were fitted with underwaterbells, which were rung to alert approaching vessels of danger in conditions of poorvisibility. In good visibility these sounds provided an indication of distance as well, byestimating the time delay between light and sound signals, as when estimating thedistance from an electrical storm by counting seconds to the thunder following a boltof lightning. These early underwater signaling systems would eventually mature intowhat we now call sonar.

1.4.1.2 The Titanic and the Fessenden oscillator

The tragic collision and subsequent sinking of RMS Titanic on the night of April 14/15, 1912 resulted in a flurry of activity and ideas directed at providing advancewarning of nearby icebergs. Lewis Richardson filed patents first for an airborneecholocation system in April 1912 and a month later for an underwater one. ReginaldFessenden (Figure 1.3) patented an electromagnetic transducer in 1913 and demon-strated its use by detecting the presence of an iceberg on April 27, 1914 at a distanceof ‘‘nearly two miles’’ (i.e., approximately 3–4 km). This device became known as theFessenden oscillator (Waller, 1989).

1.4.1.3 WW1: a sense of urgency

It took an even greater tragedy, the loss of life inflicted by U-boats during WW1, toprovide the focus of intellect and resources that would lead to the development of aworking underwater detection system. French and British efforts began in 1915, withPaul Langevin (Figure 1.4) working in Paris with Russian engineer ConstantinChilowski, while A. B. Wood worked with Harold Gerrard in Manchester. The focusof the French research was on echolocation (‘‘active sonar’’ in modern terminology),while the British team concentrated initially on listening devices known as hydro-phones (‘‘passive sonar’’).

At the outset of WW1, Lord Rutherford had assembled an extraordinary groupof physicists at his laboratory at the University of Manchester, including the house-hold names Bohr, Geiger, and Chadwick. In his autobiographical account, A. B.Wood recalls (Wood, 1965): ‘‘It would be difficult to find anywhere such a galaxy ofscientific talent, either before or since, working together in the same physics labora-tory at the same time.’’ Of particular relevance here are the arrivals of Wood himselfin 1915 and of the Canadian physicist Robert Boyle (Figure 1.4) the following year.

The Board of Invention and Research (BIR) was established in 1915, with


facilities at Hawkcraig (in Fifeshire, Scotland), and expanded in 1917 to a team ofmore than 80 scientists and technicians working at Parkeston Quay (Harwich,England) under the leadership of Professor W. H. Bragg. Amongst them were Boyleand Wood from Rutherford’s group, responsible, respectively, for research investi-gating echolocation and passive listening.

Boyle made promising initial progress with the Fessenden oscillator, such that bylate 1917 a submarine detection had been reported at a distance of 1,000 yd (910m)(Hackmann, 1984, p. 75).6 Nevertheless, this line of work was abandoned because thefrequency of Fessenden’s transmitter (1 kHz) was too low to obtain the necessaryresolution in bearing for its intended purpose of locating submarines. A high-frequency transducer was needed to achieve this.

In France, Langevin had begun to experiment with quartz early in 1917 afterobtaining a small supply from a Paris optician. Quartz is a piezoelectric materialsuitable for the radiation of high-frequency sound,7 but the unamplified received


Figure 1.4. Physicists Paul Langevin (left) and Robert William Boyle (right). The image of

Langevin is reprinted from Anon. (wp, a) and that of Boyle from http://www.100years.ualberta.

ca, last accessed October 26, 2009.

6 The yard (symbol yd) is a unit of length defined as 0.9144 meters (see Appendix B).7 Use here of the term ‘‘sound’’ is not restricted to the audible frequency range, but refers also

to ‘‘ultrasound’’, which means that the frequency is above the upper limit of normal human

hearing (i.e., 20 kHz). In general, it can also refer to sounds below 20Hz, known as

‘‘infrasound’’. Langevin’s early experiments with quartz (April 1917) were at a frequency of

150 kHz. The frequency was later lowered to 40 kHz in order to reduce absorption.

signals were found to be very weak. Fortunately, a suitable valve amplifier, designedby Leon Brillouin and G. A. Beauvais,8 was made available to Langevin soon after,enabling him to build a system by November 1917 that ‘‘gave a signalling distance ofup to six kilometres’’ (Hackmann, 1984, p. 81).

The real breakthrough came when the French and British teams started sharingtheir findings after a series of high-level meetings held in Washington, D.C. betweenMay and July 1917. Boyle visited Langevin shortly afterwards, when he would havelearnt of the French advances. On his return to England, Boyle started working onquartz transducers, and the French amplifier was made available to the British teamat Parkeston Quay. The reliance on quartz was such that, until a suitable supply wasidentified from Bordeaux, Boyle threatened to ‘‘raid the crystal exhibits in severalgeological museums’’.

Meanwhile, Langevin continued with his own work in Toulon, and by February1918 had obtained echoes from a submarine using the high-frequency (40 kHz) quartztransducers. Boyle followed suit a month later with a submarine echo from a distanceof 500 yd (about 460m). The Armistice of November 1918 led to the cancellation ofplans to fit both British and French navy ships in early 1919, but asdics (as thetechnology of high-frequency echolocation was then called) was born.9 The termsonar was coined during WW2.

The origin of the term asdics as an acronym for Anti-Submarine Division -ics,where the ‘‘ics’’ meant ‘‘activities pertaining to’’ in the same way as in ‘‘physics’’, isrecounted by Wood (1965). The alternative explanation (for the term asdic, withoutthe second ‘‘s’’) as an acronym for ‘‘Allied Submarine Detection InvestigationCommittee’’ appears to be a myth created by the British Admiralty in 1939 inresponse to a question by Oxford University Press (Hackmann, 1984, p. xxv). Duringthe initial development of the sensor at Parkeston Quay, secrecy was such that eventhe material quartz was referred to by its codename ‘‘asdivite’’.

On the subject of semantics, it is worth mentioning the change in meaning of theword ‘‘supersonic’’ after the end of WW2. Between the two world wars, this term wasused in the U.S.A. to mean ‘‘pertaining to sound whose frequency is too high to beheard by the human ear’’, synonymous with the European term ‘‘ultrasonic’’ (Klein,1968). Today the European term has been adopted worldwide, presumably as aconsequence of the modern use of ‘‘supersonic’’ to describe ‘‘faster than sound’’flight.

The first working active sonar was built in November 1918 by Boyle, a Canadianscientist working in England. Reading an account of the early history of echo rang-ing, however, one cannot help being struck by a series of key contributions made by


8 This work was assisted by a wireless expert, Paul Pichon. Having deserted from the French

army he found himself importing some American valve amplifiers to his adoptive Germany

early in WW1. Realizing the military value of these, he took them instead to France where he—

though immediately arrested—handed over his equipment to the French authorities. These

early valves provided the basis for the Beauvais–Brillouin design (Hackmann, 1984, pp. 80–81).9 Boyle’s quartz system was fitted to a trawler on November 16, 1918, five days after the end of

WW1.

French scientists, including:

— the earliest known description of the echo-ranging concept, by Mersenne (1636);— the measurement of the speed of sound in seawater, by Beudant (ca. 1816);— the discovery of piezoelectricity, by the Curie brothers and Lippmann (1880–

1881);— the development of the valve amplifier, by Beauvais and Brillouin (ca. 1916);— pioneering research on the use of quartz transducers, including the first ever

detection of an echo from a submarine, by Langevin10 (1917–1918).

To this impressive list one can add the work of a remarkable statesman named PaulPainleve (Figure 1.5). In January 1915, Chilowski had written a letter urging theFrench government to develop an underwater echolocation device as a defenseagainst U-boats. Recognizing its importance and urgency, Painleve forwardedthis letter to Langevin without delay, thus facilitating the early Langevin–Chilowskicollaboration. Painleve also saw the valuein Anglo-French co-operation, requesting ascientific exchange agreement between Franceand Britain in December 1915. Despite delayscaused by opposition from the Admiralty, theagreement, without which the co-operationbetween Langevin and Boyle might not haveflourished, was eventually approved by theBritish Government in October 1916 (Hack-mann, 1984, p. 39).

1.4.1.4 Origins of passive sonar

By comparison with active sonar, invented ina race against time between Chilowski’s 1915letter and the first successful French and Brit-ish tests in 1917, the arrival of passive sonarwas a gradual affair that lasted centuries. Its15th-century conception in Leonardo da Vin-ci’s device able to detect ships ‘‘at a greatdistance’’ was followed by a 400-year gesta-tion, including the 18th-century observationsof Benjamin Franklin (see Section 1.4.3.3),and culminating in the listening equipmentfitted to shipping vessels at the end of the


Figure 1.5. French statesman and

mathematician Paul Painleve—rep-

rinted from Anon. (wp, b). Painleve

was Minister for Public Instruction

and Inventions during the period

1915–1917, and later served two brief

periods as Prime Minister in 1917 and

1925.

10 Langevin is one of five sonar scientists after whom the Pioneers of Underwater Acoustics

Medal, awarded to this day by the Acoustical Society of America, is named. The others are

H. J. W. Fay, R. A. Fessenden, H. C. Hayes, and G. W. Pierce. In 1959, Hayes became the first

ever recipient of this medal, which was also awarded to Wood (in 1961) and to Urick (1988).

19th century to notify them of the presence of nearby lightships: in 1889, the U.S.Lighthouse Board described an invention of L. I. Blake comprising an underwaterbell and microphone receiver, and a similar system—patented in 1899 (Hersey,1977)—was developed a few years later by Elisha Gray and A. J. Mundy (Lasky,1977).11 In common with the echolocation devices of Langevin and Boyle, it wasWW1 that provided the final impetus for the birth of passive sonar. An importantdifference, though, is that underwater listening equipment was put to practical usewell before the end of the war. Portable omnidirectional hydrophones were availableas early as 1915, and directional ones followed in 1917. Towed hydrophones wereoperational before the end of WW1, and in 1918 a prototype passive-ranging systemwas fitted to an American destroyer.

British listening devices used during WW1, based on early American work, weredeveloped at BIR by Wood and Gerrard (occasionally assisted by Rutherford) atParkeston Quay and by Captain C. P. Ryan at Hawkcraig. To reduce noise, direc-tional hydrophones could be towed behind the ship in a streamlined capsule knownas a ‘‘fish’’, developed by G. H. Nash.

Ryan constructed a network of up to 18 underwater listening stations positionedstrategically in British coastal waters. These listening stations, each comprising a fieldof hydrophones, were manned with shore-based operators, who listened for distinc-tive U-boat sounds and reported their position to the nearest anti-submarine flotilla.

Some minefields were also equipped with special listening devices (magneto-phones), with which it was possible to determine the precise moment at which aU-boat was passing overhead. The mines could then be detonated remotely from ashore-based monitoring facility. According to Hackmann (2000), such minefieldswere responsible for the destruction of four U-boats towards the end of WW1, thefirst taking place on August 29, 1918.

Early in WW1, Rutherford had proposed the use of an array of multiple hydro-phones, in theory able to both amplify the signal and provide bearing information.The Royal Navy considered the proposed device too unwieldy and the idea wasdropped in Britain, but American scientists pursued it and by the end of the warhad developed the most sophisticated listening devices of that time (Hayes, 1920).This American research took place at the Naval Experimental Station in NewLondon, under the direction of Harvey Hayes.

The property of sound waves that Rutherford wished to exploit is that they retaintheir phase coherence over distances of at least several wavelengths. The first Amer-ican device to use this property was the ‘‘M-B tube’’, comprising two groups of eighthydrophones each. The (acoustic) signals from each group were combined coherentlyby a sequence of equal-length delay lines before being presented (binaurally, onecoherently summed group in each ear) to a human listener. The construction was suchthat coherent reinforcement took place from only one direction at a time, so in orderto scan over different bearings it was necessary to rotate this device in the water. Theinconvenience of the M-B tube—it needed to be lowered into the sea each time it was


11 Gray coined the term ‘‘hydrophone’’ to describe their underwater microphone, while Mundy

went on to co-found the Submarine Signal Company (now part of Raytheon) in 1901.

used—was overcome by the introduction of variable-length delay lines, which per-mitted the operator to select the direction of listening without any form of mechanicalrotation. This meant that the entire device, known as the ‘‘M-V tube’’, could be fixedto a ship’s hull, and used with the ship in motion. The M-V tube had two groups of sixhydrophones (later, two groups of ten), the signals from which were presentedbinaurally in the same way as for the M-B tube.

The capability to use the M-V tube in motion was a huge advantage, but it cameat a price—the din from a ship underway. To counter the noise problem the ‘‘U-3tube’’ (nicknamed the ‘‘eel’’), was invented. The eel comprised two groups of sixhydrophones towed behind the ship, thus benefiting from lower noise levels. TheU-3’s streamlined housing gave it the appearance of a snake or eel—hence itsnickname. The key advance that made this possible was the use of electrical insteadof acoustical delay lines, making the equipment less bulky. An experimental devicecomprising two towed eels and two ship-mounted M-V tubes was fitted to anAmerican destroyer in April 1918 (Figure 1.6). The combined system was capableof passive ranging by triangulation of the two different bearings (Hayes, 1920). Thefirst working sonar capable of localization in range and bearing was neither a Frenchnor a British invention, but an American one.

1.4.2 Sonar in its infancy (1918–1939)

1.4.2.1 Fathometers and fish finders

In peacetime, the thoughts of sonar engineers turned away from U-boats and backinitially to maritime safety, and later to fishing. The principle of acoustic echo rangingwas applied to measuring water depth, and Fessenden’s oscillator turned out to be


Figure 1.6. Installation of early U.S. passive-ranging sonar with two towed eels of length 40 ft

(12m), and 12 ft (4m) apart, and two hull-mounted M-V tubes of the same length. The eel was

towed about 300–500 ft (100–150m) behind the ship (reprinted with permission from Lasky,

1977, copyright 1977 American Institute of Physics).

ideally suited to this purpose. For this new application,12 its low operating frequencybecame an advantage because of reduced absorption, and there was no need fordirectivity because the direction to the seabed is known in advance. The first patent isattributed by Hersey (1977) to A. F. Wells as early as 1907, while Hackmann (1984)credits the first workable system to Alexander Behm in 1912.13 The first commercialecho sounders (called ‘‘fathometers’’) were designed by Fessenden at the SubmarineSignal Company, using his electromagnetic oscillator as a transmitter in combinationwith a conventional carbon microphone receiver. A recording echo sounder, enablinga permanent paper record to be kept of the echo sounder output, was invented byMarti and Langevin in 1922.

The next challenge for echo ranging was to be the detection of fish shoals.Although these produce weaker signals than the seabed, echoes from fish wererecorded by the trawler Glen Kidston in the North Sea in 1933 (Cushing, 1973). Evenbefore then, echo sounders were used by Belloc to identify the location of fish shoalsin the Bay of Biscay (Belloc, 1929a, b), and by the innovative fisherman CaptainR. Balls to find shoals of herring (Hersey and Backus, 1962, p. 499).

1.4.2.2 National research laboratories

The 1920s marked the beginning of nationally co-ordinated peacetime researchefforts in both Britain and the U.S.A., with both Wood and Hayes continuing attheir respective national research laboratories. In Britain, the Applied ResearchLaboratory (ARL) was founded in 1921, led first by B. S. Smith (1921–1927) andlater by Wood. The achievements of this group include the development of themagnetostrictive transducer in 1928 and of the recording echo sounder used onthe Glen Kidston.14

The U.S. Naval Research Laboratory (NRL) followed in 1923. The NRL SoundDivision, led by Hayes, was responsible for an oddly named listening device called the‘‘JK projector’’, installed on U.S. Navy ships in 1931 (Klein, 1968). This listeningsystem made use of Rochelle salt—a more efficient piezoelectric material thanquartz—housed in a special material known as ‘‘rho-c rubber’’, providing an impe-dance match with water while keeping the transducer dry (Rochelle salt dissolves inwater). The device was later adapted to enable its use for echo ranging also, leading toan early American active sonar known as the ‘‘QB’’, produced commercially by theSubmarine Signal Company.

1.4.2.3 Temperature and the ‘‘afternoon effect’’

Soon after the end of WW1, both British and American scientists working on therecently developed asdics sets noticed that their performance was inconsistent. A


12 The idea itself was not new, but 19th-century attempts had been unsuccessful (Drubba and

Rust, 1954; Maury, 1861; Newman and Rozycki, 1998).13 An entire chapter of Hackmann’s book is devoted to the development of echo sounders.14 Wood’s work is honored by the A. B. Wood Medal, awarded annually by the U.K. Institute

of Acoustics.

system that had achieved reliable detections to several kilometers on one day wouldperform erratically the next. More specifically, a diurnal cycle was noticed, wherebythe detection performance would deteriorate noticeably during a warm afternoon andrecover again the next morning. This so-called ‘‘afternoon effect’’ was explained inthe summer of 1937 by Columbus Iselin of the Woods Hole Oceanographic Institu-tion and L. Batchelder of the Submarine Signal Company in terms of refraction dueto unexpectedly high temperature gradients they discovered in the Cayman Sea, southof Guantanamo Bay.15 Their measurements were confirmed a year later usingSpilhaus’s recently invented bathythermograph.

1.4.3 Sonar comes of age (1939–)

1.4.3.1 WW2: a giant awakes

A detailed account of the role that sonar played in the outfolding of WW2 is given byHackmann (1984). Although alternatives to quartz had been investigated in the inter-war years, especially in the U.S.A., the asdics sets in use by the Royal Navy still reliedon the quartz technology developed during WW1. Quartz supplies were availablelocally, while essential cutting equipment and expertise were rescued from a factory inAntwerp (Belgium) shortly after the German invasion in May 1940.

The entry of the U.S.A. in WW2 shortly after the Japanese attack on PearlHarbor resulted in an investment of scientific resources in military objectives onan unprecedented scale, and sonar was no exception to this. The recently formedNational Defense Research Committee (NDRC) was restructured, while the NRLwas expanded to a strength of several thousand. American research at the Univer-sities of California and Columbia led to major developments in the theoreticalunderstanding of the propagation and scattering of underwater sound. Much of thisresearch was published in the compilation volume Physics of Sound in the Sea, editedby Lyman Spitzer Jr. (Spitzer, 1946), a document that remains of value today.

According to Hackmann (1984), the term ‘‘sonar’’ was coined by F. V. Hunt in1942 not as an acronym but as a phonetic analogue to ‘‘radar’’, while the modernexplanation as an acronym for sound navigation and ranging was invented later.Whatever its origin, Hunt’s term would eventually replace the name ‘‘asdics’’preferred by the Royal Navy.

1.4.3.2 Passive sonar in Germany

While Allied efforts focused mainly on the detection and prosecution of U-boatsusing active sonar, German scientists developed sophisticated passive sonar equip-ment known as Gruppenhorchgerat (GHG). This GHG equipment was fitted to thecruiser Prinz Eugen, providing it with effective early warning of torpedo attacks,permitting timely evasive action.


15 Sound speed was known at this time to vary with temperature.

1.4.3.3 The anomalous absorption of seawater

It has long been known that sound is able to travel remarkable distances in water. Thestatement ‘‘if you cause your ship to stop, and place the head of a long tube in thewater, and place the other extremity to your ear, you will hear ships at a greatdistance from you’’ is attributed to the 15th-century scientist Leonardo da Vinci(Urick, 1967),16 and underwater sound was used even before then by ancientAsian fishermen (Hackmann, 1984). In July 1762, Benjamin Franklin wrote ‘‘. . .the sound [of two stones struck under water nearly a mile away] did not seem faint,. . . but smart and strong, and as if present just at the ear . . .’’ The main explanationfor these observations is the rapid decrease in absorption with decreasing frequency,permitting long-range propagation of low-frequency sound. In freshwater, theabsorption of sound increases quadratically with frequency. This functional form(though not the magnitude) was consistent with 19th-century predictions of viscousabsorption by G. Stokes. After the end of WW2, oceanographers directed theirefforts to understanding the discrepancy in magnitude and the (more complicated)frequency dependence in seawater.

In a landmark publication, Liebermann (1948) explained the discrepancy withStokes’s theory by introducing the effects of dilatational viscosity. He also suggestedan ionic relaxation as a possible mechanism for the anomalous frequency dependencebelow 1MHz in seawater. In the 1950s it was realized that the salt responsible forLiebermann’s relaxation effect was magnesium sulfate, which has a relaxation timeclose to 1 ms at room temperature. The realization appears to have been a gradualone, with contributions from Leonard et al. (1949), Wilson and Leonard (1954),Fisher (1958), and Schulkin and Marsh (1962).

A lower frequency relaxation with a relaxation time of order 100 ms, due to boricacid, was suggested by Yeager et al. (1973) and confirmed later by Fisher andSimmons (1975) and Simmons (1975). The combined effect of the two relaxations,plus viscosity at high frequency (above 300 kHz), is illustrated by Figure 1.7.17

An equation for the absorption coefficient of seawater that came into widespreaduse after it appeared in the second edition of Urick’s book (Urick, 1975), in units ofdecibels per kilometer, is18

a ¼ 0:11F 2

1 þ F 2

!BðOHÞ3

þ 44F 2

4100 þ F 2

!MgSO4

þð3 � 10�4F 2ÞH2O; ð1:1Þ

where F is the acoustic frequency expressed in kilohertz. Although Equation (1.1) isknown as ‘‘Thorp’s equation’’, in fact only the first of the three terms originates from


16 Urick’s source is an unpublished report by T. G. Bell (Bell, 1962). An alternative source,

from Burdic (1984), is due to E. MacCurdy (MacCurdy, 1942). Neither Bell (1962) nor

MacCurdy (1942) have been seen by the present author.17 The possibility of a third chemical relaxation, associated with magnesium carbonate ions, is

suggested by Mellen et al. (1987).18 Urick’s original equation was expressed in units of decibels per kiloyard. The metric version

quoted here is due to Fisher and Simmons (1977).

Thorp (1967). Urick combined the viscosity and magnesium relaxation terms fromHorton (1959) with the low-frequency relaxation term introduced by Thorp.

1.4.3.4 SOFAR, SOSUS, and the Roswell Incident

1.4.3.4.1 The speed of sound in seawater

In perhaps the first-ever application of Snell’s law to underwater sound, Lichte (1919)showed that the sensitivity of compressibility to temperature, pressure, and salinitymeans that the sound speed of seawater varies with depth, and that this depth


Figure 1.7. Sound

absorption vs.

frequency in

seawater (reprinted

with permission

from Fisher and

Simmons, 1977,

copyright 1977

American Institute

of Physics).

variation influences sound propagation in a predictable manner. Such predictionsbecame a reality once accurate tables of sound speed as a function of these threeparameters became available in the 1920s (Matthews, 1927) and were later improvedby Kuwahara (1939). The essence of these tables is captured in a simple formula dueto Medwin (1975) giving the sound speed c, in meters per second, as

cðS;T ; zÞ ¼ 1,449:2 þ 4:6T þ 0:016z� 0:055T 2

þ ½ð1:34 � 0:010TÞðS � 35Þ þ 2:9 � 10�4T 3� ð1:2Þ

where T is the temperature expressed in units of degrees Celsius; S is the salinity inparts per thousand (g/kg); and z is the depth from the sea surface in meters. Undertypical ocean conditions, the first four terms are the most important ones. WhileEquation (1.2) is not accurate by modern standards, its simplicity makes it particu-larly suitable for investigating the sensitivity to temperature and pressure (parame-terized as depth), which are of interest to this historical account. Neglecting the termsin square brackets, the partial derivatives with respect to temperature and depth are

@c

@T

��S;z

4:6 � 0:110T m/s per degree Celsius ð1:3Þ

and

@c

@z

��T ;S

0:016 m/s per meter: ð1:4Þ

Putting T ¼ 10 C in Equation (1.3) gives @c=@T 3.5 m/s per C. Close to the seasurface, the most important effect usually arises from the temperature gradient,especially in summer when the temperature at the surface can be 20 C higher thanat depth. For example, using a temperature gradient of 5 C per 100m gives a soundspeed gradient of 0.18 m/s per meter due to temperature alone, much larger than thepressure effect of Equation (1.4).

1.4.3.4.2 Sound fixing and ranging: the SOFAR channel

Because of the temperature gradient referred to above, the sound speed tends todecrease with increasing depth close to the sea surface, and this process continuesuntil the temperature stabilizes at its deep-water value (ca. 2 C). At this point, thepressure effect—which works in the opposite direction—takes over, resulting in asound speed minimum at a depth that depends on the surface temperature and henceon latitude, below which sound would be refracted upwards, back towards the seasurface. A typical depth for the minimum is ca. 1 km.

Lichte realized this in 1919, but the implications of his remarkable insight werenot properly investigated until the 1940s. Maurice Ewing understood that Lichte’supward refraction would create a huge waveguide, possibly spanning entire oceans.Coupled with low absorption at low frequency, such a waveguide creates a means forcommunicating over vast distances, and the existence of the predicted channel was


confirmed in 1943 (Ewing and Worzel, 1948).19 Amongst other applications, Ewingproposed a system to aid ditched air force pilots. The scheme involved a device that,when dropped into the sea, would sink and explode at a depth close to the channelaxis. The idea was to record the arrival time of the resulting acoustic pulse at three ormore listening stations and calculate the pilot’s location by triangulation. Thisremarkable recovery system, the principle of which was demonstrated duringWW2 (Stifler and Saars, 1948), was called sofar (for sound fixing and ranging)and would later give its name to the channel itself.

1.4.3.4.3 The Sound Surveillance System (SOSUS)

During the Cold War, a series of so-called ‘‘Summer Studies’’ was held at variousAmerican universities between 1948 and 1957 to work on the major defense problemsof the time (Holbrow, 2006). The second of these, held at MIT in 1950 and directed byProfessor J. Zacharias, posed the problem of ‘‘undersea defense’’ (i.e., defense againstSoviet submarines). The proposed solution was a global network of hydrophonearrays in the sofar channel to exploit long-range waveguide propagation. Such anetwork, a natural successor to sofar and known as SOSUS, for sound surveillancesystem, was indeed designed and built. This network has since the end of the ColdWar been used for non-military purposes (Stafford et al., 1998).

1.4.3.4.4 Project MOGUL and the Roswell Incident

A U.S. Air Force report published in 1995 (Weaver and McAndrew, 1995) makes afascinating connection between SOSUS and a famous UFO story dating to 1947known as the Roswell Incident. Even before SOSUS, Ewing had proposed anambitious project to exploit an atmospheric analogue of the SOFAR channel. Hesuggested that high-altitude balloons equipped with microphones might be able todetect Soviet weapons tests. The project, known as MOGUL, went ahead, and theclaim made in the USAF report, after 50 years shrouded in secrecy, is that one of theMOGUL balloons crashed at Roswell in 1947, thus spawning the UFO story. Anillustrated account, including a general history of the SOFAR channel, is given byRichard Muller of the University of California at Berkeley (Muller, www).

1.4.3.5 Advances in detection theory and processing technology

1.4.3.5.1 Statistical detection theory

Important advances in the statistical theory of detection of signals in a noisy back-ground were made by radar and telecommunications scientists in the 1940s and 1950s(Marcum, 1947; Rice, 1948; Swerling, 1954). This seminal work, epitomized by theMarcum function, Rician distribution, and Swerling distributions, provides thefoundation for modern detection theory.


19 According to Goncharov (2008), the effect was discovered independently by L. D. Rozenberg

in 1946 and explained by L. M. Brekhovskikh in 1948.

1.4.3.5.2 FM processing and electronic scanning

In the 1950s and 1960s, new types of processing technology became feasible. Thepolar circumnavigation by USS Nautilus in 1958 was made possible by a newlydeveloped FM ( frequency modulation) sonar pulse that provided accurate navigationdata under the ice cap. The high resolution of this new technique also made it suitablefor detecting small objects such as sea mines. The sonar used by the Nautilus wasrotated mechanically. In the 1960s such mechanical devices were replaced by sophis-ticated electronic systems that could scan horizontally without the need for movingparts.

1.4.3.5.3 The computer era

The widespread availability of digital computers in the late 20th century has openedup new possibilities that would have been inconceivable in the early days of sonar. Inmodern sonar systems, beamforming, FM processing, Doppler filtering, localization,and tracking algorithms are implemented in software rather than analogue hardware.Other processing methods, relying on sophisticated computer models of soundpropagation, include time reversal acoustics, synthetic aperture sonar, matched fieldprocessing, and noise correlation (Kuperman and Lynch, 2004). Most computermodels of sound propagation available today (Jensen et al., 1994) have their rootsin one (or more) of normal mode theory (Pekeris, 1948), flux propagation theory(Weston, 1959), the fast-field method (DiNapoli, 1971), and the parabolic equationmethod (Tappert, 1977).

1.4.4 Swords to ploughshares

1.4.4.1 Oceanographic instruments

The first non-military spin-off of sonar was Fessenden’s echo sounder, and fisheriessonar followed soon after, as did early uses for seismic prospecting (Hersey, 1977).Today, an increasingly sophisticated understanding of underwater sound propaga-tion has brought new applications in acoustical oceanography, marine archeology,and deep-sea exploration. Examples are seabed classification and mapping, classifica-tion of marine flora and fauna (Fernandes et al., 2002), weather and climate observa-tion, and high-resolution acoustic imaging. Perhaps the most striking achievement ofsonar was the discovery in September 1985, by Robert Ballard and Jean LouisMichel, of the wreck of the Titanic. The wreck was found at a depth of 3,800m withthe aid of a deep-diving submersible equipped with side-scan sonar.

In the early 1990s an ambitious global experiment took place, known as theHeard Island Feasibility Test (HIFT), the purpose of which was to determine whetherit was possible to build a global acoustic thermometer by exploiting the dependenceof sound speed, and hence travel time, on temperature. A location in the southernIndian Ocean was identified, close to Heard Island, from which sound following greatcircle paths could be heard on both the east and west coasts of the U.S.A., involvingpropagation distances of up to 18,000 km (Munk et al., 1994; Collins et al., 1995).HIFT was a precursor to measurements carried out both on a large scale in the


Atlantic and Pacific Oceans and in smaller basins such as the Arabian, Barents, andMediterranean Seas (Munk et al., 1995).

1.4.4.2 Discovery of dolphin sonar and concern over the effects ofanthropogenic sound

With man’s increasingly sophisticated use of sound in the sea came also a gradualawareness that we are not alone in this use. The use of echolocation by bottlenosedolphins was first suspected in the 1940s by Arthur McBride, curator of MarineStudios (Florida). This speculation was confirmed in the 1950s by W. N. Kellogg,William Schevill, and others (Au, 1993). It is now known that many odontecetes useecholocation for both hunting and navigation (Au, 1993).

A growing concern is that whales might rely on features such as the SOFARchannel for long-distance communication, and that increasing levels of anthropo-genic sound due to shipping, underwater explosions, high-power military sonar, andeven low-power systems such as used in basin-scale acoustic thermometry, might bedisrupting their ability to do so. This concern has led to an increasing interest andawareness in the sonar of dolphins and other marine mammals (Richardson et al.,1995).

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Reunions, Conseil P, 55, 139–150 [in French].

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Applied Biology Vol. 52), Pergamon.

DiNapoli, F. R. (1971) Fast Field Program for Multilayered Media (NUSC Technical Report

4103), Naval Underwater Systems Center.

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Drubba, H. and Rust, H. H. (1954) On the first echo-sounding experiment, Annals of Science,

10(1), March, 28–32.

Etter, P. C. (2003) Underwater Acoustics Modeling and Simulation: Principles, Techniques and

Applications, Spon Press, New York.

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America Memoir, 27.

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Acoustic applications in fisheries science: The ICES contribution, ICES Marine Science

Symposia, 215, 483–492.

Fisher, F. H. (1958) Effect of high pressure on sound absorption and chemical equilibrium,

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Fisher, F. H. and Simmons, V. P. (1975) Discovery of boric acid as cause of low frequency

sound absorption in the ocean, IEEE Oceans ’75, 21–24.

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62, 558–564

Girard, M. (1877) Le son dans l’air et dans l’eau, La Nature, Revue des Sciences et de leurs

applications aux arts et a l’industrie (pp 247–250), G. Masson, Paris [in French].

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Scientific, Hackensack, NJ.

Goncharov, V. V. (2008) The development of sound propagation theory in the USSR and in

Russia, in O. A. Godin and D. R. Palmer (Eds.), History of Russian Underwater Acoustics

(pp. 71–120), World Scientific, Hackensack, NJ.

Hackmann, W. (1984) Seek & Strike: Sonar, Anti-submarine Warfare and the Royal Navy 1914–

54, HM Stationery Office, London.

Hackmann, W. (2000) Asdics at war, IEE Review, 15–19.

Hayes, H. C. (1920) Detection of submarines, Proc. Amer. Phil. Soc., LXIX, March 19, 1–47.

Hersey, J. B. (1977) A chronicle of man’s use of ocean acoustics, Oceanus, 20(2), 8–21.

Hersey, J. B. and Backus, R. H. (1962) Sound scattering by marine organisms, in M. N. Hill

(Ed.), The Sea, Ideas and Observations on Progress in the Study of the Seas, Vol. 1: Physical

Oceanography (pp. 498–539), Interscience, New York.

Holbrow, C. H. (2006) Scientists, security, and lessons from the cold war, Physics Today, 59(7),

39–44.

Horton, J. W. (1959) Fundamentals of SONAR (Second Edition). United States Naval Institute,

Annapolis.

Hunt, F. V. (1954, reprinted 1982) Electroacoustics: The Analysis of Transduction, and Its

Historical Background, American Institute of Physics, New York.

Hunt, F. V. (1992) Origins of Acoustics, Acoustical Society of America, New York.

Jensen, F. B., Kuperman, W. A., Porter, M. B., and Schmidt, H. (1994) Computational Ocean

Acoustics, American Institute of Physics, New York.

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sonic sounding, Hydrographic Review, 16, 123–140.

Lasky, M. (1977) Review of undersea acoustics to 1950, J. Acoust. Soc. Am., 61, 283–297.

Leonard, R. W., Combs, P. C., and Skidmore Jr., L. R. (1949) Attenuation of sound in

‘‘synthetic sea water’’, J. Acoust. Soc. Am., 21, 63.


Lichte, H. (1919) Uber den Einfluß horizontaler Temperaturschichtung des Seewassers auf

die Reichweite von Unterwasserschallsignalen, Physikalische Zeitschrift, 17, 385–389 [in

German].20

Liebermann, L. N. (1948) The origin of sound absorption in water and sea water, J. Acoust.

Soc. Am., 20, 868–873.

Liebermann, L. N. (1949) Sound propagation in chemically active media, Phys. Rev., 76, 1520.

MacCurdy, E. (1942) The Notebooks of Leonardo da Vinci (Chap. X), Garden City Publishing,

New York.

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memorandum RM-754), The RAND Corporation.

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Appendix (Research memorandum RM-753), The RAND Corporation.

Matthews, D. J. (1927, reprinted 1939) Tables of the Velocity of Sound in Pure Water and

Sea Water for Use in Echo-sounding and Sound-ranging (H. D. 282). Hydrographic

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(Eighth Edition). Harvard University Press.

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physics%2010/chapters%20(old)/9-SecretsofUFOs.html (last accessed September 29,

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2

Essential background

Plurality should not be posited without necessity

William of Ockham (ca. 1285–1349).

The purpose of this chapter is to introduce the basic knowledge required by the readerto understand the description of the sonar equations introduced in Chapter 3, and nomore than this. The knowledge is sub-divided into four general subject areas:oceanography, acoustics, signal processing, and detection theory. Further detailsof these four areas, omitted here for simplicity, are described in Chapters 4 andfollowing.

2.1 ESSENTIALS OF SONAR OCEANOGRAPHY

This section describes those basic physical properties of the sea and the air–seaboundary of relevance to the generation, propagation, and scattering of sound atsonar frequencies. The speed of sound and the density of water influence thegeneration and propagation of sound. These and other related parameters aredescribed in Section 2.1.1, followed by the relevant properties of air in Section2.1.2. A more comprehensive description of these oceanographic properties isprovided in Chapter 4.

Many parameters of relevance to underwater acoustics vary with temperature T ,salinity S, and hydrostatic pressure (often parameterized through the depth fromthe sea surface z). Where ‘‘representative’’ numerical values are quoted, they are

evaluated for the following conditions:

T ¼ 10 �C;

S ¼ 35;

depth z ¼ 0:

By convention the depth co-ordinate z is zero at the sea surface and increasesdownwards to the seabed.

For simplicity, in Chapters 2 and 3 the ocean is assumed to extend to infinitedepth, with uniform properties occupying the entire half-space satisfying z > 0. Forexample, the speed of sound and density are assumed independent of depth andrange. The purpose of the quantitative numerical calculations based on this idealiza-tion (see Worked Examples in Chapter 3) is to illustrate the main principles of sonarperformance modeling, rather than to provide realistic estimates of detection per-formance.More realistic examples are presented at the end of the book, in Chapter 11.

2.1.1 Acoustical properties of seawater

The two most important acoustical properties of seawater are the speed at whichsound waves travel (abbreviated as sound speed ) and the rate at which they decaywith distance traveled (the decay rate, or absorption coefficient). A third parameterthat can influence sound propagation, through its effect on interaction with bound-aries, is density. The density of seawater, and the speed and absorption of sound inseawater, all depend on salinity, temperature, and pressure. At low frequency theabsorption also depends on acidity or pH (see Chapter 4).

2.1.1.1 Speed of sound

For the representative conditions described above, the sound speed in seawater,denoted cwater, is 1490m/s. More generally, the parameters S, T , and P all vary withdepth and therefore so too does the sound speed, resulting in significant refraction. Adiscussion of these gradients and their important acoustic effects is deferred toChapters 4 and 9. Here and in Chapter 3 they are neglected for simplicity. Thewavelength � at frequency f is

� ¼ cwater=f : ð2:1Þ

2.1.1.2 Density

The density of seawater (�water) under the representative conditions introduced aboveis 1027 kg/m3. Departures of seawater density from this value are small and for mostsonar performance applications may be neglected.

2.1.1.3 Attenuation of sound

Attenuation is the name given to the process of decay in amplitude due to a combina-tion of absorption and scattering of sound. The term ‘‘absorption’’ implies conversion

28 Essential background [Ch. 2

to some other form of energy, usually heat, whereas ‘‘scattering’’ implies a redis-tribution in angle away from the original propagation direction, with no overall lossof acoustic energy.

The rate of attenuation of sound in water is less than in air and much less thanthat of electromagnetic waves in water. Low-frequency sound, of order 1Hz to 10Hz,can travel for thousands of kilometers, but high-frequency sound is attenuated morerapidly. The attenuation coefficient �water increases monotonically with frequency byabout four orders of magnitude in the frequency range from 30Hz to 300 kHz, andquadratically with frequency thereafter. For frequencies f exceeding 200Hz, it can bewritten (see Chapter 4)

�water ¼ �1f 2

f 2 þ f 21þ �2

f 2

f 2 þ f 22þ �3 f

2: ð2:2Þ

For the specified representative conditions, the three coefficients �i are1

�1 ¼ 1:40� 10�2 Np km�1;

�2 ¼ 5:58 Np km�1

and

�3 ¼ 3:90� 10�5 Np km�1 kHz�2: ð2:3Þ

The frequencies f1 and f2, explained further in Chapter 4, are known as relaxationfrequencies and for the representative conditions are equal, respectively, to 1.15 kHzand 75.6 kHz.

Numerical evaluation of Equation (2.2) gives (to the nearest order of magnitude)�water � 10�3, 10�1, and 10þ1 Np/km at 300Hz, 10 kHz, and 300 kHz, respectively. Agraph of �water vs. frequency, computed using Equation (2.2), is plotted in Figure 2.1.The reciprocal of the attenuation coefficient (i.e., ��1), plotted on the same graph,provides a rough measure of the distance that sound can travel in water if unimpededby physical obstacles. This quantity is referred to here as ‘‘audibility’’, the acousticalanalogue of optical ‘‘visibility’’, and varies between 102 mNp�1 at 300 kHz and106 mNp�1 at 300Hz. By comparison, the attenuation coefficient of green light2

is at least 10�2 Npm�1 for clear seawater, so that the optical visibility in waternever exceeds 102 mNp�1 and is usually less than 101 mNp�1. Thus, for acousticfrequencies up to 300 kHz, the audibility of sound exceeds the maximum visibility oflight by up to six orders of magnitude. This is the reason why sound waves have

2.1 Essentials of sonar oceanography 29]Sec. 2.1

1 Two sound waves are said to differ in level by 1Np if their amplitudes are in the ratio 1 : e.The neper (Np) and the related unit the decibel (dB) are defined in Appendix B.2 The sea is opaque to electromagnetic radiation with the exception of visible light, very low

frequency radio waves, and gamma rays. The extinction coefficient is a measure of the decay of

light intensity with distance, and in the present notation is equal to 2�opt,where �opt is theoptical attenuation coefficient in units of nepers per unit distance. For example, the value

quoted by (Clarke and James, 1939) of 4% per meter for the extinction coefficient in the

Sargasso Sea means that expð�2�optxÞ ¼ 0:96 when the distance x ¼ 1 m. Taking logarithms

gives 2�opt ¼ 0:04/m.

become the most successful means of probing the underwater environment. It is theraison d’etre of sonar.

2.1.2 Acoustical properties of air

Together with those of seawater, the properties of air determine the reflectioncoefficient at the air–sea boundary. The sound speed and density of air depend ontemperature (T) and pressure (P). For the representative conditions, these arecair ¼ 337m/s and �air ¼ 1.25 kg/m3. Thus, both � and c in air are considerably lowerthan their counterparts in water, which has important implications for the behaviorof underwater sound.

2.2 ESSENTIALS OF UNDERWATER ACOUSTICS

2.2.1 What is sound?

Steady-state pressure increases with increasing depth z and is equal to the total weightper unit area of water plus atmosphere supported above that depth. This quantity iscalled the static pressure (or hydrostatic pressure) and can be expressed quantitatively


Figure 2.1. Numerical value of attenuation coefficient vs. frequency of sound in seawater �(expressed in units of nepers per megameter) and of its reciprocal, ��1 (in kilometers per neper),calculated using Equation (2.2) for the specified representative conditions: S ¼ 35; T ¼ 10 �C;z ¼ 0.

in the formPstatðzÞ ¼ Patm þ PgaugeðzÞ; ð2:4Þ

where Patm is the atmospheric pressure (approximately 101 kPa); and Pgauge is theadditional pressure due to the weight of the water above depth z (the gauge pressure)

PgaugeðzÞ ¼ðz0

�waterð�Þgð�Þ d�: ð2:5Þ

Underwater disturbances result in departures P from this value, for an arbitraryposition vector x,

Ptotðx; tÞ ¼ PstatðzÞ þ Pðx; tÞ: ð2:6Þ

Once created, provided that certain basic conditions are met (Pierce, 1989), a pressuredisturbance propagates with the speed of sound, and P is known as the acousticpressure, henceforth denoted qðx; tÞ and assumed small by comparison with staticpressure.3 Such an acoustic disturbance is known as underwater sound. The studyof this sound is called underwater acoustics. The assumption of small q simplifiesthe mathematics and is generally justified because atmospheric pressure is largecompared with typical acoustic pressure fluctuations.

A brief account is given here of radiation and scattering of underwater soundfrom simple sources and for a simple geometry. First, radiation is considered from apoint source in an infinite uniform medium, with and without a perfectly reflectingplane boundary (Section 2.2.2). Then the scattering of plane waves is considered, firstfrom a point object and then from a rough surface (Section 2.2.3). The sea surface isconsidered as an example of a reflecting surface, a radiating surface, and a scatteringboundary. A more complete treatment of these phenomena is presented in Chapters 5and 8.

2.2.2 Radiation of sound

2.2.2.1 Radiation from a point monopole source

2.2.2.1.1 Spherical spreading

Consider a point monopole4 source of power W . To generate sound at a givenfrequency, the source must expand and contract at that frequency. During expansionthe source motion causes an increase in density of the surrounding fluid, with acorresponding increase in its pressure. The resulting high-pressure disturbancepropagates outwards in the form of a spherical wave, traveling at the speed ofsound cwater. The same sequence follows a contraction, except with a low-pressuredisturbance replacing the high-pressure one.

At any fixed moment in time the radiated field comprises a series of concentric‘‘rings’’ (actually spherical shells in three dimensions) of alternating high and low

2.2 Essentials of underwater acoustics 31]Sec. 2.2

3 The symbol p, introduced in Section 2.2.2, is reserved for a complex variable representing the

acoustic pressure. See footnote 5.4 A monopole source is one with a fluctuating volume, such as a pulsating bubble.

pressure. The potential of these rings to do work on the surrounding medium can beexpressed in terms of their potential energy density (Pierce, 1989)

EðPÞV ¼ q2

2Bwater

; ð2:7Þ

where EV denotes energy per unit volume; and Bwater is the bulk modulus of water, ameasure of its opposition to compression or rarefaction, analagous to the stiffness ofa spring, and equal to

Bwater ¼ �waterc2water: ð2:8Þ

The rings also contain kinetic energy, due to the particle velocity u, given by

EðKÞV ¼ �waterjuj2

2: ð2:9Þ

The superscripts ðPÞ and ðKÞ in Equations (2.7) and (2.9) denote potential and kineticenergy, respectively. If the pressure and particle velocity are in phase, it can be shownthat the kinetic and potential densities are equal (Pierce, 1989), so that the averagerate of energy flux (i.e., intensity) is

I ¼ cwater EðKÞV þ E

ðPÞV

� �¼ 2cwaterE

ðPÞV ¼ q2

�watercwater; ð2:10Þ

where the overbar indicates an average in time. Conservation of energy demands thatthe total radiated power at distance s from the source, 4s2I , be constant, which

means that the RMS pressure (i.e.,

ffiffiffiffiffiq2

q) must vary as 1=s with distance.

A point monopole source radiates omni-directionally (i.e., with equal power in alldirections). At a distance s from the source, in the assumed uniform medium theenergy is distributed uniformly on a sphere of surface area 4s2 (Figure 2.2), so thecomponent of acoustic intensity normal to the surface of the sphere at a distance s is

IðsÞ ¼ W

4s2: ð2:11Þ

Also of interest is the acoustic pressure resulting from the point source. Usingstandard complex variable notation for a diverging harmonic spherical wave ofangular frequency !, the complex pressure field p varies with time t and distance saccording to Pierce (1989)5

pðs; tÞ ¼ffiffiffi2

pp0s0

eiðks�!tÞ

s; ð2:12Þ

where p0 is the RMS pressure at a distance s0 from the source; and k is the acousticwave number, so that

k ¼ !=cwater: ð2:13Þ


5 The real part of the complex variable pðs; tÞ is the acoustic pressure qðs; tÞ. Unless otherwisestated, an expð�i!tÞ time convention is used for traveling waves throughout.

The true acoustic pressure is obtained by taking the real part of Equation (2.12), sothat

qðs; tÞ ¼ffiffiffi2

pp0s0

cosðks� !tÞs

: ð2:14Þ

From Equation (2.10) it follows that

I ¼ j pj2

2�wcw; ð2:15Þ

where the ‘‘water’’ subscript is abbreviated henceforth as ‘‘w’’. From Equations(2.11), (2.12), and (2.15) it then follows that

p0s0 ¼ �wcwW

4

� �1=2

; ð2:16Þ

or more generally (for a directional source)

p0s0 ¼ ð�wcwWOÞ1=2; ð2:17Þ

where WO indicates the radiated power per unit solid angle (the radiant intensity).It is convenient to define a steady-state propagation factor FðsÞ in terms of the

ratio of the mean square pressure at the receiver to that at a small distance ðs0Þ fromthe source, such that

FðsÞ ¼ q2

p20s20

¼ j pj2

2p20s20

: ð2:18Þ

Defined in this way, the propagation factor has dimensions [distance]�2. For aspherical wave in a medium of uniform impedance it is equal to the ratio of receivedintensity I to the radiant intensity WO.


Figure 2.2. Radiation

from a point source of

power W in free space.

The intensity at a

distance s is

I0 ¼ W=ð4s2Þ. Atdistance 2s the same

power has spread into

four times the area,

reducing the intensity by

a factor of 4.

It follows by substituting for pðs; tÞ from Equation (2.12), scaled by expð��sÞ toaccount for absorption, that the propagation factor for a point source in a uniformmedium is

FðsÞ ¼ e�2�s

s2; ð2:19Þ

where � is the sound attenuation coefficient introduced in Section 2.1.1.3.The above arguments apply to a steady-state field due to a source of constant

radiant intensity. If the power is transmitted for a short time only, it is useful to thinkin terms of the transient field resulting from the total radiated energy per unit solidangle EO. The appropriate propagation factor under these conditions is obtained byintegrating the numerator and denominator of Equation (2.18) over time instead ofaveraging them:

FðsÞ �

ðq2 dt

�wcwEO: ð2:20Þ

To summarize, the steady-state mean square pressure for a source of radiant intensityWO, from Equation (2.18), is

q2 ¼ �wcwWOFðsÞ ð2:21Þ

and for a transient field, the time-integrated pressure squared, from Equation (2.20),is ð

q2 dt ¼ �wwEOFðsÞ: ð2:22Þ

Either way, FðsÞ is given by Equation (2.19) for an omni-directional point source inan infinite uniform medium. The same equation applies also for a directional source,provided that WO (or EO) is measured in the direction of the receiver.

The behavior described by Equation (2.19), characterized by its s�2 dependencedue to the spherical nature of the expanding wave front, is known as sphericalspreading.

2.2.2.1.2 Reflection from the sea surface

Now consider the effect of placing a reflecting boundary, such as the sea surface, closeto the point source of Section 2.2.2.1.1. There are two straight-line ray paths con-necting the source to any given receiver position: the direct path and a surfacereflected one. If the source is at depth z0 below the surface (see Figure 2.3), thecontribution to the pressure field at the receiver due to the direct path is given byEquation (2.12) with a source–receiver separation equal to

s� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ ðz� z0Þ2

q: ð2:23Þ


The reflected path can be thought of as originating from an image source atheight z0 above the boundary, with image–receiver separation of

sþ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ ðzþ z0Þ2

q: ð2:24Þ

Using the method of images, the two contributions from source and image are addedcoherently to obtain the total pressure at the receiver, scaling the reflected path by thesurface reflection coefficient R

p ¼ffiffiffi2

ps0p0

eðik��Þs�

s�þ R

eðik��Þsþ

sþ

" #e�i!t: ð2:25Þ

If R is real (implying a phase change on reflection of 0 or ) it follows that

Fðsþ; s�Þ ¼e�2�s�

s2�þ R2e�2�sþ

s2þþ 2Re��ðs�þsþÞ

s�sþcosð2kxÞ; ð2:26Þ

where

x � sþ � s�2

¼ 2z0z

s� þ sþ: ð2:27Þ

Equation (2.26) can be interpreted as follows. The first and second terms on theright-hand side are associated with the energy from the direct and surface-reflectedray paths, respectively. The third term is due to interference between these two paths,resulting from the coherent addition of complex pressures. The expression is usefulfor broadband applications because the third term vanishes after averaging overfrequency.

An (equivalent) alternative version, convenient for narrowband applications, is

Fðsþ; s�Þ ¼ e��ðs�þsþÞ e�x

s�þ R

e��x

sþ

� �2

� 4R

s�sþsin2ðkxÞ

�: ð2:28Þ


Figure 2.3.

Radiation from a

point source in the

presence of a

reflecting boundary.

It is often the case that the distances sþ and s� are approximately equal, such that theproduct �x is sufficiently small to neglect the �x terms, and sþ � s� � sþ þ s�.Furthermore, for many applications the sea surface can be treated as a perfectreflector with a -phase change (i.e., R ¼ �1; see Box on p. 37). It then follows that

FcohðsÞ 4e�2�s

s2sin2

kz0z

s

� �; ð2:29Þ

where

s ¼ ðs� þ sþÞ=2: ð2:30Þ

The sequence of sinusoidal peaks and troughs predicted by Equation (2.29) is knownas a Lloyd mirror interference pattern. The ‘‘coh’’ subscript stands for ‘‘coherentaddition’’, indicating that the two contributions to the total pressure, from the directand reflected paths, respectively, are added with regard to their phase, beforesquaring. This means that the phase difference information is used for the purposeof combining the contributions from the two paths. Specifically, if Equation (2.25) iswritten in the form

p ¼ffiffiffi2

ps0 p0ðFþ þ F�Þ; ð2:31Þ

where

F� ¼ eðik��Þs�

s�e�i!t ð2:32Þ

and

Fþ ¼ Reðik��Þsþ

sþe�i!t; ð2:33Þ

it follows that

Fcoh ¼ jFþ þ F�j2: ð2:34Þ

The ‘‘incoherent’’ propagation factor is obtained by discarding the phase terms. Inother words

Finc ¼ jFþj2 þ jF�j2: ð2:35Þ

Alternatively, averaging Fcoh over (say) frequency

hFcohi ¼ hjFþj2i þ hjF�j2i þ h2jFþjjF�j cosð2kxÞi: ð2:36Þ

The first two terms are hardly affected by the average and together approximate toFinc. If the average is over several cycles of the cosine function, the third term can beexpected to average out to zero, and hence

hFcohi Finc: ð2:37Þ

For this reason, there are many situations in which a coherent sum (add and square),followed by an average over frequency, gives the same result as an incoherent sum(square and add).


The reflection coefficient at the air–sea boundary

The reflection coefficient at the sea surface is determined by the impedance of airrelative to that of water. The characteristic impedance of air for the assumedrepresentative conditions is given by

Zair ¼ �aircair ¼ 420 kg m�2 s�1;

more than three orders of magnitude smaller than that of water, which is equal to

Zwater ¼ �watercwater ¼ 1:53� 106 kg m�2 s�1:

The low impedance of air compared with that of water means that the acousticpressure required to achieve a given acoustic intensity is much smaller in air than inwater. From the continuity of pressure across the boundary it follows that thepressure on the boundary itself must also be small, and to first order this can beapproximated by the boundary condition p ¼ 0 at z ¼ 0. The only way this can beachieved for an incident plane wave of finite amplitude in water is for a reflectedwave to be generated at the surface of the same amplitude and opposite phase. Letthe horizontal and vertical wave numbers be � and , respectively, so that theincident wave can be represented by

pincident ¼ eið�x� zÞe�i!t

and the reflected wave by

preflected ¼ Reið�xþ zÞe�i!t:

By adding these two terms it can be seen that the only way the total pressurepincident þ preflected can be zero everywhere on the z ¼ 0 boundary is if R ¼ �1. Thishas two important consequences. First, the unit magnitude corresponds to 100%reflection of energy, so that sound becomes trapped in the sea, potentiallytraveling very long distances. Second, the negative sign means a phase changeof the reflected wave relative to the incident one, which results in near-perfectcancellation of acoustic pressure close to the sea surface.

In general, the reflection coefficient is a function of frequency, and of thephysical properties of the reflecting boundary. For example, the sea surface reflec-tion coefficient depends on the wave height and on the population of near-surfacebubbles created by breaking waves (see Chapters 5 and 8.)

2.2.2.2 Radiation from an infinite sheet of uniformly distributed dipoles

One of the factors that limit sonar performance is the presence of background noise inthe sea. Much of this background noise originates at the sea surface (e.g., due tobreaking waves). Consider an infinitesimal patch of sea surface with surface area Aand radiating power per unit surface area and solid angle WAO. The contributionfrom this patch to the mean square pressure at a receiver situated at a distance s (see


Figure 2.4) is

q2 ¼ �wcwWAO Ae�2�s

s2: ð2:38Þ

The sea surface behaves like a sheet of dipoles,6 with a radiation pattern proportionalto sin2 �, so that7

WAO ¼ 3

2WA sin2 �; ð2:39Þ

where � is the ray grazing angle (the angle between the ray path and the horizontal), sothat

q2 ¼ 3

2WA A �wcw

e�2�s

s2sin2 �: ð2:40Þ

The solid angle O subtended by the surface element r at the receiver, for anazimuthal increment �, is

O ¼ cos � � � ð2:41Þ


Figure 2.4. Radiation from a

sheet source element of

width r.

6 A dipole source is one made out of two out-of-phase monopole sources, placed an

infinitesimal distance apart (in practice, they must be separated by at most a small fraction

of a wavelength). An important distinction between a monopole and a dipole is that in the case

of the dipole source, there is no net change in volume. See Crocker (1997) for details.7 The constant 3=ð2Þ ensures that the power radiated per unit area, integrated over all solid

angles (into the lower half-space)Ð2WAO dO is WA. This follows from the use of

dO ¼ cos � d� d� and the resultð20

d�

ð=20

d� sin2 � cos � ¼ 2=3;

where � is the azimuth angle.

where (Figure 2.4)

� ¼ r sin �

s: ð2:42Þ

The corresponding element of area at the sea surface is

A ¼ r r � ð2:43Þso that

A ¼ s2 Osin �

; ð2:44Þ

and hence (taking the limit of infinitesimal O)

dq2

dO¼ 3

2�wcwWA sin � exp � 2�z

sin �

� �: ð2:45Þ

This is the mean square pressure per unit solid angle, at the receiver position. Nowassume that the contribution to the pressure field from each patch of the surface,covering an infinitesimal solid angle dO, is uncorrelated with all other contributions.This means that the energy contributions (mean square pressures) may be addedincoherently. The contribution from a concentric ring centered on the origin O istherefore obtained by replacing d� with 2, so that dO becomes 2 cos � d�. The totalmean square pressure is then found by integrating over �

q2 ¼ 2

ð=20

dq2

dOcos � d� ¼ 3�wcwWAE3ð2�zÞ; ð2:46Þ

where E3ðxÞ is a third-order exponential integral (see Appendix A). For small argu-ments, the limiting form may be used

limx!0

E3ðxÞ ¼ 12

ð2:47Þ

so that, in the case of negligible attenuation (�z � 1), Equation (2.46) becomes

q2 32�wcwWA: ð2:48Þ

As an example, consider the radiation of sound from the sea surface, alreadyidentified as an important source of background noise. The acoustic power radiatedby the sea surface due to wind (per unit area and bandwidth) can be written in theform8 (see Chapter 8 for details)

WAf ¼2

3�wcwK ; ð2:49Þ


8 The subscripts A and f denote derivatives with respect to area and frequency such that

Wf �dW

dfWA � dW

dAWAf �

d2W

dA df:

This is a generalization of the notation introduced previously for power and energy per unit

solid angle

WO � dW

dOEO � dE

dO.

where the parameter K varies with frequency f and wind speed v. An empiricalexpression for K , based on measurements over a wide range of frequencies is

K ¼ 1:32� 104vv2:24

1:5þ F 1:59mPa2 Hz�1 ð2:50Þ

where F is the frequency in units of kilohertz

F � f

1 kHzð2:51Þ

and vv is the wind speed in meters per second

vv � v1m/s

: ð2:52Þ

Throughout this book, standard SI units and prefixes are used so that 1 mPa(one micropascal) is equal to 10�6 Pa. See Appendix B for a complete list of SIprefixes.

2.2.3 Scattering of sound

2.2.3.1 Scattering from a small object

The likelihood that an echo from a distant object is detected depends on how muchsound is reflected (i.e., scattered) in the direction of the receiving system. The abilityof a small underwater object to scatter sound is quantified by its scattering cross-section, defined as the ratio of total scattered powerW to incoming intensity I (from aspecified grazing angle �in), of an incident plane wave

�ð�inÞ ¼W

Ið�inÞ: ð2:53Þ

Thus, � is the scattered power per unit incident intensity and has dimensions of area.A related quantity is the differential scattering cross-section, proportional to thepower per unit solid angle scattered in a specified direction9 (elevation �out, andbearing � relative to that of the incident plane wave):

�Oð�in; �out; �Þ ¼WOð�out; �Þ

Ið�inÞð2:54Þ


9 That is, the radiant intensity of the scattered sound.

so that

�ð�inÞ ¼ð�Oð�in;OoutÞ dOout; ð2:55Þ

where the shorthand Oout denotes the direction (�out; �) and dOout is an element ofsolid angle such that

dOout ¼ cos �out d�out d�: ð2:56Þ

The backscattering cross-section is defined as the differential cross-section evaluatedin the backscattering direction, multiplied by 4. In equation form10 (Pierce, 1989;Morfey, 2001):11

�backð�Þ � 4�Oð�; �; Þ: ð2:57Þ

The backscattering cross-section of a rigid sphere of radius a at high frequency(ka � 1), is (see Chapter 5)

�back ¼ a2: ð2:58Þ

2.2.3.2 Scattering from a rough surface

The echo from an underwater object can be masked by sound that happens to arriveat the same time, after it has been scattered from a rough boundary. For a roughsurface, the scattered power is proportional to the area ensonified by the incidentwave. In this situation it makes sense to define a (dimensionless) scattering coefficientas the differential scattering cross-section per unit scattering area; that is,

�OAð�in;OoutÞ �WOAðOoutÞ


The parameter �OA is also known as the scattering coefficient of a rough surface.Provided that the wind speed is low enough to neglect the influence of near-

surface bubbles, the scattering coefficient for the sea surface is approximately (exceptin directions close to that of specular reflection) (see Chapter 8)

�OAð�in; �out; �Þ CPM

16tan2 �in tan

2 �out; ð2:60Þ

where CPM is a constant equal to 0.0081. Often the scattering coefficient is written as afunction of a single angle �, in which case the backscattering direction is implied;


10 By ‘‘in the backscattering direction’’ is meant that the propagation direction of the scattered

wave is taken to be the same as that of the incident wave, except that its sense is reversed.11 An alternative definition of backscattering cross-section as the differential scattering cross-

section in the backscattering direction (i.e., omitting the factor 4 from Equation 2.57) is

sometimes used. In this book the definition of Equation (2.57) is used throughout. This point is

discussed further in Chapter 5.

that is,

�OAð�Þ � �OAð�; �; Þ CPM

16tan4 �: ð2:61Þ

Substituting for the numerical value of CPM gives

�OAð�Þ 1:61� 10�4 tan4 �: ð2:62Þ

2.3 ESSENTIALS OF SONAR SIGNAL PROCESSING

When a sound wave reaches a sonar receiver, the acoustic pressure is first convertedto an electrical voltage by an underwater microphone, or hydrophone. This voltagecould be displayed on an oscilloscope and monitored for evidence of something otherthan background noise, such as the voltage exceeding some pre-established threshold.Such a simple system might work in practice for a strong signal, while a weak onewould first need to be enhanced by signal processing. For example, the signal-to-noiseratio can be increased by filtering out sound at unwanted frequencies, or fromunwanted directions or ranges, or a combination of these. Before any such enhance-ment begins, the electrical signal is passed through an anti-alias filter and thendigitized.12 The details of subsequent processing depend on the characteristics ofthe expected signal, but almost all sonar systems use either a temporal filter to removenoise at unwanted frequencies or a spatial filter to remove noise from unwanteddirections or both. No distinction is made in the following between acoustic andelectrical signals. The justification for this is that the waveform, once digitized, can berescaled by an arbitrary constant factor to represent either the voltage or the originalpressure.

A time domain filter operation involves sampling a waveform in time andcombining successive samples in such a way as to remove any unwanted sound,whereas a spatial filter (or beamformer) samples in space instead of time. Both aredescribed below, with the purpose of introducing some basic relationships betweentime duration and frequency bandwidth (Section 2.3.1) and between spatial apertureand beamwidth (Section 2.3.2). A more complete treatment of signal processing ispresented in Chapter 6.

2.3.1 Temporal filter

Imagine a receiving system that passes frequencies between fmin and fmax and blocksfrequencies outside this range. Such a system is called a passband filter of bandwidthDf � fmax � fmin. If fmin is zero it is a low-pass filter; if fmax is infinite it is a high-passfilter. General filter theory is beyond the present scope, but it is useful to introduce


12 An anti-alias filter is one that removes high-frequency signals above a threshold that depends

on the sampling rate of the subsequent digital sampler. According to the Nyquist–Shannon

sampling theorem, the maximum acoustic frequency that can be correctly sampled is half of the

sampling rate. This maximum permissible frequency is known as the Nyquist frequency.

some basic concepts. A special kind of filter of particular interest is a discrete Fouriertransform (DFT),13 the basic properties of which are outlined below.

Let FðtÞ denote the time domain waveform of interest, sampled at discrete timestn at fixed intervals t. The DFT of FðtÞ is the spectrum Gð!Þ (see Appendix A fordetails)

Gð!Þ �XN�1

n¼0FðtnÞ expð�i!tnÞ; tn ¼ t0 þ n t: ð2:63Þ

The inverse transform is the operation that recovers the original function FðtÞ fromthe spectrum at discrete frequencies !m:

FðtÞ ¼ 1

N

XN�1

m¼0Gð!mÞ expðþi!mtÞ; !m ¼ 2

N tm: ð2:64Þ

For the special case of simple harmonic time dependence of angular frequency !

FðtÞ ¼ ei!t; ð2:65Þit follows that

Gð!mÞ ¼XN�1

n¼0exp½ið!� !mÞtn� ¼

sin ð!� !mÞDt2

�

sin ð!� !mÞt

2

� ; ð2:66Þ

where the time origin is chosen for convenience to be at the center of the sequence oftime samples (such that t0 þ tN�1 ¼ 0) and Dt is given by14

Dt ¼ N t; ð2:67Þapproximately equal to the signal duration.

If the signal is well sampled in time (such that jð!� !mÞ tj � 1), the denomi-nator of Equation (2.66) may be approximated by the argument of the sine function.The spectrum is then given by

Gð!mÞ N sincðyÞ; ð2:68Þwhere

y ¼ ð!� !mÞDt2

ð2:69Þ

and sincðyÞ is the sine cardinal function, defined (see Appendix A) as

sincðyÞ � sin y

y: ð2:70Þ

Written in this form it can be seen that the DFT operation, with output Gð!mÞ, is apassband filter, centered on !. The parameter Dt (the total time duration) determines

2.3 Essentials of sonar signal processing 43]Sec. 2.3

13 When the number of points in a DFT is a power of 2, a particularly efficient implementation

is possible. This efficient implementation is also known as a fast Fourier transform (FFT).14 The time between first and last samples is equal to Dt� t, which is approximately equal to

Dt if t is assumed small.

the frequency resolution of the filter through the argument of the sinc function.Specifically, the full width at half-maximum (fwhm), i.e., the spectral width betweenhalf-power points, is given by

!fwhm ¼ 4

Dtsinc�1

1ffiffiffi2

p� �

: ð2:71Þ

Evaluation of the inverse sinc function gives

sinc�1ð1=ffiffiffi2

pÞ 1:3916; ð2:72Þ

so the frequency resolution, as defined by

ffwhm � !fwhm2

; ð2:73Þ

is approximately equal to the reciprocal of the total time duration Dt

ffwhm ¼ 2 sinc�1ð1=ffiffiffi2

pÞ

1

Dt 0:886

Dt: ð2:74Þ

2.3.2 Spatial filter (beamformer)

Mathematically, there is no difference between spatial and temporal filtering, exceptthat spatial filtering can be carried out in more than one dimension. For simplicity,the scope is limited here to a single dimension, so the expression for a spatial DFT canbe obtained from Equations (2.63) and (2.64) by replacing the time variable ðtÞ withthe spatial one ðxÞ. It is also customary to represent the spatial ‘‘frequency’’ (the wavenumber) variable by the symbol k. Thus, the forward and inverse transforms are,respectively,

GðkÞ �XN�1

n¼0FðxnÞ expð�ikxnÞ; xn ¼ x0 þ n x; ð2:75Þ

and

FðxÞ ¼ 1

N

XN�1

m¼0GðkmÞ expðþikmxÞ; km ¼ 2

N xm: ð2:76Þ

A collection of hydrophones whose output is combined to carry out spatial filtering isknown as a hydrophone array or, if it extends in only one dimension, a line array.Consider a pressure field whose spatial distribution along such an array is of the form

FðxÞ ¼ eikx: ð2:77ÞBy an exact analogy with the time domain filter, if the origin is at the geometricalcenter of the array it follows that

GðkmÞ ¼sin ðk� kmÞ

Dx2

�

sin ðk � kmÞx

2

� ; ð2:78Þ


or (for sufficiently small hydrophone spacing x)

GðkmÞ N sincðyÞ; ð2:79Þwhere

y ¼ ðk� kmÞDx2; ð2:80Þ

and the parameter Dx isDx ¼ N x: ð2:81Þ

Here, the distance Dx plays the role of Dt in the time domain filter, and determines theresolution of the spatial filter in the wavenumber domain. It is approximately equal tothe length of the array.

The approximation Equation (2.79) requires the signal to be well sampled inspace such that

jk� kmj x � 1: ð2:82Þ

The fwhm in wave number, by analogy with Equation (2.71), is

kfwhm ¼ 4

Dxsinc�1

1ffiffiffi2

p� �

: ð2:83Þ

The output wavenumber spectrum of the hydrophone array is referred to as the arrayresponse. The squared magnitude of the normalized array response for an incidentplane wave, known as the beam pattern of the array, is

B ��GðkmÞGmax

��2 ¼ sin2 y

N 2 sin2y

N

ðsinc yÞ2: ð2:84Þ

Because of its ability to amplify selectively acoustic waves arriving from a narrowrange of angles (a ‘‘beam’’), the spatial filter is called a beamformer. To see how theangle selection process works, consider an array aligned along the x-axis (y ¼ z ¼ 0)and an acoustic plane wave traveling in a direction parallel to the x–y plane. The fieldof this plane wave as a function of space and time can be written

pðx; tÞ ¼ eikExe�i!t ¼ expðikxxÞeiðkyy�!tÞ: ð2:85Þ

Defining � as the angle between the wavenumber vector and the plane normal to thearray axis, the along-axis wavenumber component is15

kx ¼ 2

�sin �; ð2:86Þ

where � is the acoustic wavelength. Specializing to the field along the array (y ¼ 0),Equation (2.85) becomes

pðx; tÞ ¼ expðikxxÞe�i!t: ð2:87Þ

Now consider the field at an arbitrary instant in time (say t ¼ 0) and use this field as

2.3 Essentials of sonar signal processing 45]Sec. 2.3

15 Similarly, ky ¼ ð2=�Þ cos �.

input to the beamformer, so that

FðxÞ � pðx; 0Þ ¼ expðikxxÞ: ð2:88Þ

The response is Equation (2.79), with

y ¼ 2

�sin �� km

� �Dx2: ð2:89Þ

If the magnitude of km does not exceed 2=�, there exists an arrival angle � at whichthe beamformer output is maximized, corresponding to y ¼ 0. This value of � is givenby

�m ¼ arcsinkm2=�

ð2:90Þ

and is known as the beam steering angle. It is measured from the direction perpen-dicular to the array axis, known as the broadside direction. Beam patterns for twodifferent steering angles are shown in Figure 2.5.

The angular width of the beam varies with steering angle as follows. Taking afinite difference of Equation (2.86)

kx 2

�cos � �; ð2:91Þ

the fwhm beamwidth is obtained by equating the right-hand sides of Equations (2.83)


Figure 2.5. Beam patterns for L=� ¼ 5 and steering angles 0, 45 deg as indicated.

and (2.91):

�fwhm 2 sinc�1ð1=ffiffiffi2

pÞ

�

Dx cos �m

: ð2:92ÞIn radians this is

�fwhm 0:886�

Dx cos �m

rad ð2:93Þand in degrees

�fwhm 50:8�

Dx cos �m

deg: ð2:94Þ

This approximation for the beamwidth works best at angles close to the broadsidedirection. For the case of Figure 2.5, the predicted and observed half-power widthsare about 10 deg at broadside ( ¼ 0) and 14 deg at ¼ 45 deg. The approximationbreaks down at angles close to�90 deg from broadside (i.e., parallel to the array axis,known as the endfire direction), due to the singularity in the derivative d�=dkx in thatdirection. The equation for wavenumber width (Equation 2.83) is valid at any angle(see Chapter 6).

2.4 ESSENTIALS OF DETECTION THEORY

The calculation of detection probability is the whole point of sonar performancemodeling and the ultimate goal of this book. Hence, considerable attention is paid toits calculation. The end result of the processing, after all filtering, is presented to asonar operator, whose job it is to report the detection or not of a (potential) sonartarget, based on the information provided by the sonar. Depending on the signal-to-noise ratio, the probability of making a detection might be high or low, but it is nevercertain. The objective of statistical detection theory is to quantify this probability.

2.4.1 Gaussian distribution

In this section, expressions are derived for the probability of detection (denoted pd)for a simple case involving a constant signal in Gaussian noise.16 The signal isrepresented by the constant xS and noise by the variable xNðtÞ. The signal, if present,is always accompanied by a noise background, and the combination of both isrepresented by xSþNðtÞ. The parameter x can be the amplitude or energy of anacoustic wave, depending on the processing, and is referred to below as the‘‘observable’’.

The two possibilities ‘‘signal present’’ and ‘‘signal absent’’ are represented by thetotal observable xtot given by either

xtotðtÞ ¼ xSþNðtÞ ðsignal presentÞ ð2:95Þ

2.4 Essentials of detection theory 47]Sec. 2.4

16 The choice of constant signal and Gaussian noise is for mathematical convenience and does

not necessarily represent a realistic situation for a sonar system. More realistic distributions are

considered in Section 2.4.2.

or

xtotðtÞ ¼ xNðtÞ ðsignal absentÞ: ð2:96Þ

A decision-maker (the sonar operator) presented with the data sequence xtotðtÞ deemsa signal to be present whenever the value of xtot exceeds some threshold xT. To avoidtoo many false alarms it is desirable for the threshold xT to exceed the noise, in someaverage sense, but how should the noise be averaged, and by how much must thisaverage be exceeded? The answer depends, among other things, on the rate of falsealarms considered acceptable. The larger the threshold, the fewer false alarms willresult, at the expense of a reduced probability of detection.

Assuming that the operator must always choose between the two decisions‘‘signal present’’ and ‘‘signal absent’’, irrespective of the chosen threshold thereare always four possible outcomes, according to Table 2.1.

Because of the statistical fluctuations in the noise there is always a chance that thethreshold is exceeded when there is no signal, and conversely there is also a chancethat the threshold is not exceeded even when the target is present. Both situations leadto an incorrect decision, indicated in the table by gray shading. The probability ofmaking a correct ‘‘signal present’’ decision is known as the detection probability anddenoted pd. The false alarm probability pfa is the probability of making an incorrect‘‘signal present’’ decision.

One’s objective is to make the correct decision as often as possible. In otherwords, to maximize pd, implying a low threshold, while at the same time minimizingpfa, which requires a high threshold. These conflicting requirements are resolved inpractice by deciding in advance on a highest acceptable false alarm rate, and thendetermining the threshold consistent with this rate. Thus, the values of pd and pfadepend on the choice of xT as well as on the statistical fluctuations of noise and ofsignalþnoise.

The following calculations assume a randomly fluctuating noise observable xNðtÞwith Gaussian statistics, and a non-fluctuating signal so that the signal-plus-noise(xSþN) has the same statistics (the same Gaussian distribution with the same standarddeviation) as noise alone (xN). Expressions for pfa and pd are derived below for theseassumptions.


Table 2.1. Detection truth table; pd is the probability of deciding correctly

that a signal is present (‘‘detection probability’’) and pfa is the probability of

declaring a detection when there is no signal.

Threshold exceeded Threshold not exceeded

xtot > xT xtot < xT

Signal present Correct decision Incorrect decision

xtot ¼ xSþN (probability pd) (probability 1� pd)

Signal absent Incorrect decision Correct decision

xtot ¼ xN (probability pfaÞ (probability 1� pfa)

2.4.1.1 Noise only

Let the probability density function (pdf ) of the noise observable be fNðxÞ, so that themean and variance of the distribution are

xN ¼ðþ1

�1x fNðxÞ dx ð2:97Þ

and

�2 ¼ðþ1

�1ðx� xNÞ2fNðxÞ dx; ð2:98Þ

respectively. The Gaussian distribution with these properties (see Figure 2.6, uppergraph) is

fNðxÞ ¼1ffiffiffiffiffiffi2

p�exp � ðx� xNÞ2

2�2

" #: ð2:99Þ

2.4.1.2 Signal plus noise

Similarly, if the non-fluctuating signal is added

fSþNðxÞ ¼1ffiffiffiffiffiffi2

p�exp � ðx� xSþNÞ2

2�2

" #; ð2:100Þ

illustrated by the lower graph of Figure 2.6. Assuming that the observable terms addlinearly, if the signal is constant, the signal-plus-noise can be written

xSþNðtÞ ¼ xS þ xNðtÞ ð2:101Þand therefore

xSþN �ðþ1

�1x fSþNðxÞ dx ¼ xS þ xN: ð2:102Þ

Thus, the signal-plus-noise distribution has the same pdf as the noise alone but with ahigher mean value.

Suppose that a detection is declared by the operator whenever the threshold xT isexceeded. The probability of this occurring as the result of a single observation isequal to the area under the pdf curve to the right of the threshold. This area,depending on whether in reality a signal is absent or present, is either pfa or pd.In other words, respectively,

pfa ¼ð1xT

fNðxÞ dx ¼ 1

2erfc

xT � xNffiffiffi2

p�

� �ð2:103Þ

where erfcðxÞ is the complementary error function (Appendix A), or

pd ¼ð1xT

fSþNðxÞ dx ¼ 1

2erfc

xT � xSþNffiffiffi2

p�

� �: ð2:104Þ

In this treatment, a large negative result is arbitrarily not considered a thresholdcrossing. This choice might be justified if the observable is a positive definite quantity,such that the negative tail of the Gaussian has no physical meaning. If large negative



Figure 2.6. Probability density functions of noise (upper graph) and signal-plus-noise (lower)

observables. The threshold for declaring a detection, xT, is shown as a vertical dashed line. The

shaded areas are the probabilities of false alarm (upper graph) and detection (lower). The

example shown is for the case xN ¼ 2�, xSþN ¼ 5�, and xT ¼ 4�.

values were considered to be threshold crossings, the expressions for both pfa and pdwould then need to include contributions from values of x between �1 and �xT.

Both pfa and pd vary between 0 and 1. It is convenient to replace xT in Equation(2.104) by expressing it as a function of pfa (from Equation 2.103). The result is

pd ¼ 1

2erfc erfc�1ð2pfaÞ �

xSffiffiffi2

p�

�: ð2:105Þ

2.4.2 Other distributions

The analysis of sonar detection problems requires the consideration of morecomplicated distributions than that of a constant signal in Gaussian backgroundnoise. A preview of some important results from Chapter 7 is presented below. Ineach case, expressions are quoted for the false alarm probability pfa and detectionprobability pd as a function of the signal-to-noise ratio (SNR).

2.4.2.1 Coherent processing (Rayleigh statistics)

Coherent processing for Gaussian noise results in a Rayleigh distribution for thenoise amplitude A. For an amplitude threshold AT, and assuming a Rayleigh dis-tribution for the signal as well as for the noise, the false alarm and detectionprobabilities are

pfa ¼ exp � A2T

2�2

!ð2:106Þ

and

pd ¼ p1=ð1þRÞfa ; ð2:107Þ

where R is the SNR

R ¼ A2

2�2: ð2:108Þ

2.4.2.2 Incoherent processing (chi-squared statistics with many samples)

Incoherent addition of a number of Rayleigh-distributed samples results in a chi-squared (or ‘‘�2’’) distribution for the total energy. The detection and false alarmprobabilities can be expressed for this distribution in terms of special functions asdescribed in Chapter 7. If the number of samples M is sufficiently large (M > 100),these expressions simplify to those presented below. For an energy threshold ET, thefalse alarm probability becomes

pfa 1

2erfc

ffiffiffiffiffiM

2

rET

2M�2� 1

� �" #; ð2:109Þ

where � is the standard deviation of the noise samples before any averaging. The


detection probability simplifies to

pd 1

2erfc

erfc�1ð2pfaÞ �ffiffiffiffiffiffiffiffiffiffiM=2

pR

1þ R

" #; ð2:110Þ

where R is the power signal-to-noise ratio. Equation (2.110) can be rearranged for R:

R ¼ erfc�1ð2pfaÞ � erfc�1ð2pdÞffiffiffiffiffiffiffiffiffiffiM=2

pþ erfc�1ð2pdÞ

: ð2:111Þ

If M 1=2 is large compared with erfc�1ð2pdÞ, this simplifies further to

R erfc�1ð2pfaÞ � erfc�1ð2pdÞffiffiffiffiffiffiffiffiffiffiM=2

p : ð2:112Þ

The condition onM makes Equation (2.112) mainly relevant to situations involving alow SNR. If R and M are both large, there is usually no need for a detailed analysis,because in this situation the detection probability is always close to unity.

2.5 REFERENCES

Clarke, G. L. and James, H. R. (1939) Laboratory analysis of the selective absorption of light

by sea water, J. Opt. Soc. Am., 29, 43–55.

Crocker, M. J. (1997) Introduction, in M. J. Crocker (Ed.), Encyclopedia of Acoustics, Wiley,

New York.

Morfey, C. L. (2001) Dictionary of Acoustics, Academic Press, San Diego.

Pierce, A. D. (1989) Acoustics: An Introduction to Its Physical Principles and Applications,

American Institute of Physics, New York.


3

The sonar equations

If you cause your ship to stop, and place the head of a long tubein the water, and place the other extremity to your ear,

you will hear ships at a great distance from you

Leonardo da Vinci (15th century).

3.1 INTRODUCTION

The objective of this chapter is to illustrate the basic principles of sonar performancemodeling. This is achieved by deriving the most important passive and active sonarequations, each accompanied by a worked example. These worked examples areintended to be didactic rather than realistic: enough realism is included in them toillustrate the underlying principles, but no more—where there is a conflict betweensimplicity and realism then simplicity is preferred, except at the expense of theprinciple itself.1

3.1.1 Objectives of sonar performance modeling

The objective of sonar performance modeling is to quantify sonar performance,enabling a decision-maker to:

— predict the likelihood that a given sonar task, such as the detection of asubmerged object, will be carried out successfully;

1 More realistic examples are provided in Chapter 11.

— compare the effectiveness of different sonar designs in carrying out a given task;— compare the effectiveness of different strategies for carrying out a given task.

Examples of possible sonar tasks, in addition to detection, are localization,classification, evasion,2 surveillance, and communication. Irrespective of the applica-tion, sonar effectiveness must depend on the probability of making a successfuldetection each time the sonar is used. Less obvious, but equally important, is theobservation that sonar effectiveness also depends on the number of false alarms,3

because of the time and other resources wasted on investigating these. Much of sonarperformance modeling, and the main thrust of this book, is concerned with thecalculation of the probabilities of detection and false alarm for a given scenario orscenarios.

3.1.2 Concepts of ‘‘signal’’ and ‘‘noise’’

A sonar receiver is a complicated piece of equipment, typically comprising

— a hydrophone, or an array of hydrophones, to convert an underwater pressuredisturbance into an electronic one;

— a suite of signal-processing algorithms,4 to enhance the signal-to-noise ratio;— a display unit, to help the sonar operator determine whether an object of interest

(a target) is present.

Pressure fluctuations5 at the receiver can be thought of as a linear sum of two differentkinds:

— those caused by the presence of the target (the signal);— all other pressure fluctuations (the noise).

The ‘‘target’’ is any object that we wish to detect.The above definitions of signal and noise are necessarily vague, as the distinction

between them depends on details of the signal processing that have not yet beenspecified. The noise definition as ‘‘all sound that is not part of the signal’’ means that

54 The sonar equations [Ch. 3

2 Although the task of evasion is not performed directly by the sonar, the modeling of evasion

and the development of evasion tactics are nevertheless an important application of sonar

performance modeling.3 Fluctuations in noise levels alone can result in a sonar detection system erroneously reporting

the presence of a target. Such an event is known as a ‘‘false alarm’’.4 The algorithms can be implemented either in hardware or software.5 Strictly speaking, what matters is the voltage in an electrical or electronic circuit, after

filtering. For simplicity we assume for now that, to within a multiplying constant, the pressure

and voltage fluctuations are identical. The validity of this assumption requires the hydrophone

sensitivity and filter response (or at least their product) to be independent of frequency, within

the bandwidth of interest.

many different potential sound sources need to be taken into account.6 Each case isdifferent, and knowledge of which noise sources to include in a model is acquired byexperience. Common sources of ambient noise are wind and shipping. Also importantis self-noise, especially from the sonar platform.

A special kind of noise that is unique to active sonar, known as reverberation, isthe sound originating from the sonar transmitter and subsequently scattered byunderwater boundaries and obstacles other than the target, before arriving back atthe receiver. The combined effect of ambient noise, self-noise, and reverberation isknown as the background.

The sonar equation is an expression for the signal-to-noise ratio (or moregenerally signal-to-background ratio) written as a product of energy ratios, andusually expressed in decibels. The conversion to decibels turns the product of ratiosinto a sum of the logarithms of these ratios.

3.1.3 Generic deep-water scenario

For the purpose of the present chapter, attention is restricted to a specific deep-waterscenario, in which the following simplifying assumptions and approximations aremade:

— reflections from the seabed are neglected, equivalent to assuming an infinite waterdepth;

— density and sound speed are assumed to be uniform everywhere in the sea;— all background noise (including reverberation) is assumed to originate at the sea

surface.

The scenario resulting from these assumptions is not a realistic one, but it containsenough realistic features (the reflecting sea surface, a source of reverberation, andsome basic signal processing) to illustrate the main principles involved. More realisticapplications, without these simplifying assumptions, are described in Chapter 11.

A sonar equation is derived for each main category of sonar, followed by aworked example. These examples make some additional assumptions that serve tosimplify the calculations. For example, for passive sonar the background noise isassumed to arise only from wind; and the target is assumed to be located in thebroadside beam of a horizontal line array (a sequence of hydrophones placed along astraight horizontal line).

3.1.4 Chapter organization

The remainder of this chapter is divided into two main sections, one on passive sonar(Section 3.2) and one on active sonar (Section 3.3). These sections are further dividedinto sub-sections concerned with coherent and incoherent processing. In each of the

3.1 Introduction 55]Sec. 3.1

6 In some situations non-acoustic sources of noise can also be important.

four sub-sections the relevant sonar equation is derived, and illustrated by means of aworked example.

3.2 PASSIVE SONAR

3.2.1 Overview

This section is concerned with analysis of the performance of a passive sonar system,which relies on detecting sounds emitted by an underwater object (the ‘‘target’’). Inthe situation illustrated by Figure 3.1, the whale on the left (whale ‘‘A’’, representingthe target) emits a communication signal that is detected by the one on the right(whale ‘‘B’’, representing the sonar). A stylized radiated power spectrum, comprisinga series of lines (tonals) superimposed on a smoothly varying background, is illus-trated by Figure 3.2 (upper panel ). Collectively, the tonals are referred to as thenarrowband spectrum of the target, whereas the smooth background is its broadbandspectrum. The source of sound (i.e., the target) is assumed here to radiate soundcontinuously and uniformly in all directions (omni-directionally). The signal isfurther assumed to be infinite in duration and statistically stationary.

The acoustic signal is transmitted through the propagating medium (seawater)and can be detected by means of suitable receiving equipment, such as the human earor an underwater hydrophone. Some frequencies propagate better than othersthrough the same medium, so the received spectrum, while retaining the samequalitative features as the transmitted one, has a different shape, illustrated by Figure3.2 (lower panel ). By the time it arrives at the receiver, the sound power radiated bythe target has spread from a point to a finite area, so the physical parameter ofinterest is power per unit area (i.e., intensity, instead of power). Thus, the lower panelshows the spectral density of sound intensity at the receiver. In principle bothnarrowband and broadband spectra can be exploited by a suitable detector, requiringdifferent processing techniques to do so. Both are considered below, after thedefinition of some standard nomenclature.


Figure 3.1. Principles of passive detection: the sonar target (whale A) emits a sound wave that

travels through the sea and is detected by the sonar receiver (whale B).

3.2 Passive sonar 57]Sec. 3.2

Figure 3.2. Spectral density level of the radiated power at the source (upper) and intensity at the

receiver (lower).

3.2.2 Definition of standard terms (passive sonar)

3.2.2.1 Mean square pressure, sound pressure level, and the decibel

The concept of mean square (acoustic) pressure (abbreviated MSP) is one of keyimportance to sonar performance modeling and plays a central role in this book. Foracoustic pressure qðtÞ and averaging time T , the mean square pressure Q is defined as

Q � 1

T

ðT0

q2ðtÞ dt: ð3:1Þ

The result of Equation (3.1) (MSP) is independent of T for large T if qðtÞ isstatistically stationary in time.

Another important parameter is the acoustic intensity I, a vector quantity whosemagnitude I , for a plane-propagating wave, is equal to the MSP (Q) divided by thecharacteristic impedance of the medium

I � jIj ¼ Q

�c(plane wave). ð3:2Þ

For types of wave other than a plane wave, the relationship between intensity andMSP is a more complicated one, but one can define the equivalent plane wave intensity(EPWI) of any statistically stationary pressure field as the intensity of a plane wave ofthe same MSP. In other words, denoting this quantity IEPWI,

IEPWI �Q

�c(any pressure field). ð3:3Þ

In some publications the term ‘‘intensity’’ is used as a synonym for EPWI, and thesonar equation is expressed as a product of ‘‘intensity’’ ratios. In the following, adeliberate choice is made to characterize sound waves by their MSP and not EPWI, asthis avoids ambiguities associated with the choice of impedance, and is consistentwith the definition of propagation loss in common use (see Appendix B). Thedistinction becomes an important one if the impedance at the source is differentfrom that at the receiver.

Except in some special situations, neither MSP nor EPWI are proportional to thetrue acoustic intensity. The sonar equations derived in this chapter and in Chapter 11are all expressed in terms of MSP ratios and do not rely on any particular relationshipbetween sound pressure and intensity.

Sound pressure level is the mean square pressure expressed in decibels. The decibelis a logarithmic unit of power or energy (see Appendix B for details). The conversionto decibels involves the following three operations: divide by a standard referencevalue of the parameter, take the base-10 logarithm, and multiply by 10. Thus, thesound pressure level (SPL) is:

SPL � 10 log10Q

p2ref: ð3:4Þ

The reference value of MSP is p2ref and the internationally accepted value of pref foruse in underwater acoustics is one micropascal (1 mPa).


Consider a point source at the origin. Regardless of the acoustic environment andsource–receiver geometry, the mean square pressure Q at an arbitrary location x canalways be written in the form

QðxÞ ¼ p20s20FðxÞ; ð3:5Þ

where FðxÞ is the propagation factor, which is defined by Equation (3.5); and p0 isthe RMS pressure at a small distance s0 from the source.7

It is conventional to write Equation (3.5) in logarithmic form

10 log10Q

p2ref¼ 10 log

p20s20

p2refr2ref

!þ 10 log10

F

r�2ref; ð3:6Þ

where rref is a constant reference distance, with an internationally accepted value of1 meter (1m). The left-hand side of Equation (3.6) is the sound pressure level.

It is conventional to present all sonar equation terms in decibels (dB). Using theabove recipe, a power-like quantity x is converted to its dB equivalent Lx by dividingby its reference unit xref and applying the relationship

Lx � 10 log10x

xref: ð3:7Þ

For this reason the combination 10 log10 appears repeatedly. Often, for notationalconvenience, the denominator of the argument is omitted, in which case the referencevalue is quoted instead as a qualifier to the dB unit. For example, the meaning of

X � 10 log10 x dB re xref ð3:8Þ

is identical to that of Equation (3.7). Thus, the power level of a 1-watt source is10 log10(1W/1 pW)¼ 120 dB re pW. Similarly, a sinusoidal pressure wave of ampli-tude 1 Pa has a mean square pressure of 0.5 Pa2, and consequently a sound pressurelevel of 10 log10[0.5 Pa

2/(1 mPa)2]¼ 117.0 dB re mPa2. The choice of mPa2 as a refer-ence unit for SPL, adopted here in preference to the more conventional mPa, followsnaturally from Equation (3.8) (or Equation 3.7) with x equal to the mean squarepressure.

In general, a level quoted in decibels needs to be accompanied by a statement ofthe corresponding reference unit, even if an international standard exists for its value.This is partly because standards can, and do, change with time and circumstances andpartly because not all users of decibels adhere to these standards.8 The only safe


7 Equation (3.5) holds for a point source. The distance s0 from the source at which p0 is

measured must be small enough for distortions due to absorption, refraction, reflection, or

diffraction to be negligible.8 In water, reference pressures of 20 mPa and 1 mbar were both used before the modern value of

1 mPa became widespread. At the time of writing, a reference pressure of 1 mbar (and a reference

distance of 1 yd) is still in use by at least one sonar manufacturer. Further, the reference

pressure used for sound in air is 20 mPa, not 1 mPa, making it unclear which value to use in

situations involving a mixture of air and water, such as in foam caused by breaking waves.

exception to this general rule occurs with dimensionless quantities, for which it seemsreasonable to assume a reference unit of 1.

3.2.2.2 Source level

On the right-hand side of Equation (3.6) there are two terms. The first, a measure ofsource power, is known as the source level

SL � 10 log10 S0 dB re mPa2 m2; ð3:9Þ

where S0 is the product

S0 ¼ p20s20: ð3:10Þ

It is remarkable that a parameter of such fundamental importance to sonar as S0 doesnot have a widely accepted name. The term source factor is adopted here.9 It is moreconventional to define source level as the sound pressure level at a standard referencedistance (rref ) from the source (ASA, 1994; IEC, www). For a point source in freespace, the numerical value is the same provided that rref is the unit distance inwhatever units system is used (i.e., 1 meter in the SI system), but the conventionaldefinition leads to difficulties for an extended source such as a ship or an array ofsonar projectors (see Chapter 11). Because p0 varies with distance in such a way thatp0s0 is constant, the source factor is also a constant, independent of measurementposition close to the source, making it a natural physical quantity with which tocharacterize the source. Although source level is defined in terms of pressure p0, dueto the s0 scaling (through Equation 3.10), in practice it is actually a measure ofradiated power. Thus, an alternative expression for the source factor is

S0 ¼ �cWO; ð3:11Þ

where WO is the radiant intensity (power per unit solid angle).For an omni-directional source of power W , the source factor is equal to

�cW=4�. For example, if the source power is 1 watt (W ¼ 1W), then from Equation(3.9), S0 ¼ 0.122 kPa2 m2, corresponding to a source level of 170.9 dB re mPa2 m2.10

3.2.2.3 Propagation loss

The second term on the right-hand side of Equation (3.6) is (minus) the propagationloss11

PL � SL� SPL ¼ �10 log10 FðxÞ dB re m2; ð3:12Þ


9 The combination p0s0, which is the square root of the source factor, is referred to by ASA

(1989) as the ‘‘source product’’.10 For the assumed conditions (T ¼ 10 �C, S ¼ 35, at atmospheric pressure).11 In underwater acoustics a synonymous term to ‘‘propagation loss’’ is ‘‘transmission loss’’.

The term ‘‘propagation loss’’ is used here to avoid possible confusion with alternative

definitions of ‘‘transmission loss’’ from other branches of acoustics such as sound transmission

through a wall (Morfey, 2001).

or equivalently,

PL ¼ 10 log10S0Q

dB re m2: ð3:13Þ

This term plays the role of transfer function between source and receiver. The ratioS0=Q has dimensions of area, so the unit of propagation loss is dB rem2.

3.2.2.4 Noise spectrum level and array response

The mean square pressure of background noise within a specified bandwidth (usuallythe processing bandwidth of the sonar receiver) is denoted QN. Thus, the SPL ofbackground noise in the same bandwidth, known as the noise level, is

NL � 10 log10 QN dB re mPa2: ð3:14Þ

This background noise is often broadband in nature, so it is useful to consider noisespectral density (denoted QN

f ) and the corresponding noise spectral density level (ornoise spectrum level ) NLf , defined as

12

NLf � 10 log10 QNf dB re mPa2 Hz�1: ð3:15Þ

The background against which the signal is to be detected is that at the output of thebeamformer (i.e., the array response, denoted YN), not at the hydrophone. Thespectral density of this quantity (denoted YN

f ) can be written as the integral of thenoise spectral density over all solid angles O, weighted by the beam pattern BðOÞ:

YNf ¼

ðQNfOðOÞBðOÞ dO: ð3:16Þ

For the special case of isotropic noise, meaning that the magnitude of noise spectraldensity is independent of direction such that

QNfO ¼

QNf

4�; ð3:17Þ

it follows that

YNf ¼ QN

f

�O4�; ð3:18Þ

where �O is the solid angle footprint of the beam pattern (in steradians), defined by

�O ¼ðBðOÞ dO: ð3:19Þ


12 The use of the subscript f for logarithmic quantities (expressed in decibels) indicates a

spectrum level (i.e., a spectral density expressed in decibels).

3.2.2.5 Signal-to-noise ratio, array gain, and directivity index

Now consider a monochromatic signal13 whose MSP per unit solid angle is QSO. By

analogy with Equation (3.16), the array response to this signal (denoted Y S) is

Y S ¼ðQS

OBðOÞ dO: ð3:20Þ

Assuming further that this signal is in the form of an incoming plane wave from thedirection OS ¼ ð�S; �SÞ, such that

QSO ¼ QS�ðO� OSÞ; ð3:21Þ

where �ðxÞ is the Dirac delta function (see Appendix A), the array response is then

Y S ¼ðQS�ðO� OSÞBðOÞ dO ¼ QSBðOSÞ: ð3:22Þ

The value of the signal-to-noise ratio (SNR) depends on where it is measured in theprocessing chain. In particular, its value after beamforming (denoted Rarr) is differentfrom its value at the hydrophone, before any processing (Rhp). The ratio of these twoSNR values, expressed in decibels, is the array gain; that is,

AG � 10 log10 GA dB; ð3:23Þwhere

GA ¼ RarrRhp

: ð3:24Þ

Here, the hydrophone SNR (Rhp) is defined as the ratio of signal mean squarepressure (MSP) to noise MSP at the receiving hydrophone

Rhp �QS

QN; ð3:25Þ

where QN is the noise MSP integrated over the sonar processing bandwidth

QN ¼ðQNf df : ð3:26Þ

Similarly Rarr is the SNR at the output of the beamformer, for a ‘‘flat response filter’’(i.e., one whose response is independent of frequency within a specified passband14

and zero everywhere else)

Rarr �Y S

YN: ð3:27Þ

A related parameter is the directivity index (DI) of an array, which is the array gainfor the special case of a plane wave signal and isotropic noise. Unlike AG, DI is aproperty of the array and the acoustic frequency only. Thus, it is independent of


13 The single frequency assumption is relaxed in Section 3.2.4.4.14 Unless otherwise stated, this passband is understood to be the processing bandwidth

(denoted �f or Df depending on context).

medium and target properties and is usually easier to calculate. For this reason, DI isoften used as an approximation to AG in the sonar equation.

3.2.2.6 Signal gain and noise gain

The array gain can be expressed in terms of the signal gain SG:

SG � 10 log10Y S

QSdB ð3:28Þ

and noise gain NG:

NG � 10 log10YN

QNdB; ð3:29Þ

such thatAG ¼ SG�NG: ð3:30Þ

According to these definitions, SG and NG are both negative. For a well-designedbeamformer, the magnitude of NG is usually greater than that of SG, in which caseAG is positive.

3.2.2.7 Detection threshold and signal excess

The SNR threshold R50 is the value of Rarr (more generally, that of the SNR after allprocessing) required to achieve a detection probability of precisely 50%.15 (The valueof R50 depends on statistical fluctuations present in both signal and noise, and on thefalse alarm probability). When expressed in decibels, this quantity is known as thedetection threshold

DT ¼ 10 log10 R50 dB: ð3:31Þ

Unlike the amplitude threshold AT (in volts, or pascals) introduced in Chapter 2,which is the value of the signalþnoise amplitude in volts, or pascals (or energythreshold ET in V

2 s or Pa2 s) above which a ‘‘signal present’’ decision is triggered,the detection threshold (R50) is a dimensionless parameter. The signal excess isdefined as the amount by which the SNR exceeds the detection threshold, in decibels:

SE � 10 log10 Rarr �DT: ð3:32Þ

It follows from this and from the definition of array gain that

SE ¼ 10 log10 Rhp þAG �DT: ð3:33Þ

Equation (3.33) is the sonar equation. Written like this it looks simple, and in somespecial circumstances it is. To a large extent, the purpose of this book is to explainhow to calculate each of the terms on the right-hand side and corresponding ones foractive sonar (see Section 3.3). The examples and special cases considered in theremainder of Chapter 3 are deliberately simplified in order to illustrate the mainprinciples involved.


15 The symbol X50 is used to denote the value of any variable X required to achieve a detection

probability of 50%.

3.2.3 Coherent processing: narrowband passive sonar

This section is concerned with the calculation of the probability of detection ( pd) for anarrowband passive sonar. The term ‘‘narrowband’’ (abbreviated ‘‘NB’’) implies thatthe signal may be described, to a first approximation, by sound of a single frequency(see Figure 3.3). The processing considers a very narrow range of frequencies at atime, thus minimizing the noise in each processing band. The bandwidth is thenassumed to be large enough to contain the entire signal, but sufficiently small forthe noise power to be directly proportional to the bandwidth. Under these circum-stances the signal-to-noise ratio, and hence also the detection probability, increasewith decreasing bandwidth. A narrowband signal is called a tonal because of itsresemblance to a single frequency tone in music. The following sub-sections lookfirst at the sonar signal, then the background noise, signal-to-noise ratio, and finallythe probabilities of detection and false alarm. A special case involving a horizontalline array is introduced in Section 3.2.3.7, followed by a worked example for thisspecial case (Section 3.2.3.8).

3.2.3.1 Signal (single hydrophone)

For a NB system the sonar signal is the received mean square pressure QS associatedwith one of the transmitted tonals (red line in Figure 3.3). With the assumption of aninfinite water depth, there are two contributions to the received signal, one from thedirect path and one from a surface reflection. We assume that the sea surface issmooth, so that the direct and surface-reflected paths add coherently. If the tonalpower is W S it follows from Chapter 2 that

QS ¼ �cW S

4�FNBðr; zarr; ztgtÞ; ð3:34Þ

where zarr is array depth;ztgt is target depth;r is horizontal separation between array and target; and

FNB is the coherent propagation factor.

Assuming that the horizontal separation r is large compared with the productzarrztgt, where is the attenuation coefficient, this can be written:

FNB ¼ Fcohðr; zarr; ztgtÞ 4

r2e�2r sin2

kzarrztgt

r

� �: ð3:35Þ

The sine-squared behavior in Equation (3.35) is a result of alternate constructive anddestructive interference between the two paths, illustrated in Figure 3.416 for afrequency of 300Hz. The separation between successive peaks (or troughs) is�r=kzarr in depth and �r

2=kzarrztgt in range. This fringe pattern is known as a Lloydmirror interference pattern after an analogous effect from optics. For the example


16 This graph and many subsequent ones, as acknowledged in the individual captions, are

calculated using the sonar performance model INSIGHT (Ainslie et al., 1996).


Figure 3.3. Spectral density level of the transmitter source factor (upper) and mean square

pressure at the receiver (lower).

shown (the array depth is 30m), at a range of 300m the fringe spacing in depth isapproximately 25m.

3.2.3.2 Noise (single hydrophone)

We consider a broadband noise spectrum illustrated by Figure 3.5. Conceptually, thetotal noise entering the processing bandwidth �f is the area under the curve betweenthe dashed lines; that is,

QNðzÞ ¼ �fQNf ðzÞ; ð3:36Þ

where QNf is the spectral density of the ambient noise MSP. For the isovelocity case,

with infinite water depth, this can be written (see Chapter 2)

QNf ðzÞ 3�cE3ð2zÞWN

Af ; ð3:37Þ

where WNAf is the power spectral density of the noise source per unit of sea surface

area, and E3ðxÞ is a third-order exponential integral (see Appendix B). Here, theprocessing bandwidth is the analysis bandwidth of the Fourier transform, equal tothe reciprocal of the coherent processing time.


Figure 3.4. Coherent propagation loss PL ¼ �10 log10ðFcohÞ [dB re m2] vs. range r and target depth ztgtfor array depth zarr ¼ 30m and frequency f ¼ 300Hz (INSIGHT).

3.2.3.3 Signal-to-noise ratio, signal excess, and narrowband passive sonar equation

To derive the NB sonar equation we start by calculating the signal-to-noise ratio forthe array (rearranging Equation 3.24 for Y S=YN and substituting the result inEquation 3.27) as

Rarrðr; zarr; ztgtÞ �Y S

YN¼ QS

QNGA: ð3:38Þ

Recall that the ratio QS=QN is the SNR measured at a hydrophone, before anyprocessing apart from initial filtering into the frequency band �f (see Figure 3.6).

It follows from Equations (3.34), (3.36), and (3.11) that

Rarrðr; zarr; ztgtÞ ¼S0FNBðr; zarr; ztgtÞ

QNf ðzarrÞ

GA�f

: ð3:39Þ

Recall from Section 3.2.2.7 that the threshold R50 is the SNR required, after spatialand temporal filtering, to achieve a detection probability of 50%. The sonar equationis obtained by dividing Equation (3.39) through by R50, and converting to decibels inthe usual way. Specifically, the left-hand side becomes the signal excess (SE), defined


Figure 3.5. Spectral density level of background noise. The noise term QN is the contribution

in the processing bandwidth, marked �f .

as the ratio by which the SNR exceeds R50, expressed in decibels

SENB � 10 log10RarrR50

: ð3:40Þ

Thus,

SENBðr; zarr; ztgtÞ ¼ ½SL� PLðr; zarr; ztgtÞ�� ½NLf ðzarrÞ� ðAG� BWÞ� �DT; ð3:41Þ

where

SL ¼ 10 log10 S0 ðsource level: dB re mPa2 m2Þ; ð3:42Þ

PLðr; zarr; ztgtÞ ¼ �10 log10 FNBðr; zarr; ztgtÞ ðpropagation loss: dB re m2Þ; ð3:43Þ

NLf ðzarrÞ � 10 log10 QNf ðzarrÞ ðnoise spectrum level: dB re mPa2=HzÞ; ð3:44Þ

AG � 10 log10 GA ðarray gain: dBÞ; ð3:45Þ

DT ¼ 10 log10 R50 ðdetection threshold: dBÞ; ð3:46Þ

and

BW ¼ 10 log10 �f ðbandwidth: dB re HzÞ: ð3:47Þ

Equation (3.41) is the NB passive sonar equation in logarithmic form. The processing


Figure 3.6. Spectral density level of signal 10 log10 QSf (red) and noise 10 log10 Q

Nf (cyan).

bandwidth term BW, originating from the denominator of Equation (3.39), isgrouped for convenience with the array gain.17

The figure of merit (FOM) is defined as the propagation loss at which thedetection probability is 50% (i.e., FOM�PL50). From this definition it follows that

FOMNBðzarrÞ ¼ SLþ ðAG� BWÞ �NLf ðzarrÞ �DT ð3:48Þ

and the sonar equation is then

SENBðr; zarr; ztgtÞ ¼ FOMNBðzarrÞ � PLðr; zarr; ztgtÞ: ð3:49Þ

3.2.3.4 Array gain and directivity index for a horizontal line array

The array gain for a horizontal line array (neglecting the signal gain) is given by

GA ¼ QNðQN

OBðOÞ dO; ð3:50Þ

where (if the hydrophone spacing is small compared with the acoustic wavelength)

BðOÞ ¼ sin2 u

u2ð3:51Þ

and, expressing the direction in terms of the spherical co-ordinates � (elevation) and �(bearing)

u ¼ uð�; �Þ ¼ k Dx cos � sin �2

; ð3:52Þ

where Dx is the array length. Given that for a sheet dipole source, QNO is proportional

to sin � (see Chapter 2), and using dO ¼ cos � d� d�, it follows that

GA ¼

ð�=20

d� sin � cos �

ð2�0

d�ð�=20

d� sin � cos �

ð2�0

d�sin uð�; �Þuð�; �Þ

� �2: ð3:53Þ

The numerator of Equation (3.53) is �, and its denominator can be simplified bydefining the horizontal beamwidth Dð�Þ as

D�ð�Þ �ð2�0


� �2

; ð3:54Þ


17 An alternative convention is to group �f instead with the narrowband detection threshold sothat DTNB would be given instead as 10 log10ð�fR50Þ, with implied units dB re Hz. The

convention of Equation (3.41) is preferred because it makes it possible to standardize on a

single definition of DT (as 10 log10 R50) for both coherent and incoherent processing, as well as

for both passive and active sonar.

so that

GA ¼ 2�ð�=20

sin 2� D�ð�Þ d�: ð3:55Þ

The integrand of Equation (3.54) is periodic in �, with period �, which means that thebeamwidth can be written

D� ¼ 2

ð�=2��=2

d�sin u

u

� �2

: ð3:56Þ

The beamwidth calculation can be further simplified by replacing the beam patternwith a top-hat approximation of the same area18

sin u

u

� �2

Pu

�

� ; ð3:57Þ

where PðxÞ is the rectangle function (see Appendix A), so that

D� 2

ð�þ��

d� ¼ 2ð�þ � ��Þ; ð3:58Þ

where

� ¼ arcsin min 1;�

k Dx cos �

� : ð3:59Þ

It follows that

D� ¼ 4 arcsin min 1;�

k Dx cos �

� : ð3:60Þ

For a sufficiently long array (many wavelengths), Equation (3.55) may be written

GA �

2

ð�=20

sin 2� arcsin�

k Dx cos �d�

: ð3:61Þ

Approximating the arcsine function by its argument, this becomes

GA k Dx

4

ð�=20

sin � d�

; ð3:62Þ

and hence

GA k Dx4

¼ � Dx2

: ð3:63ÞThe array gain is

AG ¼ 10 log10 GA: ð3:64ÞNow consider the directivity index of the array. This can be calculated as

DI ¼ 10 log10 GD; ð3:65Þwhere GD is the array gain for isotropic noise, which can be calculated by replacing


18 See Appendix A:

ðþ1

�1

sin u

u

� �2du ¼ �.

the sin � terms in Equation (3.53) with unity. Following the steps as outlined abovefor GA then gives

GD � 10DI=10 ¼ 2Dx

: ð3:66Þ

Thus, Equation (3.63) can be written

GA �

4GD; ð3:67Þ

which means that noise directionality (for the dipole sheet noise source considered)has eroded about 22% of the array’s directivity (a degradation of 0.9 dB). Thebroadside beam of a horizontal line array is less effective at discriminating againstnoise from the vertical direction than against isotropic noise.

3.2.3.5 Probability of detection, detection threshold, and ROC curves

The probability that a sonar detects a given signal in a noisy background depends onits ability to discriminate between signal plus noise and noise alone. In turn thisdepends on the statistical fluctuations in both the signal and noise separately, aftersignal processing. As described in Chapter 2, if it is assumed that individual noisepressure samples follow a Gaussian distribution, the noise amplitude after NBprocessing will follow a Rayleigh distribution. If, further, the signal amplitude alsofollows a Rayleigh distribution, the appropriate relationship between the probabil-ities of detection ( pd) and false alarm ( pfa) is

pd ¼ p1=ð1þRarrÞfa : ð3:68Þ

Using RT to denote the SNR threshold required to achieve a specified detectionprobability equal to pT, it is convenient to write Equation (3.68) in the form

log pd ¼1þ RT1 þRarr

log pT; ð3:69Þ

where

RT ¼ log pfalog pT

� 1: ð3:70Þ

The detection threshold was introduced above as 10 log10 R50, indicating that thisthreshold is based on a detection probability of 50%. Detection thresholds based onother pT values are sometimes used. Throughout this book, if a value is not statedexplicitly, a detection threshold based on pT ¼ 0.5 is implied. Figure 3.7 shows thereceiver operating characteristic (ROC) curves calculated using Equation (3.70) in theform DT ¼ 10 log10 RT vs. pfa, for values of pT between 0.1 and 0.9. For example, thedetection threshold for a 30% detection probability and false alarm probability of10�6 is DT30 10 dB.

Of special interest is the case pT ¼ 0.5, for which it is convenient to use base-2logarithms in Equation (3.69), which then simplifies to

log2 pd ¼ � 1þ R501þ Rarr

; ð3:71Þ


where

R50 ¼ log21

2pfa: ð3:72Þ

3.2.3.6 Probability of false alarm

The SNR threshold R50 is related to the probability of false alarm ( pfa) throughEquation (3.72). The value of pfa can be estimated by relating it to the false alarm ratenfa, the total number of detection opportunities per ‘‘look’’ Ntot, and the duration ofeack look Dt (the coherent integration time)

ðnfaÞtrue ¼Ntot pfaDt

: ð3:73Þ

The number of opportunities Ntot depends on the number of sonar beams (Nbeams)and frequencies (NFFT):

Ntot ¼ NbeamsNFFT: ð3:74Þ

The true pfa value (and therefore also nfa) is determined by the threshold R50.However, this threshold should be chosen to achieve an acceptable false alarm rate.Thus, one can estimate the pfa by rearranging Equation (3.73) and removing the


Figure 3.7. ROC curves in the form 10 log10 RT vs. pfa for specified pT values and for a

Rayleigh-distributed signal in Rayleigh noise. These ROC curves are suitable for NB sonar

with a strongly fluctuating signal amplitude.

‘‘true’’ qualifier (i.e., assuming ‘‘true’’ and ‘‘acceptable’’ rates to be approximatelyequal), such that

pfa nfaNtot

Dt ¼ nfaNbeamsDf

; ð3:75Þ

where Df is the total system bandwidth

Df ¼ NFFT �f : ð3:76Þ

3.2.3.7 Special case: low-frequency tonal in the broadside beam of a horizontal linearray

As a prelude to the worked example in the following section we consider here the caseof a horizontal line array receiver operating with a target in its broadside beam. Thisspecial case is analyzed with a view to simplifying the sonar equation, such that thesignal-to-noise ratio can be expressed in terms of readily understood parameters suchas acoustic wavelength, source power, and array length. The frequency is assumed tobe low enough to justify neglecting attenuation, so (putting ¼ 0 in Equation(3.37)19) the ambient noise spectral density is

QNf 3�cE3ð0ÞWN

Af ¼ 32�cWN

Af : ð3:77Þ

It is convenient to define the signal excess in linear units as

� � RarrR50

: ð3:78Þ

For the idealized situation considered (small z, unsteered horizontal line array, largek Dx, and infinitely deep isovelocity water) the following approximation for the sonarequation then follows from Equation (3.39) for the signal-to-noise ratio—usingEquation (3.63) for the array gain and replacing FNB with Fcoh

�ðr; zarr; ztgtÞ ¼Dx�fR50

� �Fcohðr; zarr; ztgtÞ

12

W S

WNAf

!: ð3:79Þ

The first factor on the right-hand side, which is independent of frequency and hasdimensions of [distance] multiplied by [time], is a measure of the spatial (Dx) andtemporal ð�f Þ�1 aperture of the sonar, and is a useful indicator of a sonar’s effec-tiveness. The sonar performance, as measured by the detection probability alone,improves with increasing Dx or decreasing �f or decreasing R50. However, for fixedR50 this is at the expense of a greater false alarm rate, because a larger number ofbeams (or frequency bins) is needed for the increased spatial (or temporal) resolution.For fixed SNR, any reduction in R50 (the SNR threshold at which pd ¼ 1

2), will

increase pd at the expense of an increased pfa. Thus, the false alarm rate increasesin this situation also.


19 That is, neglecting the product zarr. The quantity zarrztgt=r has already been neglected inthe derivation of Equation (3.35) (see Chapter 2 for details).

The second factor in Equation (3.79) is a complicated term that depends onthe properties of the radiated tonal, the background noise, and the propagationconditions. It has dimensions [distance]�1 multiplied by [time]�1 and is stronglydependent on analysis frequency and sonar–target geometry. Use of Equation(3.79) is illustrated in Part (vii) of the following worked example.

3.2.3.8 Worked example

An underwater target in deep water, at a depth of 10m beneath the sea surface,radiates a NB signal (a tonal) at a frequency of 300Hz. The tonal power is 0.2mW,radiated uniformly in all directions. A horizontal receiving array of length 45m isplaced at a depth of 30m. The received signal is passed through a NB filter ofresolution 0.25Hz and then beamformed. The target is in the broadside beam andthe wind speed is 5m/s. For the purpose of calculating pfa, assume that 32 beams areformed for each of 1,024 frequency bins between 256Hz and 512Hz, and that one perhour is an acceptable rate of false alarms.

It is convenient to introduce the variables LS (signal level) and LN (in-beam noiselevel) as

LS � SL� PL ð3:80Þand

LN � NLf � ðAG� BWÞ; ð3:81Þ

so that the sonar equation can be written

SENB ¼ LS � LN �DT: ð3:82Þ

(i) Calculate the source level (SL) of the target using Equation (3.42).(ii) Calculate the background noise spectrum level (NLf ) using Equation (3.44).(iii) Using Equations (3.45) to (3.47), calculate the array gain (AG), detection

threshold (DT), and processing bandwidth (BW) of the sonar.(iv) What is the sonar figure of merit (FOM)?(v) Using the method of Chapter 2 or a propagation model of your choice, plot

propagation loss as a function of range PLðrÞ. What is the significance of therange at which PL¼ 78 dB re m2 (denote this range r50)?

(vi) Calculate the signal level defined by Equation (3.80). What is its value at therange r50? What is the value of the in-beam noise level (Equation 3.81)? What isthe significance of the difference between LS and LN at this range?

(vii) Using Equation (3.79), or otherwise, suggest ways in which the sonar or sonargeometry might be altered in order to improve its performance.

3.2.3.8.1 Part (i): source level SL

First use Equation (3.11) to calculate the source factor: S0 ¼ 24.4 Pa2 m2¼2.44 1013 mPa2 m2. The conversion to decibels is effected by dividing by p2refr

2ref ,

taking the base-10 logarithm, and multiplying the result by 10, giving SL¼ 133.9 dBre mPa2 m2.


3.2.3.8.2 Part (ii): noise spectrum level NLf

The noise level is calculated from the sea surface–radiated power spectral density,which follows from Chapter 2

WNAf ¼

2�

3�cK ; ð3:83Þ

where the parameter K , which is a measure of the source factor per unit area of thesea surface (see Chapter 8 for details) and has dimensions of spectral density, is

K ¼ 1:32 104 vv2:24

1:5þ F 1:59kHz

mPa2 Hz�1; ð3:84Þ

where FkHz is the numerical value of the frequency in kilohertz; and vv is the windspeed in meters per second. Substituting v ¼ 5 m/s (i.e., vv ¼ 5) into Equation (3.84)we obtain WN

Af ¼ 0:403 pWm�2 Hz�1 The noise spectrum level is then calculated,using Equations (3.44) and (3.77), as 0.926mPa2 Hz�1, or in decibels: 59.7 dB re mPa2

Hz�1. This value is independent of depth due to the assumed (isovelocity)environment and the negligible attenuation at low frequency.

3.2.3.8.3 Part (iii): AG, BW, and DT

Calculation of the bandwidth term BW is straightforward using Equation (3.47),giving �6.0 dB re Hz. For AG, Equation (3.63) gives GA ¼ 14.2, which in decibels is11.5 dB. For the detection threshold we need the SNR threshold R50, given byEquation (3.72), which depends on the desired pfa. Using Equation (3.75) for pfain the form

pfa ¼nfa

DfNbeams; ð3:85Þ

with Df ¼ 256 Hz, we obtain pfa ¼ 3:4 10�8 and hence (from Equation 3.72)R50 ¼ 23.8 (i.e., DT¼ 13.8 dB).

Answers to Parts (i), (ii), and (iii) are summarized in Table 3.1.20

3.2.3.8.4 Part (iv): figure of merit FOM

For the figure of merit, Equation (3.48) gives FOM¼ 78.0 dB re m2.

3.2.3.8.5 Part (v): propagation loss PL(r)

A graph of PLðrÞ is plotted in Figure 3.8. The 78 dB re m2 mentioned in the questionis a reference to the figure of merit, which by definition is the propagation loss


20 Intermediate calculations (see Table 3.1) are rounded to one decimal place in decibels, and to

three significant figures in the linear form. While this level of accuracy might not be justified for

all of the terms, it is advisable to prevent accumulation of errors by delaying any further

rounding (say, to the nearest decibel) until the end of the calculation.


Table 3.1. Sonar equation calculation for NB passive example. For each term in the

sonar equation a linear form and a dB form are quoted. In each case the dB form is

equal to 10 log10(linear form) so that, for example, SL ¼ 10 log10ðS0=mPa2 m2Þ and

S0 ¼ 10SL=10 mPa2 m2.

Description dB form Linear form

Symbol Value Expression Numerical value in

reference units

Source level SL 133.9 dB re mPa2 m2 S0 2.44 10þ13(Equation 3.42)

Noise spectrum level NLf 59.7 dB re mPa2 Hz�1 32 �cW

NAf 9.26 10þ5

(Equation 3.77)

Array gain AG 11.5 dB re 1 �Dx2

14.2

(Equation 3.63)

Detection threshold DT 13.8 dB re 1 R50 23.8

(Equation 3.46)

Analysis bandwidth BW �6.0 dB re Hz Df =NFFT 0.250

(Equation 3.47)

Figure 3.8. Propagation loss (blue) and figure of merit (cyan) vs. target range. The probability

of detection is 50% when the propagation loss is equal to the figure of merit, corresponding to

the intersection between the two lines.

resulting in a detection probability of 50%. The range at which this happens isknown as the detection range, which for this example is about 2.5 km.21

3.2.3.8.6 Part (vi): signal level LSðrÞ and in-beam noise level LN

A graph of LSðrÞ (i.e., SL�PL) is shown in Figure 3.9 as a solid blue line. At thedetection range (2.5 km) the signal level is ca. 56 dB re mPa2, 14 dB higher than the in-beam background (LN) of 42 dB re mPa2 (dashed). The difference is the detectionthreshold in decibels (Equation 3.82). The third line (solid, cyan) is LN þDT. Thisline intersects the signal at exactly the same range as the FOM intersects the PLðrÞcurve in Figure 3.8, thus illustrating an alternative way to calculate detection range.The two methods are equivalent, making the choice between them a matter of per-sonal preference. Either way, the intersection between blue and cyan lines separatesthe region to its left, where detections are likely ( pd > 50%), from that to its right,where they are unlikely ( pd < 50%). The point of intersection itself is the detection


Figure 3.9. Signal level LS vs. target range in blue (solid), and in-beam noise level LN in red

(dashed). The probability of detection is 50% when LS exceeds LN by the detection threshold

(DT¼ 13.8 dB), corresponding to the intersection between solid blue and cyan lines.

21 Use of different propagation models, making different inherent assumptions and

approximations, could lead to small differences in this value. For this problem it is important

to use a model that takes into account the phase difference between direct and surface-reflected

paths (such a model is sometimes referred to as a ‘‘coherent’’ model, or it might have a

‘‘coherent’’ option that can be switched on for this purpose).

range and denoted r50. At this location the signal excess (Equation 3.41) is 0 dB,which means that the detection probability is exactly 50%. Detection probability vs.range is plotted in Figure 3.10, calculated using Equation (3.71). The slight dip indetection probability at a range of 120m is caused by the interference null at thatrange (see Figure 3.8).

3.2.3.8.7 Part (vii): sensitivity to sonar parameters

Part (vii) requires an understanding of the sensitivity of the signal excess to the sonarparameters. To gain this understanding, a graph of signal excess (equal to FOMminus PL) is plotted vs. target range and depth in Figure 3.11. By definition, contoursof zero signal excess correspond to a detection probability of 50%. For r > 500m,the signal excess is seen to increase monotonically with increasing target depth in thedepth range 0m to 40 m.22 This is because the argument of the sin2 function inEquation (3.35) for Fcoh is small (about

17at 2.5 km).23 Thus, sin xmay be replaced by


Figure 3.10. Linear signal excess (Equation 3.86) and twice detection probability (Equation

3.68) vs. range for NB passive sonar (Rayleigh statistics); the two curves cross when

� ¼ 2pd ¼ 1. The range at which this happens (2.45 km) is the detection range. (The second

crossing, at 3 km, has no special significance.)

22 The � phase change at the sea surface leads to cancellation between the direct and surfacereflected paths if their path lengths are equal.23 Absorption is also small (0.0086 dB/km) and is considered negligible (<0.1 dB) for ranges upto 12 km.

its argument x, and Equation (3.79) simplifies to

� 4�2Dx33�fR50

W S

WNAf

zarrztgt

r2

� �2

; ð3:86Þ

where � is the linear signal excess given by Equation (3.78) (ratio of the actual SNR tothe minimum SNR required for detection).

As already noted, Figures (3.8) and (3.9) show that the detection range is close to2.5 km. A more precise calculation can be made by equating Rarr and R50 in Equation(3.86) and rearranging for the detection range r50:

r50 ðzarrztgtÞ1=24�2Dx33 �fR50

W S

WNAf

!1=4

: ð3:87Þ

Substituting numerical values into Equation (3.87) for a source power of 0.2mWresults in a detection range of 2.45 km. The main benefit of solving the problem in thismanner is the insight afforded by Equation (3.87) into the sensitivity of r50 to sonardepth, array length, and processing bandwidth. In particular, using Equation (3.72)for R50, Equation (3.87) can be written:

r50 / z1=2arr

Dx=�flog2ðNbeams Df =2nfaÞ

� �1=4

: ð3:88Þ


Figure 3.11. Signal excess [dB] vs. target range and depth for fixed sonar depth zarr ¼ 30m and

frequency f ¼ 300Hz. Other parameters are as Table 3.1 (INSIGHT).

According to Equation (3.88), the detection range can be increased by increasing thearray depth or array length, or by decreasing the processing bandwidth. Treating nfaas fixed, there is also a weak (logarithmic) dependence on the number of sonar beamsand the total monitored bandwidth ðDf Þ. Dependence on the last two parametersarises through their influence on the rate of detection opportunities and hence the(acceptable) false alarm probability.

3.2.4 Incoherent processing: broadband passive sonar

In the case of conventional broadband (BB) passive sonar, the received pressurewaveform is passed first through a band-pass filter (the passband of which is theprocessing bandwidth of the sonar, denoted Df ). The output of this filter is squaredand then integrated in time for a duration Dt. Such processing amounts to incoherentaddition of energy over a wide frequency band. Thus, the knoll-shaped curve ofFigure 3.2 now contributes to the signal, provided that it is within the filter passband,as illustrated by Figure 3.12.


The signal is now the total MSP integrated over the processing bandwidth Df , and isgiven for center frequency fm by

QS ¼ðfmþDf =2

m�Df =2QSf df : ð3:89Þ

We define the BB propagation factor

FBB � QS

S0; ð3:90Þ

where S0 is the source factor

S0 ¼ððS0Þf df ; ð3:91Þ

with associated spectral density

ðS0Þf ¼�c

4�W S

f : ð3:92Þ

It follows from Equation (3.90) that

SBB ¼ S0FBB: ð3:93ÞNow consider the behavior of FBB, which, for the assumed incoherent processing, canbe written in the form

FBB �

ðFcohðf ÞW S

f df

W S; ð3:94Þ

whereW S denotes that part of the signal-radiated power spectrum within the sonar-processing bandwidth. In this expression the integrand is an oscillatory function offrequency.



Figure 3.12. Spectral density level of the transmitter source factor (upper) and mean square

pressure at the receiver (lower). The sonar processing band is indicated by the solid red curve.

As previously, we consider an idealized ocean of infinitely deep isovelocitywater. There are two possible ray paths from target to sonar. These two pathsinterfere, resulting for any single frequency in a sequence of fringes due to alternateconstructive and destructive interference. The interference pattern can be written as

Fcohðf ; r; zarr; ztgtÞ ¼ 2e�2ðf Þr

r21� cos

2kzarrztgt

r

� �: ð3:95Þ

This function is plotted vs. frequency and range in Figure 3.13 (upper graph) for atarget at a depth of 150m and a receiver at depth zarr ¼ 100m. The integration inEquation (3.94) has the effect of averaging over the interference fringes, so theintegrand can be replaced by a smoothed version of itself (Figure 3.13, lower graph)without changing the result of the integration. Further, the smoothed integrand isapproximately equal to the incoherent propagation factor (scaled by the sourcespectrum), provided that the frequency is high enough to neglect the systematiccoherent cancellation (pressure node) at the sea surface , which becomes importantbelow 1 kHz in this example. Therefore, above 1 kHz Equation (3.94) may beapproximated by

FBB

ðFincð f ÞW S

f dfðW S

f df

; ð3:96Þ

where

Fincð f ; r; zarr; ztgtÞ ¼2e�2ðf Þr

r2: ð3:97Þ

Equation (3.96) can be further simplified by assuming that the source has a whitepower spectrum. In this situation, the propagation factor becomes

FBB ¼ 1

Df

ðFincð f Þ df ; ð3:98Þ

where Df is the processing bandwidth. Using Equation (3.97) and assuming that varies linearly with frequency in the sonar passband

ð f Þ ¼ m þ f ð f � fmÞ; ð3:99Þ

Equation (3.98) can be expressed in terms of the cardinal sinh function24

FBB ¼ 2

r2expð�2mrÞ sinhcðfDfrÞ: ð3:100Þ

3.2.4.2 Noise (single hydrophone)

As for the signal, the noise term is the total (noise) contribution to the MSP, in the


24 Denoted sinhc x; see Appendix A.


Figure 3.13. Propagation loss [dB re m2] vs. frequency and target range for array depth¼ 100mand target depth¼ 150m. Upper: coherent propagation loss PL ¼ �10 log10 Fcoh; lower:smoothed version of the propagation loss from the upper graph (obtained by averaging in

range) (INSIGHT).

sonar-processing bandwidth Df

QN ¼ðQNf df : ð3:101Þ

3.2.4.3 Signal-to-noise ratio, signal excess, and broadband passive sonar equation

It follows from the definition of Rhp (Equation 3.25) and of array gain (Equation3.24) that

Rarr ¼QS

QNGA: ð3:102Þ

Illustrative signal-and-noise spectra are shown in Figure 3.14. The signal term QS isthe area under the red curve, while the noise QN is the area under the noise (cyan)curve in the same frequency band (between the vertical dashed lines).

The array response25 is

Y S ¼ðBð f ;OÞQS

fO df dO ð3:103Þ

for the signal, and

YN ¼ðBð f ;OÞQN

fO df dO ð3:104Þ


Figure 3.14. Spectral density level of signal 10 log10 QSf (red) and noise 10 log10 Q

Nf (cyan).

25 For a hypothetical flat-response filter in combination with a beamformer.

for the noise. The signal excess is, by definition,

SEBBðr; zarr; ztgtÞ � 10 log10Rarrðr; zarr; ztgtÞ

R50¼ SL� PL� ðNL�AGÞ �DT;

ð3:105Þwhere

SL ¼ 10 log10 S0 ðsource level: dB re mPa2 m2Þ; ð3:106Þ

PLðr; zarr; ztgtÞ ¼ �10 log10 FBB ðpropagation loss: dB re m2Þ; ð3:107Þ

NLðzarrÞ ¼ 10 log10 QNðzarrÞ ðnoise level: dB re mPa2Þ; ð3:108Þ

AG ¼ 10 log10 GA ðarray gain: dBÞ; ð3:109Þand

DT ¼ 10 log10 R50 ðdetection threshold: dBÞ: ð3:110Þ

As for the NB case, the figure of merit is defined as

FOM � PL50: ð3:111Þ

3.2.4.4 Array gain

From the definition of array gain (Equation 3.24) it follows that

GA ¼

ðY S

f dfðYN

f df

ðQNf dfð

QSf df

: ð3:112Þ

A useful approximation is obtained by assuming that the spectral densities of boththe MSP (Qf ) and array response (Yf ) vary linearly with frequency within the sonarbandwidth. In this situation, GA may be approximated by a simple expression in-volving only the spectral densities evaluated at the center frequency fm. Specifically,

GA Gm; ð3:113Þwhere Gm is the array gain at the center frequency

Gm �Y S

f ðfmÞYN

f ð fmÞQNf ð fmÞ

QSf ð fmÞ

: ð3:114Þ


The probability of detection for incoherent processing, assuming Rayleigh statisticsfor both signal and noise amplitudes prior to incoherent processing (see Chapter 2) is

pd 1

2erfc

erfc�1ð2pfaÞ �ffiffiffiffiffiffiffiffiffiffiM=2

pRarr

1þRarr

" #; ð3:115Þ

whereM is the number of independent time samples added incoherently; and Rarr isthe SNR after beamforming (more generally, at the point in the processing where a


‘‘target present’’ or ‘‘target absent’’ decision is made). Replacing pd with the chosenprobability threshold (denoted pT) and Rarr by its value at that threshold (i.e.,Rarr ¼ RT), and rearranging for RT gives

ffiffiffiffiffiM

2

rRT erfc�1ð2pfaÞ � erfc�1ð2pTÞ

1þffiffiffiffiffiffiffiffiffiffi2=M

perfc�1ð2pTÞ

: ð3:116Þ

The right-hand side of Equation (3.116) simplifies for large M to

ffiffiffiffiffiM

2

rRT erfc�1ð2pfaÞ � erfc�1ð2pTÞ: ð3:117Þ

This asymptotic value is converted to decibels and plotted vs. pfa in Figure 3.15.For the special case pT ¼ 0.5, Equation (3.116) simplifies without further

approximation to give

R50 ffiffiffiffiffi2

M

rerfc�1ð2pfaÞ; ð3:118Þ

resulting in the following useful approximation for the detection threshold

DT50 10 log10½erfc�1ð2pfaÞ� � 10 log10ffiffiffiffiffiffiffiffiffiffiM=2

p: ð3:119Þ


Figure 3.15. ROC curves in the form 10 log10ðffiffiffiffiffiffiffiffiffiffiM=2

pRTÞ vs. pfa for fixed pT values, for a BB

signal in Rayleigh noise and M1=2 � erfc�1ð2pTÞ.

The number of independent samples M that can be obtained in time Dt is

M ¼ Dt�tN

; ð3:120Þ

where �tN is the Nyquist interval

�tN ¼ 1

2Df; ð3:121Þ

and therefore the value of M in Equation (3.119) is

M ¼ 2Df Dt: ð3:122Þ


To complete the picture, a suitable false alarm probability is needed, which can beestimated from the desired false alarm rate nfa. It is usual to divide the total availablebandwidth into smaller intervals of (say) an octave at a time. If the number of suchintervals is Nintervals, the false alarm probability is

pfa ¼nfa Dt

NintervalsNbeams: ð3:123Þ

3.2.4.7 Special case: broadband target in the broadside beam of a horizontalline array

The purpose here is to simplify the BB sonar equation in order to express it in terms offamiliar properties such as source power and array length as previously in the case ofNB sonar. This is done for a similar situation as in Section 3.2.3.7, involving ahorizontal line array. It has already been assumed (see Section 3.2.4.4) that the arrayresponse to signal and noise vary linearly with frequency so that

Rarr ¼Y S

YN¼

ðY S

f ð f Þ dfðYN

f ð f Þ dfY S

f ð fmÞYN

f ð fmÞ; ð3:124Þ

where it is understood that the signal and noise spectra are smoothly varyingfunctions of frequency.26 It then follows that

Rarr GmQSf ð fmÞ

QNf ð fmÞ

; ð3:125Þ

where Gm is given by Equation (3.114) and

QSf ð fmÞ ¼ ðS0Þf FBBð fmÞ: ð3:126Þ


26 If they are not smooth to start with, they can be locally averaged until they become so.

Substituting Equation (3.126) in Equation (3.125) gives

RarrR50

GmðS0Þf FBBð fmÞQNf ð fmÞ

1

R50; ð3:127Þ

where

ðS0Þf ¼�c

4�W S

f ð fmÞ ð3:128Þ

and, making the assumption as before that z � 1,

QNf ¼ 3�c

2WN

Af : ð3:129Þ

Converting Equation (3.127) to decibels provides a simplified form of the BB sonarequation:

SEBB ¼ SLf � PL� ðNLf �AGmÞ �DT; ð3:130Þwhere

SLf ¼ 10 log10ðS0Þf ðsource spectrum level: dB re mPa2 m2 Hz�1Þ ð3:131Þ

PLðr; zarr; ztgtÞ ¼ �10 log10 FBB ðpropagation loss: dB re m2Þ ð3:132Þ

NLf ¼ 10 log10 QNf ð fmÞ ðnoise spectrum level: dB re mPa2 Hz�1Þ ð3:133Þ

AGm ¼ 10 log10 Gm ðcenter frequency array gain: dBÞ ð3:134Þ

and

DT ¼ 10 log10erfc�1ð2pfaÞffiffiffiffiffiffiffiffiffiffiffiffi

Df Dtp ðdetection threshold: dBÞ ð3:135Þ

For the case of a surface sheet (dipole) noise source we have (from Equation 3.63)

Gm ¼ �

2

Dxm

; ð3:136Þ

where

m � c

fm: ð3:137Þ

Substituting Equation (3.136) for the array gain, the signal excess, defined as

� ¼ RarrR50

ð3:138Þ

can be written in the following convenient form

� ¼ DxR50

� �FBB12m

W Sf

WNAf

!; ð3:139Þ

as used in the worked example below.


Consider an omni-directional underwater communications transmitter at a depth of


150m. The carrier signal is a BB waveform with a white power spectrum. The signal isintercepted by a horizontal line array receiver of length 6m, at depth 100m from thesurface, and oriented with the transmitter in its broadside beam. The interceptor usesincoherent processing in the frequency band 2 kHz to 4 kHz, with an integration timeof 10 s. The power spectral density of the source in this frequency band is 100 nW/Hz.The wind speed is 5m/s.

It is useful to define the signal and noise spectrum levels, respectively, as

LSf � SLf � PL ð3:140Þ

and

LNf � BLf �AG; ð3:141Þ

so that the sonar equation becomes

SE ¼ LSf � LNf �DT: ð3:142Þ

(i) Calculate the source spectrum level (SLf ) of the transmitter using Equation(3.131). Calculate the background noise spectrum level (NLf ) using Equation(3.133).

(ii) Using Equations (3.134) and (3.135), calculate the center frequency array gain(AGm) and detection threshold (DT) of the sonar. Assume that the direction andfrequency band of the transmission are known, but not the waveform shape, andthat a false alarm rate of one per hour is acceptable.

(iii) Using the FOM method, or otherwise, calculate the maximum range at whichthe probability of detecting the communications transmission exceeds 50% for asingle (10 s) observation.

(iv) The designers of the communications system wish to halve the intercept range.Assuming that all other parameters remain unchanged, by how much must thepower be reduced?

(v) How are the answers to parts (i) to (iv) affected if the transmitter depth isdoubled to 300 m?

3.2.4.8.1 Part (i): source level SLf and noise spectrum level NLf

First use Equation (3.128) to calculate the broadband source factor, givingðS0Þf ¼ 0.0122 Pa2 m2 Hz�1.

The noise level is calculated from the sea surface–radiated power density, usingEquation (3.129), with Equations (3.83) and (3.84) for the power spectral density.The result for wind speed v10 ¼ 5 m/s isWN

Af ¼ 0:0918 pWm�2 Hz�1. The noise levelis then calculated using Equation (3.129), giving a noise spectral density ofQNf ¼ 0:209 mPa2 Hz�1.Conversions of the numerical values of ðS0Þf and QN

f to micropascals anddecibels are given in Table 3.2.


3.2.4.8.2 Part (ii): AGm and DTEquation (3.63) gives GA ¼ 19.0, and hence AGm ¼ 12.8 dB. The SNR threshold R50is given by Equation (3.118) (see also Equation 3.119 for the detection threshold),which can be written in the form (using Equation 3.122 for M)

R50 erfc�1ð2pfaÞffiffiffiffiffiffiffiffiffiffiffiffi

Df Dtp : ð3:143Þ

If the transmission direction is known, only one beam is of interest (Nbeams ¼ 1). Ifthe frequency band is known, only one octave needs to be monitored (Nintervals ¼ 1).Thus, the desired maximum false alarm rate can be achieved with (Equation 3.123)27

pfa ¼ nfa Dt 0:0028: ð3:144Þ

Putting Df ¼ 2 kHz and Dt ¼ 10 s in Equation (3.143) then gives R50 ¼ 0.0139.Answers to Parts (i) and (ii) are summarized in Table 3.2.

3.2.4.8.3 Part (iii): detection range

Detection range depends on propagation loss (PL). It is usually not appropriate touse the same calculation method for NB and BB PL. For NB it was important to takeinto account the phase of various multipaths before adding these coherently. For BB,a legitimate approach is to repeat such a coherent calculation at every frequency and


Table 3.2. Sonar equation calculation for BB passive example.


Symbol Value Expression Numerical

value in

reference

units

Source spectrum level SLf 100.9 dB re mPa2 m2 Hz�1 ðS0Þf 1.22 10þ10(Equation 3.131)

Noise spectrum level NLf 53.2 dB re mPa2 Hz�1 32 �cW

NAf 2.09 10þ5

(Equation 3.133)

Center frequency array gain AGm 12.8 dB re 1 �Dx2m

19.0

(Equation 3.134)

Detection threshold DT �18.6 dB re 1 erfc�1ð2pfaÞffiffiffiffiffiffiffiffiffiffiffiffiDf Dt

p 1.38 10�2(Equation 3.135)

27 Use of BB processing can tolerate a false alarm probability that is five orders of magnitude

larger than for NB processing, without affecting the false alarm rate. The difference is caused by

the lower frequency resolution of the BB system, resulting in fewer detection opportunities.

then average the result. However, this method is rarely used, because it is cumber-some and often unnecessary. Instead, it is more convenient to discard the phase at thebeginning and add the contributions incoherently. This is justified if the averagingprocess would have destroyed the phase information anyway, although care is neededto retain those interference effects that are persistent across the entire frequency band,and consequently remain even after averaging. (An example is the systematic can-cellation or reinforcement that occurs close to a reflecting boundary, illustrated byFigure 3.13). If the target and receiver array are both deep enough for the boundarycorrections described in Section 3.2.4.1 to be negligible, FBB may be approximated byEquation (3.100). A graph of PLðrÞ, calculated using this approximation, is plotted inFigure 3.16.28 Using Equations (3.111) and (3.113), the figure of merit is

FOMBB ¼ SLf � ðNLf �AGmÞ �DT; ð3:145Þ

which for this example is 79.0 dB re m2. The corresponding detection range read fromthe graph is about 10 km.


Figure 3.16. Propagation loss (blue) and figure of merit (cyan) vs. range. The probability of

detection is 50% when the propagation loss and figure of merit are equal, corresponding to the

intersection between solid blue and cyan lines.

28 The argument of the sinhc function in Equation (3.100), which is the change in r across thereceiver bandwidth, is a small number (0.06Np at 10 km with f Df ¼ 6:04 10�3 Np km�1).Consequently, the sinhcðxÞ function may be approximated by unity.

An alternative method to calculate the detection range is to plot the signalspectrum level vs. range and compare the resulting curve with the in-beam noiselevel. A graph of LSf ð¼ SLf � PLÞ is shown in Figure 3.17 as a solid blue line. At thedetection range (the intersection between LSf and LNf þDT, at 10.2 km) the signallevel is 21 dB re mPa2/Hz, 19 dB lower than the in-beam noise (LNf ) of 40 dB re mPa

2/Hz because, for this example, DT is negative. What this means is that a weaksignal can be detected in the presence of a strong noise background (in this casethe noise is nearly 100 times stronger than the signal). A negative detectionthreshold is a common characteristic of incoherent processing, the effect of whichis to reduce the detection threshold by removing fluctuations, without altering theSNR.29

Detection probability is given by Equation (3.115) as (using Equations 3.118 and3.122)

pd ¼1

2erfc

ffiffiffiffiffiffiffiffiffiffiffiffiDf Dt

p R50 � Rarr1þ Rarr

� �: ð3:146Þ

The result is plotted vs. range in Figure 3.18.


Figure 3.17. Signal spectrum level (LSf ) vs. range in blue (solid line), and in-beam noise

spectrum level (LNf ) in red (dashed). The probability of detection is 50% when the signal

exceeds LNf by the detection threshold (DT¼ �18.6 dB), corresponding to the intersectionbetween solid blue and cyan lines.

29 This contrasts with the purpose of coherent processing, which is to increase the SNR.

3.2.4.8.4 Part (iv): halving detection range

To answer Part (iv) it is convenient to rearrange the sonar equation in the form(see Equation 3.139)

FBBðrÞ ¼12mR50W

NAf

DxW Sf

�ðrÞ; ð3:147Þ

where � is the signal excess in linear form, defined by Equation (3.78). Using �50 todenote the figure of merit in linear form

�50 � 10FOM=10 ¼ 1

Fðr50Þð3:148Þ

and substituting � ¼ 1 in Equation (3.147), it follows that

�50 ¼DxW S

f

12mR50WNAf

: ð3:149Þ

In order for the intercept range to be halved (to r50=2 ¼ 5:1 km), �50 must decrease bya factor of FBBðr50=2Þ=FBBðr50Þ ¼ 4:96, which means that W S

f must decrease by thesame factor, to 20.1 nW/Hz.



3.146) vs. range for BB passive sonar (Gaussian statistics). The two curves cross at the detection

range, where � ¼ 2pd ¼ 1.

3.2.4.8.5 Part (v): doubling transmitter depth

For depths exceeding 20m, PL is approximately independent of transmitter depth, asillustrated by Figure 3.19. None of the other terms vary with depth for this example,so the answers to Parts (i) to (iv) are unaffected by increasing the transmitter depthto 300m.

Notice the rapid increase in PL as the receiver approaches the surface for depthsless than 20m. In this region, a coherent calculation is necessary, even for BBprocessing, because the � phase change at the sea surface results in a coherentcancellation occurring across the entire sonar bandwidth. This effect does not affectthe worked example.

3.3 ACTIVE SONAR

3.3.1 Overview

An active sonar system employs the same principle as radar: in order to detect thepresence of an underwater object (the ‘‘target’’), it transmits a short burst (or pulse) of


Figure 3.19. Propagation loss [dB re m2] vs. range and depth for the BB passive worked example. Sonar

depth zarr ¼ 100 m and frequency f ¼ 3 kHz (INSIGHT).

energy (a sound wave in the case of sonar) and listens for an echo from that object, asillustrated by Figure 3.20. As with radar, the echo contains information about theposition and nature of the target. Most obviously, the time delay between transmis-sion and reception provides a direct measure of the target range, information that isnot readily available from passive sonar data. Other practical advantages of activesonar (compared with passive sonar) include the following:

— the transmitted waveform is known, leading to large signal-processing gainsbecause it enables the receiving system to reject waveforms that do not resemblethe transmitted one;

— the signal level, and to some extent also the SNR, can be controlled by changingthe source level and other properties of the transmitted pulse.

The main performance disadvantages arising from active transmission are the loss ofstealth (the transmitted pulse is much louder, and hence easier to detect than thetarget echo) and interference due to reverberation from boundaries and other under-water scatterers. In addition, there is a growing environmental concern about theimpact on marine life (Southall et al., 2007) resulting from the use of loud underwatersound sources. Nevertheless, if there is no shortage of power and no need for stealth,then active sonar can offer a performance advantage. Concepts and terminologyspecific to active sonar are introduced below.

3.3.2 Definition of standard terms (active sonar)

For active sonar (and also for transient waveforms generally) it is useful to expressthe sonar equation in terms of quantities proportional to total energy instead ofpower (or intensity). The reason for doing this is that energy is a conserved quantity,whereas power is not. If a signal of finite duration becomes stretched in time, its

3.3 Active sonar 95]Sec. 3.3

Figure 3.20. Principles of active detection: the sonar transmitter (the dolphin) radiates a sound

wave that travels through the sea and is scattered by the target (represented by a fish) after which

it travels back to the sonar receiver.

intensity is reduced, but the total energy, in the absence of any absorption, remainsunchanged. In the following, the term ‘‘energy’’ is used as a convenient shorthand forthe time integral of the squared acoustic pressure (i.e.,

Ðq2ðtÞ dt).

3.3.2.1 Signal energy, energy source level, and total path loss

Consider a sonar transmitter, comprising a single point source, whose transmittedwaveform at a small distance s0 is q0ðtÞ.30 The pulse propagates through the ocean tothe target, from which it is reflected or scattered. The echo then propagates back31

through the same medium, returning to the sonar receiver position after a time delayequal to the two-way travel time, at which point the disturbance is denoted qSðtÞ.These quantities are related through a time-integrated version of Equation (3.5)32ð

½qSðtÞ�2 dt ¼ð½q0ðtÞ�2s20 dt F2: ð3:150Þ

Equation (3.150) serves to define F2, the two-way energy propagation factor for apoint source, as the ratio of energy (i.e., time-integrated squared pressure) at thereceiver to that at unit distance from the transmitter. The time intervals are chosen tocontain the whole of the transmitted pulse (in the case of transmitter) or target echo(for the receiver). It is assumed that the delay time is large compared with the pulseduration so that the transmitted pulse is easily separated from the target echo.

Now define the received signal energy as

QSE �

ð½qSðx; tÞ�2 dt; ð3:151Þ

and the energy source level as

SLE � 10 log10 s20

ðq0ðtÞ2 dt

� �dB re mPa2 m2 s: ð3:152Þ

Further, defining total path loss (TPL), in a manner analogous to propagation lossin the passive sonar equation, as

TPL � �10 log10 F2 dB re m2; ð3:153Þ


30 As in the case of passive sonar, this definition holds for a point source. The distance s0 must

be small enough for distortions due to absorption, refraction, reflection, or diffraction to be

negligible. In this situation the field is determined by spherical spreading. Thus,Ð½q0ðtÞ�2s20 dt is

a constant, independent of s0.31 If the receiver is in the same position as the transmitter then ‘‘back’’ just means back to the

sonar. This situation is referred to as a ‘‘monostatic’’ geometry. In some sonars the transmitter

and receiver are physically separate, in which case the geometry is ‘‘bistatic’’. For a bistatic

geometry, the return path is always different from the outward path. For a monostatic sonar

the return path can either be the same or different.32 Recall that the real acoustic pressure is denoted ‘‘qðtÞ’’, while ‘‘pðtÞ’’ is reserved for thecomplex pressure field.

it follows that

10 log10 QSE ¼ SLE � TPL: ð3:154Þ

The left-hand side of Equation (3.154) is referred to henceforth as the signal energylevel.

If the transmitted power is constant over the pulse duration �t, then Equation(3.152) simplifies to

SLE ¼ SLþ 10 log10 �t dB re mPa2 m2 s; ð3:155Þ

where SL is the source level as defined previously in terms of mean square pressure(Equation 3.42). More information about the source levels of man-made andbiological sonar can be found in Chapter 10.

3.3.2.2 Background energy and background energy level

At any instant the total background (acoustic) pressure qBðtÞ can be written as a sumof separate contributions from noise and reverberation33

qBðtÞ ¼ qNðtÞ þ qRðtÞ: ð3:156ÞIt follows that

ðqBÞ2 ¼ ðqNÞ2 þ ðqRÞ2 þ 2qNqR; ð3:157Þ

where the overbar indicates an average in time. Assuming that noise andreverberation are uncorrelated, the cross-term may be neglected, so that for a(transmission to echo) delay time � , the total background becomes

QBð�Þ ¼ QN þQRð�Þ: ð3:158Þ

Henceforth the notation QX is used to indicate the average value of the parameterðqXÞ2 over the duration of the received echo (assumed sufficiently short that thebackground can be thought of as stationary in that time). The sonar is assumedto have a natural processing bandwidth, and the relevant part of the background (forboth noise and reverberation) is that within this frequency band.

For reverberation arriving after a given delay time � , the longer the delay thefurther the sound has traveled and, in general, the more boundary reflections it hasexperienced. The sound becomes weaker at each reflection, so the reverberation QR

tends to decrease with increasing � . For as long as the reverberation remains abovethe noise, the total background is therefore also a function of delay time.

As with the signal, the background energy QBE is defined as the time integral of the

corresponding squared pressure term:

QBE �

ð½qBðtÞ�2 dt; ð3:159Þ


33 Reverberation is the sound associated with scattering of the sonar pulse by any object other

than the intended target. It can be defined as sound pressure in the absence of ambient noise,

self-noise, and the target.

or in decibels, the background energy level is

BLE � 10 log10 QBE dB re mPa2 s: ð3:160Þ

3.3.2.3 Signal-to-background ratio and array gain

The array response to the signal energy can be written

Y SE ¼ �t

ðQS


where the O subscript indicates a derivative with respect to solid angle. Similarly, thearray response to the background spectral density is

Y Bf ð�Þ ¼

ðQBfOð�;OÞBðOÞ dO; ð3:162Þ

where the f subscript indicates a derivative with respect to frequency.A distinction is made between the signal-to-noise ratio (SNR) and signal-to-

background ratio (SBR), whereby SBR is the ratio of the signal power to the totalbackground power (sum of noise plus reverberation). The SBR at the hydrophone isgiven by

Rhp �QSE

QBE

; ð3:163Þ

and after beamforming by

Rarr �Y SEð

ðY BEÞf df

: ð3:164Þ

The receiving array gain is then defined in a similar way as for passive sonar.Specifically

GA � RarrRhp

; ð3:165Þ

such that

AG � 10 log10 GA ¼ 10 log10Y SEð

ðY BEÞf df

� 10 log10QSEð

ðQBEÞf df

dB: ð3:166Þ

Making the assumption (as in Section 3.2.2.5) of an incoming plane wave signal,arriving in the main beam of the receiving array, the signal parameters Y S and QS areequal, and therefore

AG ¼ 10 log10

ððQB

EÞf dfððY B

EÞf df: ð3:167Þ


3.3.2.4 Target strength

Consider a plane wave incident on the target. Target strength (TS) is a measure ofhow much of the incident sound is scattered back in the direction of the sonar. Letqtgtin ðtÞ denote the instantaneous pressure due to the incident wave at the target,and q

tgt0 ðtÞ that of the scattered field at a small distance s0 away from it. A definition

that is valid for a point target is

TSMSP � 10 log10s20 ½qtgt0 ðtÞ�2

½qtgtin ðtÞ�2dB re m2: ð3:168Þ

The denominator and numerator can be written, respectively, in terms of the incidentintensity Iin and scattered radiant intensity WO

ðqtgtin Þ2 ¼ �cIin ð3:169Þand

s20ðqtgt0 Þ2 ¼ �cWO: ð3:170Þ

It follows from the definition of the backscattering cross section �back (see Chapter 2)that34

TSMSP ¼ 10 log10�back

4�dB re m2; ð3:171Þ

where

�backð�Þ � 4��Oð�; �; �Þ ð3:172Þand

�Oð�in; �out; �Þ ¼WOð�out; �Þ


A slight modification is needed here to make the TS definition compatible with theother parameters introduced above, all of which are defined in terms of energy.Consistency can be achieved by multiplying both the numerator and denominatorof Equation (3.168) by the pulse duration �t and replacing the resulting product withan integral:

TSE � 10 log10

s20

ð½qtgt0 ðtÞ�2 dtð

½qtgtin ðtÞ�2 dt

dB re m2: ð3:174Þ

3.3.3 Coherent processing: CW pulseþDoppler filter

A Doppler filter works by examining the spectrum of the received signal in smallfrequency bins. If the target is moving relative to the sonar, the echo is shifted infrequency relative to the transmitted pulse due to the Doppler effect and might arrivein a different Doppler bin to the reverberation. It is assumed below that a single


34 In some publications the factor 4� in the denominator of Equation (3.171) is included in thedefinition of �back (see Chapter 5).

frequency bin (or ‘‘Doppler bin’’) is found that contains all of the echo energy, butonly a small fraction of the reverberation. For the purpose of Section 3.3.3, thisfraction of the reverberation is assumed to be negligible compared with the ambientnoise background.


The signal energy is the integral of the squared pressure over the duration of the pulse�t (Equation 3.150). Assuming that the signal MSP does not change during this timeinterval, this integral may be replaced by the product

QSE ¼ QS �t; ð3:175Þ

where, in terms of the two-way propagation factor F2 (see Equation 3.150),

QS ¼ S0F2 ð3:176Þ

and S0 is the source factor given by Equation (3.10).

3.3.3.2 Background (single hydrophone)

For simplicity, Doppler processing is assumed to eliminate all reverberation, so thebackground consists of noise only. The relevant noise is that in the processingbandwidth (i.e., the width of each Doppler bin �f )

QBE ¼ ðQN

f �f Þ �t: ð3:177Þ

Therefore (assuming for a CW pulse that �f �t 1)

QBE QN

f ð3:178Þand

QNf ¼ 3�cWAf E3ð2zÞ; ð3:179Þ

where E3ðxÞ is an exponential integral of the third kind (Appendix A).

3.3.3.3 Signal-to-background ratio, signal excess, and coherent narrowband activesonar equation

The signal-to-background ratio (SBR), after beamforming, is

Rarr ¼ RhpGA; ð3:180Þwhere

Rhp ¼QSE

QBE

: ð3:181Þ

Using Equations (3.175) and (3.178) and dividing by R50 gives the sonar equation(in decibels)

SE � 10 log10RarrR50

¼ SLE � TPL � ðNLf �AGÞ �DT: ð3:182Þ


From the definitions of TSE and TPL it follows (for a point-like target) that PL andTPL are related in the following way

TPL ¼ PLTx þ PLRx � TSE; ð3:183Þwhere

PLTx � �10 log10 FTx ¼ SLE � 10 log10ð½qtgtin ðtÞ�

2 dt ð3:184Þand

PLRx � �10 log10 FRx ¼ 10 log10 s20

ð½qtgt0 ðtÞ�2 dt

� �� 10 log10 QS

E; ð3:185Þ

where FRx is the one-way propagation factor from target to receiver (Rx), definedsuch that

QS �t ¼ s20

ð½qtgt0 ðtÞ�2 dt FRx: ð3:186Þ

The meaning of the ‘‘in’’ and ‘‘0’’ subscripts above is the same as for the definition oftarget strength, Equation (3.174). In other words, ‘‘in’’ indicates the acoustic fieldincident on the target and ‘‘0’’ the scattered field at a small distance s0 from the target.

Similarly, FTx is the propagation factor from transmitter (Tx) to targetð½qtgtin ðtÞ�

2 dt ¼ S0 �t FTx: ð3:187Þ

Thus the NB active sonar equation, for a point target with coherent processing (andneglecting reverberation) is

SE ¼ ½SLE � ðPLTx þ PLRx � TSEÞ� � ðNLf �AGÞ �DT; ð3:188Þ

where

SLE � 10 log10ðS0 �tÞ ðenergy source level: dB re mPa2 s m2Þ; ð3:189Þ

PLTx ¼ �10 log10 FTx ðpropagation loss Tx to target: dB re m2Þ; ð3:190Þ

PLRx ¼ �10 log10 FRx ðpropagation loss target to Rx: dB re m2Þ; ð3:191Þ

TSE ¼ 10 log10�back4�

ðtarget strength: dB re m2Þ; ð3:192Þ

NLf ¼ 10 log10 QNf ðnoise spectrum level: dB re mPa2 Hz�1Þ; ð3:193Þ

AG ¼ 10 log10 GA ðarray gain: dBÞ; ð3:194Þ

and

DT ¼ 10 log10 R50 ðdetection threshold: dBÞ: ð3:195ÞFor noise-limited active sonar, the figure of merit can be defined as

FOM � 12ðPLRx þ PLTxÞ50 ¼ 1

2ðSLE þ TSE �NLf þAG�DTÞ: ð3:196Þ

The active sonar equation can then be written

SE ¼ 2 FOM � ðPLRx þ PLTxÞ: ð3:197Þ


3.3.3.4 Array gain

For a plane wave signal, the array gain is

GA ¼ QNðQN


where (for a continuous line array)

BðOÞ ¼ sin2 u

u2ð3:199Þ

and

u ¼ k Dx cos � sin �2

: ð3:200Þ

The operating frequency of an active sonar is typically higher than for a passive one,making it necessary to consider the effect of absorption on surface-generated noise,which then becomes

QNO ¼ 3

2��cWN

A sin � exp � 2zarrsin �

� �: ð3:201Þ

Near-horizontal contributions are penalized by the attenuation due to their longtravel paths before reaching the array. Therefore, as z increases, the main noisecontributions come increasingly from vertically above the receiving array. For ahorizontal line array, the broadside beam is less effective at filtering out noise fromdirectly above, so the resulting array gain is smaller than for the zero attenuationcase. Specifically, the gain GA of a horizontal line array in the presence of absorptionis (the ‘‘arr’’ subscript is implied)

GA ¼

ð�=20

d� sin � cos � exp � 2z

sin �

� �ð2�0

d�ð�=20

d� sin � cos � exp � 2z

sin �

� �ð2�0


� �2: ð3:202Þ

This equation is less daunting than it looks because the integrations over � areunaffected by the introduction of attenuation, and can be evaluated in the sameway as in Section 3.2.3.4, leaving

GA ¼2�

ð�=20

sin � cos � exp � 2z

sin �

� �d�

4�

kDx

ð�=20

sin � exp � 2z

sin �

� �d�

: ð3:203Þ

The effect of the exponential factors can be approximated by removing these andreplacing the lower limit of both integrals of Equation (3.203) with an angle ", whose


value increases from 0 to �=2 with increasing attenuation

GA ¼2�

ð�=2"

sin � cos � d�

4�

kDx

ð�=2"

sin � d�

: ð3:204Þ

The result is to multiply the numerator by cos2 " and the denominator by cos ",leaving a net factor of cos " overall. Thus

GA �

4G0 cos "; ð3:205Þ

with G0 from Equation (3.66).The requirement for a good choice of " is that the equationð�=2

0

f ð�Þ exp � 2z

sin �

� �d� ¼

ð�=2"

f ð�Þ d� ð3:206Þ

is satisfied for some appropriate choice of f ð�Þ. A desirable choice, with Equation(3.203) in mind, would be one of either f ð�Þ ¼ sin � cos � or f ð�Þ ¼ sin �. However, iff ð�Þ varies slowly compared with the exponential term, its precise form does notmatter much. In fact, it is convenient to choose a third form

f ð�Þ ¼ cos �

sin2 �; ð3:207Þ

as this facilitates evaluation of the integrals. With this choice, Equation (3.206) canbe solved to give

" ¼ arcsin2z

2zþ expð�2zÞ : ð3:208Þ

This result can be used with Equation (3.205) to estimate the effect of absorption onarray gain.35


As for the case of NB passive sonar, Rayleigh statistics are assumed here for bothnoise and signal amplitudes. Thus, the corresponding ROC relationships are identicalto those of Section 3.2.3.5 and are not repeated here.


The SNR threshold R50 is related to the probability of false alarm ( pfa) throughEquation (3.72). The value of pfa can be estimated by relating it to the false alarm ratenfa, the total number of detection opportunities per ‘‘look’’ Ntot, and the duration of


35 There is a problem with the application of Equation (3.208) for large z. As " approaches�=2 (meaning that the attenuation is so great that all sound is coming from directly overhead),the derivation fails. In this situation a lower limit of GA ¼ 1 is needed.

eack look Dt (the time between successive pulses)

ðnfaÞtrue ¼Ntot pfaDt

: ð3:209Þ

The number Ntot depends on the number of beams (Nbeams), Doppler cells (NDoppler),and range cells (Nranges):

36

Ntot ¼ NbeamsNDopplerNranges; ð3:210Þwhere

Nranges ¼Dt�t: ð3:211Þ

The true pfa (and therefore also nfa) depends on the threshold R50. Assuming thisthreshold to have been chosen in advance to achieve an acceptable false alarm rate,one can estimate the false alarm probability by rearranging Equation (3.73) andremoving the ‘‘true’’ qualifier (i.e., assume that the acceptable rate is achieved, sothat ‘‘true’’ and ‘‘acceptable’’ rates are equal),

pfa nfaNtot

Dt ¼ nfaNbeams Df

; ð3:212Þ

where Df is the total system bandwidth

Df ¼ NDoppler �f : ð3:213Þ

3.3.3.7 Special case: rigid spherical target in the broadside beam of a horizontalline array, CW pulse with Doppler processing

It is assumed that for a CW pulse with Doppler processing, reverberation may beneglected. In this situation, the signal-to-noise ratio (SNR) is

Rarr ¼QSE

QNE

GA ¼ QSð�ÞQN

GA; ð3:214Þ

where

QN QNf �f : ð3:215Þ

The signal MSP term is given by Equation (3.176), so that

Rarr ¼ GAS0F2QNf �f

: ð3:216Þ

The two-way propagation factor may be written, assuming a monostatic geometryand single propagation path (with spherical spreading plus absorption) to the target37


36 The frequency of the received signal differs from that of the transmitted one by an amount

proportional to the radial component of the target velocity relative to that of the sonar

platform. As this velocity is unknown, all reasonable possibilities need to be considered; hence

the NDoppler factor.37 The array and target are assumed here to be well separated from the sea surface, so that the

direct path is clearly separated from later multipaths. This direct path contribution is treated as

the signal.

and back to the sonar,

F2 ¼ F 2�back

4�; ð3:217Þ

where (neglecting surface reflections for simplicity)38

F ¼ e�2s

s2: ð3:218Þ

For a rigid sphere whose radius a is large compared with the acoustic wavelength,�back is (see Chapter 2)

�backðaÞ ¼ �a2; ð3:219Þ

and therefore, using Equation (3.192) to give the target strength,

TSE ¼ 10 log10a2

4dB re m2: ð3:220Þ

The high-frequency target strength of a rigid sphere whose radius is 2m is therefore0 dB re m2.


Consider a spherical steel target in deep water at a depth of 150m from the seasurface. A CW sonar operating at 50 kHz, with a bandwidth of 500Hz and sourcelevel of 200 dB re mPa2 m2, is placed at a horizontal distance of 1.3 km from thesphere. The sonar receiver, a horizontal line array of length 2m, is placed at a depthof 100m. The target is in the broadside beam of the receiving array.

It is convenient to introduce the signal energy level:

LSE ¼ SLE � ðPLTx þ PLRx � TSEÞ ð3:221Þ

and the in-beam noise spectrum level:

LNf ¼ NLf �AG: ð3:222Þ

With these definitions, the sonar equation becomes

SE ¼ LSE � LNf �DT: ð3:223Þ

(i) If the wind speed is 15m/s, what is the smallest sphere that could be detected bythe sonar using a pulse of duration 0.2 s?

(ii) For the sphere of Part (i), calculate the sonar figure of merit using Equation(3.196).

(iii) Calculate pdðrÞ for the same sphere. Calculate pdðzarrÞ at a range of 1.3 km.


38 In reality there is always some sound reflected from the sea surface. However, at high

frequency it is often heavily attenuated by near-surface bubbles and scattered due to sea surface

roughness. If the pulse is sufficiently short, the surface reflection can also be separated in time

from the direct arrival, such that the two paths may be treated independently.

(iv) From the answer to Part (iii), or otherwise, estimate the array depth at which thissonar is most likely to detect the sphere at a distance of 1.3 km.

For the purpose of calculating the probability of false alarm, assume an acceptablefalse alarm rate of one per hour, and that 128 beams are formed. Assume that steelacts as a perfect reflector of sound so that Equation (3.220) may be used for the TS,and that the sphere is moving fast enough to justify neglecting the reverberation. Forsimplicity, assume further that any surface-reflected sound does not contribute to thesignal (see Section 3.3.3.7).39

3.3.3.8.1 Part (i): smallest detectable sphere

The lowest value of the backscattering cross-section that results in a detection can befound by substituting Equation (3.217) in Equation (3.216) and then rearranging for�back to give

�back ¼ 4�

GA

QNf �fRarr

S0F2

: ð3:224Þ

Using Equation (3.219) (assuming �f �t ¼ 1) and requiring Rarr > R50 for detectiongives for the sphere’s radius

a >2

F

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQNf R50

GAS0 �t

s: ð3:225Þ

Using Equation (3.212) for pfa in Equation (3.72) results in

R50 ¼ log2Nbeams Df2nfa

; ð3:226Þ

which, with Df ¼ 500 Hz, Nbeams ¼ 128, and nfa ¼ 1=h, gives R50 ¼ 26.8. Equations(3.205), (3.218), and (3.179) give for the gain GA ¼ 99.1, one-way propagation factorF ¼ 5:34 10�9 m�2, and noise spectral density QN

f ¼ 0.0197mPa2/Hz. Further, theenergy source factor (defined as 10SLE=10 mPa2 m2 s) is S0 �t ¼ 20 kPa2 m2 s. Withthese values, the smallest sphere that satisfies Equation (3.225) has radius a ¼ 6.10m.

The relevant parameter values are listed in Table 3.3, expressed in standardreference units, as well as converted to decibels. For readers who prefer to workin decibels, the equivalent of Equation (3.225) is obtained by requiring SE> 0 dB inEquation (3.188):

TSE > NLf þDT� ðAG� 2PLþ SLEÞ; ð3:227Þ

from which it follows that TSE must exceed 9.7 dB re m2, consistent with the previous

calculation for the radius a.


39 See footnote 38.

3.3.3.8.2 Part (ii): figure of merit

There are two ways of calculating the FOM. The first and simplest uses the definitionof FOM from Equation (3.196), that is,

FOM ¼ PL50 ¼ �10 log10 Fðr50Þ; ð3:228Þ

with r50 ¼ 1.3 km. The value of the propagation factor from Part (i) givesFOM¼ 82.7 dB re m2. The alternative is to use the second half of Equation(3.196) with the help of Table 3.3. While the result is the same either way, thesecond method is more susceptible to rounding errors, illustrating the motivationfor retaining precision to the first decimal place for quantities expressed in decibels.

Figure 3.21 shows propagation loss plotted vs. range (upper graph) and vs. depth(lower graph), both calculated using Equation (3.218). As expected, the FOM ofapproximately 83 dB re m2 intersects the PLðrÞ curve at 1.3 km and PLðzarrÞ at100m. When the array depth exceeds about 300m, the PL starts to increase (thesignal decreases) with increasing sonar depth due to the increased path length. Thesame conclusion is reached from Figure 3.22, which compares signal and in-beamnoise levels. This behavior results in a second threshold crossing in the depth plot,close to 900m.

3.3.3.8.3 Part (iii): detection probability

Probability of detection is plotted in Figure 3.23. A value of 50% occurs at thedetection range of 1.3 km (upper) and 100m array depth (lower). Also shown is thesignal excess in linear form ð�Þ, defined in the same way as for passive sonar

� � RarrR50

: ð3:229Þ


Table 3.3. Sonar equation calculation for CW active sonar example with Doppler filter.



value in

reference

units

Energy source level SLE 193.0 dB re mPa2 m2 s S0 �t 2.00 10þ19(Equation 3.189)

Noise spectrum level NLf 42.9 dB re mPa2 Hz�1 3�cWNAf E3ðzÞ 1.97 10þ4

(Equation 3.193)

Array gain AG 20.0 dB re 1 �Dx2

cos " 99.1

(Equation 3.194)

Detection threshold DT 14.3 dB re 1 log2Nbeams Df2nfa

26.8

(Equation 3.195)


Figure 3.21. Propagation loss (blue) and figure of merit (cyan) vs. target range for array at

depth 100m (upper) and vs. array depth for target at 1.3 km (lower). The probability of detection

is 50% when the propagation loss is equal to the figure of merit, corresponding to the

intersection between solid blue and cyan lines. The target depth is 150m.


Figure 3.22. Signal level LSE (blue, solid) and in-beam noise level LNf (red, dashed) vs. target

range for array at depth 100m (upper) and vs. array depth for target at 1.3 km (lower). The

probability of detection is 50% when the signal exceeds LNf by the detection threshold,

corresponding to the intersection between solid blue and cyan lines. The target depth is 150m.



3.68) for coherent CW active sonar. The target depth is 150m. Upper: vs. range for array depth

100m—the two curves cross at the detection range, where � ¼ 2pd ¼ 1; lower: vs. array depth

for target range 1.3 km.

A more precise value of the depth at which the second threshold crossing occurs,namely 870m, can be read from Figure 3.23 (lower graph).

3.3.3.8.4 Part (iv): best depth

At depths exceeding a few hundred meters, Figure 3.21 shows that propagation lossincreases with increasing depth. The same graph shows that the figure of merit(FOM) also increases—a consequence of the decreasing in-beam noise level (LNf ).The result is that the variation in signal excess is not monotonic, with a maximumoccurring at a depth of about 430m (see Figure 3.23). A contour plot of signal excessvs. depth and range is shown in Figure 3.24. Although the predicted best depth of430m is unchanged, the detection range inferred from this graph is about 20% lessthan the value expected from the above calculations. This difference, which is attrib-uted to the use of different noise and propagation models, is typical of comparisonsbetween different sonar performance calculations with slightly different assumptions,even for nominally the same conditions.


Figure 3.24. Signal excess [dB] vs. target range and array depth for target depth 150m and sonar

frequency 50 kHz, illustrating a maximum detection range at a depth of about 430m (INSIGHT).

3.3.4 Incoherent processing: CW pulseþ energy detector

Consider now a sonar that transmits identical pulses to those of Section 3.3.3: a CWpulse of duration �t but with a simple energy detector as receiver instead of a Dopplerfilter. By ‘‘energy detector’’ is meant a receiver that squares and adds successive(equally spaced) time samples, without regard to their frequency content. Thischange does not affect the signal energy, but it has important consequences forthe background. The pulses are spaced at regular intervals Dt.


As for Section 3.3.3.1, the signal energy for a single hydrophone is

QSE ¼ QS�t; ð3:230Þ

where (using Equation 3.150)

QS ¼ S0F2; ð3:231Þand S0 is the source factor.

3.3.4.2 Background (single hydrophone)

The relevant hydrophone noise is that in the total receiver bandwidth Df , typicallyseveral orders of magnitude greater than the processing bandwidth of a Dopplerfilter. Further, the ability to discriminate between moving and stationary targets islost, meaning that reverberation also forms part of the background. (To be sure ofcontaining the whole signal the receiver bandwidth must also be large enough tocontain all reverberation.) The total background energy arriving in a time interval �tis

QBE ¼ QB�t; ð3:232Þ

where

QB ¼ðQNf df þQRð�Þ: ð3:233Þ

3.3.4.3 Signal-to-background ratio, signal excess, and incoherent activesonar equation

Although the transmitted pulse is CW, the sonar equation derived below is termed‘‘BB’’ because the receiver adds up energy incoherently across the whole sonarbandwidth. From the definition of AG, the signal-to-background ratio (SBR) is

Rarr ¼ RhpGA; ð3:234Þwhere

Rhp ¼QSE

QBE

: ð3:235Þ


Substituting Equations (3.230) and (3.232) in Equation (3.235), it follows fromEquation (3.234) (using Equation 3.231 for QS) that

Rarr ¼S0 �t F2

�t QBð�Þ=GA; ð3:236Þ

where F2 is the two-way propagation factor. The sonar equation is obtained in theusual way by dividing by R50 and converting to decibels:

SE � 10 log10RarrR50

¼ SLE � TPL� ðBLE �AGÞ �DT: ð3:237Þ

This equation has the same form as its NB counterpart, Equation (3.182), except thatNLf is replaced by BLE. The expressions for some of the individual terms are alsodifferent. As in the case of NB active sonar, TPL is given by

TPL ¼ PLTx þ PLRx � TSE: ð3:238Þ

Thus the BB active sonar equation for a point target and with incoherent processing,including reverberation is

SE ¼ ½SLE � ðPLTx þ PLRx � TSEÞ� � ðBLE �AGÞ �DT: ð3:239Þ

The individual sonar equation terms are given by

SLE � 10 log10ðS0 �tÞ ðenergy source level: dB re mPa2 m2 sÞ; ð3:240Þ

PLTx ¼ �10 log10 FTx ðpropagation loss Tx to target: dB re m2Þ; ð3:241Þ

PLRx ¼ �10 log10 FRx ðpropagation loss target to Rx: dB re m2Þ; ð3:242Þ

TSE ¼ 10 log10�back

4�ðtarget strength: dB re m2Þ; ð3:243Þ

BLE ¼ 10 log10½�t QBð�Þ� ðbackground energy level: dB re mPa2 sÞ; ð3:244Þ

AG ¼ 10 log10 GA ðarray gain: dBÞ; ð3:245Þ

and

DT ¼ 10 log10 R50 ðdetection threshold: dBÞ: ð3:246Þ

A figure of merit can be defined as previously for coherent processing, although theconcept is a less useful one in the presence of reverberation because the background isa function of distance to the target. The result is

FOMðrÞ � 12ðPLRx þ PLTxÞ50 ¼ 1

2½SLE þ TSE � BLEðrÞ þAGðrÞ �DT�; ð3:247Þ

and hence

SEðrÞ ¼ 2 FOMðrÞ � ½PLRxðrÞ þ PLTxðrÞ�: ð3:248Þ


Finally, in analogy with Equation (3.244), it is useful to introduce energy levels ofnoise and reverberation separately:

NLE � 10 log10 �t

ðQNf df

� �ð3:249Þ

and

RLE � 10 log10½�t QRð�Þ�: ð3:250Þ

3.3.4.4 Array gain

The array gain depends on the relative importance of noise and reverberation andtherefore depends on the delay time � in the following manner

GAð�Þ ¼QN þQRð�Þð

½QNO þQR

Oð�Þ�BðOÞ dO; ð3:251Þ

where (for a continuous line array)

BðOÞ ¼ sin2 u

u2ð3:252Þ

and

u ¼ k Dx cos � sin �2

: ð3:253Þ

The reverberation arriving after time � does so at a well-defined grazing angle �� ,determined by

sin �� ¼2zarrc�

; ð3:254Þ

such that

QROð�Þ ¼

QR

2� cos ��ð�� Þ: ð3:255Þ

The array response to reverberation, calculated using the method of Section 3.2.3.4, isððQR

OÞBðOÞ dO ¼ 1

2�QRD�ð��Þ; ð3:256Þ

where D� is given by Equation (3.60). For a long array, away from the endfiredirection:

D� ¼ 4 arcsin�

k Dx cos �; ð3:257Þ

so that ððQR

OÞBðOÞ dO ¼ 2QR

�arcsin

�

k Dx cos ��; ð3:258Þ


which may (assuming the argument of the arcsine function to be small) beapproximated further by ð

ðQROÞBðOÞ dO

2 Dx2QR

� cos ��: ð3:259Þ

It follows from Equation (3.205) for the noise gain thatððQN

O ÞBðOÞ dO ¼

2 Dx4QN

� cos ": ð3:260Þ

Substitution of this result and Equation (3.259) in Equation (3.251) yields

GAð�Þ ¼ G0QN þQRð�Þ

4

�

QN

cos "þ QRð�Þ2 cos ��

" # : ð3:261Þ

The gain against noise is the directivity factor multiplied by ð�=4Þ cos ", as previously(Section 3.3.3.4). The gain against reverberation includes an additional factor2 cos �� . Thus, if the reverberation is from a distant scatterer, such that �� is closeto zero, the gain against reverberation (in the broadside beam, and neglectingabsorption of wind noise) is approximately twice the gain against surface-generatedwind noise. This is because the broadside beam is narrower in azimuth than in theelevation direction.


The calculation of detection probability follows that for active sonar with coherentprocessing, with Rayleigh statistics assumed for both background and signal ampli-tudes. The corresponding ROC relationships are identical to those of Section 3.2.3.5and are not repeated here.


Following Section 3.2.4.6, the false alarm probability can be estimated from thedesired false alarm rate nfa

nfa Ntot pfaDt

; ð3:262Þwhere

Ntot ¼ NbeamsNranges: ð3:263Þ

Range resolution is determined by the pulse duration �t, so that the total number ofrange cells is

Nranges ¼Dt�t; ð3:264Þ

and hence

pfa ¼nfa �t

Nbeams: ð3:265Þ


3.3.4.7 Special case: point target in the broadside beam of a horizontal line array,CW pulse with incoherent processing

Consider now the same special case as previously for NB, except with incoherentreceiver processing. In this situation it is no longer legitimate to neglect reverberation,so the signal-to-background ratio (SBR) after beamforming is

Rarr ¼QSE

QBE

GA ¼ QSð�ÞQN þQRð�ÞGAð�Þ; ð3:266Þ

where

QN ¼ðfmþDf =2

fm�Df =2QNf df ; ð3:267Þ

fm is the center frequency; and Df is the total receiver bandwidth. If the spectrum QNf

varies linearly with frequency within the sonar bandwidth, the integral simplifies to

QN ¼ QNf ð fmÞ Df : ð3:268Þ

The signal and reverberation MSP terms are given by

QSð�Þ ¼ S0F2 �

back

4�ð3:269Þ

and

QRð�Þ ¼ S0Fð�Þ2�OAð�Þ �Að�Þ; ð3:270Þwhere

�Að�Þ ¼ 2�c�

2

c �t

2ð3:271Þ

is the scattering area and

�OAð�Þ ¼ ðCPM=16�Þ tan4 � ð3:272Þ

is the surface scattering coefficient, evaluated in the backscattering direction.The time variable � is proportional to the distance s

� ¼ ð2=cÞs ð3:273Þ

and the grazing angle � is given by

tan2 � ¼ z2arrs2 � z2arr

: ð3:274Þ

Substituting back into Equation (3.266), with Equation (3.218) for the propagationfactor, and assuming small angles (such that zarr � s), the SBR is

Rarr ¼4GAð2s=cÞ

�

�back

16s4 e4sQNf ð fmÞ Df =S0 þ CPMcz

4arr �t=s

3: ð3:275Þ



(i) For otherwise the same problem as Section 3.3.3.8—Parts (i)–(iii)—what isthe effect on (a) signal level, (b) in-beam background level, and (c) detectionthreshold of replacing the Doppler filter with an energy detector?

(ii) What is the detection range for the energy detector if the sonar is kept at itsoriginal depth of 100m?

(iii) Calculate pdðrÞ for a sonar depth of 100m and pdðzarrÞ at a range of 0.9 km.It is convenient to introduce the in-beam background energy level

LBE � BLE � AG: ð3:276Þ

The signal level takes the same form as for coherent processing (Equation 3.221)

LSE ¼ SLE � ðPLTx þ PLRx � TSEÞ ð3:277Þ

so that the sonar equation becomes

SE ¼ LSE � LBE �DT: ð3:278Þ

3.3.4.8.1 Part (i): signal, background, and detection threshold

(a) signal: the signal level, as defined by Equation (3.277) and plotted in Figure 3.25,is independent of receiver processing and therefore identical to that of Figure3.22. Small differences arise in the depth plots only because these are evaluated ata range of 900m instead of 1300m previously.

(b) background: the in-beam noise level is 20 dB higher than for the same problemwith Doppler processing (compare Figure 3.26 with Figure 3.22); at short range(up to 400m) the contribution to the background from reverberation is higherstill; the reverberation peaks at a depth of about 900m, causing an inflexion inthe total background.

(c) detection threshold: the incoherent processing results in a 100-fold reduction inthe number of detection opportunities per ping, which (if pfa is kept fixed) resultsin a decrease in DT50 of 1.2 dB compared with Doppler processing.

3.3.4.8.2 Part (ii): detection range

The net result of the large increase in background level and small decrease indetection threshold is a decrease of about 19 dB in long-range signal excess, withan even larger decrease at short range due to reverberation. Consequently, thedetection range is reduced to 0.9 km (the intersection of LSE and L

BE þDT in Figure

3.25, or of PL and FOM in Figure 3.27).

3.3.4.8.3 Part (iii): detection probability

Figure 3.28 shows detection probability vs. range at 100m depth and vs. depth at900m range. The increased reverberation at short range (Figure 3.27) manifests itselfhere as a dip in the detection probability for ranges less than 200m. While thepresence of reverberation must affect sonar performance (by reducing the detection



Figure 3.25. SignalLSE and (in-beam) backgroundLBE levels vs. target range at an array depth of

100m (upper) and vs. array depth at range 900m (lower). The probability of detection is 50%

when LSE exceeds LBE by the detection threshold (DT¼ 13.0 dB), corresponding to the inter-

section between solid blue and cyan lines.


Figure 3.26. Total background BLE, background components NLE, RLE, and in-beam

background level LBE vs. target range at an array depth of 100m (upper) and vs. array depth

at range 900m (lower).


Figure 3.27. Propagation loss vs. target range for array at depth 100m (upper) and vs array

depth for target at 0.9 km (lower). The probability of detection is 50%when propagation loss is

equal to the figure of merit, corresponding to the intersection between solid blue and cyan lines.


Figure 3.28. Linear signal excess and twice detection probability (Equation 3.68) for incoherent

CW active sonar. Upper: vs. range for array depth 100m—the two curves cross at the detection

range, where � ¼ 2pd ¼ 1; lower: vs. array depth for target range 900m.

probability), it has no effect here on the predicted detection range. This is becausereverberation has decreased to a negligible contribution at that range. In thissituation, the detection is said to be noise-limited.

3.4 REFERENCES

Ainslie, M. A., Harrison, C. H., and Burns, P. W. (1996) Signal and reverberation prediction

for active sonar by adding acoustic components, IEE Proc. Radar, Sonar, Navig., 143(3),

190–195. [Special issue on recent advances in sonar.]

ASA (1989) American National Standard: Reference Quantities for Acoustical Levels, ANSI

S1.8-1989 [ASA 84-1989, Revision of ANSI S1.8-1969(R1974)], Acoustical Society of

America, New York.

ASA (1994) American National Standard: Acoustical Terminology, ANSI S1.1-1994 [ASA 111-

1994, Revision of ANSI S1.1-1960(R1976)], Acoustical Society of America, New York.

Horton, J. W. (1959) Fundamentals of SONAR (Second Edition), United States Naval Institute,

Annapolis, MD.

IEC (www) Electropedia (IEV online), Acoustics and electroacoustics/IEV 801 (International

ELectrotechnical Commission), available at http://www.electropedia.org/iev/iev.nsf (last

accessed June 23, 2009).


Southall, B. L., Bowles, A. E., Ellison, W. T., Finneran, J. J., Gentry, R. L., Greene Jr., C. R.,

Kastak, D., Ketten, D. R., Miller, J. H., Nachtigall, P. E., Richardson, W. J., Thomas

J. A., and Tyack, P. L. (2007) Marine mammal noise exposure criteria: Initial scientific

recommendations, Aquatic Mammals, 33(4), 411–521.


Table 3.4. Sonar equation calculation for CW active sonar example with incoherent energy

detector.



value in

reference

units

Energy source level SLE 193.0 dB re mPa2 m2 s S0 �t 2.00 10þ19(Equation 3.240)

Noise spectrum level NLf 42.9 dB re mPa2 Hz�1 3�cWNAf E3ð2zÞ 1.97 10þ4

(Equation 3.193)

Array gaina AG 20.0 dB re 1 G0½QN þQRð�Þ�4

�

QN

cos "þ QRð�Þ2 cos ��

� � 99.1

(Equation 3.245)

Detection threshold DT 13.0 dB re 1 R50 ¼ log2Nbeams

2nfa �t20.1

(Equation 3.246)

a The numerical values quoted correspond to evaluation of the stated formula in the limit of large � .

Part II

The Four Pillars

4

Sonar oceanography

All science is either physics or stamp collecting

Ernest Rutherford (ca. 1910)

This is the first of four chapters dealing further with each of the four main subjectsintroduced in Chapter 2. The purpose is to describe them in sufficient detail to equipthe reader with the necessary knowledge to carry out predictions of sonar per-formance in realistic situations. The first subject of the four, and that of the presentchapter, is oceanography. The remaining three are underwater acoustics (seeChapter 5), sonar signal processing (Chapter 6), and detection theory (Chapter 7).

By ‘‘sonar oceanography’’ is meant a description of those properties of the sea,its contents, and its boundaries of relevance to sonar. As implied by the introductoryquotation from Lord Rutherford, the material is presented as a collection of em-pirical facts—an organized ‘‘stamp collection’’ of the acoustical properties of the sea.Readers are encouraged to treat this chapter as they might an encyclopedia, skim-ming through it on first reading, and returning to consult it as often as necessary tolook up details for later chapters.

Wherever possible, simple equations are given for key parameters such as thespeed of sound in seawater. More accurate (and more complicated) expressions areavailable if required (Fisher and Worcester, 1997; Fofonoff and Millard, 1983; Leroy,2001). For application to sonar performance modeling, very high accuracy is rarelyneeded, justifying some simplifications made in the interests of clarity, withoutsacrificing realism.

The following are described:

— the bulk physical and chemical properties of seawater that affect the propagationof underwater sound (Section 4.1);

— the properties of the acoustically significant contents of the sea, especially thosewith an air-filled enclosure (Section 4.2);

— the properties of wind-generated waves and associated bubble clouds (Section4.3);

— the acoustically significant parameters of the seabed (Section 4.4).

4.1 PROPERTIES OF THE OCEAN VOLUME

4.1.1 Terrestrial and universal constants

A parameter that appears repeatedly in this chapter is the acceleration due to gravityg. Except where stated otherwise, the value used for g is 9.806 65 m/s2, correspondingto one standard gravity. Other important parameters are atmospheric pressure, forwhich a standard atmosphere is used, defined as (IAPSO, 1985)1

PSTP � 101:325 kPa; ð4:1Þand the freezing point of water at one standard atmosphere, given by

YSTP ¼ 273:15 K: ð4:2ÞAlso needed is a value for the Boltzmann constant K :

K ¼ 1:38065 � 10�23 J/K: ð4:3Þ

4.1.2 Bathymetry

An important characteristic of the sea is its depth. In the oceans, water depth valuesbetween 2 km and 5 km are common, increasing to about 10 km in the deepest oceantrenches. The continental shelves have a water depth of typically 20 m to 200 m.Regions of intermediate water depth (200–2000 m) are relatively rare and usuallylimited to the continental slopes. Water depths traditionally of interest to sonarperformance modeling are mostly between 20 m and 5 km, with increasing emphasison the shallow end of this range.

The lateral variations in water depth within a geographical region are referred toas the bathymetry of that region. A global bathymetry map, available from thewebsite of the University of California, San Diego,2 is shown in Figure 4.1. The dataare derived from satellite measurements of geographical variations in Earth’s gravityfield (see Smith and Sandwell, 1997).

4.1.3 Factors affecting sound speed and attenuation in pure seawater

Seawater covers more than two-thirds of Earth’s surface. Each of the three mainoceans, the Pacific, Atlantic and Indian Oceans, has a distinct oceanographic sig-nature determined, for example, by its temperature and salinity. All three are

126 Sonar oceanography [Ch. 4

1 The subscript STP denotes ‘‘standard temperature and pressure’’.2 http://topex.ucsd.edu/marine_topo (last accessed February 16, 2008).

interconnected via the Southern (or Antarctic) Ocean. The properties of these oceansthat affect the propagation speed (c) or rate of attenuation (�) of underwater soundare described below. Advice for calculating c and �, respectively, is provided inSections 4.1.4 and 4.1.5.

4.1.3.1 Density and static pressure

The static pressure Pw (denoted Pstat in Chapter 2) increases monotonically withdepth, starting from atmospheric pressure at the surface, and is given in terms of thedensity � and acceleration due to gravity g (a function of depth here) by

PwðzÞ ¼ Patm þðz

0

�ð�Þgð�Þ d�: ð4:4Þ

Leroy (1968) provides the following convenient expression for gðzÞ as a function oflatitude �

gðzÞ ¼ ð9:7805 m s�2Þð1 þ 5:28 � 10�3 sin2 �Þ þ ð2:4 � 10�6 s�2Þz: ð4:5Þ

Similarly, the density profile can be written in terms of the pressure, salinity, andtemperature profiles (Pierce, 1989, p. 34)

��ðzÞ ¼1027 þ 4:3 �10�7PPwðzÞ þ 0:75½SðzÞ�35� 0:16½TTðzÞ�10 � 0:004½TTðzÞ�102:ð4:6Þ

Here, ��, PPw, and TT are dimensionless variables, equal to the numerical values ofdensity, hydrostatic pressure, and temperature, when expressed in units of kilograms

4.1 Properties of the ocean volume 127]Sec. 4.1

Figure 4.1. Global bathymetry map from NOAA, derived from satellite measurements of the

gravity field. Scale in meters above sea level (reprinted from Sandwell et al., www).

per cubic meter, pascals, and degrees Celsius, respectively. In other words

��

1 kg m�3; ð4:7Þ

PPw � Pw

1 Pa; ð4:8Þ

and

TT � Y �YSTP

1 K; ð4:9Þ

where Y is the absolute temperature

YðTÞ ¼ T þYSTP: ð4:10Þ

More generally, the notation xx is used throughout for the numerical value of thevariable x when expressed in SI units. For example, the variable vv introduced inChapter 2 is the wind speed expressed in meters per second.

The integral of Equation (4.4) cannot be calculated directly, because it is animplicit equation, with the pressure that we are seeking to evaluate appearing itselfinside the integrand. However, for realistic conditions in seawater, Leroy derives thefollowing quadratic approximation for the variation of pressure with depth andlatitude (Leroy, 1968):3

PPwðzÞ ¼ 98066:5½1:04 þ 0:102506ð1 þ 5:28 �10�3 sin2 �Þzzþ 2:524 �10�7zz2; ð4:11Þ

where

zz � z

1 m: ð4:12Þ

The pressure at any given depth z increases with increasing latitude � due to theincreasing gravitational force, which is at its greatest close to the poles. If the latitudeis not known, the sin2 � term can be approximated by its average value of 0.5.

4.1.3.2 Temperature

Apart from the polar regions, the temperature of the deep ocean has an almostconstant value of about 2 C (see Figure 4.2). There is little seasonal variation, soonly the annual average is shown. The Atlantic Ocean is about one degree warmerthan the Indian and Pacific Oceans. This is a consequence of the oceanic thermohalinecirculation, a global conveyor belt that begins its cycle as surface water in the NorthAtlantic.4

Surface temperature is higher and less uniform in space and time (Figure 4.3).


3 Leroy gives separate equations for the Baltic and Black Seas, but for most applications the

general equation is sufficient.4 Strong winds in the North Atlantic evaporate the surface water, simultaneously cooling it and

increasing its salinity. The combined effect is a sudden increase in density that causes the

surface water to sink and begin a long journey south as fresh North Atlantic deep water

(NADW). Because of its recent contact with the atmosphere, the NADW is warmer than deep

water in other oceans (Brown et al., 1989).

Seasonal changes are particularly noticeable in temperate seas. For example, theMediterranean region, including the Black Sea, exhibits changes in surface tempera-ture of up to about 15 C between summer and winter. Similar variations can be seenin the northwest corner of the Pacific (the Yellow Sea and Sea of Japan). Between thewarm near-surface water and the cold deep water there is a region of rapidly decreas-ing temperature with depth, known as a thermocline. Variations of sound speed withdepth resulting from the temperature profile have a strong influence on underwatersound propagation and hence on sonar performance.

A temperature profile can be measured using a conductivity–temperature–depth(CTD) probe, although the use of this type of probe might require a stationarymeasurement platform. An alternative is an expendable bathythermograph (XBT),which can be deployed while the measurement ship is in motion. The main benefit ofthe CTD probe is that, by combining the conductivity and temperature data, thesalinity can also be calculated. Two typical deep-water temperature profiles areshown in Figure 4.4, representative of deep-water profiles in the northwest Pacific(see Figure 4.5) and northeast Atlantic (Figure 4.6), respectively.

4.1.3.3 Salinity

The traditional definition of salinity is a ratio by mass of dissolved salts in water.Today this traditional definition, referred to as absolute salinity, is superseded bypractical salinity. Practical salinity is defined as a dimensionless (and unitless) ratio interms of the conductivity of the salt-water solution (IAPSO, 1985), in such a way thatits value is almost identical to that of absolute salinity expressed in parts per thousandby mass (i.e., grams per kilogram).


Figure 4.2. Annual average temperature map at depth 3 km, from the World Ocean Atlas

(WOA, 1999). The deep-water temperature of major non-polar oceans is between 1 C and

3 C. Green indicates land (or water depth less than 3000 m).

Compared with that of temperature, the effect of salinity on sound speed isusually small; a simple estimate of its value is often sufficient. According to Fisherand Worcester (1997), three-quarters of the ocean has a salinity within 1.0 of itsmedian value of 34.69. This point is illustrated by Figure 4.7, showing that the salinityof the major oceans falls mostly between 34.5 and 35.0. Hence, if no better informa-tion is available for salinity, a default value of 34.7 is suggested. However, there aresystematic differences between the average salinity of major oceans (see Table 4.1 andFigure 4.8) and this information can be used to obtain improved estimates. Noticeespecially the high salinity prevalent in the deep Atlantic Ocean (Figure 4.7), neces-


Figure 4.3. Geographical variations in surface temperature for northern winter (upper graph)

and northern summer (lower) (from WOA, 1999). Green indicates land.


Figure 4.4. Temperature profiles from WOA (1999) for locations in the northwest Pacific

Ocean (thin curves: 19N, 150E) and northeast Atlantic (thick curves: 42N, 13W). Upper:

full profile; lower: zoom of upper 300 m.


Figure 4.5. Bathymetry map for the northwest Pacific Ocean from NOAA, derived from

satellite measurements of the gravity field (reprinted from Sandwell et al., www). For scale

see Figure 4.1.

Figure 4.6. Bathymetry map for the north Atlantic Ocean from NOAA, derived from satellite

measurements of the gravity field (reprinted from Sandwell et al., www). For scale see Figure 4.1.

sary to maintain a uniform density at a depth of 3000 m, compensating for the highertemperature there relative to the other oceans (Figure 4.2).

Seasonal variations in salinity are difficult to discern even at the sea surface(Figure 4.9). Nevertheless, where significant departures of salinity from its median


Figure 4.7. Annual average salinity map at depth 3 km, from WOA (1999). White indicates

land (or water depth less than 3000 m).

Table 4.1. Average salinity and potential temperature by major ocean basin

(Worthington, 1981).a

Ocean Mean salinity Mean potential temperatureb/C

North Pacific 34.57 3.13

South Pacificc 34.63 3.50

Indianc 34.79 4.36

North Atlantic 35.09 5.08

South Atlanticc 34.84 3.81

Southern 34.65 0.71

World Ocean 34.72 3.51

a The difference in salinity between the Pacific and Atlantic (and a similar difference intemperature already described), is well documented (Worthington, 1981).b Potential temperature is the temperature the water would have if transportedadiabatically to a standard depth or reference pressure (usually atmospheric)(Brown et al., 1989).c Excluding Southern Ocean.

value do occur, these need to be considered for an accurate prediction of attenuation(see, for example, Equation 4.33). For most locations a reasonable estimate can beobtained from climatology. In special situations,5 the effect of salinity on sound speedcan also be important. Salinity profiles for the same situations as Figure 4.4 areshown in Figure 4.10.

Salinity can be measured with a CTD probe. Alternatively, if only temperature ismeasured in situ (e.g., using an XBT), the salinity profile can be calculated from anestimate of the density profile using the procedure outlined below. It is assumed thatrepresentative T and S profiles are available from a nearby CTD cast (or, if no CTDdata are available, from climatological data). The method is based on the presump-tion that the density profile does not change significantly between the CTD and XBTmeasurements. The force of gravity can usually be relied on to stabilize the densityprofile, even if the sea is in a state of flux. An illustration of this point is found in theupper graph of Figure 4.11. Even though salinity and temperature profiles in the deepAtlantic Ocean are different from those in the deep Pacific, the density profilesbetween 1 km and 5 km are almost identical.

The first step in the procedure is to estimate the density profile using Equation(4.6) from representative temperature and salinity profiles. We call this density profile


Figure 4.8. Temperature salinity (T–S) diagram for the World Ocean (adapted from

Worthington, 1981, MIT Press, reprinted with permission).#

5 Regions of exceptionally low salinity (e.g., the Baltic Sea) can occur due to the influx of

freshwater from rivers, precipitation or melting ice, and of high salinity (e.g., Persian Gulf ) due

to freezing or evaporation.

�estðzÞ and use it as an estimate of in situ density. An estimate of the in situ salinityprofile can then be obtained by rearranging Equation (4.6):

SestðzÞ ¼ 35 þ 43��estðzÞ � 4

3f1027 þ 4:3 � 10�5PPwðzÞ � 0:16½TTmeasðzÞ � 10

� 0:004½TTmeasðzÞ � 102g ð4:13Þ

where TmeasðzÞ is the in situ temperature measurement; and PwðzÞ is the pressureprofile from Equation (4.11).

Other variants of this procedure are also possible. For example, if in situ soundspeed and temperature profiles are both known (the sound speed can be measureddirectly using a velocimeter), salinity can be estimated by rearranging Equation 4.21.


Figure 4.9. Seasonal variations in surface salinity (from WOA, 1999). White indicates land.


Figure 4.10. Salinity profiles from WOA (1999) for locations in the northwest Pacific Ocean

(thin curves: 19N, 150E) and northeast Atlantic (thick curves: 42N, 13W). Upper: full

profile; lower: zoom of uppermost 300 m.


Figure 4.11. Density profiles from WOA (1999) for locations in the northwest Pacific Ocean

(thin curves: 19N, 150E) and northeast Atlantic (thick curves: 42N, 13W). Upper: full

profile; lower: zoom of uppermost 300 m.

4.1.3.4 Acidity (pH)

The absorption of underwater sound is sensitive to the pH of seawater. Severaldifferent pH scales are in use, each resulting in a different numerical value for pHunder identical conditions (see Appendix B for their definitions). This problem can bemitigated by presenting acidity data in terms of a parameter ‘‘K ’’, whose precisedefinition is adjusted according to the scale used in such a way that its numerical valueis approximately independent of that scale (Brewer et al., 1995). Pioneering work byMellen et al. (1987) with the U.S. National Bureau of Standards scale ( pHNBS) usedthe definition

KNBS ¼ 10ðpHNBS�8Þ: ð4:14Þ

Other scales in use include the total proton scale ( pHT), with

KT ¼ 10ðpHT�7:858Þ ð4:15Þ

and the seawater scale ( pHSWS)

KSWS ¼ 10ðpHSWS�7:85Þ: ð4:16Þ

These three pH scales are related approximately as follows

pHNBS pHSWS þ 0:15 ð4:17Þand

pHT pHSWS þ 0:01; ð4:18Þ

from which it can be seen that the differences between the three K definitions, of up toa factor of order 100:002 (about 1.005), are small and for the present purpose may beneglected. For this reason, no further distinction is made between them and thesubscript is dropped. (The subscript is retained for pH scales, as differences betweenthese are significant). To avoid the risk of confusion between these scales, theconvention is adopted here to report acidity values in terms of K rather than pH.

Modern use of the NBS scale for seawater is discouraged by Brewer et al. (1995)and Millero (2006). Instead, Brewer et al. (1995) use the seawater scale ( pHSWS),which is recommended by a UNESCO report (Dickson and Millero, 1987). Morerecent literature (Wedborg et al., 1999; Ternon et al., 2001) adopts the total protonscale ( pHT). In the following, where a pH value is mentioned, the SWS scale is used,accompanied by a conversion to NBS in order to facilitate comparison with earlyliterature.

Figure 4.12 shows the global contours of K from Mellen et al. (1987). The NorthPacific is renowned for its low attenuation at low frequency, caused by low pH values(i.e., low K) close to the sound channel axis (see Figure 4.12).6 Figure 4.13 showssimilar maps for the Arctic Ocean and Figure 4.14 shows K profiles for major oceans,calculated using GEOSECS data from Mellen et al. (1987). Modern (higher resolu-


6 The reason for the low pH is a 10 % higher concentration of carbon (in the form of carbonic

acid) than in the Atlantic. The carbon originates from CO2 formed by the respiration and

decomposition of living matter (Brown et al., 1989, p. 113). The CO2 concentration reaches a

peak at a depth of about 1 km, resulting in a minimum of pH at that depth (Millero, 2006).

tion) measurements for the equatorial Atlantic Ocean are reported by Ternon et al.(2001).

The lowest reported value of K (based on Figure 4.12) is 0.5, meaning that pHSWS

is greater than 7.55 (i.e., pHNBS > 7.70). More precisely, the pH of seawater is mostlyin the range 7.55 < pHSWS < 8.15 (i.e., 7.7 < pHNBS < 8.3), which is slightly alkaline.A good global default value for pHSWS is 7.85, increasing to 8.10 close to the seasurface.7

4.1.3.5 Viscosity

Shear viscosity (commonly abbreviated as ‘‘viscosity’’) can be described in simpleterms as resistance to shear flow, of the kind encountered when stirring honey.Formally it is a constant of proportionality relating viscous stress to the rate of strainof a fluid (Morfey, 2001). Water has a much lower shear viscosity than honey, but it isnevertheless sufficient to have a noticeable effect on the attenuation of high-frequencysound.

The shear viscosity of seawater, denoted S, is equal to 1.4 mPa s at a temperatureof 10 C, salinity 35, and atmospheric pressure. It varies with temperature from about1.9 mPa s at T ¼ 0 C to 0.9 mPa s at 30 C (Horne, 1969, p. 96). The effect of salinityis small by comparison, viscosity decreasing by no more than 10 % with a reductionin salinity from 40 to 5. The effect of pressure is also relatively small, with a reductionof viscosity of at most 5 % from an increase in pressure to 50 MPa (500 atmospheres).Dependence on salinity S and absolute temperature Y is given by the empiricalformula (Francois and Garrison, 1982a)8

SðS;YÞ ¼ 0:924½1 þ 0:0018ðS � 35ÞYY expð2431=YYÞ nPa s: ð4:19Þ

A related parameter is bulk viscosity (also known as volume viscosity or compressionviscosity). The measured value of bulk viscosity is given by Liebermann (1948) as

B

S

¼ 2:2: ð4:20Þ

4.1.4 Speed of sound in pure seawater

The speed of sound in pure seawater is a complicated function of salinity S,temperature T , and pressure P (Fofonoff and Millard, 1983; Fisher and Worcester,1997; Leroy, 2001). A useful summary of seawater properties, including on-linecalculators for the speed of sound in seawater and freshwater, is provided by NPL(2007a). A simplified formula for the speed of sound in seawater, suitable for sonar


7 The pH of the sea surface is predicted to drop at a rate of up to 0.05 per decade during the 21st

century (Caldeira and Wickett, 2005; Raven et al., 2005).8 For a more complicated expression, including the effect of pressure, see Matthaus (1972).

performance modeling, is given by Mackenzie (1981)

ccðS;T ; zÞ ¼ 1448:96 þ 4:591TT � 0:05304TT 2 þ 2:374 � 10�4TT 3

þ ð1:340 � 0:01025TTÞðS � 35Þ þ 0:01630zz

þ 1:675 � 10�7zz2 � 7:139 � 10�13TTzz3; ð4:21Þ


Figure 4.12.

GEOSECS global

K contours at sea

surface (left) and at

depth 1 km (right)

(reprinted from

Mellen et al., 1987).

where, as previously, the circumflex indicates the numerical value of each variableexpressed in SI units. Thus, evaluation of the right-hand side of Equation (4.21) givesthe sound speed in units of meters per second.9 A graph of sound speed vs. depth isknown as a sound speed profile (see Figure 4.15 for an example) The expression is


9 The temperature TT is in degrees Celsius, not kelvin.


Figure 4.13. GEOSECS

Arctic K contours at the

sea surface (upper:

left¼winter; right¼summer) and at depth 1 km

(lower left); the 1 km

bathymetry contour is also

shown (lower right)

(reprinted from Mellen et

al. (1987).

valid (to within a precision of �0.07 m/s) in the following parameter ranges:

�2 < TT < þ30

25 < S < 40

0 < zz < 8000:

9>>>=>>>;

ð4:22Þ

The dependence on pressure is parameterized in Equation (4.21) by means of thedepth z. This is made possible by the near-universal relationship between pressureand depth described previously (see Equation 4.11).

A slightly more complicated equation (with 14 terms instead of 8), given by Leroyet al. (2008), retains its precision even in extreme situations such as the Black Sea,Baltic Sea, Arctic Ocean, and Mariana Trench. Leroy’s 2008 equation is

ccðS;T ; z; �Þ ¼ 1402:5 þ 5TT � 0:0544TT 2 þ 2:1 � 10�4TT 3

þ ð1:33 � 0:0123TT þ 8:7 � 10�5TT 2ÞS

þ ½0:0156 þ 1:2 � 10�6ð�deg � 45Þ þ 3 � 10�7TT 2 þ 1:43 � 10�5Szz

þ 2:55 � 10�7zz2 � ð7:3 � 10�12 þ 9:5 � 10�13TTÞzz3; ð4:23Þ

where �deg is the latitude in degrees; and the pressure-dependent terms are written as apower series in depth to facilitate comparison with Equation (4.21). Leroy et al.


Figure 4.14. GEOSECSK profiles for major oceans derived using Nutall’s algorithm (Mellen et

al., 1987).


Figure 4.15. Sound speed profiles calculated using Equation (4.21) with TðzÞ and SðzÞ from

WOA (1999), for locations in the northwest Pacific Ocean (thin curves: 19N, 150E) and

northeast Atlantic (thick curves: 42N, 13W). Upper: full profile; lower: zoom of uppermost

300 m.

(2008) demonstrate the accuracy of their equation (to within �0.2 m/s) for so-called‘‘Neptunian’’10 waters in the range

�1 < TT < þ21

12 < S < 40:5

0 < zz < 12000:

9>>=>>; ð4:24Þ

Equations (4.21) and (4.23) do not include the influence of bubbles on the speed ofsound. A discussion of this important effect is deferred to Chapter 5. Sound speedcalculated using Equation (4.21) is plotted vs. depth for the northwest Pacific andnortheast Atlantic locations considered previously.

4.1.5 Attenuation of sound in pure seawater

The limiting factor for long-range sonar performance can sometimes be the absorp-tion of sound in seawater. At very high frequency, of order 1 MHz and higher, soundabsorption is caused mainly by water viscosity, which depends mainly on temperatureand salinity. At lower frequencies (up to about 300 kHz) chemical relaxations areimportant, resulting in an additional dependence on pressure and acidity (parameter-ized through the parameter K as described in Section 4.1.3.4).

The presence of bubbles can have a significant effect on the attenuation ofsound.11 Also important, especially in coastal regions, is the possible presence oflarge numbers of fish. Discussion of the effects of both bubbles and fish on attenua-tion is deferred to Chapter 5. The present focus is on the effects of S, T , z, and K .

The amplitude attenuation coefficient � introduced in Chapter 2 was expressed inunits of nepers per unit distance (e.g., Np/km). It is conventional to express thisquantity in units of decibels per unit distance (e.g., dB/km), for which the symbol a isused here. The two quantities are related by

a=(dB km�1) ¼ ð20 log10 eÞ�=(Np km�1): ð4:25Þ

Denoting the individual contributions due to viscous and chemical relaxation effectsas avisc and achem, respectively, the total (decibel) absorption coefficient in pureseawater due to viscosity and chemical relaxations can be written

awater ¼ avisc þ achem: 4:26Þ

It is convenient to write the viscous and chemical terms in the following forms

avisc ¼ Avisc f2 ð4:27Þ


10 The colorful adjective ‘‘Neptunian’’ refers to those water masses interconnected with the

World’s oceans. The term originates from the notion that King Neptune would have been

unable to reach landlocked seas and lakes, which are thus excluded from his ‘‘kingdom’’.11 The word ‘‘attenuation’’ is used here as a synonym of ‘‘extinction’’; in general, including the

effects of both absorption and scattering.

and

achem ¼ AB

f 2

f 2 þ f 2B

þ AMg

f 2

f 2 þ f 2Mg

; ð4:28Þ

where fB and fMg are the respective relaxation frequencies of boric acid (B(OH)3) andmagnesium sulfate (MgSO4), which vary with temperature and salinity according to

fB ¼ ð0:78 kHzÞ � S

35

� �1=2

eTT=26 ð4:29Þ

and

fMg ¼ ð42 kHzÞ � eTT=17: ð4:30Þ

The coefficients of Equations (4.27) and (4.28) are given by

Avisc ¼ 4:9 � 10�4 exp � TT

27þ zz

17000

!" #dB km�1 kHz�2; ð4:31Þ

AB ¼ 1:06 � 10�4 ffBK0:776 dB km�1; 4:32Þ

and

AMg ¼ 5:2 � 10�4 1 þ TT

43

!S

35

� �ffMg e�zz=6;000 dB km�1: ð4:33Þ

The above empirical equations, from Ainslie and McColm (1998), are based on themore complicated expressions of Francois and Garrison (1982b).12 They agree withFrancois–Garrison in the following parameter ranges

200 < ff < 106

0:5 < K < 2 ð7:55 < pHSWS < 8:15Þ ði.e., 7:7 < pHNBS < 8:3Þ

8 < S < 40

�2 < TT < þ30

0 < zz < 3000

9>>>>>>>>>=>>>>>>>>>;

ð4:34Þ

and were used to calculate the coefficients of the formula quoted in Chapter 2 for therepresentative conditions T ¼ 10 C, S ¼ 35, z ¼ 0, and K ¼ 1.

A more accurate simplification of the Francois–Garrison equation is given byvan Moll et al. (2009). An alternative equation, with particular emphasis onaccuracy in the frequency range 10 kHz to 120 kHz, is given by Doonan et al.(2003). Attenuation models are reviewed in NPL (2007b).


12 The term K 0:776 in Equation (4.32) is written by Ainslie and McColm (1998) as eðpH�8Þ=0:56.

The applicable pH scale is not stated (the author was unaware at that time of the ambiguity),

but is the same scale as used by Francois and Garrison (1982b). In the absence of a statement to

the contrary it is assumed here that Francois and Garrison used the same pH scale as Mellen et

al. (1987), which, according to Brewer et al. (1995), is the NBS scale. The conversion from pH

to K to obtain Equation (4.32) was therefore made using Equation (4.14).

At very low frequency, of order 100 Hz and below, measurements show a residualattenuation, of unknown origin, and between 0.0002 dB/km and 0.004 dB/km inmagnitude (Francois and Garrison, 1982b). This residual is not included in the aboveequations. At slightly higher frequency (say, 300 Hz to 3 kHz), attenuation increasesrapidly with increasing pH (Figure 4.16, upper graph, cyan curves). The low value ofK in the north Pacific (see Section 4.1.3) results in exceptionally low absorption oflow-frequency sound in the deep-sound channel there (Figure 4.16, lower graph, solidblue curve).

The blue curve (upper graph) is a reference curve calculated using therepresentative parameters of Chapter 2. At high frequency, the attenuation coefficientis sensitive to temperature, salinity, and pressure. The effect of varying temperaturebetween 0 C and 20 C is illustrated by the dashed red lines (upper graph). At 100 kHzthe attenuation coefficient at 20 C is about 60 % higher than at 0 C.

Figure 4.16 shows the predicted attenuation coefficient for various realistic oceanconditions as listed in Table 4.2. Notice the low values of attenuation in the Baltic,caused by a combination of low temperature and very low salinity, and high values inthe Red Sea (where temperature and salinity are both high).

Figure 4.17 shows fractional sensitivity sð f Þ to a parameter x, calculated usingthe expression

sð f Þ ¼ x0

að f ; x0Þ@að f ;xÞ@x

��x¼x0

; ð4:35Þ

where x is any one of T , S, K, and z, and x0 is the representative value of theappropriate parameter except for the case of depth, for which z0 ¼ 1 km is chosen.

4.2 PROPERTIES OF BUBBLES AND MARINE LIFE

Gas bubbles form an acoustically important part of the sea’s constituents, as they areinvolved in the creation, scattering, and destruction of sound. Animals and plants,many of which contain a gas enclosure of some kind, are also important. Theacoustical properties of both are reviewed here.13

4.2.1 Properties of air bubbles in water

4.2.1.1 Properties of air under pressure

The (adiabatic) sound speed in air can be calculated as a function of temperature Tas:

cairðTÞ ¼ ½�airRairYðTÞ1=2 ð4:36Þ


13For a discussion of the influence of solid suspensions, see Chapter 5. Suspensions can usually

be ignored in the open sea, but are sometimes important in coastal waters, especially near river

outlets.

4.2 Properties of bubbles and marine life 149]Sec. 4.2

Figure 4.16. Seawater attenuation coefficient vs. frequency calculated using Equations (4.26) to

(4.33). Upper graph: sensitivity to temperature and acidity; lower graph: curves for different

oceans with parameters from Table 4.2.

where the specific heat ratio of air is

�air �ðCPÞair

ðCVÞair

¼ 1:4011; ð4:37Þ

Rair is the gas constant of air, equal to 287 J kg�1 K�1; and Y is the absolute


Table 4.2. Seawater parameters used for evaluation of attenuation

curves plotted in Figure 4.16 (adapted from Ainslie and McColm,

1998).

Ka S T/C z/km

Arctic Ocean 1.58 30 �1.5 0.0

Atlantic Ocean 1.00 35 4.0 1.0

Baltic Sea 0.79 8 4.0 0.0

Pacific Ocean 0.50 34 4.0 1.0

Red Sea 1.58 40 22.0 0.2

a The parameter K is defined in Section 4.1.3.4.

Figure 4.17. Fractional sensitivity sð f Þ of seawater attenuation (Equation 4.35) to temperature

(T), salinity (S), acidity (parameterized through K), and depth ðzÞ. The attenuation coefficient

að f Þ is calculated using Equations (4.26) to (4.33).

temperature. The ideal gas law gives the equilibrium air density under pressure

�airðzÞ ¼�STPYSTP

PSTP

PairðzÞYðzÞ ; ð4:38Þ

where �STP is the density of air at standard temperature and pressure (STP), equal to1.29 kg m�3.

The rate at which heat can be transported in a gas bubble is controlled by thethermal diffusivity of the gas. The higher the diffusivity, the more quickly heat can bedissipated and the more acoustical energy is lost to heat when the bubble pulsates. Itis defined by Morfey (2001)14 as

Dair �Kair

�airðCPÞair

; ð4:39Þ

where ðCPÞair is the specific heat capacity of air, equal to 1.005 J g�1 K�1 (Leacock,2003); and Kair is the thermal conductivity of air, equal to 0.0249 W m�1 K�1 at atemperature of 10 C. For other temperatures it can be calculated using (Pierce, 1989,p. 513)

KairðTÞ ¼ K0

YðTÞY0

�3=2 Y0 þYA expð�YB=Y0Þ

YðTÞ þYA exp½�YB=YðTÞ ; ð4:40Þ

where Y0, YA, and YB are constant temperatures, given by

Y0 ¼ 300:0 K; ð4:41Þ

YA ¼ 245:4 K; ð4:42Þand

YB ¼ 27:6 K: ð4:43Þ

The remaining constant, K0, is the value of Kair at temperature Y0, equal to2.624� 10�2 W m�1 K�1. The value of Dair at 10 C and atmospheric pressure isabout 20 mm2/s.

4.2.1.2 Properties of water that affect the behavior of a pulsating bubble

The pressure inside a submerged gas bubble exceeds hydrostatic pressure by anamount that depends on the surface tension of water. The attractive force betweenwater molecules results in a tension at the surface of the bubble. The surface tension can be defined as the force acting tangentially to the (air–water) interface, per unitlength of that interface. For clean water it is equal to 0.072 N/m. The surface tensionof slightly dirty bubbles is lower (a value of 0.036 N/m is quoted by Thorpe, 1982),whereas for bubbles in salt water it is slightly higher.


14 Alternative definitions are sometimes encountered. Weston (1967) uses specific heat at

constant volume instead of at constant pressure, and Stephens and Bate (1966, p. 765) omits the

specific heat factor altogether.

4.2.1.3 Properties of bubbly water

A presentation of the acoustical properties of bubbly water (such as sound speed andattenuation of the air–water mixture) is deferred to Chapter 5.

4.2.2 Properties of marine life

4.2.2.1 Basic physiological properties

4.2.2.1.1 Zooplankton

Greenlaw and Johnson (1982) give expressions for the volume V of individualeuphausiids, decapods, and copepods, as a function of their length L. For example,the average for all euphausiids is

V ¼ ð5:75 � 10�3 mm3Þ L

1 mm

� �3:10

; ð4:44Þ

and for a species of decapod (Sergestes similis)

V ¼ ð3:74 � 10�3 mm3Þ L

1 mm

� �3:00

: ð4:45Þ

For arthropods, Stanton et al. (1987) provide expressions relating animal length Land volume V to their weight. Combining them gives the following relationshipbetween length and volume

V ¼ 7:7 mm3 þ ð4:06 � 10�4 mm3Þ L

1 mm

� �2:295

: ð4:46Þ

4.2.2.1.2 Fish

The single most important physiological property of fish, from an acoustical point ofview, is the presence or absence of a gas-filled bladder. It is known that entire familiesof fish, such as gadoids and clupeoids, possess such a bladder. The effect of the gas-filled enclosure is to enhance the scattering properties of the fish, with a particularlydramatic effect close to the resonance frequency of the enclosure.

Two types of bladdered fish can be distinguished. Some, known as physostomes,are equipped with a connecting tube between the bladder and the gut, enabling theexchange of air between these two organs. Others, called physoclists, have a com-pletely closed bladder. According to MacLennan and Simmonds (1992), all gadoids(e.g., cod or haddock) are physoclists, and all clupeoids (e.g., herring) are physo-stomes. Other classes of fish, such as mackerel, have no bladder at all. Appendix Ccontains a list of species, compiled from various sources, with information for eachspecies concerning the presence or absence of a bladder, and the type of bladder ifpresent. For a valuable and comprehensive Internet resource describing thetaxonomy and physiology of fish generally, see fishbase (Froese and Pauly, 2007).15


15 The bladder is absent in the gadoids Melanonus and Squalogadus (Froese and Pauly, 2007).

Assuming a fish has a bladder, the volume and surface area of the bladder can beestimated from the length L of the fish by means of the equations (Haslett, 1962)

Vbladder 3:40 � 10�4L3 ð4:47Þand Weston (1995)

Sbladder 0:0291L2: ð4:48Þ

The volume of the whole fish is approximately (Haslett, 1962)

Vfish 0:0083L3: ð4:49Þ

An estimate of the surface area of the fish can be made by a simple geometrical scalingof the form16

Sfish Sbladder

Vfish

Vbladder

� �2=3

¼ 0:238L2: ð4:50Þ

4.2.2.1.3 Marine mammals

Typical values of mass m and length L are given for selected species of marinemammal in Table 4.3. The ratio m=L3 is also included, providing information relatingto the aspect ratio of the animal. A prolate spheroid of volume V and length L has abreadth-to-length aspect ratio X given by (see Table 4.4)

X ¼ffiffiffiffiffiffiffiffiffi6V

�L3

r: ð4:51Þ

The final column of Table 4.3 shows this aspect ratio X , estimated for each species byreplacing the volume V with that of the animal m=�. The value of X varies between0.11 (using m=L3 7 kg/m3, for the franciscana dolphin and sperm whale) and 0.26(m=L3 37 kg/m3, for the northern sea lion, walrus, and elephant seal).

4.2.2.2 Acoustical properties

4.2.2.2.1 Fish flesh

The response of a fish bladder to sound is similar to that of an air bubble, butinfluenced in a non-trivial way by the surrounding fish flesh. The sound speed anddensity of fish flesh exceed those of seawater by a few percent, as summarized in Table4.5.

The elasticity of fish flesh is determined by the (complex) shear modulus �. Thereal part determines the pressure Pe exerted by the bladder wall on the gas contents(Andreeva, 1964)

Pe ¼4

3�a

Reð�Þ: ð4:52Þ

The imaginary part of � determines losses due to vibration of the flesh. The value of �is subject to considerable uncertainty but a typical value, attributed by Love (1978) to


16 Use of Equation (4.50) implies an assumption that the bladder and fish are of similar shape.


Table 4.3. Mass, length, and aspect ratio of selected sea mammals (based on Pabst et al., 1999).

Speciesa Mass Length mL�3/ Aspect

m/kg L/m kg m�3 ratiob

X

Northern fur seal (female) (Callorhinus ursinus) 30 1.4 10.9 0.14

Franciscana dolphin (Pontoporia blainvillei) 32 1.7 6.5 0.11

Sea otter (Enhydra lutris) 45 1.5 13.3 0.16

Harbor seal (Phoca vitulina) 140 1.9 20.4 0.19

Amazonian manatee (Trichechus inunguis) 450 3.0 16.7 0.18

Bottlenose dolphin (Tursiops truncatus) 650 4.0 10.2 0.14

Northern sea lion (Eumetopias jubatus) 1,100 3.2 33.6 0.25

Walrus (Odobenus rosmarus) 1200 3.2 36.6 0.26

Elephant seal (Mirounga leonina, Mirounga angustirostris) 5000 5.0 40.0 0.27

Steller’s sea cow (Hydrodamalis gigas)c 10000 7.0 29.2 0.23

Sperm whale (male) (Physeter macrocephalus) 45000 18.5 7.1 0.11

Right whale (Lissodelphis borealis, Eubalaena glacialis) 90000 17.7 16.2 0.17

a Thumbnail images Garth Mix, GMIX Designs. Reprinted with permission.b Of an equivalent prolate spheroid with the same length and mass (see Equation 4.51).c Extinct species.

#

Andreeva (1964), is

� ¼ ð300 þ 90iÞ kPa; ð4:53Þ

consistent with a value of Pe close to 300 kPa. Weston (1995) suggests a smaller valuefor Pe in the range 50 kPa to 100 kPa, based on the measurements of Love (1978) andLøvik and Hovem (1979).

An alternative model introduced by Love (1978) treats fish flesh as a viscous fluidmedium described by a viscosity parameter �, defined as

� ¼ 43S þ B: ð4:54Þ

Nero et al. (2004) suggest for fish flesh a value of

� ¼ 50 Pa s: ð4:55Þ


Table 4.4. Volume and surface area of ellipsoids with semi-axes a � b � c.

Shape Volume Surface area Ellipticity

General ellipsoid, Expressable in terms of an

semi-axes a, b, and c 43�abc elliptic integral N/A

(Weisstein, www)

Prolate spheroid,a

semi-major axis a, 43�ab

2 2� b2 þ abarcsin e

e

� �1 � b2

a2

� �1=2

semi-minor axes b,

ellipticity e

Oblate spheroid,b

semi-major axes a, 43�a

2c 2� a2 þ c2

2eloge

1 þ e

1 � e

� �1 � c2

a2

� �1=2

semi-minor axis c,

ellipticity e

Sphere, radius a 43�a3 4�a2 0

a A prolate spheroid has one major axis and two minor ones, like an airship.b An oblate spheroid has two major axes and one minor one, like a flying saucer.

Table 4.5. Acoustical properties of fish flesh.

Species Sound speed Density ratio Reference

ratio c=cw �=�w

Cod (Gadus morhua) 1.050 1.040 Clay and Horne (1994)

Unspecified fish with swimbladder 1.033 1.023 Love (1978)

4.2.2.2.2 Whale tissue

The acoustical properties of whale tissue, as reported by Miller and Potter (2001) andJaffe et al. (2007), are summarized in Table 4.6.

4.2.2.2.3 Zooplankton

The density and sound speed of krill (euphausiid) flesh are correlated with animallength. Chu and Wiebe (2005) give the following correlation equations, valid for krilllength L exceeding 25 mm (i.e., LL > 0:025):

ckrill

cw

¼ 1:009 þ 0:50LL ð4:56Þ

�krill

�w

¼ 1:002 þ 0:54LL: ð4:57Þ

A summary of the acoustical properties of euphausiids is given in Table 4.7 and forother zooplankton in Table 4.8 (see also Lavery et al., 2007).

4.2.2.3 Population estimates

4.2.2.3.1 Fish in the North Sea: population density and case study

An order of magnitude estimate of the average areic17 mass of all marine fauna isabout 10 g/m2 in both deep and shallow water. It is difficult to improve on thisestimate except for well-studied locations. Despite this uncertainty, the effects canbe large and should therefore be considered. Population estimates for the North Seaare provided as an example although, at the time of writing, the data on which theestimates are based are already 20 years out of date.

The total North Sea biomass is estimated (Sparholt, 1990; Yang, 1982) to be


Table 4.6. Acoustical properties of whale tissue. Attenuation measurements are standardized for ease

of comparison by dividing them by frequency and presenting in units of dB/(m kHz). The actual

measurement frequencies are 100 kHz (Miller and Potter, 2001) and 10 MHz (Jaffe et al., 2007).

Species Tissue type c=m s�1 �=kg m�3 �

f

�dB m�1 Reference

kHz�1

Atlantic northern Skin 1700 1200 Miller and Potter (2001)

right whale Blubber 1600 900 0.09 Miller and Potter (2001)

(Eubalaena glacialis)

Florida manatee Connective 1680–1710 1030–1150 0.3–0.6 Jaffe et al. (2007)

(Trichechus manatus tissue

latirostris) Blubber 1520–1530 960–1060 0.5–0.8 Jaffe et al. (2007)

Muscle 1600–1630 1020–1070 0.3–0.5 Jaffe et al. (2007)

17 Following Taylor (1995), the adjectives ‘‘areic’’ and ‘‘volumic’’ are used, respectively, to

mean ‘‘per unit area’’ and ‘‘per unit volume’’.

about 10 Tg (1 teragram is equal to 1012 grams, or 1 million metric tons). Assumingfor the North Sea a total surface area of 575 000 km2 and volume 42 300 km3, theaverage areic and volumic biomass densities for the North Sea are 17 g/m2 and 0.24 g/m3, respectively. Data by individual species appear in Table 4.9. Additional informa-tion about North Sea fish can be found in Knijn et al. (1993). Also worth mentioningare the argentines (Argentina spp.), pelagic physostomes of length about 13 cm,common in the Norwegian Deep. The estimated biomass of argentines in the Nor-wegian Deep is 0.4 Tg (Yang, 1982).

The geographical distribution of two important North Sea species is shown inFigure 4.18, including an indication of the variation with season (summer vs. winter)and fish size (adults vs. juveniles). The data show that, during the 1980s, herringwere common in both summer and winter throughout the North Sea except in the


Table 4.7. Acoustical properties of euphausiids (from Simmonds and

MacLennan, 2005).

Species Sound speed ratio Density ratio

c=cw �=�w

Euphausia pacifica 1.005–1.015 1.035–1.040

Euphausia superba 1.028� 0.002 1.021–1.040

Thysanoessa raschii 1.010 1.027

Thysanoessa spp. 1.025 1.026–1.044

Meganyctiphanes norvegica 1.035 1.029–1.048

Table 4.8. Values of zooplankton density and sound speed ratios (from Clay

and Medwin, 1998, Table 9.5 and Simmonds and MacLennan, 2005, p. 277).

Class Sound speed ratio Density ratio

c=cw �=�w

Amphipods 1.000–1.009 1.055–1.088

Cladocerans 1.011–1.017

Copepods 1.006–1.012 1.023–1.049

Decapods 0.997–1.006

Mysids 1.075

Cephalopods 1.007 1.003

Cod eggs 1.017–1.026 0.979–0.992

Table 4.9. North Sea fish population estimates by species, in order of decreasing biomass B. Estimates apply to

various periods between 1977 and 1992.

Species B/Tg Reference L50/ma Estimated Description

(Knijn North Sea

et al., population/

1993) millions

Sandeel 1.82 Sparholt (1990)b (0.20) 26 700 Demersal (no bladder)

(Ammodytes spp.)

Herring 1.33 Sparholt (1990) 0.24 11 300 Pelagic (physostome)

(Clupea harengus)

Norway pout 1.20 Sparholt (1990) 0.13 64 100 Pelagic (physoclist)

(Trisopterus esmarkii)

Dab (Limanda limanda) 0.74 Yang (1982)c 0.12 50 300 Demersal (bladder

absent in adults)

Grey gurnard 0.64 Yang (1982) 0.19 11 000 Demersal

(Eutrigla gurnardus)

Plaice 0.55 ICES (1994)d 0.33 1800 Demersal flatfish

(Pleuronectes platessa) (bladder absent in

adults)

Haddock 0.50 ICES (1994) 0.30 2200 Pelagic (physoclist)

(Melanogrammus aeglefinus)

Starry ray (Raja radiata) 0.45 Yang (1982) 0.47 500 Demersal

Mackerel 0.44 Sparholt (1990) (0.30) 1900 Pelagic (no bladder)

(Scomber scombrus)

Whiting 0.40 ICES (1994) 0.20 5900 Pelagic (physoclist)

(Merlangius merlangus)

Cod (Gadus morhua) 0.35 ICES (1994) 0.70 100 Pelagic (physoclist)

Saithe (Pollachius vireus) 0.30 ICES (1994) (0.45) 400 Pelagic (physoclist)

Silvery pout 0.25 Yang (1982) (0.06) 135 800 Pelagic (physostome)

(Gadiculus argenteus)

Long rough dab 0.23 Yang (1982) 0.17 5500 Demersal flatfish

(Hippoglossoides (bladder absent in

platessoides) adults)

Horse mackerel 0.22 Yang (1982) 0.24 1900 Pelagic (no bladder)

(Trachurus trachurus)

Sprat (Sprattus sprattus) 0.20 Sparholt (1990) 0.10 23 500 Pelagic (physostome)

a The length L50 is the fish length ‘‘at which 50 % of the individuals sampled in that length class are sexually maturing/mature’’(Knijn et al., 1993) (values in brackets are estimates).b Three-year average (1983–1985).c Two-year average (1977–1978).

northernmost region, and that most were juveniles (with L < L50). By comparison,Norway pout were present mainly in the north and northwest, with a population thatwas more evenly balanced between adults and juveniles.


A biogeographical database, including information from marine mammal sightings,is available from the Ocean Biogeographic Information System (obis, www). Theabundance of different species varies enormously. Global population estimates fromBowen and Siniff (1999) for various periods in the 1980s and 1990s include

— for pinnipeds, between 12 000 000 crabeater seals18 or 6 000 000 harp seals tofewer than 500 Mediterranean monk seals;

— for cetacea, between 2 000 000 sperm whales or 2 000 000 spinner dolphins19 to500 Indus river dolphins and fewer than 1000 northern right whales;

— for sirenians, between 100 000 dugongs or 100 000 sea otters to just 2500 Floridamanatees.

4.3 PROPERTIES OF THE SEA SURFACE

If the sea surface is perfectly calm, its only effect on underwater sound in water is toreflect it like a mirror. The consequences of a disturbance (e.g., due to local wind,currents, precipitation, and distant storms), described in more detail in Chapters 5and 8, include

— scattering of sound at a roughened air–sea boundary;— generation, scattering, refraction, and absorption of sound by near-surface

bubbles created by breaking waves and other mechanisms.

The most important single parameter required to model these effects is local windspeed (denoted v). Knowledge of wind speed permits estimating surface roughnessand near-surface bubble population.

4.3.1 Effect of wind

Visible effects of wind can be described qualitatively by means of the Beaufort windforce scale, ranging from force 0 (calm) to force 12 (a hurricane). Many versions ofthe Beaufort scale exist, the most widely used being WMO code 1100.20 However,WMO 1100 is known to be biased in the sense that wind speeds estimated visuallyusing WMO 1100 are systematically lower than the true wind speed for v less than

4.3 Properties of the sea surface 159

18 The crabeater seal is described by Bowen and Siniff (1999) as ‘‘probably the most abundant

marine mammal in the world’’.19 Eastern tropical Pacific population.20 WMO: World Meteorological Organization.

12 m/s (for higher wind speeds, the opposite is true). Kent and Taylor (1997) comparevarious alternatives to WMO 1100 and show that, of the scales considered, those dueto da Silva et al. (1994) and to Lindau (1995) agree best with observation. Table 4.10shows the revised Beaufort scale of da Silva et al. (1994), preferred here over Lindau


Figure 4.18. Geographical distribution in the North Sea by season, 1985–1987: herring . . .(reprinted from Knijn et al., 1993).

(1995) because, in addition to an average value of v for each wind force, an indicationis given of the likely spread in v. The first column shows the Beaufort force numberand corresponding WMO description. An alternative description used for the samescale, referred to by Bowditch (1966) as the ‘‘seaman’s term’’, is included for compar-

4.3 Properties of the sea surface 161]Sec. 4.3

Figure 4.18 (cont.). . . . and Norway pout (reprinted from Knijn et al., 1993).


Table 4.10. WMO Beaufort wind force scale and estimated wind speed v20 due to da Silva et al. (1994) at a

measurement height of 20 m. The final column shows the wind speed v10 converted to a standard height of

10 m, assuming air temperature is equal to water temperature (see Figure 4.19).

Beaufort force Photograph Appearance at sea if fetch and duration of Estimated Wind speed

WMO description reproduced the blow have been sufficient to develop wind speed at standard

Seaman’s term from the sea fully height

(Bowditch 1966) NOAA (http) (WMO, 1970) v20/m s�1 v10/m s�1

0 Sea like a mirror 0.0–1.0 0.0–1.0

Calm

Calm

1 Ripples with the appearance of scales are 1.0–3.0 1.0–2.9

Light air formed, but without foam crests

Light air

2 Small wavelets, still short but more 3.0–4.6 2.9–4.3

Light breeze pronounced; crests have a glassy

Light breeze appearance and do not break

3 Large wavelets; crests begin to break; 4.6–6.8 4.3–6.4

Gentle breeze foam of glassy appearance; perhaps

Gentle breeze scattered white horses

4 Small waves, becoming longer; fairly 6.8–9.8 6.4–9.2

Moderate breeze frequent white horses

Moderate breeze

5 Moderate waves, taking a more 9.8–12.0 9.2–11.3

Fresh breeze pronounced long form; many white

Fresh breeze horses are formed (chance of some spray)

6 Large waves begin to form; the white 12.0–15.0 11.3–14.1

Strong breeze foam crests are more extensive

Strong breeze everywhere (probably some spray)


Beaufort force Photograph Appearance at sea if fetch and duration of Estimated Wind speed

WMO description reproduced the blow have been sufficient to develop wind speed at standard

Seaman’s term from the sea fully height

(Bowditch 1966) NOAA (http) (WMO, 1970) v20/m s�1 v10/m s�1

7 Sea heaps up and white foam from 15.0–17.8 14.1–16.5

Near gale breaking waves begins to be blown in

Moderate gale streaks along the direction of the wind

8 Moderately high waves of greater length; 17.8–21.0 16.5–19.5

Gale edges of crests begin to break into the

Fresh gale spindrift; the foam is blown in well-

marked streaks along the direction of the

wind

9 High waves; dense streaks of foam along 21.0–24.2 19.5–22.5

Strong gale the direction of the wind; crests of waves

Strong gale begin to topple, tumble and roll over;

spray may affect visibility

10 Very high waves with long overhanging 24.2–27.8 22.5–25.9

Storm crests; the resulting foam, in great

Whole gale patches, is blown in dense white streaks

along the direction of the wind; on the

whole, the surface of the sea takes a white

appearance; the tumbling of the sea

becomes heavy and shock-like; visibility

affected

11 Exceptionally high waves (small and 27.8–31.4 25.9–29.0

Violent storm medium sized ships might be for a time

Storm lost to view behind the waves); the sea is

completely covered with long white

patches of foam lying along the direction

of the wind; everywhere the edges of the

wave crests are blown into froth; visibility

affected

12 The air is filled with foam and spray; sea >31.4 >29.0

Hurricane completely white with driving spray;

Hurricane visibility very seriously affected

ison. Some of these descriptions are ambiguous unless the Beaufort force is alsoquoted. For example, the same word ‘‘storm’’ can mean either force 10 or force11 depending on whether the WMO or seaman’s term is intended. A more completephysical description is provided in columns 2 and 3 in the form of an image (NOAA,http) and text (WMO, 1970). Finally, columns 4 and 5 contain the likely spread ofwind speeds (at two different measurement heights of 20 and 10 m, denoted v20 andv10 respectively, and averaged over 10 min in time) associated with these conditions.Figure 4.19 shows the conversion factors between these two measurement heights.

The defining characteristics of the Beaufort scale are the physical descriptionsunder the heading ‘‘appearance at sea’’. All other parameters, including quoted windspeed values, have the status of derived or likely parameters for the stated appear-ance. Wind speed is given here in units of meters per second, in keeping with theadoption of SI units throughout this book. For wind speed values reported in knots(kn), the conversion is (see Appendix B)

1 kn ¼ ð1852=3600Þ m=s 0:5144 m=s:

In an attempt to address the shortcomings of code 1100, which is based on measure-ments made in the late 19th and early 20th centuries, in 1970 the WMO published anupdated Beaufort Scale (dubbed ‘‘proposed new Code 1100’’ and referred to by Kent


Figure 4.19. Wind speed scaling factors to convert from a 20 m reference height to the standard

reference height of 10 m (Dobson, 1981). The legend shows the temperature difference

jTair � Twaterj in C. Red curves indicate stable conditions (Tair > Twater); blue curves indicate

unstable conditions (Tair < Twater).

and Taylor, 1997 as CMM IV21), intended for scientific use (WMO, 1970). Windspeed estimates based on the WMO 1100 and WMO CMM-IV scales are compared inTable 4.11 with those of the modern values of da Silva from Table 4.10. Treating daSilva’s scale as a reference, WMO 1100 underestimates wind speed by up to 1 m/s forBeaufort force 1–5, and overestimates it by up to 3 m/s for force 7-11, while theCMM-IV scale generally tends to underestimate wind speed. This bias (the differencebetween columns 5 and 6 of Table 4.11) increases with increasing wind force up to amaximum of about 1.8 m/s for force 9 and above. Kent and Taylor conclude that‘‘the operationally used WMO1100 seemed to be better than the CMM-IV scalerecommended for scientific use.’’

Another widely used measure of sea surface conditions is sea state, which is ameasure of the height of surface waves rather than wind speed, although the two arerelated. As with the Beaufort scale, there is no single, universally accepted definition,but the one known as WMO code 3700 is in widespread use (see Table 4.12).


Table 4.11. Comparison of wind speed estimates for Beaufort force 1–11 based on WMO code

1100 and CMM-IV with those of da Silva. All are average values in meters per second except the

shaded column, which shows wind speed ranges in knots for the original WMO code 1100.

Beaufort WMO 1100 WMO 1100 da Silva da Silva WMO CMM-IV

force (NOAA, http) (Table 4.10) (Table 4.10) (WMO, 1970)

spread/kn av./m s�1 av./m s�1 av./m s�1 av./m s�1

meas.

height: 10 m 10 m 10 m 20 m 20 m

1 1–3 1.0 2.0 2.0 2.0

2 4–6 2.6 3.6 3.8 3.6

3 7–10 4.4 5.4 5.7 5.6

4 11–16 6.9 7.8 8.3 7.9

5 17–21 9.8 10.3 10.9 10.2

6 22–27 12.6 12.7 13.5 12.6

7 28–33 15.7 15.3 16.4 15.1

8 34–40 19.0 18.0 19.4 17.8

9 41–47 22.6 21.0 22.6 20.8

10 48–55 26.5 24.2 26.0 24.2

11 56–63 30.6 27.5 29.6 28.0

21 CMM IV stands for ‘‘Commission for Maritime Meteorology IV’’.

4.3.2 Surface roughness

Sea surface roughness is determined by the spectrum of surface waves propagatingalong it. An overview of surface waves and associated spectra, including the effect ofwind fetch, is provided by Robinson (2004). Two surface roughness spectra aredescribed in Sections 4.3.2.1 and 4.3.2.2, both applicable to open ocean conditions.Of these, the Pierson–Moskowitz spectrum is in modern use, while the Neumann–Pierson spectrum is needed for comparison with results from older literature.

The RMS slope � of the sea surface has been measured optically by Cox andMunk (1954) and related empirically to the wind speed. Their equation relating thesetwo parameters is22

�2 ¼ ð3 þ 5:12vv10Þ � 10�3: ð4:58Þ

4.3.2.1 Pierson–Moskowitz spectrum

The statistics of sea surface waves can be represented by a spectrum due to Piersonand Moskowitz (1964). The Pierson–Moskowitz (PM) wave height spectral density


Table 4.12. Definition of sea state (WMO code 3700).

Sea state code WMO code 3700 (NODC, www)

Description Significant wave heighta

in meters (exact)

0 Calm (glassy) 0

1 Calm (rippled) 0–0.1

2 Smooth (wavelets) 0.1–0.5

3 Slight 0.5–1.25

4 Moderate 1.25–2.5

5 Rough 2.5–4

6 Very rough 4–6

7 High 6–9

8 Very high 9–14

9 Phenomenal >14

a See Equation (4.65).

22 The actual measurement height for wind speed was 12.5 m (41 ft). For simplicity, it is

assumed here that the difference between the wind speeds at 10 m and 12.5 m may be neglected.

(of squared displacement from the mean surface) vs. surface wave frequency O, forwind speed between 0 m/s and 20 m/s, is of the form

SðOÞ ¼ CPM

g2

O5exp �BPM

gOv20

� �4

�; ð4:59Þ

where23 v20 is the wind speed at an anemometer height of 20 m; g is acceleration due togravity; and BPM and CPM are dimensionless constants given by

BPM ¼ 0:74 ð4:60Þand

CPM ¼ 0:0081: ð4:61Þ

The RMS roughness is the square root of the variance of the sea surface elevationabout its mean value. Thus (Chapman, 1983)

�2PM ¼ CPM

4BPM

v420

g2: ð4:62Þ

Substituting for numerical values, this becomes

�2PM ¼ DPMv

420; ð4:63Þ

where

DPM ¼ 2:85 � 10�5 m�2 s4: ð4:64Þ

A common descriptor of the sea surface is significant wave height Hsig, often referredto simply as ‘‘wave height’’. This parameter is historically defined as the averagepeak-to-trough height of the highest third of all waves. With this definition Hsig isgiven to a good approximation by four times the RMS roughness, whereas theAmerican Meteorological Society defines Hsig as precisely this value, that is,

Hsig � 4�: ð4:65Þ

Another descriptor sometimes used is the mean peak-to-trough wave height �HH, givenapproximately by

�HH 2:5�: ð4:66Þ

Using Equation (4.62) one can estimate the RMS roughness and hence the waveheight for each Beaufort force, shown in Table 4.13 for Beaufort force 0 to 7. Theinverse operation converts wave height to wind speed, as shown in Table 4.14 for seastates 0 to 6.

4.3.2.2 Neumann–Pierson spectrum

An alternative spectrum that is of historical importance, as it is used in much earlytheoretical work on surface wave scattering, is the Neumann–Pierson (NP) spectrum,


23 The actual wind speed measurement height of 19.5 m is rounded here to 20 m.


Table 4.13. Beaufort wind force: relationship between wind speed and wave height.

Beaufort WMO description Wind speed Wave height Approximate

force (Table 4.10) (Pierson–Moskowitz sea state

spectrum) equivalent

v10=m s�1 v20=m s�1 �=m Hsig=m

0 Calm 0–1.0 0–1.0 0.000–0.006 0.000–0.024 0–1

1 Light air 1.0–2.9 1.0–3.0 0.006–0.050 0.024–0.20 1–2

2 Light breeze 2.9–4.3 3.0–4.6 0.050–0.11 0.20–0.44 2

3 Gentle breeze 4.3–6.4 4.6–6.8 0.11–0.24 0.44–0.97 2–3

4 Moderate breeze 6.4–9.2 6.8–9.8 0.24–0.51 0.97–2.03 3–4

5 Fresh breeze 9.2–11.3 9.8–12.0 0.51–0.77 2.03–3.10 4–5

6 Strong breeze 11.3–14.1 12.0–15.0 0.77–1.19 3.10–4.77 5–6

7 Near gale 14.1–16.5 15.0–17.8 1.19–1.68 4.77–6.72 6–7

Table 4.14. Sea state: relationship between wave height and wind speed.

Sea state Description Significant RMS Wind speed of Approximate

(WMO code wave roughness corresponding Pierson– Beaufort

3700) height Moskowitz spectrum forceequivalent

Hsig/m �/m v10=m s�1 v20=m s�1

0 Calm (glassy) 0.00 0 0.0 0.0 0

1 Calm (rippled) 0.00–0.10 0–0.025 0.0–2.1 0.0–2.2 0–1

2 Smooth (wavelets) 0.10–0.50 0.025–0.12 2.1–4.6 2.2–4.8 1–2

3 Slight 0.50–1.25 0.12–0.31 4.6–7.2 4.8–7.7 3–4

4 Moderate 1.25–2.50 0.31–0.62 7.2–10.2 7.7–10.8 4–5

5 Rough 2.50–4.00 0.62–1.00 10.2–12.9 10.8–13.7 5–6

6 Very rough 4.00–6.00 1.00–1.50 12.9–15.7 13.7–16.8 6–7

given by (Neumann and Pierson, 1957, 1966)

SðOÞ ¼ ANP

1

O6exp �2

gOv5

� �2

�; ð4:67Þ

where ANP is a constant equal to 2.4 m2 s�5; and v5 is the wind speed at an anem-ometer height24 of 5 m.

The RMS wave height roughness corresponding to the NP spectrum is given by(Ainslie, 2005)

�2NP ¼ ANP

3�1=2

211=2

v5g

� �5

: ð4:68Þ

Substituting for numerical values yields

�2NP ¼ DNPv

55; ð4:69Þ

where

DNP ¼ 3:11 � 10�6 m�3 s5: ð4:70Þ

4.3.3 Wind-generated bubbles

Wind-generated bubbles close to the sea surface are caused primarily by breakingwaves. The number of whitecaps, and hence the bubble population density, is highlycorrelated with wind speed. Bubble density is highest close to the sea surface anddecreases with increasing depth away from the sea surface. Some measurements ofbubble population density, from Trevorrow (2003), are shown in Figure 4.20.

The calculation of sound speed and attenuation (see Chapter 5) requires as inputthe bubble population density nðaÞ as a function of wind speed, depth, and bubbleradius, in principle vs. position in three-dimensional (3D) space. A detailed 3D modelof near-surface bubble distribution, including the bubble plumes, is given by Novariniet al. (1998). A range-averaged model, retaining only depth dependence, is describedbelow.

The following recipe, based on measurements by Johnson and Cooke (1979), isdue originally to Hall (1989) and modified by Novarini as described by Keiffer et al.(1995). The resulting bubble population model, referred to henceforth as the‘‘Hall–Novarini’’ model, is

nða; zÞ ¼ n0uðv10ÞDðz; v10ÞGða; zÞ; ð4:71Þ

where n0 is a constant equal to

n0 ¼ 1:6 � 1010 m�4: ð4:72Þ

The other factors are uðv10Þ, Dðz; v10Þ, and Gða; zÞ which describe the dependence,respectively, on wind speed at 10 meters v10, depth from the surface z, and bubble


24 The wind speed measurement height is not specified explicitly by Neumann and Pierson

(1957), but the implied value is approximately 5.5 m, rounded here to 5 m.

radius a. The first of these three factors depends only on the wind speed

uðvÞ ¼ v13 m/s

� �3

: ð4:73Þ

The depth factor D also includes some wind speed dependence through the e-foldingdepth LðvÞ

Dðz; vÞ ¼ exp � z

LðvÞ

�; ð4:74Þ

where

LðvÞ ¼0:4 m v � 7:5 m/s

0:4 m þ 0:115 sðv� 7:5 m/sÞ v > 7:5 m/s.

�ð4:75Þ

Finally, the bubble radius distribution G varies with depth according to

Gða; zÞ ¼

0 a < amin

½arefðzÞ=a4 amin � a4 arefðzÞ½arefðzÞ=ax arefðzÞ < a � amax

0 a > amax,

8>>><>>>:

ð4:76Þ


Figure 4.20. Measurements of the near-surface population density of wind-generated bubbles

vs. bubble radius; wind speed 12 m/s (reprinted from Trevorrow, 2003, American Institute of

Physics).

#

where

arefðzÞ ¼ 54:4 mm þ 1:984 � 10�6z; ð4:77Þ

amin ¼ 10 mm; ð4:78Þ

amax ¼ 1000 mm; ð4:79Þand

x ¼ xðzÞ ¼ 4:37 þ z

2:55 m

� �2: ð4:80Þ

The bubble radius appears only in Gða; zÞ, so the void fraction can be written

UðzÞ ¼ð1

0

VðaÞnða; zÞ da ¼ n0uðv10Þ exp � z

Lðv10Þ

�IðzÞ; ð4:81Þ

where VðaÞ is the volume of a single bubble; and IðzÞ is the integralÐ10 VðaÞGða; zÞ da. It follows that

IðzÞ ¼ 43�a4

refðzÞ loge

arefðzÞamin

þ 1 � ½arefðzÞ=amaxxðzÞ�4

xðzÞ � 4

( ): ð4:82Þ

An especially important parameter is the surface void fraction, as this controls thesound speed and, through Snell’s law, the angle of acoustic interaction with the seasurface. For the Hall–Novarini bubble population model it is given by

Uð0Þ ¼ n0uðv10ÞIð0Þ; ð4:83Þwhere

Ið0Þ ¼ 43�a4

0 loge

a0

amin

þ 1 � ða0=amaxÞ0:37

0:37

" #ð4:84Þ

anda0 ¼ arefð0Þ: ð4:85Þ

Thus,

Uð0Þ ¼ 2:04 � 10�6uðv10Þ: ð4:86Þ

4.4 PROPERTIES OF THE SEABED

The seabed is a complicated layered medium whose acoustical properties vary withdepth on length scales from a few millimeters to hundreds of meters. For simplicity itcan be convenient to represent this layered medium by means of a uniform seabedwith representative depth-averaged properties. However, to be useful the averagemust be over a depth scale relevant to the frequency of interest—typically a fewwavelengths. In the following, some representative depth-averaged properties areprovided, with guidance on the frequency range for which they are applicable. Adistinction is made between consolidated and unconsolidated sediments as follows.

4.4 Properties of the seabed 171]Sec. 4.4

4.4.1 Unconsolidated sediments

The individual grains of surface sediments are usually interconnected only looselyand are able to slide relative to one another. A common assumption is that suchunconsolidated sediments are unable to support a large shear stress and can becharacterized by their compressional properties alone. This contrasts with harderconsolidated sediments and rocks, whose ability to propagate shear waves is notnegligible (see Section 4.4.2).

The most common unconsolidated marine sediments are clastic sediments, whichare made up from tiny fragments of weathered rock. Other sediment types arechemical sediments (made from salts previously dissolved in seawater) and biogenicsediments (made of decomposing plants and animals). Here the main emphasisis placed on clastic sediments as these are the most abundant. According toBuckingham (2005), many of the properties of unconsolidated sediments can beobtained from the knowledge of a single parameter: sediment porosity. The basicrelationships derived by Buckingham are described in Chapter 5. The emphasis herein Chapter 4 is on measured values of geoacoustic parameters, and the empiricallydetermined relationships between them.

4.4.1.1 Pure samples and porosity

Pure sediment samples (i.e., thosewhose grains all have the same size)can be classified according to graindiameter as in Table 4.15. Sub-divisions of these sediment types areintroduced in the discussion of mixedsizes below. For many applications, anunconsolidated sediment with a suffi-ciently low shear speed may beapproximated as a fluid medium.

Acoustical behavior can be char-acterized to a large extent by the planewave reflection coefficient Rð�Þ. For a fluid, Rð�Þ depends on density �sed, soundspeed csed, and attenuation coefficient �sed. The depth variation of these parameters issuch that the characteristic impedance �sedcsed tends to increase with increasing depthin the sediment.

Grain size is a useful descriptor of the acoustical properties of the seabed becauseit is strongly correlated with sound speed and density and, for a given sediment type,is independent of depth. The term ‘‘grain size’’, denoted M, is a logarithmic measureof grain diameter d defined as

M � �log2

d

dref

; ð4:87Þ

wheredref � 1 mm: ð4:88Þ


Table 4.15. Sediment type vs. grain diameter.

Sediment type Grain diameter

Clay <4 mm

Silt 4 mm to 62.5 mm

Sand 62.5 mm to 2 mm

Gravel >2 mm

The porosity of a fluid–solid mixture is the ratio of the volume occupied by the fluidto the total volume of fluid plus solid. Porosity is related to density via

�sed ¼ �w þ ð1 � Þ�grain; ð4:89Þ

where the grain density �grain is approximately 2680 kg/m3 (Hamilton and Bachman,1982). Therefore, using 1027 kg m�3 for �w,

¼ 1:62 � 0:622�sed

�w

: ð4:90Þ

Like grain size, sediment porosity is correlated with its density and sound speed.The grains become more tightly packed with increasing pressure, and consequentlyporosity decreases with increasing depth into the sediment.

4.4.1.2 Mixed samples and the ‘‘phi’’ scale

Naturally occurring sediments are mixtures of different sediment types, with differentgrain diameters. They can be characterized by an average value of the logarithmicgrain size, denoted Mz, and defined as (Folk, 1966)

Mz � � 1

3log2

d16

dref

þ log2

d50

dref

þ log2

d84

dref

� �; ð4:91Þ

where the subscript denotes the percentile by weight (e.g., d50 is the median graindiameter). The parameter Mz is referred to in the literature as ‘‘mean grain size’’. Byconvention the averaging is done in log space, so that Mz is a measure of thegeometric mean diameter and not the arithmetic mean. This can be seen by rewritingthe definition as

Mz ¼ �log2

dGM

dref

; ð4:92Þ

where

dGM ¼ ðd16d50d84Þ1=3: ð4:93Þ

Grain size is often expressed in so-called ‘‘phi units’’ (Appendix B). If the right-handside of Equation (4.92) is equal to x, this is written as Mz ¼ x�, meaning that themean grain size in phi units is x. Table 4.16 lists sub-divisions of sediment type usingthe Udden–Wentworth25 classification scheme as presented by Krumbein and Sloss(1963). Also included in the table are the names of typical mixed samples using theShepard (1954) and Folk (1954) classification schemes. The Shepard scheme groupssediment types according to the relative proportions of sand, silt and clay in a sample.For example, ‘‘sandy silt’’ is mostly silt but with a significant proportion of sand,whereas ‘‘sand–silt–clay’’ indicates roughly equal proportions of all three. Folk’sclassification scheme is similar, but more suited for classifying coarser sediments


25 The sediment classification scheme by grain size is attributed by Krumbein and Sloss (1963)

to Wentworth (1922), but the naming of silts and clays owes at least as much to Udden (1914).

The Udden–Wentworth scheme is in widespread use for the classification of marine sediments.


Table 4.16. Definition of sediment grain sizes and qualitative descriptions.

Sediment description Grain size Grain Typical mixed sample of the same

(Udden–Wentworth) parameter diameter mean grain sizea

Mð�Þ d/mm

Shepard Folk

Boulder gravel < �8 >256 Gravel

Large cobble gravel �8 to �7 128–256 Gravel

Small cobble gravel �7 to �6 64–128 Gravel

Very large pebble gravel �6 to �5 32–64 Gravel

Large pebble gravel �5 to �4 16–32 Gravel

Medium pebble gravel �4 to �3 8–16 Gravel

Small pebble gravel �3 to �2 4–8 Sandy gravel

Granule gravel �2 to �1 2–4 Muddy sandy gravel

Very coarse sand �1 to 0 1–2 Sand Gravelly sand

Coarse sand 0 to 1 12 –1 Sand Gravelly sand

Medium sand 1 to 2 14–1

2Sand Muddy gravelly sandb

Fine sand 2 to 3 18–1

4Silty sand Gravelly muddy sand

Very fine sand 3 to 4 116

–18

Silty sand Muddy sand

Coarse silt 4 to 5 132

– 116

Sandy silt Sandy gravelly mudb

Medium silt 5 to 6 164– 1

32 Silt Gravelly sandy mud

Fine silt 6 to 7 1128

– 164

Sand–silt–clay Sandy mud

Very fine silt 7 to 8 1256

– 1128

Clayey silt Mud

Coarse clay 8 to 9 1512

– 1256

Silty clay Mud

Medium clay 9 to 10 11024

– 1512

Clay Mud

Fine clay 10 to 11 12048– 1

1024 Clay Mud

a Guide of typical sediment mixtures only. Mean grain size is not enough on its own to determine either theFolk or Shepard class unambiguously.b The terms ‘‘muddy gravelly sand’’ and ‘‘sandy gravelly mud’’ are not included in Folk’s classification butthere seems no obvious reason for excluding them. See Krumbein and Sloss (1963, p. 158) for a moregeneral scheme if needed.

as it takes into account the proportion of gravel, and makes no distinction betweensilt and clay, adopting the term ‘‘mud’’ for both.

The Folk and Shepard classification schemes are both based on triangles andhence limited to a maximum of three primary sediment types (gravel, sand, and mudfor one, and sand, silt, and clay for the other). Thus the user of either one must choosebetween ignoring the (acoustically important) distinction between silt and clay orignoring the gravel content altogether. An alternative classification scheme, based ona tetrahedron and thus allowing all four primary sediment types, is suggested byKrumbein and Sloss (1963).

4.4.1.3 Near-surface (high-frequency) properties

The boundary between water and sediment is characterized by a transition betweenthe bulk properties of water and those of the sediment, across a distance of a fewmillimeters or centimeters (Lyons and Orsi, 1998; Pouliquen and Lyons, 2002; Tanget al., 2002; Tang, 2004). At high frequency (above 10 kHz) the properties of thetransition layer become significant. APL-UW (1994) presents empirical parametersrepresentative of the average properties of the top few centimeters, partly based onthe work of Hamilton (Hamilton, 1972; Hamilton and Bachman, 1982) and applic-able to frequencies between 10 kHz and 100 kHz (APL-UW, 1994; Sternlicht and deMoustier, 2003). For this reason the subscript ‘‘HF’’ is used to denote these near-surface properties. The equations and values are given in Table 4.17 for grain sizes�1 � Mz � þ9 (correcting a typographical error in the equation for sound speedfrom Sternlicht and de Moustier, 2003).26 The dimensionless ratios are approximatelyindependent of salinity, temperature, and pressure. The attenuation coefficients (inNp/m) are, to a first approximation, proportional to frequency. This results in aconstant value in nepers per wavelength (conventionally abbreviated Np/�) or dec-ibels per wavelength (dB/�).27 The table presents values of attenuation � in units ofdB/�. These values can be related to � in Np/m according to

�sed ¼ 20 log10 ecsed�sed

f: ð4:94Þ

4.4.1.4 Bulk (medium frequency) properties

Bulk geoacoustic parameters representative of the uppermost few meters of sedimentare described by Hamilton (1972) and Bachman (1985). These are applicable tointermediate acoustic frequencies approximately in the range 1 kHz to 10 kHz, and


26 More recent work by Richardson is described by Jackson and Richardson (2007).27 Despite appearances, the symbols ‘‘Np/�’’ and ‘‘dB/�’’ both represent dimensionless units.

This can be confirmed by inspection of Equation (4.94), the right-hand side of which is

dimensionless. Therefore, so too must be the left-hand side. The paradox is resolved by

realizing that the notation ‘‘dB/�’’ is used as shorthand for a decibel per wavelength which

is the same as a decibel per meter multiplied by the wavelength in meters. In other words

1 ‘‘dB/�’’¼ � � 1 dB/m, which is dimensionless.


Table 4.17. Default HF geo-acoustic parameters (10–100 kHz). Near-surface sediment

properties vs. grain size.

Sediment Representative Sound speed Density Attenuation Porosity

description grain size ratio ratio coefficient fraction

(see Table 4.16) Mð�Þ cHF=cw �HF=�w �HFðdB=�Þ HF

�1 1.3370 2.492 0.91 0.07

Very coarse sand �0.5 1.3067 2.401 0.89 0.13

0 1.2778 2.314 0.87 0.18

Coarse sand 0.5 1.2503 2.231 0.87 0.23

1 1.2241 2.151 0.88 0.28

Medium sand 1.5 1.1782 1.845 0.86 0.47

2 1.1396 1.615 0.86 0.62

Fine sand 2.5 1.1073 1.451 0.85 0.72

3 1.0800 1.339 0.92 0.79

Very fine sand 3.5 1.0568 1.268 1.00 0.83

4 1.0364 1.224 1.07 0.86

Coarse silt 4.5 1.0179 1.195 1.15 0.88

5 0.9999 1.169 0.67 0.89

Medium silt 5.5 0.9885 1.149 0.36 0.91

6 0.9873 1.149 0.20 0.91

Fine silt 6.5 0.9861 1.148 0.16 0.91

7 0.9849 1.147 0.13 0.91

Very fine silt 7.5 0.9837 1.147 0.10 0.91

8 0.9824 1.146 0.09 0.91

Coarse clay 8.5 0.9812 1.145 0.08 0.91

9 0.9800 1.145 0.08 0.91

cHF

cw

¼1:2778 � 0:056452Mz þ 0:002709M2

z �1 � Mz < 1

1:3425 � 0:1382798Mz þ 0:0213937M2z � 0:0014881M3

z 1 � Mz < 5:3

1:0019 � 0:0024324Mz 5:3 � Mz � 9

8><>:

�HF

�w

¼2:3139 � 0:17057Mz þ 0:007797M 2

z �1 � Mz < 1

3:0455 � 1:1069031Mz þ 0:2290201M2z � 0:0165406M3

z 1 � Mz < 5:3

1:1565 � 0:0012973Mz 5:3 � Mz � 9

8><>:

�HFðdB=�Þ ¼ 1:490cHF

cw

0:4556 �1 � Mz � 0

0:4556 þ 0:0245Mz 0 � Mz < 2:6

0:1978 þ 0:1245Mz 2:6 � Mz < 4:5

8:0399 � 2:5228Mz þ 0:20098M2z 4:5 � Mz < 6:0

0:9431 � 0:2041Mz þ 0:0117M2z 6:0 � Mz < 9:5

0:0601 9:5 � Mz

8>>>>>>>>><>>>>>>>>>:

are referred to here as ‘‘medium frequency’’ (MF) parameters. Sound speed anddensity can be estimated using the following correlations from (Bachman, 1985)

ccMF ¼ 1952 � 86:3Mz þ 4:14M 2z � 1:5% ð0:81 <Mz < 9:70Þ

��MF ¼ 2380 � 172:5Mz þ 6:89M 2z � 7:5% ð0:81 <Mz < 10:69Þ:

)ð4:95Þ

These equations give sound speed in m/s and density in kg/m3, for standardconditions involving atmospheric pressure, a temperature of 23 C, and salinity 35.Thus, to obtain the ratio of these parameters to the corresponding ones in water(needed for calculating the plane wave reflection coefficient) it is necessary to divideby the value of the same parameter in water and for the same conditions (i.e.,cw ¼ 1529.4 m/s and �w ¼ 1024.2 kg/m3).

The attenuation coefficient, in units of dB/�, is approximately independent offrequency in the range 1 kHz to 100 kHz (Kibblewhite, 1989). This means that a valueof �HF can be converted to MF by multiplying it by the ratio of sound speedscMF=cHF. Results and equations for �1 � Mz � þ10 are given in Table 4.18.(The equations leave cMF and �MF, and by extension also �MF, undefined for0:50 � Mz � 0:81. It is reasonable to interpolate � and c linearly in Mz for inter-mediate values.) Equation (4.95) is intended for computing c or �, given the grain sizeMz. Bachman (1985) advises against its use for any other purpose, including thereverse conversion from c or � to Mz. Instead Bachman provides separate correlationequations for this and other similar conversions.

A useful measure of uncertainty is provided by the percentage error fromEquation (4.95). This is the standard error computed by Bachman for his data set.Similar uncertainties apply to high-frequency equations (for cHF and �HF). Theuncertainty in the values of �, for both HF and MF cases, is an order of magnitudelarger.

4.4.1.5 Low-frequency properties

A vertical sound speed gradient normally exists in the seabed on a depth scale of 1 mto 100 m, affecting sound propagation at frequencies of 1 kHz or below. Althoughmedium-frequency (MF) parameter values can sometimes be used as default low-frequency parameters, there are significant complications. Some of these complica-tions, discussed below, result in an increase in the reflection coefficient, and others in adecrease. Typical effects in deep water are quite different from those in shallow water,so they are treated separately.

4.4.1.5.1 Deep water

Deep sea sediments comprise very fine particles, typically of sizes corresponding toclayey silt or silty clay (see Table 4.16), the sound speed of which is close to and oftenless than that of seawater. Sound speed then increases with increasing depth in thesediment, with gradient dc=dz of order 1/s (Hamilton, 1979, 1980, 1987). The low-frequency reflection coefficient is enhanced by this sound speed gradient because ofthe additional contribution to the reflected field from refracted sound. In deep water



Table 4.18. Default MF geo-acoustic parameters (1–10 kHz). Bulk sediment properties vs.

grain size.

Sediment Representative Sound speed Density Attenuation Porosity

description grain size ratio ratio coefficient fraction

(see Table 4.16) Mð�Þ cMF=cw �MF=�w �MFðdB=�Þ MF

�1 1.3370 2.492 0.91 0.07

Very coarse sand �0.5 1.3067 2.401 0.89 0.13

0 1.2778 2.314 0.87 0.18

Coarse sand 0.5 1.2503 2.231 0.87 0.23

1 1.2226 2.162 0.87 0.28

Medium sand 1.5 1.1978 2.086 0.88 0.32

2 1.1743 2.014 0.88 0.37

Fine sand 2.5 1.1522 1.945 0.89 0.41

3 1.1314 1.879 0.96 0.45

Very fine sand 3.5 1.1120 1.817 1.05 0.49

4 1.0939 1.758 1.13 0.53

Coarse silt 4.5 1.0772 1.702 1.22 0.56

5 1.0619 1.650 0.71 0.60

Medium silt 5.5 1.0479 1.601 0.38 0.63

6 1.0352 1.555 0.21 0.65

Fine silt 6.5 1.0239 1.513 0.17 0.68

7 1.0140 1.474 0.13 0.70

Very fine silt 7.5 1.0054 1.439 0.11 0.73

8 0.9982 1.407 0.09 0.75

Coarse clay 8.5 0.9923 1.378 0.08 0.76

9 0.9877 1.353 0.08 0.78

Medium clay 9.5 0.9846 1.331 0.09 0.79

10 0.9827 1.312 0.09 0.81

cMF

cw

¼cHF=cw �1 � Mz � 0:5

1:2763 � 0:05643Mz þ 0:002707M 2z 0:81 <Mz < 9:70

(

�MF

�w

¼cHF=cw �1 � Mz � 0:5

2:3237 � 0:16842Mz þ 0:006727M 2z 0:81 <Mz < 10:69

(

�MFðdB=�Þ ¼ cMF=cw

cHF=cw

�HFðdB=�Þ

the sediment layer is usually several hundred meters thick, which means that for mostsonar frequencies the interaction with the solid earth crust (a layer of igneous rockbeneath sediment and sedimentary rock) may be ignored.

In addition to the sound speed gradient, a second complication is that theattenuation coefficient, in units of dB/�, is known to be lower at frequencies below1 kHz than above this frequency (Kibblewhite, 1989; Potty et al., 2003). Finally, bothdensity and attenuation coefficient vary with depth, although the effect of thesegradients on low-frequency sound is minor compared with that of the sound speedgradient.

4.4.1.5.2 Shallow water

In shallow water, the effect of a sound speed gradient is generally less important thanfor deep water, and this is for two main reasons. The first is that in shallow water thepredominant sediment type is sand, whereas in deeper water it is more likely to beclay. The acoustical properties of sand (primarily its high sound speed) are such thatlittle sound penetrates into the sediment and this means that the sound speed gradientin the sediment is relatively unimportant. The second reason is that the sedimentthickness in shallow water is typically less than in deep water, so for the same gradientthere is a smaller contrast between the sound speed at the top of the sediment and thatat the bottom.

The main complication in shallow water arises from the interaction of sound withharder layers of igneous or sedimentary rock beneath the sediment. These rocks mustbe treated as solids that support shear waves.28 The speed of propagation of shearwaves is called the shear speed, and denoted cs. In such conditions the rigidity ofthe sediment layer, though relatively low, also becomes important. According toHamilton (1987) the shear speed of sediments can be parameterized in the form

ccs ¼ Azzx; ð4:96Þ

where A depends on the grain size; and x is a constant, approximately equal to 0.31.An empirical fit to expressions from Hamilton (1987) for different sediment types is

A ¼ 79 þ 41 expð�0:4MzÞ ðMz � �1Þ: ð4:97Þ

Because of their low rigidity, unconsolidated sediments support shear waves onlyweakly, resulting in very low shear wave speeds. Typical values of the ratio cs=cp areless than 0.1 near the sediment surface for unconsolidated sediments, increasing to 0.4at a depth of 1000 m (Hamilton, 1979). The appropriate correlation equations are


28 Shear waves can arise when the medium has non-zero rigidity, such that there exists a

restoring force for particle displacements that do not involve a local change in volume (see

Chapter 5). In a shear wave the displacement of individual particles away from their mean

position is normal to the wave propagation direction. This contrasts with an ordinary sound

wave (compressional wave), for which the displacement is always parallel to the propagation

direction.

(Hamilton, 1979, 1980)

ccs ¼

3:884ccp � 5757 1512 < ccp � 1555:15

1:137ccp � 1485 1555:15 < ccp � 1657:13

0:47cc2p � 1:136ccp þ 991 1657:13 < ccp � 2150:59

0:78ccp � 962 2150:59 < ccp,

8>>><>>>:

ð4:98Þ

where the precise transition values have been adjusted slightly to ensure that cs

approximates to a continuous function of cp. The shear waves are also heavilyattenuated, with a typical value for the attenuation coefficient of between3 dB m�1 kHz�1 and 30 dB m�1 kHz�1, and no obvious correlation with grain size(Kibblewhite, 1989, Fig. 11). A discussion of possible frequency dependence in theseparameters is deferred to Chapter 5.

4.4.2 Rocks

Once a layer of sediment has settled on the seabed, it is gradually covered with morelayers. On a geological time scale, physical and chemical changes take place thatconvert the sediment material into a rigid structure that can no longer be separatedinto its individual grains. This process is known as lithification. The solid materialresulting from lithification is called sedimentary rock. Other rock types are igneousrocks (made from re-solidified molten rock) and metamorphic rocks (very hardmaterial, such as quartzite or marble, produced under extreme conditions of tem-perature and pressure from sedimentary or igneous rock).

Metamorphic rocks are rarely encountered near the sea floor and we thereforeconcentrate here on sedimentary and igneous rocks. Sedimentary rocks are furtherclassified according to the size of the constituent sediment grains as listed in Table4.19 (Amateur Geologist, www).

4.4.2.1 Wave speed—density correlation equations

Rocks are characterized by their high rigidity, and they cannot reasonably berepresented by means of a fluid model. Instead they must be treated as solids that


Table 4.19. Names of sedimentary rocks resulting from the lithification of

different sediment types.

Grain size Original sediment type Resulting sedimentary rock

M < �1 Gravel Conglomerate

�1 <M < 4 Sand Sandstone

4 <M < 8 Silt Siltstone

M > 8 Clay Claystone or shale

support shear waves. The usual sound speed (i.e., the propagation speed of thelongitudinal wave, denoted cp) is known as compressional speed.

Figure 4.21 shows measurements of cp and cs vs. density for all rocks, fromLudwig et al. (1970). For rocks of relatively low density (sedimentary rocks and


Figure 4.21. Compressional and shear speed (in km/s) vs. density of rocks (in g/cm3) (reprinted

from Ludwig et al., 1970, Wiley).#

some igneous rocks, of density �rock in the range 1900 kg/m3 to 2400 kg/m3), thevariation of both wave speeds with density is approximately quadratic:

cp

1000 m s�1¼ 16:23 � 16:61

�rock

1000 kg m�3þ 4:798

�rock

1000 kg m�3

� �2

cs

1000 m s�1¼ 2:17 � 3:67

�rock

1000 kg m�3þ 1:515

�rock

1000 kg m�3

� �2

:

9>>>=>>>;

ð4:99Þ

For rocks whose density exceeds 3000 kg/m3 (mostly metamorphic rocks) thedependence is nearly linear:

cp

1000 m s�1¼ 2:63

�rock

1000 kg m�3

� �� 0:63

cs

1000 m s�1¼ 1:47

�rock

1000 kg m�3

� �� 0:27

:

9>>>=>>>;

ð4:100Þ

Combining the linear form of Equation (4.100) (denoted clin below) with thequadratic one of Equation (4.99) (denoted cquad), an overall fit can be obtained thatis valid for �rock � 1900 kg/m3

call ¼ ðc�8quad þ c�8

lin Þ�1=8 �500 m s�1: ð4:101Þ

Equation (4.101) is applicable to both compressional (p) and shear (s) waves. Graphsof call are plotted for both p-wave and s-wave speeds in Figure 4.22.

For the special case of basalt (an igneous rock common in the oceanic crust), alarge number of measurements is presented by Christensen and Salisbury (1975), whofit their data (for a pressure of 50 MPa) to the following empirical equations, valid fordensity values in the range �rock ¼ 2100–3000 kg/m3

cp

1000 m s�1¼ 2:33 þ 0:081

�rock

1000 kg m�3

� �3:63

� 0:03

cs

1000 m s�1¼ 1:33 þ 0:011

�rock

1000 kg m�3

� �4:85

� 0:03

:

9>>>=>>>;

ð4:102Þ

The Christensen–Salisbury curves for basalt are also shown in Figure 4.22.

4.4.2.2 Typical parameter values

In this section, typical values of geoacoustic parameters are given for a number ofimportant types of sedimentary and igneous rocks. Values are rounded to the nearest50 kg/m3, 50 m/s, and 0.05 dB/�. The variation around these representative valuescan be large as exemplified by Figure 4.21. The main sources used to construct Table4.20 are Carmichael (1982) for wave speeds and Jensen et al. (1994) for attenuation.Other source references are Christensen and Salisbury (1975) for selected propertiesof basalt, plus Assefa and Sothcott (1997) and Hamilton (1979).


4.4.3 Geoacoustic models

The lower the acoustic frequency, the deeper the sound penetrates into the seabed.Thus for accurate modeling of low-frequency sound propagation, a description ofseabed acoustic properties (e.g., sound speed and attenuation, shear speed and


Figure 4.22. Fit to Ludwig’s data for all rocks (including basalts, Equation 4.101) and

Christensen–Salisbury equations for basalts (Equation 4.102).

Table 4.20. Representative geoacoustic parameters for typical sedimentary and igneous rocks,

in order of increasing density.

Type of rock �/(kg/m3) cp/(m/s) �p/(dB/�) cs/(m/s) �s/(dB/�)

Mudstone 1500 2050 0.15 600 0.40

Chalk 2200 2400 0.20 1000 0.50

Sandstone 2400 4350 0.10 2550 0.25

Basalt 2550 4750 0.10 2350 0.20

Granite 2650 5750 0.10 3000 0.20

Limestone 2700 5350 0.10 2400 0.20

attenuation, and density) vs. depth is needed. The collection of such profiles for agiven location is known as a geoacoustic model. For example, a geoacoustic modelrepresentative of the continental shelf might comprise a thin sand sediment over alayer of sandstone, and a granite crust beneath. A typical deep-water model could bea thick layer of clay over mudstone, with a basalt crust below these. Either or both ofthe sediment and sedimentary rock layers can be absent, especially in shallow water(as in the case of exposed granite). For examples, see Hamilton (1980).

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5

Underwater acoustics

We know very little about the murmur of the brook, the roar ofthe cataract, or the humming of the sea

Marcel Minnaert (1933)

5.1 INTRODUCTION

While the scope of this chapter is not limited to bubble acoustics, this statement byMinnaert, published at the height of the quantum revolution, as part of his classicarticle ‘‘On musical air-bubbles and the sounds of running water’’ (Minnaert, 1933),is a stark reminder that we do not understand fully even the mundane. Nevertheless,important advances have been made in the intervening years, and the present chapterdocuments current theoretical knowledge of reflection, scattering, attenuation, anddispersion of underwater sound,1 starting with a derivation of the wave equations forfluid and solid media in Section 5.2. The plane wave reflection coefficients for fluid–fluid and fluid–solid boundaries are described in Section 5.3, including the effects oflayering. Section 5.4 deals with the scattering of sound from rigid and non-rigidbodies and from rough boundaries. Finally, the dispersive effect of impurities, inthe form of bubbly water or suspended sediment, is described in Section 5.5.

1 The propagation of sound in the sea is strongly influenced by small variations of sound speed

with depth. This behavior has important consequences for the propagation of ocean acoustic

signals of all kinds, including ambient noise and reverberation. A separate chapter (Chapter 9)

is devoted to ocean acoustic propagation.

5.2 THE WAVE EQUATIONS FOR FLUID AND SOLID MEDIA

Underwater sound is a manifestation of acoustic pressure waves traveling through thesea. These waves comprise successive regions of compression and rarefaction in whichthe local density is, respectively, slightly higher and slightly lower than the equilib-rium density. The particle motion giving rise to these density changes is subject to arestoring force, which is determined by a parameter known as the bulk modulus ofwater. Because water is a fluid, to a large extent the theoretical study of underwatersound concerns itself with solutions to the wave equation for a fluid medium. Thereare also times when sound reflects from solid objects such as the seabed. In order tounderstand such interactions it is sometimes necessary to consider a second type ofparticle motion, known as shear or transverse motion, that does not result in a localchange in density. The restoring force associated with shear motion is determined bythe shear modulus. For a fluid medium, the shear modulus is zero. The purpose ofthis section is to provide a mathematical description of both kinds of motion in termsof bulk and shear moduli, and to explain how these are related to other acousticalparameters such as the speed of sound.

5.2.1 Compressional waves in a fluid medium

5.2.1.1 Equations of motion

The behavior of a compressible fluid is described by two fundamental equations ofmotion. The first, a statement of conservation of mass, can be written

@�

@tþrEð�uÞ ¼ 0; ð5:1Þ

where � is the fluid density; and u is the particle velocity. The second is a statement ofNewton’s second law (force equals mass times acceleration), which simplifies, if theforces of gravity and viscosity are neglected, to the following equation due to Eulerrelating particle acceleration to the gradient of the pressure P (Pierce, 1989).

1

�rPþ @u

@tþ ðuErÞu ¼ 0: ð5:2Þ

Expanding in powers of particle velocity (whose magnitude is assumed small com-pared with the speed of sound), and retaining only the lowest order terms, theseequations become

@�

@tþ �0rEu � 0 ð5:3Þ

and1

�0

rPþ @u

@t� 0; ð5:4Þ

where �0 is the equilibrium density.Taking the time derivative of Equation (5.3) and the divergence of Equation

(5.4), and eliminating the velocity terms gives the following linear second-order

192 Underwater acoustics [Ch. 5

differential equation

rE 1

�0

rP� �

� 1

�0

@2�

@t2¼ 0: ð5:5Þ

5.2.1.2 Bulk modulus and the acoustic wave equation

Bulk modulus is a parameter that quantifies the restoring force of a fluid to a localchange in volume or density. If a material is subject to a dilatation D, defined as thefractional increase in volume

D ¼ �V

V0

; ð5:6Þ

there is an associated change in pressure �P. Assuming a linear relationship betweenthem, the bulk modulus is the constant of proportionality B in the equation

�P ¼ �BD: ð5:7Þ

Thus, B is like the stiffness of a spring in Hooke’s law, a measure of the material’selastic strength, and equal to the reciprocal of the fluid’s compressibility K

K � 1=B: ð5:8Þ

From the definition of B it follows that

B ¼ �dP

d�: ð5:9Þ

Substituting this expression into Equation (5.5) gives

BrE 1

�0

rP� �

� @ 2P

@t2¼ 0: ð5:10Þ

Equation (5.10) is the linear wave equation satisfied by a pressure wave in acompressible fluid medium of density �0 and bulk modulus B. The correspondingwave equation satisfied by the dilatation D is

rE 1

�0

rðBDÞ� �

� @2D

@t2¼ 0: ð5:11Þ

Having neglected second-order and higher order terms, the scope is limited hereafterto the regime of linear acoustics. Leighton (2007a) describes the effects of discardednon-linear terms and the conditions under which they might become significant.

5.2.1.3 Compressional wave speed

Equation (5.10) is the linear wave equation for a compressible fluid medium whosedensity varies with position. The generic equation for a field variable F describing awave traveling at speed c is

r2F� 1

c2

@2F@t2

¼ 0: ð5:12Þ

5.2 The wave equations for fluid and solid media 193]Sec. 5.2

Comparing Equation (5.10) with Equation (5.12) (and assuming the equilibriumdensity �0 to be spatially uniform) it can be seen that the speed of sound c is relatedto density and bulk modulus according to

c ¼ffiffiffiffiffiB

�0

s: ð5:13Þ

5.2.2 Compressional waves and shear waves in a solid medium

Whether in a fluid or solid, motion that results in local changes in density (throughcompression or rarefaction) is opposed by a force proportional to bulk modulus. Ina solid, a second type of motion is possible, known as shear or transverse motion,that does not result in density changes. Waves associated with compressional andtransverse motion in a solid are the subject of this sub-section.

5.2.2.1 Shear modulus and the wave equations for a solid

Transverse motion in a solid is opposed by a restoring force proportional to thedisplacement. For this kind of force, the constant of proportionality (i.e., the ratio ofshear stress to shear strain) is known as the shear modulus (or rigidity modulus) anddenoted �. Kolsky (1963) derives the following differential equation relating thedisplacement vector x to the dilatation D

�0

@2x

@t2¼ ðBþ 1

3�ÞrDþ �r2

x; ð5:14Þ

valid if B, �, and �0 are all independent of position. The wave equation for thedilatation follows from Equation (5.14) by taking its divergence and noting that

rEx ¼ D: ð5:15ÞThe result is

�0

@2D

@t2¼ ðBþ 4

3�Þr2D: ð5:16Þ

Traveling-wave solutions to Equation (5.16) are known as compressional waves (also‘‘longitudinal waves’’ or ‘‘p waves’’), and are illustrated by the upper panel of Figure5.1. Taking the curl of Equation (5.14) instead of its divergence results in thefollowing equation for curl x

�0

@ 2

@t2ðr ^ xÞ ¼ �r2ðr ^ xÞ; ð5:17Þ

which is the wave equation describing the propagation of shear motion (i.e., motiondue to lateral displacements that are not accompanied by a change in volume).Traveling wave solutions to Equation (5.17) are known as shear waves (also‘‘transverse waves’’ or ‘‘s waves’’), as illustrated by the lower panel of Figure 5.1.


5.2.2.2 Lame parameters, Young’s modulus, and Poisson’s ratio

It is often the case that the elastic properties of materials are described in terms of theso-called Lame parameters, denoted � and �. Of these, � is the shear modulusintroduced in Section 5.2.2.1, while � is related to bulk and shear moduli accordingto

� ¼ B� 23�: ð5:18Þ

5.2 The wave equations for fluid and solid media 195]Sec. 5.2

Figure 5.1. Illustration of compressional (p) and shear (s) wave propagation (from Leighton,

2007a, Elsevier, reprinted with permission).#

Other parameters sometimes encountered are Young’s modulus E (the ratio oflongitudinal stress to longitudinal strain) and Poisson’s ratio (the ratio of lateralcontraction to the longitudinal extension of the material). These are related to theLame parameters via (Kolsky, 1963)

E ¼ 3�þ 2�

�þ �� ð5:19Þ

and

¼ �

2ð�þ �Þ : ð5:20Þ

5.2.2.3 Compressional and shear wave speeds

Comparing Equations (5.17) and (5.16) with Equation (5.12), it can be seen that thewave speeds of p and s waves are, respectively,

cp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBþ 4

3�

�0

sð5:21Þ

and

cs ¼ffiffiffiffiffi�

�0

r: ð5:22Þ

In a common approximation known as the ‘‘fluid sediment’’ model, the oceansediment is modeled as if it were a fluid by assuming that the effects of shear wavesare negligible. The name of this model can be misleading because it seems to implythat sediment rigidity is neglected. However, even when cs is small, the rigidity term inEquation (5.21) provides an important correction to the sediment compressionalwave speed. Thus, what is neglected is usually not � but the energy associated withshear motion. The p-wave and s-wave speeds of ocean sediments can be estimatedusing the model described in Section 5.5.2.

For most materials, an extension along one axis results in a compression in adirection perpendicular to that axis, implying that Poisson’s ratio must be positive,and requiring in turn that Lame’s � parameter also be positive. Writing thecompressional speed as

cp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�þ 2�

�0

s; ð5:23Þ

and requiring a positive shear modulus, it follows from Equation (5.22) that cs and cpare related according to

c2s

c2p

<1

2: ð5:24Þ

Thus, a p wave in a solid medium always travels faster than an s wave in the samesolid. If a given event (say, an earthquake in Earth’s crust) generates both types ofwave simultaneously, a distant receiving station detects the arrival of the p wavebefore that of the s wave. The terms ‘‘p wave’’ and ‘‘s wave’’ were originally coined as


abbreviations of ‘‘primary wave’’ (meaning the wave that arrives first) and ‘‘second-ary wave’’. In modern use they are associated with ‘‘compressional wave’’ and ‘‘shearwave’’.

A further condition (Equation 5.31) follows from the requirement for thecomplex bulk modulus to have a negative imaginary part (otherwise a pure compres-sion would violate the principle of conservation of energy). A derivation of thiscondition follows. The complex bulk modulus can be defined as

~BB � �~cc2p � 4

3 �~cc2s ; ð5:25Þ

where ~ccp and ~ccs are the complex p-wave and s-wave speeds defined as

~ccp �cp

1 þ i�pcp=!ð5:26Þ

and

~ccs �cs

1 þ i�scs=!; ð5:27Þ

where cs are cp are the wave speeds of s and p waves; and �s and �p are thecorresponding attenuation coefficients in nepers per unit distance. They are equalto the imaginary part of the respective complex wavenumber of the shear wave

ks ¼!

csþ i�s ð5:28Þ

and compressional wave

kp ¼ !

cpþ i�p: ð5:29Þ

The complex bulk modulus is then

~BB ¼�c2

p

ð1 þ i�pcp=!Þ2� 4

3

�c2s

ð1 þ i�scs=!Þ2: ð5:30Þ

Assuming the imaginary parts to be small, the requirement for ~BB to have a negativeimaginary part leads to the inequality

�s

�p

c3s

c3p

<3

4: ð5:31Þ

5.3 REFLECTION OF PLANE WAVES

Many sources of underwater sound can be approximated to first order as points thatgenerate spherical waves. After traveling some distance these spherical waves expandand the wavefront curvature is reduced. Eventually the curvature becomes negligibleand for some applications the wave may then be approximated by a plane wave,simplifying the analysis. The reflection of plane waves from plane boundaries is thepresent subject.

5.3 Reflection of plane waves 197]Sec. 5.3


5.3.1 Reflection from and transmission through a simple fluid–fluid or fluid–solid

boundary

5.3.1.1 Amplitude reflection coefficient

Consider a plane wave of amplitude Ainc incident on a plane boundary separating twouniform fluid half-spaces2

pinc ¼ Ainc expði 1z� i�xÞ e�i!t: ð5:32Þ

If the wavenumber and ray grazing angle3 in the medium of the incident and reflectedwaves are denoted, respectively, k1 and �1, the horizontal and vertical wavenumberscan be written, respectively,

� ¼ k1 cos �1 ð5:33Þand

1 ¼ k1 sin �1: ð5:34Þ

If the reflected wave has amplitude Aref such that

pref ¼ Aref expð�i 1z� i�xÞ e�i!t; ð5:35Þ

the plane wave amplitude reflection coefficient is defined as the ratio of the twoamplitudes

R � Aref

Ainc

: ð5:36Þ

The simplest non-trivial case is that of a plane boundary between two uniform fluidmedia. Denoting the sound speed and density of each layer by ci and �i, where i ¼ 1or 2, the reflection coefficient is (Brekhovskikh and Lysanov, 2003)

R ¼ � � 1

� þ 1; ð5:37Þ

where � is the impedance ratio

� ¼ �2 1

�1 2

; ð5:38Þ

and the horizontal and vertical wavenumbers are related by

2i þ �2 ¼ k2

i ; i ¼ 1 or 2 ð5:39Þ

where (for an angular frequency !),

k1 ¼!

c1; ð5:40Þ

and

k2 ¼!

c2þ i�2: ð5:41Þ

2 For the remainder of this chapter the complex variable p is used to represent the acoustic

pressure as described in Chapter 2.3 The angle between the wave vector and the horizontal plane.


The expression for R given by Equation (5.37) is known as the Rayleigh reflectioncoefficient. The true grazing angle of the refracted wave is (using Snell’s law)

ð�2Þtrue ¼ arccosc2c1

cos �1

� �: ð5:42Þ

It is convenient to represent this angle by means of the closely related complex angle

�2 ¼ arccosð�=k2Þ; ð5:43Þ

the real part of which is approximately equal to the true grazing angle given byEquation (5.42). The purpose of the imaginary part of k2 is to simulate the effect of adecaying wave in layer 2.

If layer 2 is a solid, Equation (5.37) still applies if � is generalized to(Brekhovskikh and Godin, 1990)

� ¼ �p cos2 2�s þ �s sin2 2�s; ð5:44Þ

where the ‘‘p’’ and ‘‘s’’ subscripts indicate properties of the compressional and shearwaves, respectively, in the solid. Specifically, the impedance ratios are then given by

�X ¼ �2 1

�1 X

ðX ¼ p or sÞ ð5:45Þ

and �X denotes the (complex) grazing angle of the p wave or s wave in layer 2 suchthat

�X ¼ arccos�

!=cX þ i�X

� �; ð5:46Þ

where cX and �X are the corresponding wave speed and attenuation coefficient,respectively. Finally, each of the two vertical components of wavenumber X isrelated to the magnitude of the corresponding wavenumber kX ¼ !=cX according to

2X ¼ k2

X � �2; ð5:47Þor, equivalently,

X ¼ kX sin �X: ð5:48Þ

5.3.1.2 Amplitude transmission coefficients

In addition to the reflected compressional wave there is also a transmitted one. In thecase of a fluid–solid boundary there is further a transmitted shear wave. Amplitudetransmission coefficients for both types of wave are given in this section.

It is convenient to describe the incident displacement field xinc in terms of a scalarpotential �inc, such that the corresponding displacement is equal to the gradient ofthis potential

xinc ¼ r�inc; ð5:49Þ

and similarly for the reflected field

xref ¼ r�ref : ð5:50Þ

A potential defined in this way represents a compressional wave. In general, thetransmitted field in a solid comprises a shear wave as well as the usual compressional

one. This situation can be represented by including in the analysis a vector potentialt, such that the total (compressional plus shear) displacement is (Achenbach, 1975;Miklowitz, 1978)4

xtrans ¼ r�trans þr ^t: ð5:51Þ

If the incident wave is in the x–z plane, any transverse displacement associated withthe shear wave is confined to this plane, which means that the transmitted shear waveis associated with the y component of t, denoted y.

Two transmission coefficients are needed to characterize the transmitted field,one for each of the two types of transmitted waves (p and s). The compressional wavetransmission coefficient, denoted Tp, is defined as the ratio of the amplitude of �trans

to that of �inc, including any phase change in the same way as for the pressure wavesconsidered in Section 5.3.1.1. The result is (Miklowitz, 1978; Brekhovskikh andGodin, 1990)

Tp ¼ �2�1�p cos 2�s

�2ð1 þ �Þ ; ð5:52Þ

where the impedance ratio � is given by Equation (5.44); �p and �s by Equation (5.45);and the shear wave grazing angle �s by Equation (5.46) (with X¼ s). In the same way,the shear wave transmission coefficient Ts is the ratio of the amplitude of y to that of�inc, and is equal to

Ts ¼ �2�1�s sin 2�s

�2ð1 þ �Þ : ð5:53Þ

5.3.1.3 Energy reflection and transmission coefficients

In the following, a new set of reflection and transmission coefficients is introduced,defined as ratios of energy instead of amplitude. This is useful because energy is aconserved quantity, making it possible to derive simple relationships between thecoefficients.

Consider, as previously, a plane wave traveling in a fluid medium and incident ona plane fluid–solid boundary. The incident energy flux is partly reflected and partlytransmitted. Making the simplifying assumption that the shear modulus � of the solidis real, it is possible unambiguously to associate certain proportions of the reflectedand transmitted energy with each of the various reflected and transmitted paths:namely, the reflected p wave, the transmitted p wave, and the transmitted s wave.5

The energy reflection coefficient (i.e., the proportion of incident normal energy fluxcarried by the reflected wave) is then given by

V ¼ jRj2: ð5:54Þ

The p-wave energy transmission coefficient (the proportion of incident normal energy


4 The potentials � and t satisfy the same wave equations, respectively, as div x and curl x.5 If � has a non-zero imaginary part (implying the presence of an attenuation mechanism), such

unambiguous association is no longer possible because the total energy sum includes a term

proportional to Imð�Þ and the product of p and s amplitudes (Ainslie and Burns, 1995).

flux carried by the transmitted p wave) is

Wp ¼ jTpj2�2 Re p

�1 Re 1

ð5:55Þ

and, similarly, for the s wave

Ws ¼ jTsj2�2 Re s

�1 Re 1

: ð5:56Þ

Conservation of energy demands that incoming and outgoing energy flux be equal,and therefore

V þWp þWs ¼ 1: ð5:57Þ

The V and W terms, as defined above, are ratios of energies, and consequently arereal-valued parameters. In general, the various amplitude coefficients (R, Tp, and Ts)can take complex values, and there exists a similar relationship to Equation (5.57)that relates the closely related complex parameters (Deschamps and Changlin, 1989;Ainslie and Burns, 1995):

~VV � R2 ¼ � � 1

� þ 1

� �2

; ð5:58Þ

~WWp � T 2p

�2 p

�1 1

¼ 4 cos2 2�s

ð� þ 1Þ2

�2 1

�1 p

ð5:59Þ

and

~WWs � T 2s

�2 s

�1 1

¼ 4 sin2 2�s

ð� þ 1Þ2

�2 1

�1 s

: ð5:60Þ

These complex coefficients are not energy ratios, but they have the remarkableproperty that

~VV þ ~WWp þ ~WWs ¼ 1; ð5:61Þ

thus providing a useful check on the calculation of individual reflection andtransmission coefficients. In contrast with Equation (5.57), Equation (5.61) holdseven if Im � 6¼ 0.

5.3.2 Reflection from a layered fluid boundary

Consider now a more complicated boundary involving a uniform fluid transitionlayer sandwiched between uniform fluid half-spaces above and below. For example,the transition layer might be a uniform sediment between water and substrate. This isa special case of the situation illustrated by Figure 5.2 with the substrate shear speedcs equal to zero.

If the incident wave has unit amplitude, the reflection coefficient R is equal to theamplitude of the reflected wave. From the figure this is found by adding the following


infinite series

R ¼R12þT12½R23 expð2i 2hÞ f1þR21½R23 expð2i 2hÞ þR221½R23 expð2i 2hÞ 2þ� � �gT21;

ð5:62Þ

where Rij and Tij denote partial reflection and transmission coefficients (ratios ofdisplacement potentials) for a wave incident on layer j from layer i. Thus,

Rij ¼�i j � 1

�i j þ 1ð5:63Þ

and

Tij ¼�i�jð1 þ RijÞ; ð5:64Þ

where

�i j ¼�j i�i j

ði; j ¼ 1, 2, or 3Þ: ð5:65Þ

In Equation (5.62), the factor R23 always appears multiplied by the phase termexpð2i 2hÞ. This is because a reflection from the lower boundary is always accom-panied by a two-way transit of the sediment layer. Thus, it is convenient to define thefactor S23 as

S23 ¼ R23 expð2i 2hÞ: ð5:66Þ

This factor is the reflection coefficient of the lower (sediment–substrate) boundary,modified by the two-way phase shift relative to the origin at the upper one. In fact, itis the sediment–substrate amplitude reflection coefficient for a depth origin at thewater–sediment boundary. Substituting Equation (5.66) in Equation (5.62) it follows


Figure 5.2.

Example of

layered boundary

comprising a

uniform fluid

transition

sediment layer

between two

uniform half-

spaces above

(water) and below

(solid substrate).

that

R ¼ R12 þT12S23T21

1 � R21S23

: ð5:67Þ

Two equivalent forms are

R ¼ R12 þS23ð1 þ R12Þð1 þR21Þ

1 � R21S23

ð5:68Þ

and

R ¼ R12 þ S23

1 � R21S23

: ð5:69Þ

Equation (5.68) continues to hold even if the sediment layer is not uniform, providedthat the following expressions are used for the individual reflection coefficients(Ainslie, 1996):

R12 ¼�12 � f þð0Þ�12 þ f þð0Þ

; ð5:70Þ

R21 ¼�21 f

�ð0Þ � 1

�21 fþð0Þ þ 1

; ð5:71Þ

and

S23 ¼pþ2 ðhÞ�23 f

þðhÞ � 1

p�2 ðhÞ�23 f�ðhÞ þ 1

; ð5:72Þ

where pþ2 ( p�2 ) is the downward (upward) traveling pressure field in the sediment; and

f �ðzÞ ¼ dp�2 ðzÞ=dz�i 2ðzÞp�2 ðzÞ

: ð5:73Þ

An example of particular interest, because an analytical solution is known for p2ðzÞ,involves the density profile due to (Robins, 1991)

�2ðzÞ ¼�2ð0Þ

cosh�z

2

� ��

��2ð0Þsinh

�z

2

� �� 2; ð5:74Þ

combined with a linear k2 profile in the sediment:

k2ðzÞ2 ¼ k2ð0Þ2ð1 � 2qzÞ: ð5:75Þ

The parameters q, �, and � are constants controlling the derivatives of density andsound speed at depth z ¼ 0. In particular, � is given by

�2 ¼ 3

�2

d�

dz

� �2

� 2

�

d2�

dz2: ð5:76Þ

The sound speed and attenuation profiles are related to the wavenumber by

c2ðzÞ ¼!

Re k2ðzÞð5:77Þ


and

�2ðzÞ ¼40�

loge 10

Im k2ðzÞRe k2ðzÞ

; ð5:78Þ

where the units of � are decibels per wavelength. For this special case, the functionsp2ðzÞ and f ðzÞ are given in terms of the Airy functions AiðxÞ and BiðxÞ (see AppendixA) by

p�2 ðzÞ ¼�2ðzÞ�0

� 1=2 Ai½� ðzÞ � i Bi½� ðzÞ

Ai½� ð0Þ � i Bi½� ð0Þ ð5:79Þ

and

�i 2ðzÞ f �ðzÞ ¼1

2�2ðzÞd�2ðzÞ

dz� 0ðzÞAi 0½� ðzÞ � i Bi0½� ðzÞ

Ai½� ðzÞ � i Bi½� ðzÞ ; ð5:80Þ

where

ðzÞ ¼ GðzÞ2

ð2qk20Þ2=3

ð5:81Þ

and

GðzÞ2 ¼ 2ðzÞ2 � �2

4: ð5:82Þ

An alternative recursive approach for evaluating the reflection coefficient of anarbitrarily layered fluid sediment, by means of multiple uniform sub-layers, isdescribed by Jensen et al. (1994).

5.3.3 Reflection from a layered solid boundary

If one or more of the layers is a solid (in the sense of having a non-negligible shearspeed), calculation of the reflection coefficient can become considerably more com-plicated. The simplest case arises if only the substrate is solid, with the sediment layerstill a fluid, which is the situation depicted in Figure 5.2. For this case, Equation(5.68) still holds provided that Equation (5.44) is used for �23 in Equation (5.72). Ifthe sediment layer is also a solid, Equation (5.67) generalizes to the matrix equation

R ¼ R12 þ T12ð1 � S23R21Þ�1S23T21; ð5:83Þ

where

Ri j ¼Rppi j

Rspi j

Rpsi j

Rssi j

!; ð5:84Þ

Ti j ¼T ppi j

T spi j

T psi j

T ssi j

!; ð5:85Þ

S23 ¼R

pp23 exp i�pp

Rsp23 exp i�sp

Rps23 exp i�ps

Rss23 exp i�ss

!: ð5:86Þ

and the two-way phase terms are

�XY ¼ ð X þ YÞh: ð5:87Þ


In these equations, the partial p–p and p–s coefficients RXYi j ;T

XYi j are the ratios of

velocity (or displacement) potentials as described by Miklowitz (1978). For anexample of their application, see Ainslie (1995).

For the case of multiple solid layers the above method can be generalized with arecursive solution due to Kennett (1974) (see also Chapman, 2004, pp. 263–268). Analternative calculation method, described by Schmidt (1988), is implemented in thewidely used OASES-OASR model (Schmidt, ca. 2000).

5.3.4 Reflection from a perfectly reflecting rough surface

The reflective properties of a randomly rough—but otherwise perfectly reflecting—surface are determined by the statistics of a rough surface, characterized by the RMSheight displacement6 of the surface �, and its correlation length L. Together with thegrazing angle � and acoustic wavenumber k, roughness determines the Rayleighparameter Q for a plane wave

Q ¼ 2k� sin �: ð5:88Þ

If the correlation length is small, on the scale of a wavelength (kL� 1), then theeffects of rough surface scattering are usually small, scaling with the fourth power offrequency and the second power of correlation length. For this reason only large-scale roughness, such that kL� 1, is considered below.

If a plane wave is reflected from a rough but otherwise plane surface, the reflectedwave is a distorted version of the incident one. Small phase differences are introduceddue to the interaction of the incident plane wave with different parts of the distortedboundary, and these phase differences manifest themselves as slight imperfections inreflected wavefronts.7 If the roughness height is small compared with the acousticwavelength, the wavefronts are still recognizable as approximately planar, but withsmall imperfections mirroring those of the rough surface. If, further, the roughnesshas a random nature, one can define the coherent reflection coefficient Rc as anensemble average of the ratio of the reflected wave amplitude to that of the incidentwave. This ensemble average is taken over an infinite number of realizations of therandomly rough surface.

5.3.4.1 Perturbation theory (small Q)

This section follows Brekhovskikh and Lysanov (2003), who, for the case of small Qand large correlation length, use the method of ‘‘small perturbations’’ to obtain theresult

jRcj2 ¼ 1 � 4k2�2Y sin �; ð5:89Þ


6 That is, the RMS departure from mean surface height, known sometimes as surface

‘‘roughness’’.7 The Rayleigh parameter is a measure of these phase differences (Brekhovskikh and Lysanov,

2003).

where Y is a dimensionless parameter related to the 2D wavenumber spectrum G2ðsÞof sea surface (or seabed) displacement. Specifically, if the 2D wavenumber vector sof the roughness spectrum has magnitude � and bearing �, then Y is related to the2D spatial roughness spectrum G2ðsÞ according to

Y � ��2

ðG2ðsÞ sin2 �� 2�

kcos � cos �� 2

k2

!1=2

ds; ð5:90Þ

where the notation ds indicates a 2D element of ‘‘area’’ in wavenumber space suchthat ds ¼ � d� d�, and the limits of integration are over all real values of theintegrand.

The roughness spectrum is related to the spatial correlation function Bð�Þ via the2D Fourier transform

G2ðsÞ ¼1

4�2

ðBð�Þ e�isEo do; ð5:91Þ

and is normalized such that ðG2ðsÞ ds ¼ �2: ð5:92Þ

If �0 and � are the incident and scattered grazing angles, and � is the bearing of thescattered path, relative to that of the incident one, then the Bragg scattering vector is

s ¼ kcos � cos ’� cos �0

cos � sin ’

� �: ð5:93Þ

For example, the specular direction (� ¼ �0; � ¼ 0) corresponds to s ¼ 0.

5.3.4.1.1 Near-grazing (kL sin2 �� 1)

The limit of large kL and small kL sin2 � is an important one in underwater acoustics,because the propagation geometry can result in grazing angles close to zero. Theconsequence of assuming a large correlation length is that the spectrumGð�Þ vanishesat large distances from the wavenumber origin, and consequently the contribution tothe integral from the third of the three bracketed terms in Equation (5.90) becomesnegligible. If, further, the grazing angle � is sufficiently small compared with ðkLÞ�1=2,the first term is also negligible, leaving

Y � ��2

ð10

d�

ð3�=2

�=2

d� �G2ðsÞ � 2�

kcos �

� �1=2

: ð5:94Þ

Assuming G2ðsÞ to be separable, such that

G2ðsÞ ¼ 2�G1ð�ÞKð�Þ; ð5:95Þ

where G1ð�Þ is the 1D spectrum; and Kð�Þ is a dimensionless function describing the


azimuthal dependence of the 2D spectrum, normalized according toðþ��Kð�Þ d� ¼ 1; ð5:96Þ

it follows that

Y � 2�

�2

2

k

� �1=2 ð3�=2

�=2

ð�cos �Þ1=2Kð�Þ d�

ð10

�3=2G1ð�Þ d�: ð5:97Þ

For an isotropic spectrum, that is, with

Kð�Þ ¼ 1=2�; ð5:98Þthis becomes

Y ¼ 2�EPT

�2

2

k

� �1=2 ð1

0

G1ð�Þ�3=2 d�; ð5:99Þ

where EPT is a constant defined as

EPT � 1

2�

ðþ�=2��=2

ffiffiffiffiffiffiffiffiffiffifficos �

pd� ð5:100Þ

and equal to

EPT ¼ffiffiffi4

�

rGð3

4Þ

Gð14Þ� 0:3814: ð5:101Þ

As an example, consider the special case of an isotropic Gaussian roughness spectrumwith correlation radius L

Bð�Þ ¼ �2 exp � �2

L2

!; ð5:102Þ

so that the roughness spectrum takes the form (using Equation 5.91)

G1ð�Þ ¼�2L2

4�expð� 1

4�2L2Þ: ð5:103Þ

It then follows from Equation (5.99) that

Y ¼ EPTL2ffiffiffiffiffi

2kp

ð�3=2 expð� 1

4�2L2Þ d�: ð5:104Þ

Substituting for the integralð�3=2 expð� 1

4�2L2Þ d� ¼ Gð5=4Þ

2

2

L

� �5=2

; ð5:105Þ

this becomes

Y ¼ 1

�kL

� �1=2

Gð3=4Þ: ð5:106Þ

The coherent reflection coefficient is obtained by substituting this expression backinto Equation (5.89) for Rc.


5.3.4.1.2 Non-grazing (kL sin2 �� 1)

For larger angles—that is, large compared with ðkLÞ1=2—the second bracketed termof Equation (5.90) may be neglected. Retaining only the first of the three terms, itfollows that

Y ¼ ��2 sin �

ðG2ðsÞ ds; ð5:107Þ

and thereforeY ¼ sin �; ð5:108Þ

irrespective of the roughness spectrum.

5.3.4.2 Heuristic extension for large Q

According to Brekhovskikh and Lysanov (2003), the above perturbation theoryholds for small values of the Rayleigh parameter Q. For large Q, the Kirchhoffapproximation may be used to obtain:

Rc ¼ðþ1

�1expð�2ik� sin �Þwð�Þ d�: ð5:109Þ

If a Gaussian distribution is assumed for the sea surface height displacement �

wð�Þ ¼ 1ffiffiffiffiffiffi2�

p�

exp � � 2

2�2

!; ð5:110Þ

and putting jR0j ¼ 1 for a perfect reflector, it follows that (Eckart, 1953)

jRj2 ¼ expð�Q2Þ: ð5:111ÞThis equation can be compared with the equivalent expression for small Q, obtainedby substitution of Equation (5.108) in Equation (5.89)

jRcj2 ¼ 1 �Q2 � expð�Q2Þ ðQ2 � 1Þ: ð5:112Þ

Thus, for this case (i.e., for a Gaussian-distributed displacement, and in the limit oflarge kL sin2 �) the expression jRcj2 ¼ e�Q

2

may be used for any Q, large or small. Aheuristic extension of this is to assume that Equation (5.89) may be replaced with

jRcj2 � expð�4k2�2Y sin �Þ; ð5:113Þfor any value of the argument (i.e., for any kL sin2 � and for any surface statistics),with Y from Equation (5.90) or (depending on the magnitude of the kL sin2 �parameter) one of the two limits given by Equations (5.99) and (5.108).

5.3.5 Reflection from a partially reflecting rough surface

So far, scattering has been considered from a perfect reflector (i.e., one that reflects100 % of the incident energy). Assume now that some energy is transmitted across theboundary, such that the reflection coefficient of the surface, if perfectly smooth,would be R0.


Further, let Rc denote the coherent reflection coefficient corresponding to the trueroughness. Brekhovskikh and Lysanov (2003, p. 205) show that for a Gaussiansurface elevation, in the Eckart (i.e., large Q) limit:

Rc ¼ R0 e�Q2=2: ð5:114Þ

A heuristic generalization of this expression, permitting both small and large valuesof Q, is

Rc ¼ R0 expð�2k2�2Y sin �Þ: ð5:115Þ

5.4 SCATTERING OF PLANE WAVES

5.4.1 Scattering cross-sections and the far field

The ability of an object to scatter sound can be characterized by its total scatteringcross-section �tot, defined for an incident plane wave as the ratio of total scatteredpower W to the magnitude of the incident intensity, I0

�tot �W=I0: ð5:116ÞIf the power is not re-directed uniformly in all directions it is useful to quantify thescattered power per unit solid angle in a given direction. This can be done in terms ofthe differential scattering cross-section �O, defined as the ratio of scattered radiantintensity8 (WO) to the incident intensity

�O �WO=I0; ð5:117Þwhere the radiant intensity is understood to be measured in the far field of thescattering object.9

A related parameter is the backscattering cross-section (abbreviated BSX)10

�backð�Þ � 4��Oð�; �; �Þ; ð5:118Þwhere the meaning of the arguments is explained in Chapter 2. If the power isscattered uniformly in all directions, then �back is a constant (independent of angle�), and equal to �tot. The scattering cross-sections �O and �back are therefore relevantto the far field.

5.4 Scattering of plane waves 209]Sec. 5.4

8 Radiant intensity, denoted WO, is defined as power per unit solid angle.9 The field point is said to be in the far field of the object if the scattered pressure and particle

velocity fields are in phase, such that the scattered intensity is purely radiative (Morse and

Ingard, 1968).10 This definition of backscattering cross-section, from Pierce (1989) and ASA (1994), is

adopted consistently throughout this book. The reader’s attention is drawn to the potential for

confusion with an alternative convention that is common in work related to fisheries acoustics

(Clay and Medwin, 1977; MacLennan et al., 2002) that omits the factor 4� in Equation (5.118).

With the alternative convention (indicated by the subscript ‘‘alt’’), the backscattering cross-

section is defined as �backalt ð�Þ � �Oð�; �; �Þ, while the definition of total scattering cross-section

(Equation 5.116) is unchanged.

Theoretical expressions for BSX are given below for various solid objects(Section 5.4.2) and fluid ones (Section 5.4.3), including spherical gas bubbles andfish. Scattering from rough boundaries is considered in Section 5.4.4. Measurementsof target strength (a logarithmic measure of BSX) of real submerged objects aresummarised in Chapter 8, as well as the scattering strength of the sea surface andseabed, and the volume backscattering strength of scatterers that are extended inthree dimensions.

5.4.2 Backscattering from solid objects

Exact solutions for the scattering cross-section are known for a limited number ofsimple shapes such as a sphere or cylinder (Dragonette and Gaumond, 1997). Figure5.3 shows the magnitude of the form function, given by

j f ðkaÞj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�backðkaÞ�a2

s; ð5:119Þ

plotted vs. dimensionless radius (or frequency) for two hard spheres (one perfectlyrigid and one made of tungsten carbide) in the uppermost two graphs. The remaininggraphs show the same function calculated for spheres made of various metals. Thewave speeds and density of tungsten carbide and the metals are listed in Table 5.1.For the tungsten carbide sphere, the departure of the form factor from that of a rigidsphere is small for ka < 5. The remainder of this section concentrates on approximatesolutions for scattering from rigid bodies of various simple shapes, including thesphere.

5.4.2.1 Small rigid object of approximately spherical shape

For objects that are small compared with the acoustic wavelength in water, thescattering cross-section is approximately determined by the volume of the objectV and the acoustic wavelength �—independent of the shape of the object. For thecase of a bistatic scattering angle � (defined as the angle between incident andscattered wave vectors), the differential scattering cross-section is (Urick, 1983)

�Oð�Þ ¼�2V 2

�4ð1 � 3

2cos �Þ2: ð5:120Þ

Substituting � ¼ � to give the backscatter direction and using Equation (5.118) givesthe following equation for the BSX11

�backLF ¼ 25�3V

2

�4: ð5:121Þ

Equation (5.121), derived by Rayleigh for a perfect sphere, is assumed here to applymore generally to slightly deformed spheres and in particular to ellipsoids with a


11 The subscript is used here to indicate special cases, such as applicability to a low-frequency

(LF) or high-frequency (HF) limit.


Rigid

Brass

Aluminum

Steel

Tungsten carbide

Figure 5.3. Form function

j f ðkaÞj vs. ka for a rigid sphere

(uppermost graph), a tungsten

carbide sphere (second graph),

and spheres made of various

metals in water (lower graphs).

For the form function of

cylinders made of the same

materials, see Dragonette and

Gaumond (1997) (reprinted

from Dragonette and

Gaumond, 1997, Wiley,

with permission).

#

moderate aspect ratio. Expressions for the volume of various ellipsoids are given inChapter 4.

5.4.2.2 Large rigid object

For objects that are large in comparison with the acoustic wavelength in water, theBSX is determined by the size and shape of the object. Making use of the Kirchhoffapproximation, Neubauer (ca. 1982, p. 18) provides a simple facet summation for-mula that can be used to compute the scattering cross-section of any shape at highfrequency. For some simple shapes it is possible to express the result in closed form,and Table 5.2 summarizes the most important results from Urick (1983).

For the first entry in the table (the arbitrary convex shape), there are someimportant special cases that are worthy of further attention. First, consider anellipsoid whose semi-major axes are a1, a2, and a3. For a plane wave incident parallelto the axis of length a3, the radii of curvature in question are

A1 ¼a2

1

a3

ð5:122Þ

and

A2 ¼a2

2

a3

: ð5:123Þ

Thus, the BSX of a large ellipsoid ensonified along the a3 axis is

�backHF ¼ �

a1a2

a3

� �2

; ð5:124Þ

which reduces trivially to the result for a sphere of radius a

�backHF ¼ �a2; ka� 1; ð5:125Þ

For the case of a convex object ensonified at random aspect, the BSX depends on thetotal surface area of the object. The surface area of an ellipsoid is considered inChapter 4.


Table 5.1. Compressional speed cp, shear speed cs, and density

� used to calculate the form factors for the four metals shown in

Figure 5.3 (Dragonette and Gaumond, 1997).

Material cp/m s�1 cs/m s�1 �/kgm�3

Tungsten carbide 6860 4185 13800

Brass 4700 2110 8600

Steel 5950 3240 7700

Aluminum 6376 3120 2710

5.4.2.3 Rigid object of arbitrary size

A simple approximate expression for an object of arbitrary size can be obtained bycombining the low-frequency result from Section 5.4.2.1 with the high-frequency onefrom Section 5.4.2.2, using

1

�backð f Þ� 1

�backLF

þ 1

�backHF

: ð5:126Þ


Table 5.2. Backscattering cross-sections of large rigid objects (Urick, 1983).

Shape �backHF

4�

Conditions

Arbitrary convex object with principal

radii of curvature A1 and A2. (The

curvature is measured at the leading

edge, where the plane wave first A1A2

4kA1 � 1; kA2 � 1

intersects the convex surface, making

a tangent with it). Important special

cases, such as a perfect sphere, are

described in the text.

Cylinder of radius a and length L,

ensonified at an angle � to the normalaL2

2�

sinðkL sin �ÞkL sin �

� 2cos2 � ka� 1

to the cylinder axis

Arbitrary-shaped flat plate of area S, kl � 1, where l is

ensonified at normal incidenceS

�

� �2

the smallest

dimension

Rectangular plate of sides a and b,

ensonified at an angle � to the normal.

The direction of the incident andab

�

� �2 sinðka sin �Þka sin �

� 2cos2 � ka > kb� 1

scattered ray paths is in the plane

containing the side a

Circular plate of radius a, ensonified �a2

�

� �2 2J1ðka sin �Þka sin �

� 2cos2 � ka� 1

at an angle � to the normal

Infinite cone of half-angle ,

ensonified at an angle � to the cone’sð�=�Þ2 tan4

ð1 � sin2 �=cos2 Þ3� <

axis

Arbitrary convex object with surface kl � 1, where l is

area S, ensonified at random aspectS

16�the smallest radius

(ensemble average cross-section) of curvature

Circular plate of radius a, ensonified

at random aspect (ensemble averagea2

8ka� 1

cross-section)

As an example, for a rigid object of arbitrary convex shape of volume V and radii ofcurvature A and B, this is

�back � �1

ABþ �2

5�V

!2

" #�1

: ð5:127Þ

In particular, for a sphere of radius a:

�back � �a2

1 þ 3�2

20�2a2

!2: ð5:128Þ

5.4.2.4 Sand grains of irregular shape and arbitrary size

Thorne and Meral (2008) propose the following simple semi-empirical expression forthe form factor of irregular sand particles in water

j f ðkaÞj � ðkaÞ2

1 þ 0:9ðkaÞ21� 0:35 exp � ka � 1:5

0:7

� �2

� � �1þ 0:5 exp � ka� 1:8

2:2

� �2

� � �:

ð5:129Þ5.4.3 Backscattering from fluid objects

5.4.3.1 Small fluid object of arbitrary shape

For a small fluid object of volume V , density �, and bulk modulus B, and immersed inwater, the differential scattering cross-section is

�Oð�Þ ¼�2V 2

�41 � �wc

2w

B� 3ð�=�w � 1Þ

1 þ 2�=�w

cos �

!2

; ð5:130Þ

where � is the acoustic wavelength in water. Substituting � ¼ � gives the monostaticresult for the backscattering cross-section (BSX)

�backLF ¼ 4�3 V

2

�41 � �wc

2w

Bþ 3ð�=�w � 1Þ

1 þ 2�=�w

!2

: ð5:131Þ

This expression reduces trivially to Equation (5.121) in the limit �w=�! 0. Anotherimportant special case is that of a gas-like object (i.e., one with negligible density andhigh compressibility):

�backLF ¼ 4�3 V

2

�4

�wc2w

B

!2

: ð5:132Þ

5.4.3.2 Large fluid object

In general, some of the energy incident on any real object is reflected and some istransmitted into the interior of the object. Assuming that the transmitted part does


not contribute to the scattered field, the BSX may be approximated by multiplyingthe rigid body result by jRj2, where

R ¼ � � 1

� þ 1; ð5:133Þ

and

� ¼ �c

�wcw: ð5:134Þ

The applicable results are summarized in Table 5.3.

5.4.3.3 Fluid object of arbitrary size

A simple approximation for BSX, of the same form as Equation (5.126), is

1

�backð f Þ� 1

�backLF

þ 1

�backHF :

ð5:135Þ

This expression is not applicable to cases featuring a strong resonance such asscattering from a gas bubble, which is considered next. Approximate results for afluid sphere, prolate spheroid, straight cylinder, and bent cylinder are given byStanton (1989).

5.4.3.4 Gas bubble

A gas bubble in water, once disturbed from its rest state by a compression orrarefaction, will pulsate at its natural frequency (Morfey, 2001), provided that it isnot subject to further forcing. If forced to pulsate at this frequency (by an incomingacoustic wave) the bubble resonates, meaning that it undergoes high-amplitudeoscillations or pulsations in response to the force.12


Table 5.3. Backscattering cross-sections of large fluid objects.

Shape �backHF

4�

Conditions

Arbitrary convex object with principal jRj2 A1A2

4kA1 � 1; kA2 � 1

radii of curvature A1 and A2

Cylinder of radius a and length L,

ensonified at an angle � to the normal jRj2 aL2

2�

sinðkL sin �ÞkL sin �

� 2cos2 � ka� 1

to the cylinder axis

Arbitrary convex object with surface kl � 1, where l is

area S, ensonified at random aspect jRj2 S

16�the smallest radius

(ensemble average cross-section) of curvature

12 At resonance, the bubble’s shape is likely to depart from spherical symmetry, but the

radiated field is nevertheless well approximated by considering only the changes in volume

(Leighton, 1994, p. 203).

The frequency of maximum response is known as the resonance frequency of thebubble and denoted !res. Whether or not a bubble is resonating, its motion is damped,meaning that some of the vibrational energy of the bubble is lost to its surroundings.If the amount of damping is small, the resonance frequency is approximately equal tonatural frequency. The concept of resonance is essential for understanding theresponse of a gas bubble to an acoustic signal.

The approach used here to analyze bubble response is a low-frequency one in thesense that kwa� 1 is required, where kw is the acoustic wavenumber in water. This isnot a big constraint because most bubbles are small compared with the acousticwavelengths of interest. For a single resonating air bubble close to atmosphericpressure the kwa product is always small—a consequence of the high compressibilityof air compared with that of water.

For a spherical bubble of radius a, the BSX (for kwa� 1) can be approximatedin the form (Weston, 1967)13

�back � 4�a2

½1 � !resðaÞ2=!2 2 þ �ða; !Þ2: ð5:136Þ

Resonance occurs when the term in square brackets vanishes and the BSX becomesvery large, limited only by the damping coefficient �. Full expressions for the reson-ance frequency and damping coefficient are rather involved and thus deferred to thetreatment of bubble resonance in Section 5.5.3.4. Here, simplified expressions arepresented that are valid for moderately large bubbles (radius exceeding about100 mm). For such bubbles, resonance frequency is equal to the Minnaert frequency(Minnaert, 1933):

!0ðaÞ ¼3 aPw

�wa2

� �1=2

: ð5:137Þ

The product of wavenumber and bubble radius at resonance, as determined by theMinnaert frequency at atmospheric pressure, is a constant that appears repeatedly inthe discussion concerning bubble acoustics. This constant is denoted D0 and definedby the formula

D0 �3 aPSTP

�wc2w

� �1=2

� 0:01367: ð5:138Þ

If the value of � in Equation (5.136) is small, a more precise condition for resonance isthat ! and !res be equal. Thus, at resonance, Equation (5.136) becomes

�backres ¼ 4�a2Q2; ð5:139Þ

where Q is the Q-factor, defined as 1=�ð!resÞ. In practice, the damping coefficient mayoften be replaced in Equation (5.136) by 1=Q even away from resonance because �2 is


13 Ainslie and Leighton (2009) point out that Equation (5.136) is ambiguous without explicit

definitions of !res and �. An alternative expression without this ambiguity is given in Section

5.5.3.6.

then small compared with other terms in the denominator of Equation (5.136). Inother words

�back � 4�a2

½1 � !resðaÞ2=!2 2 þ 1=Q2: ð5:140Þ

The simplest form of damping is caused by re-radiation: as the bubble pulsates it actsas a source of sound in its own right, the radiated sound carrying energy away fromthe bubble. Damping can also be caused by absorption (i.e., conversion of soundenergy to heat or, in some special situations, to light, see Leighton, 1994).14 Close toresonance, for a gas bubble in water, the dominant form of absorption is due to thetransport of heat inside the bubble. Absorption due to the viscosity of water providesadditional damping for small bubbles.

The total reciprocal Q-factor can be written as the sum of three separatecontributions due to re-radiation, thermal conduction, and viscous losses:

1

Q¼ 1

Qrad

þ 1

Qtherm

þ 1

Qvisc

; ð5:141Þ

where radiation and thermal Q-factors are given by

1

Qrad

¼ !0a

cwð5:142Þ

and

1

Qtherm

¼ 3ð a � 1Þa

Da

2!0

� �1=2

: ð5:143Þ

Parameters Da and a are the thermal diffusivity and specific heat ratio of air,respectively.

The third Q-factor, for viscous losses, is given by Leighton (1994, p. 190) for thecase of negligible bulk viscosity via

1

Qvisc

¼ 4�S

�w!0a2; ð5:144Þ

where �S is the shear viscosity of seawater. With the introduction of bulk viscosity,which in seawater contributes at least as much to the damping as shear viscosity (seeChapter 4), this expression for Qvisc becomes (Love, 1978, Eq. (9))

1

Qvisc

¼ 3�

�w!0a2; ð5:145Þ

where � is a viscosity parameter, defined as

� � 13�S þ �B: ð5:146Þ


14 A flash of light is sometimes observed associated with the sound creation mechanism of the

snapping shrimp (see Chapter 8).

The low-frequency and high-frequency limits of Equation (5.136) (but always withinthe frequency regime satisfying ka� 1) are:

�backLF ¼ 4�a2ð!=!0Þ4 ð5:147Þ

and

�backHF ¼ 4�a2: ð5:148Þ

5.4.3.5 Dispersed bubbles

Consider a cloud of bubbles of different sizes, spread uniformly in space. Let nðaÞ dabe the number of bubbles per unit volume whose radius is between a and aþ da, suchthat the backscattering cross-section per unit volume of the bubble cloud (thevolumic15 BSX)

d�back

dV� �back

V ¼ð1

0

�backðaÞnðaÞ da; ð5:149Þ

where �back is given by Equation (5.136). For bubbles large enough to pulsateadiabatically, but still small in the sense of ka� 1, this simplifies to

�backV � 4�

ð10

a2

f1 � ½!0ðaÞ=! 2g2 þ �2nðaÞ da; ð5:150Þ

where !0 is given by Equation (5.137) and � may be replaced by 1=Q from Equation(5.141). Corrections to !0 and � for smaller bubbles are described in Section 5.5.3.4.

A simplifying approximation that is sometimes made to Equation (5.150) is toassume that the contribution from near-resonant bubbles dominates the integral. Theresult is (Medwin and Clay, 1998)

�backV � 2�2Qa3

0nða0Þ; ð5:151Þwhere

a20 ¼

3 aPw

�w!2: ð5:152Þ

This approximation, though appealing, is not valid for situations involving awide spectrum of bubble sizes. Nevertheless, it can still be useful to calculate thecontribution to BSX from resonant bubbles in this way.

5.4.3.6 Single fish (with bladder)

Horne and Clay (1998) review methods for computing the BSX from fish and othermarine organisms, including a sophisticated model of scattering from fish flesh andbladder due to Clay and Horne (1994). A simpler approach is adopted here by first




expressing total BSX as a sum of contributions from flesh and bladder

�backtotal ¼ �back

bladder þ �backflesh : ð5:153Þ

In practice, if a bladder is present, the contribution from the bladder usuallydominates the scattering, so the BSX may then be approximated as

�backtotal � �back

bladder: ð5:154ÞThe scattering cross-section of a fish bladder can be calculated in much the same wayas for a gas bubble, modified due to the additional tension from flesh elasticity (Pe)and corrected for the non-spherical shape of the bladder. The result can be approxi-mated as (Weston, 1967)

�backbladder �

4�a2S

½1 � ð!0=!Þ2 2 þQ�2; ð5:155Þ

where

!20 ¼ 4�aS

a½PwðzÞ þ Pe �wVbladder

; ð5:156Þ

and aS is an equivalent bladder radius defined as

aS �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSbladder

4�

r; ð5:157Þ

such that the surface area of a sphere of that radius, 4�a2S, is equal to the bladder

surface area. The resonance frequency f0 in cycles per unit time (i.e., f0 ¼ !0=2�) is

f0ðLÞ ¼K0

ffiffiffiffiffiffiffiffiffiffiPðzÞ

pL

; ð5:158Þ

where P is the dimensionless pressure inside the bladder16

PðzÞ � PwðzÞ þ Pe

PSTP

ð5:159Þ

and K0 is a constant, with dimensions of length times frequency, equal to

K0 ¼ D0cw1

3�

aSL2

Vbladder

!1=2

� 78:9 Hz m: ð5:160Þ

The physical significance of this constant is the resonance frequency of a fishmultiplied by its length, at the sea surface, with a (hypothetical) completely limpbladder (i.e., whose shear modulus � is negligible). The measurements of f0 by Løvikand Hovem (1979) are consistent with a total bladder pressure ðPw þ PeÞ of about250 kPa.


16 The term Pe was introduced in Chapter 4 as the ‘‘pressure Pe exerted by the bladder wall on

the gas contents.’’ More precisely, Pe is the sum of two terms: one equal to the increase in

bladder pressure due to the bladder wall tension at a fixed bladder size (denoted E); the other

describing the variation of this pressure with the volume of the cavity (dE=dV). The details are

described in Section 5.5.3.4.2.

In Equation (5.159), the pressures PSTP, Pw, and Pe are, respectively, onestandard atmosphere (approximately 101 kPa), the hydrostatic pressure at depth z,and the contribution to the pressure inside the bladder due to the elasticity of thebladder membrane. The elasticity term is uncertain, with values between 50 kPaand 300 kPa used in different publications. A value of 75 kPa is adopted here (seeChapter 4).

The square root dependence on pressure in Equation (5.158) assumes thatchanges in depth are slow compared with the time required for the fish to adjustto new conditions (such that the parameter K0 is a constant). Assuming a constantbladder mass for rapid changes (instead of the constant volume implied by Equation5.160) results in a 5/6 power law dependence on pressure (Weston, 1995). The timerequired for the fish to make this adjustment is greater for the physoclist than for thephysostome17 because the latter is able to exchange gas relatively quickly with its gut(or by releasing bubbles if rising; Weston, 1995). According to Løvik and Hovem(1979), the adjustment time for coalfish (Pollachius virens, a physoclist) is between 12and 24 hours.

Equation (5.158) can be written in the alternative form

L0ð f Þ ¼K0

ffiffiffiffiffiffiffiffiffiffiPðzÞ

pf

; ð5:161Þ

where L0 is the length of fish whose bladder resonance frequency is f . The (reciprocal)Q-factor for a single fish can be written

1

Qfish

¼ 1

Qrad

þ 1

Qtherm

þ 1

Qflesh

; ð5:162Þ

where the first two terms take the same form as for large bubbles:

1

Qrad

¼ KradPðzÞ1=2 ð5:163Þ

and1

Qtherm

¼ Ktherm

L1=2PðzÞ�1=4; ð5:164Þ

where the constants Krad and Ktherm are (see Equation 5.138)

Krad ¼ D0 � 0:0137 ð5:165Þand

Ktherm ¼ 3ð a � 1Þ2aS=L

Da

�K0

� �1=2

� 0:0089 m1=2: ð5:166Þ

The third term in Equation (5.162) contains the Q-factor for fish flesh, replacing thatfor water viscosity (Weston, 1967, p. 69; 1995, p. 10):

1

Qflesh

¼ KfleshPðzÞ�1; ð5:167Þ


17 See Appendix C for an explanation of these terms.

where (assuming Im � ¼ 90 kPa)

Kflesh ¼ Im �

�2K 20�w

ðaS=LÞ�2 � 0:61: ð5:168Þ

At resonance, flesh damping is an order of magnitude larger than thermal andradiation terms. It is also the most uncertain of the three in magnitude, because ofthe uncertainty in the value of Imð�Þ.

From Equation (5.162), the overall Q-factor for a single fish can be written

1

Qfish

¼ KradP1=2 þ Ktherm

L1=2P�1=4 þ KfleshP

�1: ð5:169Þ

A typical value for Qfish at atmospheric pressure, obtained by substituting P ¼ 2 andL ¼ 0.1 m, is 3. The theoretical (reciprocal) Q-factor is dominated by the flesh term,which is inversely proportional to pressure. If the value of P is increased to 10, Qfish

increases to about 9. According to Andreeva (1964), acoustic radiation dominates thedamping at depths exceeding about 200 m.

5.4.3.7 Single fish (without bladder)

In the absence of a gas-filled bladder,18 scattering is from flesh alone and can bewritten

�backflesh � 1

½�backLF �1 þ ½�back

HF �1; ð5:170Þ

where the low-frequency limit (from Equation 5.131 and assuming the difference inimpedance between water and flesh to be small) is

�backLF ¼ �3 8Vfish

�2

� �2

jRðLFÞj2 ð5:171Þ

and the high-frequency limit (using the result from Table 5.3 for a convex shape atrandom aspect) is

�backHF ¼ Sfish

4jRðHFÞj2: ð5:172Þ

The low-frequency reflection coefficient is

jRðLFÞj2 � 1

4

�c

cwþ ��

�w

� �2

; ð5:173Þ

where �c and �� are the difference in sound speed and density between average fishflesh properties and those of water. For a bladderless fish it is reasonable to assume


18 Appendix C contains a list of species of fish indicating whether or not they have a bladder.

Common examples of bladdered fish are cod and herring. An example of a bladderless species

is mackerel.

that the density contrast is small. Specifically, using Love’s values of

�c

cw¼ 0:033 ð5:174Þ

and��

�w

¼ 0:022 ð5:175Þ

the reflection coefficient is

jRðLFÞj2 � 0:00076: ð5:176Þ

The high-frequency reflection coefficient can be determined empirically frommeasurements of the target strength of bladderless fish. A typical value, inferredfrom measurements of the target strength of mackerel (see Chapter 8), is

jRðHFÞj2 � 0:0045: ð5:177Þ

5.4.3.8 Dispersed fish (with bladder)

For fish that are dispersed (i.e., uniformly distributed in space), the contributionsfrom individual fish can be integrated in the same way as for bubbles. Specifically, letnðLÞ dL be the number of fish per unit volume whose length is between L and Lþ dL.The total volumic BSX is then

�backV ¼

ð10

�backbladderðLÞnðLÞ dL; ð5:178Þ

where the BSX is given by Equation (5.155).Compared with a cloud of gas bubbles, a group of fish is likely to have a relatively

narrow size distribution, centered on some value of fish length, say Lgroup. Thisdistribution is represented in terms of a Gaussian of width Lgroup/Qgroup, that is,following Weston (1995)

nðLÞ ¼ NVQgroup

Lgroup

exp ��Q2group

L

Lgroup

� 1

� �2

� ; ð5:179Þ

such that NV is the total population density

NV ¼ð1

0

nðLÞ dL: ð5:180Þ

Thus, the presence of nðLÞ in the integrand has the effect of broadening the resonanceby an amount that depends on Qgroup. The resonance peak can be approximated by aDirac delta function

�backbladderðLÞ � 4�a2

SL0ð f ÞQfish �ðL� L0ð f ÞÞ; ð5:181Þ

where the length L0 is the length of fish whose bladder would resonate at a givenfrequency f (see Equation 5.161). Substituting Equations (5.179) and (5.181) intoEquation (5.178) yields an approximation that relies on the fish being close to


resonance

�backV � 4�a2

SQfishQgroupNVL0ð f ÞLgroup

exp ��Q2group

L0ð f ÞLgroup

� 1

� �2

� : ð5:182Þ

If no better information is available concerning fish length distribution, Weston(1995) suggests a default value of Qgroup ¼ 2.

5.4.3.9 Dispersed fish (without bladder)

The volumic BSX for dispersed bladderless fish can be written

�backV ¼

ð10

�backfleshðLÞnðLÞ dL; ð5:183Þ

with �O from Equation (5.170). If all fish are of approximately the same length Lgroup,such that

nðLÞ � NV �ðL� LgroupÞ; ð5:184Þit follows that

�backV ¼ �back

fleshðLgroupÞNV : ð5:185Þ

5.4.3.10 Aggregated fish (with bladder)

If, instead of being distributed uniformly in space (the dispersed model of Section5.4.3.8), the fish form discrete aggregations, or ‘‘shoals’’, an estimate for the BSX ofsuch a shoal of bladdered fish can be obtained from the volumic BSX, �back

V . Specific-ally, if fish density within the shoal is sufficiently low, total shoal BSX may beapproximated as the product of �back

V and shoal volume Vshoal. For a high-densityshoal, it seems reasonable to propose an upper limit on the scattering cross-sectiondetermined by its surface area, resulting in the following tentative expression

1

�backshoal

� 4

Sshoal

þ 1

Vshoal�backV

; ð5:186Þ

valid for both low and high fish density, with �backV from Section 5.4.3.8.

5.4.3.11 Aggregated fish (without bladder)

The reasoning leading to Equation (5.186) applies just as well to a shoal ofbladderless fish. Thus,

1

�backshoal

� 4

Sshoal

þ 1

Vshoal�fleshV

; ð5:187Þ

with �fleshV equal to �back

V from Section 5.4.3.9.

5.4.4 Scattering from rough boundaries

For scattering from a rough boundary, the main parameter of interest is the scatteringcoefficient �AO, which is the differential scattering cross-section �O per unit area (i.e.,


the areic differential scattering cross-section), and sometimes denoted mS. The behav-ior close to the direction of specular reflection is very different from other directions,so this case is treated separately.

5.4.4.1 Non-specular term

For incident and scattered grazing angles �0 and �, the scattering coefficient awayfrom the specular direction is (Brekhovskikh and Lysanov, 2003)

�AOð�0; �; �Þ ¼ 4k4G2ðs0Þ sin2 �0 sin2 �; ð5:188Þ

where G2 is the spatial roughness spectrum (see Section 5.3.4), evaluated here for theBragg scattering vector s0. For simplicity, attention is limited to isotropic spectra, sothat only the magnitude of the vector s is important, and G2ðsÞ may be replaced bythe 1D spectrum G1ð�Þ, where

� ¼ jsj: ð5:189Þ

In particular, for the backscattering direction,

�AOð�Þ � �AOð�; �; �Þ ¼ 4k4G1ð2k cos �Þ sin4 �; ð5:190Þ

where the use of a single subscript in �AOð�Þ is used hereafter to indicate evaluation ofthe scattering coefficient in the backscattering direction (� ¼ �0; � ¼ �). The param-eter �AOð�Þ is known as the backscattering coefficient. No distinction is made between� and �0 in this situation because they are identical.

Equation (5.188) is a general result, applicable to any roughness spectrum. Forthe special case of a Gaussian roughness spectrum of correlation radius L (seeEquation 5.103), it follows from Equation (5.188) that

�AOð�0; �; �Þ ¼1

�ðk�Þ2ðkLÞ2 sin2 �0 sin2 � exp ��2L2

4

!: ð5:191Þ

Specializing to the backscattering case, we have

� ¼ 2k cos �0 ð5:192Þand hence

�AOð�Þ ¼1

�ðk�Þ2ðkLÞ2 sin4 � exp½�ðkLÞ2 cos2 � : ð5:193Þ

5.4.4.2 Near-specular term

Brekhovskikh and Lysanov (2003, pp. 207–209) consider rough boundary scatteringfor the case of isotropic roughness with a normally distributed slope and at highfrequency. For this situation they derive an expression for the near-specularscattering coefficient, which can be written in the form (Ellis and Crowe, 1991)

�AOð�0; �; �Þ ¼ ð1 þ DOÞ2 exp �DO2�2

� �; ð5:194Þ


where (in this section) � refers to the RMS roughness slope of the seabed,

DO ¼ cos2 �0 þ cos2 �� 2 cos �0 cos � cos �

ðsin �0 þ sin �Þ2ð5:195Þ

and

¼ Rð�0Þ2

8��2: ð5:196Þ

For in-plane scattering, Equation (5.194) becomes

�AOð�0; �; �Þ ¼

sin4 �exp � 1

2�2 tan2 �

� �; ð5:197Þ

where

� ¼ð�0 þ �Þ=2 � ¼ �

ð�0 þ �� Þ=2 � ¼ 0.

�ð5:198Þ

Finally, the backscattering coefficient is

�AOð�Þ ¼1

8��2

Rð�Þ2

sin4 �exp � 1

2�2 tan2 �

� �: ð5:199Þ

5.5 DISPERSION IN THE PRESENCE OF IMPURITIES

The speed and attenuation of sound in seawater can be affected by impurities,especially if these contain any gas. The simplest situation, considered in Section5.5.1, involves a dilute suspension of solid particles. The situation is complicatedsignificantly if there is the possibility of contact between solid grains, as considered inSection 5.5.2. The effects of bubbles and bladdered fish are considered in Section5.5.3. All of these effects take as a baseline the sound speed and attenuation of pureseawater, covered in Chapter 4.

5.5.1 Wood’s model for sediments in dilute suspension

Consider a mixture of a substance x, of density �x, and bulk modulus Bx, with water.If the fraction of x by volume is U, the density of the mixture is

�m ¼ ð1 �UÞ�w þU�x: ð5:200Þ

Similarly, the bulk modulus Bm of the mixture is given by (Wood, 1941)

1

Bm

¼ 1 �UBw

þ U

Bx; ð5:201Þ

where Bw is the bulk modulus of water

Bw ¼ �wc2w: ð5:202Þ

5.5 Dispersion in the presence of impurities 225]Sec. 5.5

The speed of sound in the mixture is then given by

cm ¼ffiffiffiffiffiffiffiBm

�m

sð5:203Þ

or, equivalently,

c�2m ¼ ½ð1 �UÞ�w þU�x ½ð1 �UÞB�1

w þUB�1x : ð5:204Þ

The last form is known as Wood’s equation.If one or both of Bw and Bx are complex, indicating the presence of an

attenuation mechanism,19 the right-hand side of Equation (5.204) becomes complex.This complex sound speed is denoted ~ccm to distinguish it from the real value cm. Thetwo variables are related to each other according to

~ccm � cm1 þ i�cm=!

; ð5:205Þ

so that

cm ¼ 1

Reð1=~ccmÞð5:206Þ

and� ¼ ! Imð1=~ccmÞ: ð5:207Þ

The corresponding complex wavenumber is then

k ¼ !

~ccm: ð5:208Þ

Though derived originally with a mixture of two fluids in mind, Wood’s equation isapplied above without modification to the case of a solid in dilute suspension. Ageneralization that allows for the presence of a solid frame is derived by Gassmannand extended further by Biot and Stoll. The interested reader is referred to Buchananet al. (2004) or Jackson and Richardson (2007) for details of these models. Analternative approach that continues to treat sediment grains as individual particles,but allows contact between them, is described in Section 5.5.2. A further complica-tion, not considered here, is the possible presence of gas in the sediment (Andersonand Hampton 1980a, b; Leighton, 2007b).

5.5.2 Buckingham’s model for saturated sediments with intergranular contact

Application of Wood’s model requires that sediment particles be in a dilutesuspension. The acoustical behavior of a material in which the individual grainsare in contact, as when they are deposited on the seabed, is described by Buckingham(2000, 2005).

An important parameter in determining the acoustical properties of marinesediments is the sediment porosity �. Assuming the sediment to be saturated with


19 The attenuation of sound due to suspended sediment is described by Richards (1998) and

Richards et al. (2003).

water, porosity is the ratio by volume of water to the total (waterþ sediment grain)medium. The density of the water–grain mixture is

�m ¼ ��w þ ð1 � �Þ�g; ð5:209Þ

where �g is the density of solid grain.Buckingham shows that for grain–grain interactions, complex compressional and

shear wave speeds (in the sense of Equation 5.205) are, respectively

~ccpðzÞ ¼ c0 1 þ !!nPðzÞ þ 4

3SðzÞ

B0

e�in�=2

� 1=2

ð5:210Þ

and

~ccsðzÞ ¼ c0!!nSðzÞB0

� 1=2

e�in�=4; ð5:211Þ

where

!! � 2�ff ð5:212Þ

is the angular frequency in radians per second; c0 is the sound speed of a non-interacting mixture as given by Wood’s formula (Equation 5.203)

c20 ¼ B0=�m ð5:213Þ

and1

B0

¼ �

Bw

þ 1 � �

Bg

: ð5:214Þ

The exponent n in Equations (5.210) and (5.211) is described as the ‘‘strain-hardeningindex’’. The parameters P and S (denoted p and s by Buckingham, 2005) arerelated to the bulk modulus and shear modulus of the combined porous mediumcomprising sediment grain and water, and are referred to by Buckingham as thecompressional and shear rigidity coefficients. The main depth dependence arisesthrough these rigidity coefficients. The equations that describe this dependence ondepth z in the sediment are

PðzÞ ¼ P0�ðd; zÞ ð5:215Þand

SðzÞ ¼ S0�ðd; zÞ; ð5:216Þ

where � is a dimensionless function of grain diameter d, porosity �, and depth z

�ðd; zÞ ¼ 1 � �ðd; zÞ1 � �0

d

d0

z

z0

� 1=3

; ð5:217Þ

where parameters �0, d0, and z0 are reference values of the variables �, d , and z. Theseand other parameters required for Buckingham’s model are listed in Table 5.4.

5.5.3 Effect of bubbles or bladdered fish

Despite the reference to fish in the title of this section, it is mostly about the propertiesof sound in bubbly water. However, much of the theory can be applied with little


modification to dispersed populations of bladdered fish as demonstrated in Weston’sclassic 1967 article and illustrated in Section 5.5.3.6.

5.5.3.1 Dispersion in bubbly water

Commander and Prosperetti (1989) and Hall (1989) derive equations for the complexeffective sound speed of a mixture of water and bubbles. Hall’s approach, based onWood’s equation, results in the following expression for ~ccm in terms of the densities ofair (�g) and water (�w) and gas volume fraction U

ð1=~ccmÞ2 ¼ ½ð1 � UÞ�w þU�a ½ð1 �UÞKw þ DK ; ð5:218Þ

where Kw is the compressibility of bubble-free seawater such that

Kw ¼ 1

Bw

¼ 1

�wc2w

: ð5:219Þ


Table 5.4. Water and solid grain sediment parameter values needed for Buckingham’s grain-

shearing model. Values for the density and bulk modulus of water, and the density of solid grain

are from Chapter 4. All other parameter values are from Buckingham (2005).

Symbol Description Value

�w Density of water 1027 kg/m3

Bw Bulk modulus of water 2.28GPa

�g Density of solid grain 2680 kg/m3

Bg Bulk modulus of solid grain 36GPa

P0 Compressional rigidity coefficient of porous medium at depth z0,

of grain diameter d0, and porosity �0 388.8 MPa

S0 Shear rigidity coefficient of porous medium at depth z0, of grain

diameter d0, and porosity �0 45.88 MPa

n Strain-hardening index 0.0851

z0 Reference depth for P0 and S0 30 cm

d0 Reference grain diameter for P0 and S0 1mm

�0 Reference porosity for P0 and S0 0.377

The contribution to compressibility from the air fraction (denoted DK) is given by20

DK ¼ 4�

�w

ðanðaÞ

!20 � !2 þ 2i�!

da; ð5:220Þ

where !0, a parameter closely related to bubble resonance frequency, can be writtenin terms of its bulk modulus Bb

!20 ¼

3 Re Bbða; !Þ�wa

2; ð5:221Þ

and the damping factor21 � is given by

2�

!¼ 3 Im Bbða; !Þ

�wa2!2

þ !a

cmþ 4�

�wa2!: ð5:222Þ

The bulk modulus Bb is given (anticipating a result from Section 5.5.3.2) by

Bb ¼ GPa �2�

3a: ð5:223Þ

Neglecting the density of air ( �a � �w), and assuming the gas fraction to be small(U � 1), it follows from Equation (5.218) that

cw~ccm

� �2

� 1 þ DKKw

: ð5:224Þ

The validity of Equation (5.224) does not depend on DK=Kw being small, providedthat U is small.

It is convenient to define the apparent void fraction � as

� � Pw DK ; ð5:225Þ

so that Equation (5.224) can be written

cw~ccm

� �2

� 1 þ �ðzÞKwPwðzÞ

: ð5:226Þ

A useful simplification is obtained by approximating the denominator of theintegrand of Equation (5.220) by !2

0, requiring frequency ! to be small. UsingEquation (5.221) for !0, the parameter � is then

� ¼ðPw

Re Bb

VðaÞnðaÞ da; ð5:227Þ

where VðaÞ is the volume of a single spherical bubble of radius a. Before consideringthe bulk modulus in detail in the next section, for isothermal pulsations (consistentwith the low-frequency approximation already made), and neglecting surface tension,


20 The same result follows by linearizing the non-linear theory of bubble dynamics (Leighton et

al., 2004).21 The damping factor and damping coefficient are related according to Equation (5.308).

it can be replaced in Equation (5.227) by Pw, so that � is real and equal to the gasfraction U, and hence

cwcm

� �2

� 1 þ U

KwPw

: ð5:228Þ

Substituting numerical values for seawater gives, for the stated conditions,

cwcm

� �2

� 1 þ 2:250 � 104U: ð5:229Þ

At very low frequency (see Section 5.5.3.4.1 for details) all bubbles behaveisothermally. A more common situation, at slightly higher frequency, is a mixtureof different types of behavior, with the smallest bubbles pulsating isothermally andthe larger ones adiabatically. If the frequency is increased further, eventually theapproximation leading to Equation (5.227) breaks down as the resonance frequencyof the larger bubbles is approached.

5.5.3.2 Bulk modulus Bbða; !Þ

The bulk modulus of a bubble of volume V is defined as

Bb � �V dQw

dV; ð5:230Þ

where dQw denotes the change in externally applied pressure required to effect aninfinitesimal change dV in the bubble volume. The notation Qw denotes instanta-neous pressure, and is used to avoid possible confusion with the hydrostatic pressurePw, which is equal to the equilibrium value of Qw.

Similarly, let QaðRÞ be the pressure inside the bubble at the moment itsinstantaneous radius is R. This internal pressure is related to the instantaneousapplied pressure QwðRÞ via the surface tension � according to

QwðRÞ ¼ QaðRÞ �2�

R: ð5:231Þ

Differentiating Qw and substituting the result into Equation (5.230), it follows that

Bb ¼ �VðaÞ dQa

dV� 2�

3a: ð5:232Þ

Gas pressure is assumed to vary according to the ideal gas law, which for adiabaticconditions is

QaV a ¼ constant; ð5:233Þ

where a is the specific heat ratio of air. More generally, the exponent can take a valuebetween 1 and a depending on the speed of the change. In equation form, thisstatement can be written

QaVG ¼ constant; ð5:234Þ

where the exponent G is known as the polytropic index. For small bubbles and lowfrequency (leading to isothermal conditions), G is close to unity, whereas for large


bubbles and high frequency there is no opportunity for heat transfer, and Gapproaches a. With G defined in this way, Equation (5.223) follows from Equation(5.232). More generally, using E to denote the difference in pressure between gas andliquid, that is,

EðRÞ ¼ QaðRÞ �QwðRÞ; ð5:235Þ

regardless of the physical mechanism responsible for the pressure difference, andtherefore

Bb ¼ GPa þa

3

dE

dR

� �R¼a

: ð5:236Þ

This expression becomes relevant if the bubble is replaced by a fish bladder keptunder pressure by an elastic membrane.

5.5.3.3 Effect of surface tension on small bubbles at low frequency

If surface tension is not negligible, its effect is to decrease the compressibility of anair–water mixture. This decrease in compressibility can be quantified by making theassumption that the frequency is low enough for the bubbles to pulsate isothermally,the bubble bulk modulus Bb is related to the static pressure inside and outside thebubble according to

Re Bb ! Pa �2�

3a¼ Pw þ 4�

3a: ð5:237Þ

It is useful to define a distance a� as the bubble radius for which the contributions tothe internal bubble pressure from surface tension and hydrostatic pressure are equal.In other words

a� �2�

Pw

; ð5:238Þ

equal to 1.4 mm for clean bubbles at atmospheric pressure. Substituting for Bb inEquation (5.227) then gives for the apparent void fraction

� ¼ð

1

1 þ 2a�=3a

� �VðaÞnðaÞ da: ð5:239Þ

Using the Hall–Novarini bubble model for nðaÞ, this can be written (followingChapter 4)

� ¼ IðzÞn0uðv10ÞDðz; v10Þ; ð5:240Þwhere

IðzÞ ¼ð

1

1 þ 2a�=3a

� �VðaÞGða; zÞ da: ð5:241Þ

The depth distribution is described by the function Gða; zÞ, given by

Gða; zÞ ¼ aref

a

� �pða;zÞ; ð5:242Þ



where

pða; zÞ ¼4 a � aref

xðzÞ a > aref

�; ð5:243Þ

and

x ¼ 4:37 þ z

2:55 m

� �2: ð5:244Þ

Equation (5.241) can be written as the sum of two separate integrals

IðzÞ ¼ 4�

3a4

ref

ðaref

amin

1

aþ 23 a�

daþ 4�

3axref

ðamax

aref

a4�x

a þ 23 a�

da: ð5:245Þ

The integration runs from amin (the lower limit of the first integral, equal to 10 mm forthe Hall–Novarini model) to amax (the upper limit of the second, equal to 1000 mm).The value of aref is intermediate between these and varies with depth.

The second integral (from aref to amax) can be expressed formally in terms of thehypergeometric function 2F1 (see Appendix A). Alternatively, a useful approximationis obtained by using the first-order expansion

1

1 þ 2a�=3a¼ 1 � 2a�

3aþO a2

�

a2

!; ð5:246Þ

valid for sufficiently large bubbles, satisfying a�=a� 1. Using this expansion in thesecond integral of Equation (5.245) only,22 it follows that

3

4�a4ref

IðzÞ ¼ loge

aref þ 23a�

amin þ 23a�

þ 1�ðaref=amaxÞx�4

ðx� 4Þ � 2a�3aref

1�ðaref=amaxÞx�3

ðx� 3Þ þO a2�

a2ref

!:

ð5:247Þ

5.5.3.4 Bubble resonance

It is sometimes useful to be able to express resonance frequency as a function ofbubble radius, or the resonant bubble radius as a function of ensonification frequency.The Minnaert relationship (Equation 5.137) may be used for large bubbles and lowresonance frequency, but for small bubbles (high resonance frequency) there areimportant corrections caused by heat conduction and surface tension. The purposeof this section is to quantify these corrections.

The equation describing the motion of a spherical bubble of (instantaneous)radius R in an incompressible medium of density �w and shear viscosity �S, known

22 The approximation is not made in the first term because, although for the Hall–Novarini

bubble population model the ratio a� =amin is small, in general it might not be.


as the Rayleigh–Plesset equation, is (Leighton, 1994, p. 305)23

�w R €RRþ 3 _RR2

2

!¼ QwðRÞ � Pw � 4�S

_RR

R� p0ðtÞ; ð5:248Þ

where p0ðtÞ is the acoustic pressure at infinity (the forcing term); and Qw indicatestotal (static plus acoustic) pressure. Using the gas law—Equation (5.234) (withEquation 5.231)—this becomes

�w R €RRþ 3 _RR2

2

!¼ Pa

a

R

� �3G� 2�

R� Pw � 4�S

_RR

R� p0ðtÞ; ð5:249Þ

where Pa is the equilibrium value of Qa

Pa ¼ QaðaÞ: ð5:250ÞEquation (5.249) is a non-linear equation that is difficult to solve without furtherapproximation. It simplifies if it is assumed that the amplitude of the oscillations issmall compared with the bubble radius. Specifically, writing

RðtÞ ¼ aþ "ðtÞ; ð5:251Þif j"j � a it follows that

��wa2€""� 4�S _"" � 3Bb" � ap0ðtÞ; ð5:252Þ

where Bb is the bulk modulus.Thus, for the simple harmonic motion of frequency !, with p0 ¼ A expð�i!tÞ and

" ¼ "A expð�i!tÞ, the response "A is related to the forcing amplitude A according to

Aa

"A¼ �wa

2!2 þ 4i!�S � 3Bb: ð5:253Þ

It is convenient to define the variables

Z � piðx ¼ 0Þ_RR

¼ iA

!"Að5:254Þ

and

O2u � !2 þ i

!Z

�wa; ð5:255Þ

so that Equation (5.253) takes the form

�wað!2 � O2uÞ"A ¼ A: ð5:256Þ

Using Equation (5.253), Equation (5.255) can be written

O2u ¼ !2

u � 2i�!; ð5:257Þwhere24

!2u � Re O2

u ¼3 Re Bb

�wa2

ð5:258Þ

23 The effects of vapor pressure are neglected here.24 The variable !u is approximately equal to the bubble resonance frequency.

and � is the damping factor25

� � � Im O2u

2!¼ 4�S!� 3 Im Bb

2�wa2!

: ð5:259Þ

Thus,

!2u ¼

3Pa Re G� 2�

a�wa

2; ð5:260Þ

or more generally, using Equation (5.236) for the bulk modulus,

!2u ¼

3Pa Re Gþ adE

dR�wa

2: ð5:261Þ

The condition for resonance is a maximum in the magnitude of the ratio "A=A, which(if � is sufficiently small) occurs when ! and !u are equal. For the remainder of thischapter, the term resonance frequency, denoted !res, is used to mean the frequency atwhich !u is equal to !. If !u were a constant (independent of frequency), that constantwould equal !res. For large bubbles this is indeed the case, with the polytropic indexthen equal to the specific heat ratio. The resonance frequency derived in this approx-imation is the Minnaert frequency !0 of Equation (5.137). For smaller bubbles, G andhence also !u is a function of frequency, thus complicating the calculation of !res. Fora hypothetical case with negligible thermal diffusivity Da, the effect of surface tensioncan be taken into account without difficulty in the adiabatic limit. In this situation,Equation (5.258) simplifies to

!2u

!20

¼ 1 þ 1 � 1

3 a

� �a�a; ð5:262Þ

where a� is given by Equation (5.238). Using the subscript ‘‘ad’’ to denote theresonance frequency (or bubble radius) for adiabatic conditions, it follows fromEquation (5.262) that

!2adðaÞ!2

0

¼ 1 þ 1 � 1

3 a

� �a�a: ð5:263Þ

Equation (5.263) is useful for gas bubbles that are large enough to neglect thermalconduction but small enough for surface tension to become significant. For airbubbles in seawater, this combination arises rarely, if ever.26 Nevertheless, Equation(5.263) serves to define the parameter !ad, used later.


25 In general, there is an additional contribution to � due to acoustic radiation. No such term

appears in Equation (5.259) due to the assumption here of an incompressible medium.26 For large bubbles (satisfying a� a� ), Equation (5.263) may still be used, but there is little

point as the Minnaert frequency is usually a better approximation.

A more realistic scenario involves very small bubbles such that the pulsations areisothermal. In this situation Equation (5.260) becomes

!2u ¼

!20

a

1 þ 2

3

a�a

� �; ð5:264Þ

which is also independent of frequency. The isothermal resonance frequency istherefore

!isoðaÞ ¼!0ffiffiffiffiffi a

p 1 þ 2

3

a�a

� �1=2

: ð5:265Þ

It is often the case that the pulsations are neither purely adiabatic nor purelyisothermal. In this situation the bubble resonance frequency is higher than !iso

and lower than !ad.

5.5.3.4.1 Polytropic index G

The polytropic index is a dimensionless complex number whose real part takes valuesbetween unity for isothermal oscillations to the specific heat ratio a for adiabaticones. It is a function of both ! and a.27 Specifically (Devin, 1959; Hall, 1989)

Gð!Þ ¼ a

1 þ ð1 � iÞxtanh½ð1 � iÞx � 1

� �3ið a � 1Þ

2x2

; ð5:266Þ

where

xða; !Þ ¼ a!

2Da

� �1=2

: ð5:267Þ

The parameter x can be written

x ¼ a

aDð!Þ; ð5:268Þ

where aD is the thermal diffusion length, given by

aD ¼ 2Da

!

� �1=2

: ð5:269Þ

At atmospheric pressure the diffusion length is equal to about 80 mm at a frequency of1 kHz, and 8 mm at 100 kHz. A pulsating bubble develops an isothermal layer of gas,at the edge of the bubble and in contact with the surrounding water. The thickness ofthis layer is approximately one thermal diffusion length aD. If aD is small comparedwith the bubble radius (corresponding to large x) there is negligible heat transfer, andoscillations are adiabatic so that G � a. If it is large (small x), heat is transferred tothe interior of the bubble, and the oscillations are isothermal.


27 For large bubbles, the polytropic index is a function of the product a!1=2; for small ones

there is a more complicated dependence because of the effect of surface tension on pressure

(and hence on gas thermal diffusivity).

An alternative form, showing the frequency dependence of x explicitly, is

x ¼ !

!DðaÞ

� 1=2

; ð5:270Þ

where !D, by analogy with aD, is referred to as the thermal diffusion frequency and isgiven by

!D ¼ 2Da

a2: ð5:271Þ

Diffusion frequency is a transition or threshold frequency between isothermal andadiabatic behavior. It is the frequency at which the diffusion length is equal to thebubble radius. Its value varies by four orders of magnitude between 6Hz for largebubbles (a � 1 mm) and about 60 kHz for small ones (a � 10 mm).

A useful approximation to Equation (5.266) is obtained by assuming that x islarge:

G � a

1 þ 3ð a � 1Þ2x

1 þ i 1 � 1

x

� �� : ð5:272Þ

From this it follows that

a

Re G� 1 þ �

xþ �

x

� �2 ð1 � 1=xÞ2

1 þ �=xð5:273Þ

and

a

Im G� �

1 þ �

x

� �2þ �2

x21 � 1

x

� �2

�

x1 � 1

x

� � ; ð5:274Þ

where

� ¼ 32ð a � 1Þ � 0:602: ð5:275Þ

5.5.3.4.2 Resonance frequency

Given a bubble of known size, at what frequency does it resonate? If the bubbleradius exceeds about 1mm, the answer is at the Minnaert frequency, given byEquation (5.137). For smaller bubbles, two corrections are needed, one arising fromthermal conduction inside the bubble, which reduces the resonance frequency, andanother due to surface tension, which increases it. Because the two mechanisms workin opposite directions, their net effect rarely exceeds a 10 % correction except formicroscopic bubbles.

General solution for arbitrary bubble radius. For a given arbitrary bubble radiusa, the frequency ! at which !uða; !Þ is equal to ! is the resonance frequency !res. Thisstatement can be written in the following equation form,

!uða; !resÞ ¼ !res: ð5:276Þ


In the general case, bubble resonance frequency can be found by writing Equation(5.276) as:

a

!2res

!20

¼ 1 þ a�a

� �Re Gð!resÞ �

a�3a: ð5:277Þ

The parameter Re G varies between its isothermal and adiabatic values (1 and a),which means that the right-hand side of Equation (5.277) depends only weakly on thefrequency variable (!res). This suggests an iterative solution with first guess equal toone of !iso or !ad. In other words

ð! ð jþ1Þres Þ2 ¼ !2

0

a

1 þ a�a

� �Re Gð! ð jÞ

resÞ �a�3a

h i; j � 0 ð5:278Þ

with either

!ð0Þres ¼ !ad; ð5:279Þ

or

!ð0Þres ¼ !iso: ð5:280Þ

The choice of !iso as a seed tends to work best for small bubbles (a < 30 mm) whereas!ad works better for larger ones. The solid blue line in Figure 5.4 shows the convergedresult after repeated application of Equation (5.278), normalized by dividing by theMinnaert frequency. Also shown are the adiabatic and isothermal resonance fre-quencies !ad and !iso, and an alternative approximation derived below (Equation5.283).

The parameter values used are listed in Table 5.5. The thermal diffusivity of air is(see Chapter 4)

Da �Ka

�aðCPÞa: ð5:281Þ

Thermal conductivity Ka and specific heat capacity ðCPÞa are treated as constants,given by Table 5.5. However, the density of air ( �a) is a function of pressure andhence, because of surface tension, of bubble radius. Although the effect of this changein diffusivity with bubble radius on the resonance frequency is small, the correction tothe damping coefficient is significant (see Section 5.5.3.5.3 and Leighton, 1994).

Approximation for bubbles exceeding radius 20 �m. For bubbles whose radius islarger than about 20 mm, the difference between !ad and !0 is small, so in thissituation the iteration can be simplified by using !0 as a seed instead of !ad. Withthis simplification, the first iteration of Equation (5.278) gives

!2res � !2

0

1 þ ð1 � 1=ð3 aÞÞa� =a1 þ �ð!D=!0Þ1=2

: ð5:282Þ

The second iteration, dropping all terms involving a�=a except the first-order one,gives

!2res � !2

0

1 þ ð1 � 1=ð3 aÞÞa� =a1 þ �ð!D=!0Þ1=2 þ ð5�2=4Þ!D=!0

; ð5:283Þ



Figure 5.4. Approximations to the resonance frequency (normalized by dividing by the

Minnaert frequency !0) for air bubbles in water at atmospheric pressure (upper graph) and

at a depth of 90m, corresponding to 10� atmospheric pressure (lower): ‘‘pert.’’¼Equation

(5.283); ‘‘conv.’’¼ converged iterative solution to Equation (5.278); ‘‘Minnaert’’¼Equation

(5.137); ‘‘adiab.’’¼Equation (5.263); ‘‘isoth.’’¼Equation (5.265).

where !D is the thermal diffusion frequency;and � is given by Equation (5.275). The lar-gest omitted terms are of order ð!D=!0Þ3=2

and ða� =aÞð!D=!0Þ1=2 and may be neglectedif the bubble radius is larger than about20 mm, as illustrated by Figure 5.4. Alsoapparent from Figure 5.4 is that, for the con-ditions considered and bubbles of radiusgreater than 3 mm, the error incurred byusing the Minnaert frequency is less than10 %. Finally, a simple approximation forresonance frequency, with an error of lessthan 3% across the entire range of bubblesizes considered, is obtained by taking thelarger of !iso and Equation (5.283).

Fish bladder resonance. A fish bladder can be modeled acoustically as a largedeformed bubble under tension. The formula for resonance is found by writing thebulk modulus in the form

Bb ¼ aðPw þ EÞ þ Ve

dE

dV

� �V¼Ve

; ð5:284Þ

where Ve denotes the equilibrium bladder volume; and EðVÞ is the difference in staticpressure across the bladder wall (cf. Equation 5.236). Substituting the formula fromChapter 4

Vbladder ¼ 3:40 � 10�4L3 ð5:285Þ

into Equation (5.258) gives resonance frequency as

!2res ¼

3 aðPw þ PeÞ�wa

2; ð5:286Þ

where

Pe ¼ E þ 2:43 � 10�4 dE

dVL3: ð5:287Þ

5.5.3.4.3 Resonant bubble radius

A more likely scenario than a single bubble is one involving a cloud of bubbles ofdifferent sizes. The question addressed here is, if such a cloud is ensonified with aplane wave of arbitrary frequency, which bubbles will resonate? If the frequency islower than about 10 kHz, the answer is those whose radius is equal to the Minnaertradius, given by rearranging Minnaert’s equation (Equation 5.137) in the form

a0 ¼ a0ð!Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffi3 aPw

�w!2

s; ð5:288Þ


Table 5.5. Values of physical constants

(from Chapter 4) used for the evaluation

of the bubble resonance characteristics

in Figures 5.4 and 5.5.

Parameter Value used

a 1.4011

Ka 24.9mW m�1 K�1

ðCPÞa 1.005 J g�1 K�1

� 0.072 Nm�1

(i.e., the adiabatic resonant radius if the restoring force is due to water pressureonly). For higher frequency, two corrections are needed, one arising from thermalconduction inside the bubble, which reduces resonant bubble radius, and anotherdue to surface tension, which increases it. Because the two mechanisms work inopposite directions, their net effect rarely exceeds a 10 % correction for sonar fre-quencies.

The answer is those whose radius ares satisfies the equation

!uðares; !Þ ¼ !: ð5:289Þ

Limits for adiabatic and isothermal conditions. If the conditions are adiabatic, theequation for the resonant radius (substituting G ¼ a in Equation 5.289) is

a3ad � a2

0aad � ð1 � 1=ð3 aÞÞa20a� ¼ 0; ð5:290Þ

where a0 is the Minnaert radius given by Equation (5.288). The solution to Equation(5.290) can be written

a2adð!Þ ¼

4a20

3cos2 1

3arccos

ffiffiffiffiffi27

p

21 � 1

3 a

� �a�a0

" #( ): ð5:291Þ

The equations for isothermal conditions can be solved in exactly the same way as theadiabatic case, except with G ¼ 1. The resulting cubic equation for resonant radius is

a3iso �

a20

a

aiso �2

3

a20

a

a� ¼ 0; ð5:292Þwhose solution is

a2isoð!Þ ¼

4a20

3 a

cos2 1

3arccos

ffiffiffiffiffiffiffi3 a

p a�a0

� �� : ð5:293Þ

General solution for arbitrary frequency. To solve for the bubble radius atresonance, Equation (5.289) can be written in the form

a

a2res

a20

¼ 1 þ a�ares

� �Re GðaresÞ �

a�3ares

: ð5:294Þ

The solution is between aiso and aad. A general iterative solution to Equation (5.294)is

ða ð jþ1Þres Þ2 ¼ a2

0

a

1 þ a�

að jÞres

!Re Gða ð jÞ

resÞ �a�

3a jres

" #; j � 0; ð5:295Þ

with either

að0Þres ¼ aad; ð5:296Þ

or

að0Þres ¼ aiso ð5:297Þ

as seed. Resonant bubble radius calculated in this way is shown in Figure 5.5 as afunction of frequency (dark blue line). Also shown are the adiabatic and isothermal



Figure 5.5. Approximations to the resonant radius (normalized by dividing by the Minnaert

radius a0) for air bubbles in water at atmospheric pressure (upper graph) and at a depth of 90m,

corresponding to about 10� atmospheric pressure (lower): ‘‘pert.’’¼Equation (5.299);

‘‘conv.’’¼ converged iterative solution to Equation (5.295); ‘‘Minnaert’’¼Equation (5.288);

‘‘adiab.’’¼Equation (5.291); ‘‘isoth.’’¼Equation (5.293).

values aad and aiso, and an alternative approximation valid for frequencies up toabout 100 kHz (Equation 5.299). The Minnaert radius (dashed cyan line) issurprisingly accurate up to frequencies of several megahertz.

Approximation for frequencies up to 100 kHz. The use of the general solution isoften unnecessary. If the frequency is not too high, an accurate solution can beobtained using one or two iterations. A single iteration, using a0 as a seed, yields

a2res � a2

0

1 þ ð1 � 1=ð3 aÞÞa�=a0

1 þ �aD=a0

: ð5:298Þ

The second iteration, dropping all terms involving a�=a0 except the first-order one,gives28

a2res � a2

0

1 þ ð1 � 1=ð3 aÞÞa�=a0

1 þ �aD=a0 þ ð3�2=2ÞðaD=a0Þ2: ð5:299Þ

The largest omitted terms are of order ðaD=a0Þ3 and ða�=a0ÞðaD=a0Þ and may beneglected if the frequency is lower than about 100 kHz. The first-order surface tensionterm contributes order 1% to the numerator at 20 kHz and 10 % at 200 kHz. Finally,a simple approximation, with an error of less than 3% across the entire frequencyrange considered, is obtained by taking the larger of aiso and Equation (5.299).

5.5.3.5 Damping factor

5.5.3.5.1 Thermal and viscous damping

Equation (5.257) can be written in the form

O2u ¼ !2

u � 2ið�visc þ �thermÞ!; ð5:300Þwhere

�therm ¼ �thermð!Þ ¼ � 3 Im Bb

2�wa2!

ð5:301Þ

and

�visc ¼2�S

�wa2; ð5:302Þ

where �therm and �visc are the thermal and viscous damping factors.These equations are derived for the Rayleigh–Plesset model, which assumes that

the liquid medium containing the bubble is incompressible. An incompressible med-ium does not support acoustic waves, so there is no mechanism to carry sound awayfrom the bubble and thus no radiation damping. A more subtle point is that, bydefinition, an incompressible medium does not support changes in volume. As aresult, only shear viscosity (i.e., no bulk viscosity) is included in the viscous dampingterm. For a more complete description that includes the contribution to dampingfrom acoustic radiation and from volume viscosity, see Section 5.5.3.5.3.


28 In Equation (5.299), the parameter aD is a function of pressure and hence of bubble radius.

The error incurred by evaluating aD at atmospheric pressure is usually small.

5.5.3.5.2 Radiation and thermal damping

Consider the response of a bubble of equilibrium radius a, centered at the origin, to aplane wave traveling parallel to the x-axis, such that the incident acoustic pressure is

p0 ¼ A exp iðkx� !tÞ; ð5:303Þ

whereA is the amplitude, assumed constant. The scattered wave ps is assumed to havespherical symmetry:

ps ¼C

rexp iðkr� !tÞ ðr � aÞ; ð5:304Þ

where r is the distance from the origin; and C is a constant with dimensions pressuretimes distance. The scattering cross-section � can then be expressed in the form

� ¼ 4�jC=Aj2: ð5:305Þ

An expression for the ratio C=A, derived by Ainslie and Leighton (2009), is

C

A� ae�ika

!2u

!2ka� Im G

Re G

� �� 1 � i�

; ð5:306Þ

where

� ¼ !2u

!2ka� Im G

Re G

� �: ð5:307Þ

Substituting Equation (5.306) in Equation (5.305) gives the scattering cross-sectionwith radiation and thermal damping. A more complete expression is given in Section5.5.3.6.

5.5.3.5.3 Total damping

The total damping coefficient �tot for a gas bubble in water is related to the dampingfactor �tot according to

�tot ¼2

!�tot þ

!2u

!2� 1

!ka: ð5:308Þ

For the special case of radiation damping only (i.e., if 2�tot=! ¼ ka), it follows that

�rad ¼ !2u

!2ka; ð5:309Þ

and therefore

�tot � �rad ¼ 2

!�tot � ka; ð5:310Þ

providing a conversion between damping coefficient and damping factor for non-acoustic forms of damping.

Equation (5.308) can be written in the form

�totð!Þ ¼ �radð!Þ þ �thermð!Þ þ �viscð!Þ; ð5:311Þ


in which the two main contributions are losses due to re-radiation of sound29 (�rad,Equation 5.309), and dissipation of heat due to the finite thermal conductivity of air(�therm), which is (see Equation 5.307)

�therm ¼ �!2u

!2

Im GRe G

: ð5:312Þ

The third contribution, resulting from the viscosity of water (�visc), is important onlyfor bubbles smaller than about 10 mm (or very high–frequency sound). The basic formof this term for the case of an incompressible liquid can be derived from the dampingfactor (Equation 5.302) as

�visc ¼2

!�visc ¼

4�S

�w!a2: ð5:313Þ

For a compressible liquid there is an additional contribution due to bulk viscosity(Love, 1978), such that

�visc ¼3ð�B þ 4

3�SÞ

�w!a2

: ð5:314Þ

The variation of the damping coefficient with frequency is described by Medwin(1977), except with the opposite frequency dependence for the radiation dampingterm �rad to that presented here. The discrepancy is explained by Ainslie and Leighton(2009).

5.5.3.5.4 Q-factors

It is sometimes convenient to express the damping coefficient in terms of its reciprocalat resonance, the so-called Q-factor. Viscous and radiation loss terms are inverselyproportional to frequency and hence can be written

�visc ¼1

Qvisc

!res

!

� �; ð5:315Þ

and

�rad ¼ 1

Qrad

!res

!

� �ð5:316Þ

where

1

Qvisc

¼3ð�B þ 4

3�SÞ

�w!resa2

ð5:317Þ

and

1

Qrad

¼ !resa

cw: ð5:318Þ


29 That is, the release as an acoustic wave of the energy invested in bubble pulsations.

The frequency dependence of �therm is more complicated. For sufficiently largebubbles:30

�therm ¼ 1

Qtherm

!res

!

� �5=2; ð5:319Þ

where

1

Qtherm

¼ 3ð a � 1Þ2

!D!res

� �1=2

: ð5:320Þ

At resonance it is usually the case that !=!D is larger than 3, so that theapproximation of Equation (5.272) may be used. In the low-frequency limit(isothermal conditions), Equation (5.266) may be written

Gð!Þ ¼ 1 � 2i

15

a � 1

a

!

!D: ð5:321Þ

Combining the low-frequency and high-frequency forms, it follows from Equation(5.312) that

�therm ¼ ð a � 1ÞPaðaÞ�w!

2a2�

2

5 a

!

!D

!

!D� 1

9 a

2

!D!

� �1=2 !

!D� 1.

8>>><>>>:

ð5:322Þ

5.5.3.6 Scattering, extinction, and absorption cross-sections

The scattering cross-section �s of an object relates the total power scattered by theobject to the intensity of an incident plane wave, and can be written

�s ¼ð�O dO; ð5:323Þ

where the integral is over all scattered solid angles O; and �O is the differentialscattering cross-section. Some of the incident energy is absorbed (i.e., converted toheat rather than scattered), as quantified by the absorption cross-section �a. Ainslieand Leighton (2009) derive a general purpose expression for �s. Converting to thepresent notation, their Eq. (43) is

�s ¼4�a2

½!2u=!

2 � 1 � ð�tot � �radÞ" 2 þ �2tot;

ð5:324Þ

with the Andreeva–Weston model for the damping coefficient � (Ainslie andLeighton, 2009).


30 Equation (5.319) is valid if the bubble radius is large compared with both the diffusion length

aD (Equation 5.269) and the parameter a� , related to the surface tension (Equation 5.238).

Total acoustic attenuation (due to scattering plus absorption) is determined bythe extinction cross-section �e (Weston, 1967; Ainslie and Leighton, 2009):

�e

�s

¼ 2�tot

!"�1

m ; ð5:325Þ

where

"m ¼ !a

cm: ð5:326Þ

Parameters � and � are related via Equation (5.310). Hence, the extinction andabsorption cross-sections can be calculated, respectively, as

�e ¼ 1 þ �tot � �rad

"m

� ��s ð5:327Þ

and

�a � �e � �s ¼�tot � �rad

"m

�s: ð5:328Þ

At low frequency, a small object like a bubble scatters sound equally in all directions.Thus, �O may be approximated by

�O � �back

4�; ð5:329Þ

where �back is the backscattering cross-section (BSX) of a single bubble, equal in thissituation to the total scattering cross-section �s. Using this equation for �O togetherwith Equation (5.327) (and Equation 5.323), the extinction cross-section for a singlebubble can be written

�e ¼ �back 1 þ �tot � �rad

"m

� : ð5:330Þ

Thus, with thermal and viscous absorption:

�e ¼ �back 1 þ �therm þ �visc

"m

� ; ð5:331Þ

where �back can be calculated using Equation (5.136). Thermal and viscous dampingterms are given by Equations (5.301) and (5.314).

The extinction cross-section of a fish bladder can be calculated in the same way asfor a gas bubble, by replacing the viscosity term with the appropriate dampingcoefficient for flesh damping and making allowance for non-sphericity. The result,following Weston (1995), is

�e ¼ �backbladderðL; !Þ 1 þ !2

0

!2

�therm þ �flesh�rad

" #; ð5:332Þ

where !0 is the resonance frequency (see Equation 5.156); and �back may beapproximated by Equation (5.155). The damping terms can be expressed in terms


of their respective Q-factors

�thermðL; !Þ ¼1

QthermðLÞ!0ðLÞ!

� 5=2

; ð5:333Þ

�fleshðL; !Þ ¼1

QfleshðLÞ!0ðLÞ!

� 2

; ð5:334Þ

and

�radðL; !Þ ¼1

QradðLÞ!0ðLÞ!

: ð5:335Þ

The three Q-factors are given, respectively, by Equations (5.164), (5.167), and (5.163).

5.6 REFERENCES

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Ainslie, M. A. (1995) Plane-wave reflection and transmission coefficients for a three-layered

elastic medium, J. Acoust. Soc. Am., 97, 954–961. [Erratum, J. Acoust. Soc. Am., 105, 2053

(1999).]

Ainslie, M. A. (1996) Reflection and transmission coefficients for a layered fluid sediment

overlying a uniform solid substrate, J. Acoust. Soc. Am., 99, 893–902.

Ainslie, M. A. and Burns, P. W. (1995) Energy-conserving reflection and transmission

coefficients for a solid–solid boundary, J. Acoust. Soc. Am., 98, 2836–2840.

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Anderson, A. L. and Hampton, L. D. (1980a) Acoustics of gas-bearing sediment. I:

Background, J. Acoust. Soc. Am., 67, 1865–1889.

Anderson, A. L. and Hampton, L. D. (1980b) Acoustics of gas-bearing sediment. II:

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Andreeva, I. B. (1964) Scattering of sound by air bladders of fish in deep sound-scattering

ocean layers, Akust. Zh., 10, 20–24 [English translation in Sov. Phys. Acoust., 10, 17–20

(1964)].

Anon. (1946) Physics of Sound in The Sea (NAVMAT P-9675, p. 462). National Defense

Research Committee, Washington, D.C.

ASA (1994) American National Standard, Acoustical Terminology, ANSI S1.1-1994 (ASA 111-

1994, revision of ANSI S1.1-1960 (R1976)). Acoustical Society of America, New York.

Brekhovskikh, L. M. and Godin, O. A. (1990). Acoustics of Layered Media I: Plane and

Quasi-Plane Waves, Springer-Verlag, Berlin.

Brekhovskikh, L. M. and Lysanov, Yu. P. (2003) Fundamentals of Ocean Acoustics (Third

Edition). AIP Press Springer-Verlag, New York.

Buchanan, J. L., Gilbert, R. P., Wirgin A., and Xu, Y. S. (2004) Marine Acoustics: Direct and

Inverse Problems, Society for Industrial and Applied Mathematics, Philadelphia.

Buckingham, M. J. (2000) Wave propagation, stress relaxation, and grain-to-grain shearing in

saturated, unconsolidated marine sediments, J. Acoust. Soc. Am., 108, 2796–2815.

Buckingham, M. J. (2005) Compressional and shear wave properties of marine sediments:

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Chapman, C. H. (2004) Fundamentals of Seismic Wave Propagation, Cambridge University

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Clay, C. S. and Horne, J. K. (1994) Acoustic models of fish: The Atlantic cod (Gadus morhua),

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Clay, C. S. and Medwin, H. (1977) Acoustical Oceanography: Principles and Applications,

Wiley, New York.

Commander, K. W. and Prosperetti, A. (1989) Linear pressure waves in bubbly liquids:

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in water, J. Acoust. Soc. Am., 31, 1654–1667.

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diffraction from underwater targets, in M. J. Crocker (Ed.), Encyclopedia of Acoustics

(pp 469–482), Wiley, New York.

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6

Sonar signal processing

Mathematics may be compared to a mill of exquisite workmanship, whichgrinds you stuff of any degree of fitness; but, nevertheless, what you get out

depends on what you put in; and as the grandest mill in the world will notextract wheat-flour from peascods, so pages of formulae will not get a

definite result out of loose data.

Thomas Henry Huxley (ca. 1894)

Once a pressure disturbance has been converted to an electric current, it can beprocessed in various ways to enhance the signal-to-noise ratio before being presentedto an operator for interpretation. In a modern sonar system, the processing is mostlycarried out by a digital computer, which means that one of the first steps must be aconversion from an analogue signal to a digital one. Even before this, an analoguelow-pass filter, known as an anti-alias filter, is used to remove frequencies that exceedthe specification of the analogue-to-digital converter (ADC).

The purpose of digital signal processing is partly to filter out as much unwantednoise as possible. Both active and passive sonars are designed to find signals in apredetermined frequency band, so noise outside this band is not normally consideredto contribute to the noise level term in the sonar equation (NL). The noise left afterthis filtering is referred to as the noise ‘‘in the sonar bandwidth’’.

Assuming that the sonar has more than one receiving hydrophone, the next stepafter this initial filtering, for both active and passive systems, is to filter out unwantedangles by beamforming as summarized in Chapter 2. The calculation of array gain forpassive sonar is the main subject of Section 6.1. For active sonar a further gain can beachieved by means of a special process that exploits knowledge of the shape of thetransmitted pulse, enabling rejection of any noise that does not resemble that pulse.This process, known as matched filtering, and the resulting gain, is the main subject ofSection 6.2.

6.1 PROCESSING GAIN FOR PASSIVE SONAR

Time domain and spatial domain filters are both used for passive sonar processing.The gain from time domain filtering is by convention considered as a reduction in thenoise level. The subject of this section is the gain from spatial filtering, which is largelydetermined by the beam pattern of the receiving array, described in Section 6.1.1.This is followed by a discussion of the directivity index, which is a function of thebeam pattern, in Section 6.1.2. The array gain, discussed in Section 6.1.3, depends onthe beam pattern in a similar way as the directivity index, and on the directionalproperties of the signal and noise fields. Array gain (AG) is arguably the most difficultterm of the passive sonar equation to calculate precisely. However, often there is noneed to do so, because it is straightforward to approximate its effect by replacingit with the directivity index (DI), which is equal to AG in certain idealizedcircumstances.

6.1.1 Beam patterns

Angular discrimination using omni-directional hydrophones can be achieved byconstructing an array with a horizontal or vertical aperture, or both, with the indi-vidual hydrophones placed a fraction of a wavelength apart. Such an array has abeam pattern BðOÞ, which is the squared magnitude of the array output in response toan acoustic plane wave, normalized by dividing by its maximum value in angle.

6.1.1.1 Steered line array

The beamformer output from Chapter 2 can be generalized by multiplying each termin the sum by a scaling factor wðxÞ, so that (using H here for the beamformer outputinstead of G, to avoid confusion with the array gain and directivity index)

H �XN�1

n¼0

wðxnÞFðxnÞ expð�ikxn sin Þ; ð6:1Þ

where FðxÞ is the incident pressure field;1 and is the steering angle (denoted �m inChapter 2), defined as

� arcsinkm

k; ð6:2Þ

and km is the steer direction in wavenumber space. The effect of the scaling factor,known as a shading function or window function2 is considered in Sections 6.1.1.1.2 to6.1.1.1.4. Before that, the beam pattern of an unshaded array is described.

252 Sonar signal processing [Ch. 6

1 In a modern sonar, the hydrophone signal is converted to voltage and then digitized. If all

processes are linear the final digital representation is proportional to the original pressure.2 Alternative names are weighting function or taper function. See Harris (1978) and Nuttall

(1981) for two in-depth reviews of the properties of many different shading functions.

6.1.1.1.1 Unshaded

Before considering the effects of shading, it is useful to revisit the properties of anunshaded array, equivalent to the trivial case wðxÞ ¼ 1 in Equation (6.1). Moreprecisely

wðxÞ ¼ Pðx=LÞ; ð6:3Þ

where Pð�Þ is the rectangle function, equal to unity if j�j < 1=2 and zero otherwise(see Appendix A). This window is known as a rectangular or Dirichlet window. For aplane wave of unit amplitude, the beamformer output is then (Chapter 2)3

HðkmÞ � N sincðuÞ; ð6:4Þwhere

u ¼ kðsin �� sin ÞL

2: ð6:5Þ

From Equation (6.5) it can be seen that is the value of the look angle � at which thesinc argument u passes through zero. In other words, it is the look direction4 in whichthe beamformer output is greatest. The broadside direction corresponds to ¼ 0(i.e., to an unsteered array) and the fore and aft endfire directions are given by ¼ ��=2.

Recall that the beam pattern is the normalized squared magnitude of thebeamformer output, so that

Bð�Þ � sinc2 uð�Þ: ð6:6Þ

The function Bð�Þ is plotted in Figure 6.1 for an array of length five wavelengths,with steering angles of 0 deg, 30 deg, 60 deg, and 90 deg (solid lines). The corre-sponding graphs for negative are obtained from these curves by taking their mirrorimages in the � ¼ 0 axis.

If the array axis is aligned horizontally (in which case it is known as a horizontalline array), it is convenient to replace the single angle �, representing the lookdirection, with the elevation and azimuth by writing

sin � ¼ cos sin ð6:7Þand hence

u ¼ �L

�ðcos sin � sin Þ; ð6:8Þ

where � is the acoustic wavelength at the array. The reason for doing so is that and are sometimes more natural co-ordinates for describing the search geometry (e.g., might be the bearing of a distant sonar contact).

From Figure 6.1 it is apparent that the main beam gets broader as the steeringangle is increased from 0deg to 90 deg. The width in wavenumber is given by

6.1 Processing gain for passive sonar 253]Sec. 6.1

3 The notation ‘‘sinc’’ is shorthand for the sine cardinal function (sinc x ¼ sin x=x) (see

Appendix A).4 That is, the direction of maximum sensitivity to an incoming plane wave, relative to the array

axis.


Figure 6.1. Sinc beam patterns for L=� ¼ 5 and steering angles 0, 30, 60, 90 deg anticlockwise

from upper left (solid). The dashed curves illustrate the effect of shading (see Section 6.1.1.1.2).

(Chapter 2)

�kfwhm ¼ 4

Lsinc�1 1ffiffiffi

2p� �

: ð6:9Þ

This is the full-width at half-maximum (fwhm). The expression holds for all beams,but is of limited use in this form. To understand its implications for the width of thebeams in real physical space, an angular width is more appropriate, as derived below.

For non-zero steering angles the beams lose their symmetry, becoming increas-ingly asymmetrical as they approach endfire. In this situation it is appropriate to talkof two half-widths rather than a single full-width, as the beams are unequal in extentto either side of the main peak. From Equation (6.9), the half-power half-width5 (i.e.,half-width at half-maximum) to broadside of the peak, denoted ��, is determined bythe equation

sin �m � sinð�m � ��Þ ¼ Y; ð6:10Þwhere

Y � sinc�1ð1=ffiffiffi2

pÞ

�

�

L� 0:4430

�

L: ð6:11Þ

Similarly, the half-width to endfire (��þ) is found from

sinð�m þ ��þÞ � sin �m ¼ Y: ð6:12Þ

For the endfire beam itself (�m ¼ �=2), Equation (6.10) simplifies to

ð��Þef ¼ arccos Y: ð6:13Þ

The array of Figure 6.1 has a length of 5�, so at endfire its half-width (��) fromEquation (6.13) is 24 deg, consistent with the graph. Similarly for the broadside beam(�m ¼ 0), Equation (6.10) or Equation (6.12) gives

ð��Þbs ¼ arcsin Y: ð6:14Þ

Equations (6.10) and (6.12) can be combined into a single more compact expressionto give

�� ¼ arcsinðY � sin �mÞ � �m: ð6:15Þ

Because the angle �� is defined as the half-power half-width, for it to exist at all thebeam pattern must drop below 0.5 somewhere in the real range of angles, and for thisit is necessary for the array to exceed a certain minimum length. A necessary con-dition for the existence of ��þ (and a sufficient one for ��) is that Y be less thanunity. Necessary and sufficient conditions for a continuous line array are

Y < 1 � sin �m ðfor ��þÞ ð6:16Þand

Y < 1 þ sin �m ðfor ��Þ: ð6:17Þ

For an array whose length exceeds �=2, the existence of �� (half-width to broadside)


5 The half-power full-width (i.e., f.w.h.m.) is sometimes referred to as the ‘‘3 dB’’ width,

because 10 log10ð1=2Þ � �3.01 dB is approximately equal to �3 dB.

is ensured, but the half-width to endfire exists only for beams that are not too close tothe endfire direction.

The sinc function has a main peak at u ¼ 0 and secondary ones close to oddmultiples of �=2. The secondary peaks, called sidelobes, are unwanted because theyresult in undesirable sensitivity to sound from directions other than the signal direc-tion. They are caused by the abrupt start and end of the array, and can be reduced bysmoothing these edges. That is, by reducing the contribution from hydrophones closeto the two ends of the array relative to those at the center. This procedure is known asarray shading or array weighting.

To illustrate how shading helps control sidelobe levels, consider the choice offunction wðxÞ in Equation (6.1). The simplest way of removing the step at x ¼ �L=2is to introduce a linear variation in amplitude, meaning that wðxÞ decreases linearlyfrom 1 at x ¼ 0 to 0 at �L=2. The resulting function is known as a triangular windowbecause of the triangular shape of the taper. Thus,

wtriðxÞ ¼ 1 � 2jxjL

� �Pðx=LÞ; ð6:18Þ

with the resulting beam pattern,

btriðuÞ ¼ sinc4 u

2: ð6:19Þ

A lower case ‘‘b’’ is used here for the beam pattern expressed as a function of thevariable u (Equation 6.5). This is to distinguish it from the upper case ‘‘B ’’ for thebeam pattern as a function of angle �. The functions bðuÞ and Bð�Þ are thereforerelated via

Bð�Þ ¼ b½uð�Þ : ð6:20Þ

6.1.1.1.2 Cosine shading (cosn)

Another function that tapers in a simple way to zero is a half-cycle of a cosine,perhaps raised to a power n such that

wcosnðxÞ ¼ cosn �x

L

� �Pðx=LÞ: ð6:21Þ

Beam patterns for the special cases n ¼ 1 and n ¼ 2 are

bcosðuÞ ¼�2

16sinc u þ �

2

� �þ sinc u � �

2

� �h i2ð6:22Þ

and

bcos2ðuÞ ¼ fsinc u þ 12 ½sincðu þ �Þ þ sincðu � �Þ g2: ð6:23Þ

The cosine squared window (i.e., Equation 6.21 with n ¼ 2) is also known as a Hannwindow.6 The associated beam pattern, given by Equation (6.23), is shown in Figure6.1 (dashed lines) for various steer angles.


6 The window is named after Julius von Hann. The term ‘‘Hann window’’ is preferred here over

its synonym ‘‘Hanning window’’ to avoid possible confusion with ‘‘Hamming window’’.

Beam patterns for rectangular, cosine, and cosine-squared (Hann) windows areplotted in Figure 6.2, this time for a 10-wavelength array. As the severity of shadingincreases (corresponding to increasing the value of n in Equation 6.21), the height ofthe sidelobes decreases (the desired effect) while the width of the main lobe increases(an unwanted side-effect, as it decreases the angular resolution of the beam).

The nulls for rectangular and Hann shading occur at identical angles, wheneverthe argument u is equal to a non-zero integer multiple of �. For cosine shading thenulls are shifted by �=2. The width of the main beam and the number of sidelobesdepend on the length of the array in wavelengths. Specifically, the longer the array inwavelengths, the narrower the beam and the larger the number of sidelobes.

6.1.1.1.3 Cosine on a pedestal (Hamming family)

The cos2 window of Section 6.1.1.1.2 can be generalized by placing it on a pedestal ofheight ". Because of the simple relationship between cos 2 and cos2 , this type ofwindow is also known as a raised cosine. The general case taper function for thiswindow is

wðxÞ ¼ "Pðx=LÞ þ ð1 � "Þwcos2ðxÞ; ð6:24Þ

where wcos2ðxÞ is given by Equation (6.21) with n ¼ 2. The correspondingbeamformer output is

H ¼ "Hrect þ ð1 � "ÞHcos2 ; ð6:25Þ


Figure 6.2. Beam patterns 10 log10 Bð�Þ for continuous line array of length L=� ¼ 10 for

unshaded array (cyan), cosine window (dashed red), and Hann window (blue).

where

Hrect ¼ N sinc u ð6:26Þand

Hcos2 ¼ N

2½sinc u þ 1

2sincðu þ �Þ þ 1

2sincðu � �Þ : ð6:27Þ

The resulting beam pattern is

bðuÞ ¼ sinc u þ 1 � "

2ð1 þ "Þ ½sincðu þ �Þ þ sincðu � �Þ

2

: ð6:28Þ

This family of windows includes a special case known as the Hamming window,7

obtained with " ¼ 0.08:

wHammingðxÞ ¼ 0:08Pðx=LÞ þ 0:92wcos2ðxÞ; ð6:29Þ

with corresponding beam pattern

bHamming ¼ sinc u þ 23

54½sincðu þ �Þ þ sincðu � �Þ

2

: ð6:30Þ

The generic window described by Equation (6.24), with arbitrary " between 0 and 1, isreferred to below as the ‘‘Hamming family’’. In addition to the Hamming windowitself, the Hamming family includes as members the rectangular and cos2 windows(obtained with " ¼ 1 and " ¼ 0, respectively).

Figure 6.3 shows Bð�Þ for the Hamming family with various values of thepedestal height ", including the Hamming window. The effect of reducing " from1.00 to 0.08 is to reduce the peak sidelobe level at the expense of an increased beamwidth. When " is reduced further than this the sidelobe levels increase. Thus, theHamming window (" ¼ 0.08) is close to an optimum from this point of view. Theprecise value of " that minimizes the highest sidelobe level is 0.076711, the beampattern for which is shown as a solid cyan line. Graphs of bðuÞ for selected " values areplotted in Figure 6.4, alongside their respective shading functions.

6.1.1.1.4 Tukey shading (raised cosine spectrum)

The Tukey family of windows provides an alternative generalization of therectangular and Hann windows, with the cosine function displaced laterally insteadof vertically. The two halves of a Hann window are compressed and pulled outtowards each of the two ends of the window, leaving a gap in the middle. The Tukeywindow is obtained by padding this gap with ones. This approach removes theundesirable discontinuities associated with a rectangular window (thus reducingsidelobe levels), while improving the resolution compared with the Hann window.The Tukey shading function (correcting a typographical error in Eq. (38) of


7 Named after Richard Hamming.

Harris, 1978) is

wTukeyðxÞ ¼

1 jxj � "L

2

cos2 ðjxj � "L=2Þ�ð1 � "ÞL "

L

2< jxj � L

2

0 jxj > L

2

8>>>>>>><>>>>>>>:

: ð6:31Þ

The corresponding beam pattern is

bTukeyðuÞ ¼cos½ð1 � "Þu=2

1 � ð1 � "Þ2ðu=�Þ2sinc

u

2

2

: ð6:32Þ

The Tukey window is used for shading sonar pulses in the time domain. Forcommunications signals, it is used in the frequency domain, where it is known asthe raised cosine spectrum (Proakis, 1995); the resulting (predictable and periodic)zero crossings in the time domain (corresponding to u ¼ n�, for integer n 6¼ 0) areexploited to minimize intersymbol interference (van Walree, pers. commun., 2009).

6.1.1.1.5 Summary

The main purpose of shading is to reduce the sidelobes. The numerical values ofthe reduced sidelobe levels for various shading functions are listed in Table 6.1.


Figure 6.3. Beam patterns 10 log10 Bð�Þ for continuous line array of length L=� ¼ 10 with

raised cosine shading and " values as marked.

Additional properties for some of these windows are listed by Harris (1978). Harris(1978) and Nuttall (1981) describe further windows not included in Table 6.1, someof which, such as the Barcilon–Temes, Blackman–Harris, Dolph–Chebyshev, andKaiser–Bessel windows, feature particularly low sidelobe levels.

An unwanted side-effect of shading is a broader main beam than for theunshaded case, as illustrated by Figure 6.2. By tapering the contributions from theedges, the apparent length of the array is reduced, so its angular resolution is alsoreduced. The fwhm of the broadside beam is also included in Table 6.1, relative to itsvalue for an unshaded array. To convert these relative values to absolute beamwidths,they need to be multiplied by the fwhm of an unshaded array, which for the broadsidebeam of a long array is

��fwhm � 2 sinc�1ð1=ffiffiffi2

pÞ

�

�

L: ð6:33Þ

Substituting numerical values gives

��fwhm � 0:8859�

Lrad; ð6:34Þ

or

��fwhm � 50:76�

Ldeg: ð6:35Þ

6.1.1.2 Unsteered planar arrays

6.1.1.2.1 Piston arrays

The beam pattern of an unshaded circular array of diameter D is (Tucker and Gazey,1966, p. 180)

bðuÞ ¼ ½2J1ðuÞ=u 2; ð6:36Þ

where J1ðuÞ is a first-order Bessel function of the first kind (Appendix A), theargument u is

u ¼ ð�D=�Þ sin ð6:37Þ

and is the angle from the circle’s axis of symmetry. The function bðuÞ is plotted inFigure 6.5.

The basic properties of a circular array with Taylor shading are listed in Table6.2. The half-power beamwidth (fwhm) of an unshaded circular array is

�fwhm ¼ 2 arcsin�

�Du0

� �; ð6:38Þ

where u0 is the value of u for which bðuÞ is equal to 12

u0 � 1:614: ð6:39Þ



Figure 6.4. Hamming family shading patterns (left) and beam patterns (right) for continuous

line array with various " as labeled: " ¼ 0:2; " ¼ 0:08; . . .


Figure 6.4 (cont.) . . . " ¼ 0.06 and " ¼ 0.

Table 6.1. Summary of properties for various taper functions.

Window Highest Sidelobe Half-power Shading Main sourcesidelobe fall-off beamwidth factor

level (dB per (relative to FS at(dB) octavea) unshaded) broadside

Rectangle (Dirichlet) �13.3 6.0 1.00 1.00 Harris (1978)

Triangle �26.5 12.0 2.05 0.75 Harris (1978)

Cosine family (cosn)n ¼ 0 �13.3 6.0 1.00 1.00 Dirichlet windown ¼ 1 �23.0 12.0 1.35 0.81 Harris (1978)n ¼ 2 (Hann) �31.5 18.1 1.63 0.67 Harris (1978)n ¼ 3 �39 24.1 1.87 0.58 Harris (1978)n ¼ 4 �47 30.1 2.10 0.52 Harris (1978)

Hamming family" ¼ 1:0 �13.3 6.0 1.00 1.00 Dirichlet window" ¼ 0:3 �26.8 6.0 Figure 6.4" ¼ 0:2 �31.6 6.0 Figure 6.4" ¼ 0:1 (20 dB pedestal) �40.1 6.0 1.45 0.75 Cheston and Frank (1990)" ¼ 0:08 (Hamming) �42.7 6.0 1.47 0.74 Harris (1978)" ¼ 0.076711 �43.19 6.0 Figure 6.3" ¼ 0:04 �36.2 6.0b Figure 6.4" ¼ 0:0 �31.5 18.1 1.63 0.67 Hann window

Tukey windows" ¼ 1.00 �13.3 6.0 1.00 1.00 Dirichlet window" ¼ 0.75 �14 18.1 1.14 0.91 Harris (1978)" ¼ 0.50 �15 18.1 1.30 0.82 Harris (1978)" ¼ 0.25 �19 18.1 1.48 0.74 Harris (1978)" ¼ 0.0 �31.5 18.1 1.63 0.67 Hann window

Riesz �21 12.0 1.31 0.83 Harris (1978)

Riemann �26 12.0 1.42 0.77 Harris (1978)

de la Vallee-Poussin �53 24.1 2.05 0.52 Harris (1978)

Bohman �46 24.1 1.93 0.56 Harris (1978)

Blackman �58.1 18.1 1.90 0.58 Harris (1978), Nuttall (1981)

Taylorn ¼ 3 �26 1.18 0.9 Cheston and Frank (1990)n ¼ 5 �36 1.33 0.8 Cheston and Frank (1990)n ¼ 8 �46 1.47 0.73 Cheston and Frank (1990)

a The terminology ‘‘per octave’’ arises from the alternative use of these windows in the time domain to construct passbandfilters (Harris, 1978). In the present context it means ‘‘per doubling of the argument u’’ as defined in Equation (6.5) orEquation (6.8).b 6 dB/octave is the theoretical fall-off rate of the Hamming family in the limit of large u if " 6¼ 0. However, if " is non-zerobut still small, the large u limit might not be reached, in which case the fall-off rate of practical interest is close to that of theHann window, about 18 dB/octave.

6.1 Processing gain for passive sonar 265

At high frequency (D � �) this can be approximated by

�fwhm � �

D

� �� 58:9 deg; ð6:40Þ

or

�fwhm � �

D

� �� 1:03 rad: ð6:41Þ

The above equations apply to an unsteered 2D circular plate.

Figure 6.5. Beam pattern 10 log10 bðuÞ of unshaded circular array.

Table 6.2. Summary of beam properties for selected shading (circular

arrays) (based on Cheston and Frank, 1990).

Window Highest sidelobe Half-power Shading factor

level beamwidth FS

(dB) (relative to

unshaded array)

Uniform �17.57 1.00 1

Taylor n ¼ 3 �26.2 1.10 0.91

Taylor n ¼ 5 �36.6 1.21 0.77

Taylor n ¼ 8 �45 1.31 0.65

6.1.1.2.2 Rectangular arrays

The beam pattern of a rectangular array whose sides have length L1 and L2 is (Tuckerand Gazey, 1966, p. 177, from Eq. 6.22)

B ¼ sinc2ða1 cos Þ sinc2ða2 sin Þ; ð6:42Þwhere

a1 ¼ �L1 sin

�ð6:43Þ

and

a2 ¼ �L2 sin

�; ð6:44Þ

where and are the polar azimuth and elevation angles, respectively. Specifically,the azimuth is the angle between the axis parallel to the side of length L1, measuredin the plane of the array. The projector axis is normal to the plane of the array. Theangle is measured from this axis, reaching a value of �=2 in the plane of the array.

Special cases of Equation (6.42) include the square array of sides L (i.e.,L1 ¼ L2 ¼ L)

B ¼ sinc2ða cos Þ sinc2ða sin Þ; ð6:45Þwhere

a ¼ �L sin

�ð6:46Þ

and the line array of length L (L1 ¼ L;L2 ¼ 0)

B ¼ sinc2ða cos Þ; ð6:47Þwhere

a ¼ �L sin

�: ð6:48Þ

6.1.2 Directivity index

The directivity index (DI) is a measure of the angular resolution of an array. It can bedefined as

DI � 10 log10 GD; ð6:49Þ

where GD is the directivity factor

GD ¼ 4�

�O; ð6:50Þ

and �O is the solid angle ‘‘footprint’’ of the beam pattern:

�O �ð

4�

BðOÞ dO ¼ð2�

0

d

ðþ�=2��=2

d cos Bð; Þ: ð6:51Þ

The cases of a steered line array (Section 6.1.2.1) and an unsteered planar array(Section 6.1.2.2) are considered next.


6.1.2.1 Steered line array

To find the footprint of a steered line array, consider first the solid angle subtended bya circular ring between angles � and �þ d� from the array axis, which is 2� cos � d�

˘.

The contribution to the footprint is then obtained by multiplying this solid angle bythe beam pattern, so that the total footprint for an unshaded line array is

�O ¼ 2�

ðþ�=2��=2

d� cos �sin2 u

u2; ð6:52Þ

where u is given by Equation (6.5). Changing the integration variable from � to uyields

�O ¼ 2�

L

ðð�L=�Þð1�sin Þ

ð�L=�Þð�1�sin Þdu

sin2 u

u2: ð6:53Þ

It is convenient to introduce the function

�ðxÞ �ðx

0

dusin2 u

u2¼ Sið2xÞ � sin2 x

x; ð6:54Þ

where SiðxÞ is the sine integral function (Appendix A)

SiðxÞ �ðx

0

dusin u

u: ð6:55Þ

It follows that

�O ¼ 4

G0

��G0

2ð1 � sin Þ

� �þ �

�G0

2ð1 þ sin Þ

� � ; ð6:56Þ

where G0 is the high-frequency limit of GD for the broadside beam8

G0 ¼2L

�: ð6:57Þ

At high frequency, the integration limits of Equation (6.53) tend to �1, in whichcase the integral is equal to �. An exception occurs with the endfire case, for whichone of the integration limits is zero and the integral drops to �=2. From this it followsthat in the high-frequency limit the directivity index tends to 10 log10ð4L=�Þ nearendfire, and to 10 log10ð2L=�Þ for all other steer directions. The low-frequency limit isalways 10 log10ð1Þ ¼ 0 dB.

Generally speaking, the smaller the angular footprint (i.e., the larger DI), the lessnoise will enter the beam, and the better the array is likely to perform. Near endfirethe footprint halves in size, thus eliminating half of the noise (and doubling the signal-to-noise ratio) if the noise is isotropic. However, the optimum steer direction of ahorizontal line array is often not close to endfire, partly because the noise field israrely isotropic, but rather has strong peaks in predictable directions. For example,the low-frequency noise field tends to be dominated by contributions from distant


8 Further, G0 is also equal to the high-frequency limit of GD for all steering angles except those

close to endfire.

sources, propagating at angles close to horizontal ( � 0). An HLA has its besthorizontal resolution at broadside and consequently at low frequency the signal-to-noise ratio (SNR) tends to be largest for contacts close to the broadside beam.At high frequency the strongest noise source is likely to be the sea surface immedi-ately above the sonar, resulting in a peak in the noise field from that direction. For anHLA, this would lead to a higher SNR for endfire contacts at high frequency.

The directivity factor increases from 1 for a very short array (L � �) to G0 for along one, and can be written (for the broadside case)

GD ¼ �

2�ð�G0=2ÞG0; ð6:58Þ

where �ðxÞ is defined by Equation (6.54). This function increases monotonically fromzero at x ¼ 0 to �=2 for x ! 1. For small x, �ðxÞ is approximately equal to x.

The directivity index (DI ¼ 10 log10 GD) is plotted in Figure 6.6 as a function ofG0 using Equation (6.58) for G, with Equation (6.54) for �ðxÞ (solid blue line). Thedashed line is an alternative approximation calculated using

�ðxÞ � x

1 þ 2x

�tanh

�

18x

� � : ð6:59Þ


Figure 6.6. Directivity index DI ¼ 10 log10 GD for an unsteered continuous line array vs.

normalized array length 2L=� (¼ G0), evaluated using Equation (6.58) with Equation (6.54)

(solid blue line) or Equation (6.59) (dashed red); the third curve is the high-frequency

approximation GD ¼ G0 (solid cyan).

Also plotted is the high-frequency approximation calculated using G � G0. It isapparent that the high-frequency approximation is in error for 2L=� < 1.5, whereasEquation (6.59) retains reasonable accuracy at all frequencies.

Figure 6.7 shows a graph of the normalized directivity factor GD=G0, plotted (indecibels) vs. steering angle. This normalized gain is close to unity (0 dB) for a long andunshaded array, except when steered close to endfire. Close to the endfire direction(�90 deg), the ratio increases to 2 (i.e., 3 dB) for long arrays, as can be expected fromthe discussion following Equation (6.53).

An unwanted side-effect of shading, caused by broadening of the main beam, is asmall reduction in DI compared with an unshaded array of the same length. Forisotropic noise, this would usually also result in a reduction in the SNR. The benefitof shading is the cancellation of noise peaks that might otherwise enter through asidelobe.

This degradation can be quantified in terms of a shading degradation factor FS,defined as the ratio of the directivity factors with and without shading, that is,

FS �

ð4�

Brect dOð4�

B dO; ð6:60Þ


Figure 6.7. Normalized directivity index vs. steering angle for Hann-shaded and unshaded

arrays of length L=� ¼ 0.5 (——), 5 (– –), 50 (� � � � �), and 500 (– � –). The broadside direction

is at ¼ 0.

so that

DI ¼ DIrect þ 10 log10 FS; ð6:61Þ

where DIrect is the directivity index of an unshaded array steered at the same angle.Table 6.1 lists values of FS for unsteered shaded arrays. The theoretical high-

frequency value for Hann shading at broadside is 2/3 (i.e., �1.8 dB). Figure 6.8 showsthat, at least for arrays longer than 5 wavelengths, departures from this value aresmall except in the immediate vicinity of endfire. There is no new information inFigure 6.8, as each curve is just the ratio between pairs of curves from Figure 6.7. Itspurpose is to illustrate the behavior of the shading degradation on its own, withoutthe complication of the variation with steering angle of the directivity index.

6.1.2.2 Unsteered planar array

The solid angle footprint of a baffled planar array of beam pattern Bð; Þ is

�O ¼ð2�

0

d

ð�=20

d sin Bð; Þ; ð6:62Þ

where is the angle measured from the normal9; and is the azimuth angle about the


Figure 6.8. Shading factor (in decibels) vs. steering angle for Hann-shaded and unshaded arrays

of length L=� ¼ 0.5 (——), 5 (– – –), 50 (� � � � � �), and 500 (– � –). The broadside direction is at

¼ 0.

9 That is, the normal to the plane of the array.

normal. Apart from this change to the definition of , Equation (6.62) is the same asEquation (6.51). For a circular array of diameter D, it becomes

�O ¼ 2�

D

ð�D=�

0

du tan ðuÞbðuÞ; ð6:63Þ

where

sin ðuÞ ¼ �

�Du ð6:64Þ

and bðuÞ is given by Equation (6.36). For a large array (D � �), Equation (6.63)simplifies to

�O � 8�2

�D2

ð10

½J1ðuÞ 2

udu: ð6:65Þ

Using the standard integral (see Appendix A),ð10

J1ðxÞx

� �2

x dx ¼ 1

2; ð6:66Þ

it follows that

GD ¼ 4�

�O� �2D2

�2: ð6:67Þ

More generally, the directivity index of a large baffled planar array of area S (andextending many wavelengths in both dimensions) is given by (Barger, 1997)

DI � 10 log10 4�S

�2

� �: ð6:68Þ

The above expressions are for a baffled array. The significance of the baffling is that itreduces the footprint by a factor of 2 and hence doubles the directivity factor. Theequivalent expression for an unbaffled pulsating plate would therefore be

DIunbaffled � 10 log10 2�S

�2

� �: ð6:69Þ

6.1.3 Array gain

6.1.3.1 Definition

The gain in signal-to-noise ratio achieved by spatial filtering (also known asbeamforming) is called array gain (AG). This term is defined (see Chapter 3) as

AG � 10 log10

Rarr

Rhp

; ð6:70Þ


where Rhp and Rarr are the signal-to-noise ratios before and after beamforming,respectively. Specifically, Rhp is the SNR at the hydrophone

Rhp ¼ QS

QN; ð6:71Þ

where QS and QN denote the mean square pressure of the signal and noise. Similarly,Rarr is the SNR at the output from the beamformer10

Rarr ¼Y S

Y N; ð6:72Þ

where Y S and Y N are the array response to signal and noise.Of all terms in the passive sonar equation, array gain is the hardest to calculate

precisely. The reason for this is apparent from Equation (6.70), which shows that thesignal-to-noise ratio must be calculated not just once, but twice, with and withoutthe effects of the beamformer.11 For each calculation of SNR, all remaining terms ofthe sonar equation except DT are needed. If the array beam pattern is BðOÞ, the signaland noise terms after beamforming (assuming a narrowband signal) are12

Y S ¼ð

QSOBðOÞ dO ð6:73Þ

and

Y N ¼ðð

QNfOðOÞBðOÞ dO df : ð6:74Þ

Similarly, the signal and noise terms before beamforming, the ratio of which gives Rhp

in Equation (6.71), are

QS ¼ð

QSO dO ð6:75Þ

and

QN ¼ðð

QNfOðOÞ dO df : ð6:76Þ

Because of the complications associated with using AG, it is common to approximatethis parameter by making simplifying assumptions about the directionality of signaland noise. Specifically, the signal tends to come from a single predominant direction,whereas the noise tends to come from all around. In the limit of a plane wave signaland isotropic noise, the AG becomes equal to the directivity index (DI), which is thesubject of Section 6.1.2. The shape of the beam pattern, described in Section 6.1.1, isneeded for calculations of both AG and DI. However, whereas detailed knowledge of


10 Strictly speaking, there is not one output from a beamformer, but several, one for each beam

(i.e., one for each steering angle) and each with a different array gain. It is presumed that one of

the beams is steered in the direction of the target and hence has a higher SNR than the others.

By convention it is the beam with the highest SNR that determines the array gain.11 Thus, AG is not only a property of the sonar, but also a complicated function of the

propagation conditions, noise sources, and the target.12 Broadband signals are considered in Section 6.1.4.

the sidelobe levels is important for AG, for DI it is often sufficient to parameterize thebeam pattern in terms of the footprint of the main beam alone.

It is convenient to define GA as 10AG=10, with GA written as a ratio of signal gainGS to noise gain GN:

GA ¼ GS=GN; ð6:77Þwhere

GS ¼

ðQS

OBðOÞ dO

QSð6:78Þ

and

GN ¼

ðQN

OBðOÞ dO

QN: ð6:79Þ

Both GS and GN are less than 1. The hope (and expectation) is that the value of GS

exceeds that of GN, such that a net gain results overall. Four special cases areconsidered below.

6.1.3.2 Special cases (noise gain for horizontal line array)

The noise gain of an array (and hence also the total array gain) depends on theanisotropy of the noise field. This dependence is examined here by considering thenoise gain for four different noise fields (including the isotropic case).

For each case, GN is calculated. If the signal gain is known, the array gainfollows using Equation (6.77). For example, in the simplest case the signal can beapproximated as a plane wave, meaning that GS � 1, and then

GA � 1=GN: ð6:80Þ

6.1.3.2.1 Noise gain for isotropic noise

The first case considered is the trivial one involving no anisotropy. For the broadsidebeam of a long line array it can be shown (see Chapter 3) that

GN ¼ �=2L: ð6:81Þ

This result is almost independent of steering angle except close to endfire, for whichthe noise gain halves (causing the array gain to double) to (see Section 6.1.2.1)

GN ¼ �=4L: ð6:82ÞIn general,

GN ¼ 1

GD

¼ �O4�

; ð6:83Þ

where �O is given by Equation (6.56).These results are not limited to a horizontal line array, but apply to a line array of

any orientation. Further, if the signal is a plane wave, the array gain and directivityindex are identical for the case of isotropic noise.


6.1.3.2.2 Noise gain for horizontal isotropic noise

If all of the incoming noise is restricted to the horizontal plane, but is independent ofthe azimuth angle, the angular distribution of the noise field can be written

QNO ð; Þ ¼

QN

4��ðÞ: ð6:84Þ

Therefore, the noise gain is

GN ¼

ðQN

4��ðÞBðOÞ cos d d

QN: ð6:85Þ

Carrying out the integral first, Equation (6.85) becomes

GN ¼ D2�

; ð6:86Þwhere

D �ð2�

0

Bð0; Þ d: ð6:87Þ

This integral is a measure of the width of the beam pattern in the horizontal plane. Itcan be written

D ¼ð2�

0

sin2 u

u2d; ð6:88Þ

where

u ¼ kL

2ðsin � sin Þ: ð6:89Þ

Following Chapter 3, Equation (6.88) can be approximated by13

D � 2

ðþ�=2��=2

Pðu=�Þ d; ð6:90Þ

and hence

D � 2

ðþ�

d ¼ 2ðþ � �Þ; ð6:91Þ

where

� ¼ �arcsin min 1;�

kL� sin

� �: ð6:92Þ

It follows that

D2

� arcsin min 1;�

kLþ sin

� �þ arcsin min 1;

�

kL� sin

� �: ð6:93Þ

For a short array (kL < �), this approximation gives the correct limiting value of

D � 2� ð6:94Þ


13 See Appendix A: ðþ1

�1

sin u

u

� �2

du ¼ �:

and hence (using Equation 6.86)

GN � 1: ð6:95Þ

For a long array, the behavior depends on the proximity of to the endfire direction(��=2). If j j is small (not close to endfire)

D ¼ 2 arcsin sin þ �

kL

� �� 2 arcsin sin � �

kL

� �: ð6:96Þ

At high frequency (�=kL � 1), and using the result (valid for small ")

arcsinðx þ "Þ � arcsinðx � "Þ � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � x2

p "; ð6:97Þ

Equation (6.96) becomes

D2

� 2�

kL cos ; ð6:98Þ

and hence (away from endfire)

GN � 2

kL cos : ð6:99Þ

Thus, in horizontal isotropic noise, AG exceeds DI (Equation 6.57) by 2 dB(¼ 10 log10ð�=2Þ) in the broadside beam.

The width of the endfire beam can be found using Equation (6.96) with ¼ �=2:

D2

¼ �

2� arcsin 1 � �

kL

� �; ð6:100Þ

which simplifies for large kL to

D2

�ffiffiffiffiffiffi2�

kL

r: ð6:101Þ

The noise gain at endfire is therefore

ðGNÞef ¼ffiffiffiffiffiffiffiffiffi2

�kL

r: ð6:102Þ

6.1.3.2.3 Noise gain for a uniform sheet of dipole noise sources

The case of a uniform sheet of dipoles is considered next. This situation wasconsidered in Chapter 3 for the broadside beam. The analysis is now extended forarbitrary steering angle.

The noise gain for a horizontal line array is given by Equation (6.79) with (if thehydrophone spacing is small compared with the acoustic wavelength)

BðOÞ ¼ sin2 u

u2ð6:103Þ


and

uð; Þ ¼ �L

�ðcos sin � sin Þ: ð6:104Þ

Given that, for a dipole (see Chapter 2 for details), QNO is proportional to sin , and

the solid angle is proportional to cos according to

dO ¼ cos d d; ð6:105Þit follows that

GN ¼

ð�=20

d

ð2�

0

dsin u

u

� �2

sin cos ð�=20

d

ð2�

0

d sin cos

: ð6:106Þ

The denominator of Equation (6.106) is �. If the numerator is denoted N, the noisegain can be written

GN ¼ N

�; ð6:107Þ

where

N ¼ð�=2

0

DðÞ sin cos d ð6:108Þ

and

DðÞ �ð2�

0

sin u

u

� �2

d: ð6:109Þ

If D is independent of , Equation (6.107) simplifies to

GN ¼ D2�

: ð6:110Þ

More generally, following Section 6.1.3.2.2, Equation (6.109) can be approximatedby

D � 2

ðþ�=2��=2

Pu

�

� �d; ð6:111Þ

and hence

D � 2

ðþ�

d ¼ 2ðþ � �Þ; ð6:112Þ

where

sin � ¼1 �� > 1

�� 1 < �� < 1

�1 �� < �1

8<: ; ð6:113Þ

and

�� ¼ ��=kL þ sin

cos : ð6:114Þ

For sufficiently large kL, and provided also that the elevation angle and steering


angle are not too large,14 such that �� is between �1 and þ1,

D2

� arcsin�=kL þ sin

cos þ arcsin

�=kL � sin

cos : ð6:115Þ

Using Equation (6.97), Equation (6.115) becomes

D2

� 2�

kLðcos2 � sin2 Þ�1=2: ð6:116Þ

Substituting this approximation into Equation (6.108) and integrating over realvalues of the resulting integrand gives for the numerator of Equation (6.107)

N � 4�

kL

ðarccosðsin Þ

0

ðcos2 � sin2 Þ�1=2 sin cos d: ð6:117Þ

The solution to the indefinite integral isððcos2 � sin2 Þ�1=2 sin cos d ¼ �ðcos2 � sin2 Þ1=2 ð6:118Þ

and hence

GN � 4

kLcos : ð6:119Þ

At broadside this expression predicts twice the gain of the corresponding case forhorizontal noise (Equation 6.99). Further, the noise gain decreases towards endfireaccording to Equation (6.119), whereas for horizontal noise it increases. Both differ-ences can be understood qualitatively by realizing that the noise from a dipole sheetof noise sources originates mainly from overhead, a direction to which the array issensitive at broadside and not at endfire.

For the endfire case ( ¼ �=2), Equation (6.113) gives

þ ¼ �

2ð6:120Þ

and

� ¼ arcsin min 1;1 � �=kL

cos

� �: ð6:121Þ

It follows that

cosD2

� min 1;1 � �=kL

cos

� �; ð6:122Þ

or (expanding for small D and small �=kL)

D2

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimax 0;

2�=kL � sin2

cos2

� �s: ð6:123Þ


14 Neither may approach �=2.


GN ¼ 2

�

ðarcsinffiffiffiffiffiffiffiffiffiffi2�=kL

p

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�=kL � sin2

qsin d: ð6:124Þ

If kL is large, is small in the range of integration, and hence

GN � 2

3�

�

L

� �3=2

: ð6:125Þ

6.1.3.2.4 Noise gain for multiple point sources of noise

The noise is sometimes best characterized as a sum of one or more incoming planewaves, each contributing Ni to the differential spectral density from a specific direc-tion represented by Oi, such that

QNO ¼ N1 �ðO1Þ þ N2 �ðO2Þ þ � � �

¼X

i

Ni �ðOiÞð6:126Þ

and hence ðQN

OBðOÞ dO ¼X

i

NiBðOiÞ: ð6:127Þ

It then follows from Equation (6.79) that

GN ¼

Xi

NiBðOiÞXi

Ni

: ð6:128Þ

6.1.4 BB application

The definition of beam pattern results in a function of angle for a specified frequency,so the concept is a narrowband one. Nevertheless, the definition of AG (and hence ofDI) in terms of the ratio of two SNR values applies more generally, as follows.Substituting for Rhp and Rarr in Equation (6.70), using

Rhp ¼

ðQS

f ð f Þ dfðQN

f ð f Þ df

ð6:129Þ

and

Rarr ¼

ðY S

f ð f Þ dfðY N

f ð f Þ df

; ð6:130Þ


gives

AG � 10 log10

ðY S

f ð f Þ dfðY N

f ð f Þ df

ðQN

f ð f Þ dfðQS

f ð f Þ df

0BB@

1CCA: ð6:131Þ

At each frequency f , it is understood that Yð f Þ is calculated using the appropriatebeam pattern at that frequency.

6.1.5 Time domain processing

In Chapter 3, two types of time domain processing were considered for passive sonar.These were coherent averaging, resulting in the narrowband passive sonar equation,and incoherent averaging, resulting in the broadband equivalent. These two types ofpassive sonar processing are discussed briefly below. An alternative type of coherentprocessing, beyond the present scope, involves (for example) the correlation of thereceived waveform with itself to exploit shape information contained in one part ofthe signal to enhance the SNR in another part.

6.1.5.1 Coherent averaging

Coherent averaging (or coherent integration) over time has the effect of reinforcing astable signal relative to random noise. In the frequency domain the effect of coherentaveraging over a time duration T is that of a filter whose bandwidth is 1=T . Thisresults in a gain in SNR represented by the bandwidth term (BW) in the sonarequation (see Chapter 3).

The gain in SNR is contingent on both the tonal bandwidth and any fluctuationsin tonal frequency being small compared with the processing bandwidth 1=T . Anadditional benefit of NB processing arises because a high resolution in frequencymakes it possible to characterize a signal in terms of a sequence of tonals. Thisprovides an acoustic signature that can be used to aid the classification process.

6.1.5.2 Incoherent averaging

Incoherent averaging (or incoherent integration) does not alter the SNR, but insteaddecreases the fluctuations in both signal and noise. This makes a sonar operator lesslikely to mistake a noise fluctuation as signal and hence permits detection at a lowerSNR. Thus, the gain is achieved not by increasing SNR but by decreasing the SNRthreshold required for detection (known as the detection threshold ). Details aredeferred to Chapter 7.

6.2 PROCESSING GAIN FOR ACTIVE SONAR

Consider a generic active sonar system whose processing chain consists of abeamformer followed by a time domain filter. For such a sonar, the processing gain

6.2 Processing gain for active sonar 279]Sec. 6.2

is the sum in decibels of the array gain (AG) and the gain of this filter. Of particularinterest is a special kind of filter known as a matched filter, described further inSections 6.2.1 to 6.2.6. Compared with passive sonar, calculation of AG, consideredtogether with the total processing gain in Section 6.2.7, is complicated here by thepresence of reverberation.

6.2.1 Signal carrier and envelope

Active sonar signals are usually designed as relatively small perturbations, inamplitude or phase, to a well-defined sinusoidal wave form known as the carrierwave. Denoting the carrier frequency !0, the total signal (carrier plus modulation) ofsuch a signal can be written

xðtÞ ¼ AðtÞ cos½!0t þ FðtÞ : ð6:132Þ

The functions AðtÞ and FðtÞ vary in complexity depending on the task to be carriedout by the sonar. For simplicity they are considered initially to vary slowly bycomparison with cos !0t and !0t, respectively. For a simple CW sonar, the phaseterm would be constant and the amplitude would vary in a simple manner like (say) aGaussian or rectangle function. A more complex variation (modulation) is needed inorder to carry out the transmission of underwater messages. In general, the twofunctions can be chosen by the designer to optimize either the processing gain(i.e., to maximize the detection probability) or the information content of an echofrom an underwater target (e.g., to maximize resolution in range or frequency). Forapplications concerning the acoustic transmission of a message, the modulation is acoded representation of that message.

The significance of the amplitude AðtÞ and phase FðtÞ is described below, firstintuitively (Section 6.2.1.1) and then formally (Section 6.2.1.2). The formal derivationfollows that of Burdic (1984, pp. 197–198).

6.2.1.1 Intuitive concept

Consider some real signal xðtÞ and write it as the real part of a complex one zðtÞxðtÞ ¼ Re zðtÞ; ð6:133Þ

with the imaginary part to be determined. For example, if xðtÞ is a sinusoidal functionof the form

xðtÞ ¼ A cosð!0t þ FÞ; ð6:134Þ

with A and F both real constants, for zðtÞ one intuitively thinks of a complexexponential

zðtÞ ¼ A expð2�if0t þ iFÞ; ð6:135Þwhere

f0 ¼!0

2�: ð6:136Þ

Now consider the more general case of Equation (6.132) (i.e., with either or both of A


and F varying with time). A heuristic generalization of Equation (6.135), if the timevariation is slow, is

zðtÞ ¼ AðtÞ exp½2�if0t þ iFðtÞ : ð6:137Þ

6.2.1.2 Formal methodology: analytic signals and the Hilbert transform

Now consider the spectrum Xð f Þ of an arbitrary real signal xðtÞXð f Þ ¼ I½xðtÞ ; ð6:138Þ

where the operator I½xðtÞ indicates the Fourier transform of the function xðtÞ (seeAppendix A). The result is a complex spectrum containing both positive and negativefrequencies. By comparison, the Fourier transform of zðtÞ (Equation 6.137) isconcentrated around the positive frequency f0, with negligible contributions fromnegative frequencies.

In order to construct a spectrum similar to the one obtained intuitively, it isnecessary to remove the negative frequencies. This is achieved by zeroing the negativepart of the spectrum and doubling the positive part,15 that is,

Yð f Þ ¼ 2Hð f ÞXð f Þ; ð6:139Þ

where Hð f Þ is the Heaviside step function (Appendix A). In the time domain thisbecomes

yðtÞ ¼ I�1½2Hð f ÞXð f Þ ; ð6:140Þ

or, equivalently, the Fourier transform product rule (see Appendix A) gives

yðtÞ ¼ 2I�1½Hð f Þ � I�1½Xð f Þ ; ð6:141Þ

where the symbol � denotes a convolution operation.Using the result

2I�1½Hð f Þ ¼ �ðtÞ þ i

�t; ð6:142Þ

it follows that

yðtÞ ¼ I�1½2Hð f ÞXð f Þ ¼ �ðtÞ þ i

�t

� �� xðtÞ ¼ xðtÞ þ i

�

ðþ1

�1

xð�Þt � �

d�: ð6:143Þ

In this equation yðtÞ is known as the analytic signal, the imaginary part of which is theHilbert transform of xðtÞ, denoted xxðtÞ. Thus,

yðtÞ ¼ xðtÞ þ ixxðtÞ; ð6:144Þ

where xxðtÞ is the integral16

xxðtÞ ¼ 1

�

ðþ1

�1

xð�Þt � �

d�: ð6:145Þ


15 This operation removes the term expð�i!tÞ and transforms expðþi!tÞ into 2 expðþi!tÞ. In

this way, cosð!tÞ is converted into expðþi!tÞ.16 Cauchy principal value.

With this prescription, the cosine function transforms into a sine:

1

�

ðþ1

�1

cosð!0�Þt � �

d� ¼ sinð!0tÞ ð6:146Þ

and hence, for a real signal cosð!0tÞ, the analytic signal is

yðtÞ ¼ cosð!0tÞ þ i sinð!0tÞ ¼ expði!0tÞ; ð6:147Þ

precisely as required. The factor 2 introduced on the right-hand side of Equation(6.139) ensures that the sine wave in the imaginary part (of Equation 6.147) has thecorrect amplitude.

It is convenient to write the analytic signal in the following exponential form:

yðtÞ ¼ �ðtÞ expði!0tÞ; ð6:148Þ

where �ðtÞ is known as the envelope function

�ðtÞ ¼ aðtÞ exp½iðtÞ : ð6:149Þ

The amplitude and phase terms, both real, are then recovered as

aðtÞ ¼ ½x2ðtÞ þ xx2ðtÞ 1=2 ð6:150Þand

ðtÞ ¼ arctanxxðtÞxðtÞ � !0t: ð6:151Þ

This procedure provides a formal recipe for generating aðtÞ and ðtÞ unambiguously.Recall that xðtÞ is an arbitrary function of time, so there is no longer a restriction onamplitude and phase to vary slowly. In Burdic’s words ‘‘If �ðtÞ is a narrow-bandfunction, relative to f0, it will have properties we intuitively associate with anenvelope. Otherwise, it is simply a convenient mathematical representation.’’

The analytic signal contains twice the energy of the original real signal. This canbe seen from Equation (6.139)ðþ1

�1jYð f Þj2 df ¼ 4

ðþ1

0

jXð f Þj2 df ð6:152Þ

and hence ðþ1

�1jYð f Þj2 df ¼ 2

ðþ1

�1jXð f Þj2 df : ð6:153Þ

6.2.2 Simple envelopes and their spectra

Some examples of simple signal envelopes �ðtÞ are given in Tables 6.4 (envelopeamplitude) and 6.5 (phase). The effective pulse duration (Burdic, 1984) is

�eff �

ðþ1

�1j�ðtÞj2 dt

� �2

ðþ1

�1j�ðtÞj4 dt

: ð6:154Þ


The spectra of these pulses, calculated using

Mð f Þ ¼ I½�ðtÞ ; ð6:155Þ

are listed in Table 6.3. The effective bandwidth is defined in a similar way as effectivepulse duration (Burdic, 1984, p. 229)

�eff �

ðþ1

�1jMð f Þj2 df

� �2

ðþ1

�1jMð f Þj4 df

: ð6:156Þ

The normalization convention adopted for envelopes and spectra isðþ1

�1j�ðtÞj2 dt ¼

ðþ1

�1jMð f Þj2 df ¼ 1: ð6:157Þ

Using this normalization, Equations (6.154) and (6.156) simplify to

�eff ¼1ðþ1

�1j�ðtÞj4 dt

ð6:158Þ

and

�eff ¼1ðþ1

�1jMð f Þj4 df

: ð6:159Þ

The product �eff�eff is included in Table 6.3. This product is closely related to the gaindue to time domain processing (see Section 6.2.5).

The instantaneous angular frequency (in radians per unit time) can be defined asthe rate of change of phase . Dividing this quantity by 2� gives the instantaneousfrequency finst in cycles per unit time. Thus,

finst �1

2�

d

dtðtÞ: ð6:160Þ

The parameter �ðtÞ is related in a similar way to the phase acceleration

�ðtÞ � 1

2�

d2

dt2ðtÞ; ð6:161Þ

and is referred to henceforth as the ‘‘frequency rate’’. The properties of two kinds ofsimple amplitude modulation, Gaussian and rectangular, are listed in Table 6.4 andthose of three kinds of phase modulation in Table 6.5. Together, they provide the sixcombinations of Table 6.3. The first of the three phase modulations (CW or con-tinuous wave) is a trivial one, with no modulation. The other two are linear frequencymodulation (LFM) and hyperbolic frequency modulation (HFM);17 these are so called


17 An alternative name is linear period modulation (LPM), so called because the instantaneous

period TinstðtÞ �1

f0 þ finstðtÞvaries linearly with time t: TinstðtÞ ¼

1 � ð�0=f0Þtf0

.


Table

6.3.

Sum

mary

offr

equen

cydom

ain

pro

per

ties

of

sim

ple

pulse

envel

opes

.

Am

plitu

de

Phase

Spec

trum

Mðf

Þ� e

ff� e

ff

CW

21=4�

1=2

eff

expð��

f2�

2 effÞ

1

Gauss

ian

LF

M2

1=4�

1=2

eff

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffi1�

i�0�

2 eff

qex

p�

�f

2�

2 eff

1�

i�0�

2 eff

��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffi1þ�

2 0�

4 eff

q

HF

Ma

�f 0

fþ

f 0

af

fþ

f 0

f 0 �0

��

ffiffiffiffiffiffiffi j�0j

pex

p2�if 0 �0

f 0lo

ge

fþ

f 0

f 0�

f

��

�� ex

p�i 4sg

n�

0

��

�j�

0j�

2 eff

ðj�0j� e

ff�

f 0Þ

CW

T1=2sincð�

fTÞ

3/2

Rec

tangula

rL

FM

�ffiffiffiffiffiffiffiffi

ffiffiffi1

j�0jT

sex

p�

i�f

2

�0

�� ex

p�i 4sg

n�

0

�� P

f

�0T

��

�j�

0jT

2

This

appro

xim

ation

isvalid

inth

elim

itof

ala

rge

tim

e-bandw

idth

pro

duct

ðj�0jT

2�

1Þ.

The

exact

spec

trum

isgiv

enby

Equation

(6.1

78).

HF

Ma

�f 0

fþ

f 0

ffiffiffiffiffiffiffiffiffiffiffi

1

j�0jT

sex

p2�if 0 �0

f 0lo

ge

fþ

f 0

f 0�

f

��

�� ex

p�i 4sg

n�

0

�� P

f�

f c

B

��

�j�

0jT

2

1þ�

2 0T

2

12f

2 0T

his

expre

ssio

nis

valid

inth

elim

itof

ala

rge

tim

e-bandw

idth

pro

duct

ðj�0jT

2�

1Þ.

The

cente

rfr

equen

cyf c

and

spre

ad

Bare

defi

ned

inSec

tion

6.2

.2.3

.A

nim

pro

ved

appro

xim

ation

isgiv

enby

Equation

(6.2

16).

For

an

exact

solu

tion

see

Kro

szcz

ynsk

i

(1969).

aSta

tionary

phase

appro

xim

ation.

because the instantaneous frequency varies, respectively, linearly and hyperbolicallywith time. Both LFM and HFM modulations include CW as a special case with�0 ¼ 0, where �0 is the frequency rate evaluated at the time origin, t ¼ 0. The symbolT is used to denote the total duration of the pulse. Thus, T is the elapsed time duringwhich the envelope function is non-zero. For a rectangular envelope, T and �eff areidentical.

The instantaneous frequency for an HFM pulse is singular at time f0=�0,imposing a maximum pulse duration of 2f0=j�0j. In combination with a Gaussianenvelope, the more stringent requirement �eff � f0=j�0j must be satisfied.

It is useful to introduce the concept of ‘‘frequency spread’’ B, defined as thedifference between the largest and smallest value of the instantaneous frequencyduring effective pulse duration. Assuming a monotonic variation in finst this gives

B ¼ finst þ �eff2

� �� finst � �eff

2

� �� : ð6:162Þ

The rectangular envelope (see Table 6.4) is proportional to the factor Pðt=TÞ, wherePðxÞ is the rectangle function, equal to unity if jxj < 1

2, and zero otherwise (Appendix

A). This factor is therefore equal to 1 for t between �T=2 and þT=2 and 0 at all other


Table 6.4. Summary of time domain properties

of simple pulse shapes (envelope).

Description Envelope amplitude aðtÞ

Gaussiana 21=4

�1=2eff

exp �� t2

� 2eff

� �

Rectangular T�1=2Pðt=TÞa The parameter �eff is related to � used by Burdic(1984) through the equation � ¼ �eff=

ffiffiffiffiffiffi2�

p.

Table 6.5. Summary of time domain properties of simple pulse shapes (phase).

Description Phase ðtÞ Instantaneous frequency Frequency rate

finstðtÞ �ðtÞ(Equation 6.160) (Equation 6.161)

CW 0 0 0

LFMa ��0t2 �0t �0

HFM �2�f0f0

�0loge 1 � �0

f0t

� �þ t

� �f0

1 � �0

f0t� f0

�0

1 � �0

f0t

� �2

a The parameter �0 is related to k used by Burdic (1984) through the equation k ¼ ��0.

times. For example, the envelope function for the LFM pulse can be written (seeEquation 6.149)

�ðtÞ ¼ T�1=2 expði��0t2Þ jtj < T=2

0 jtj > T=2

(: ð6:163Þ

The FM pulses are further parameterized by �0, which is the frequency rate at timet ¼ 0. The LFM pulse has a constant frequency rate so that �ðtÞ is equal to �0 at alltimes. For fixed �0, the phase of the HFM pulse depends weakly on the carrierfrequency.

The bandwidth and duration of a pulse are related by a form of uncertaintyprinciple that states that their product must be of order unity or greater. A moreprecise version of this statement can be made in terms of the variance in time andfrequency

ð�tÞ2 ¼ð

t2j�ðtÞj2 dt ð6:164Þ

ð�f Þ2 ¼ð

f 2jMð f Þj2 df : ð6:165Þ

In terms of these parameters, the uncertainty principle is (Woodward, 1964)

�f �t � 1

4�; ð6:166Þ

where equality is achieved with a Gaussian CW pulse, for which

ð�tÞ2 ¼ � 2eff

4�ð6:167Þ

and

ð�f Þ2 ¼ 1

4�� 2eff

: ð6:168Þ

Equations (6.167) and (6.168) follow from the definite integral (see Appendix A)ð1�1

x2 expð�Ax2Þ dx ¼ffiffiffi�

p

2A3=2: ð6:169Þ

6.2.2.1 CW spectra

Spectra for the CW pulse take particularly simple forms, as shown in Table 6.3. Theyfollow immediately from the Fourier transforms of the Gaussian and rectanglefunctions (Appendix A).

Making use of the standard integrals (see Appendix A)ðþ1

�1expð�u2Þ du ¼

ffiffiffi�

pð6:170Þ

and ðþ1

�1

sin u

u

� �4

du ¼ 2�

3; ð6:171Þ


the effective bandwidths (Equation 6.159) for the Gaussian and rectangular envelopesare

�eff ¼1=�eff Gaussian

3=ð2TÞ rectangular

: ð6:172Þ

For the rectangular envelope, Equation (6.172) can be compared with the 3 dB widthfor this window (from Chapter 2)

�ffwhm � 0:886

T; ð6:173Þ

so for this case the effective bandwidth exceeds the 3 dB width by about 70%.For a CW pulse, instantaneous frequency is not a useful concept. Its value is zero

throughout the duration of the pulse and hence the frequency spread as defined byEquation (6.162) is also zero.

6.2.2.2 LFM spectra

The frequency spread of an LFM pulse is

B ¼ j�0j�eff : ð6:174Þ

6.2.2.2.1 Gaussian envelope

The LFM spectrum for a Gaussian envelope can be derived in the same way as for theCW spectrum (see Table 6.3). It is convenient to define a dimensionless spectralamplitude as

Sð f Þ � Mð f ÞMð0Þ

��: ð6:175Þ

The squared modulus of Mð f Þ is known as the power spectrum.18 The dimensionlesspower spectrum, Sð f Þ2, of a Gaussian LFM pulse is plotted in Figure 6.9.

For a broadband pulse, the frequency spread B is closely related to the effectivebandwidth �eff . Specifically, for a Gaussian LFM pulse the relationship can bequantified by writing �eff from Table 6.3 in the form

�eff ¼ B 1 þ 1

B2� 2eff

� �1=2

: ð6:176Þ

Thus, if the product B�eff is sufficiently large, B and �eff are approximately equal.Equation (6.176) can also be written in a way that is more appropriate for small B�eff ,

�eff ¼1

�effð1 þ B2� 2

effÞ1=2; ð6:177Þ

demonstrating that in this limit the bandwidth is equal to the reciprocal of the pulseduration, as for a CW pulse.


18 Alternative terms are autospectral density and power-spectral density.

6.2.2.2.2 Rectangular envelope

For a rectangular envelope the spectrum can be written

Mð f Þ ¼ffiffiffiffiffiffiffiffiffiffiffi

1

j�0jT

sexp ��i f 2

�0

!Pð f Þ; ð6:178Þ

where Pð f Þ is defined as

Pð f Þ � 1ffiffiffi2

pðuþ

u�

exp si�

2x2

� �dx ¼ Esðu�; uþÞ; ð6:179Þ

with

u� ¼ffiffiffiffiffiffiffij�0j2

r� 2f

�0

� T

� �ð6:180Þ

ands ¼ sgn �0: ð6:181Þ

The function Esð�; �Þ is given by the following linear combination of Fresnelintegrals CðxÞ and SðxÞ (see Appendix A)

Esð�; �Þ ¼Cð�Þ � Cð�Þ þ si½Sð�Þ � Sð�Þ ffiffiffi

2p : ð6:182Þ

The resulting power spectrum is plotted in Figure 6.10.


Figure 6.9. Power spectrum 10 log10½Sð f Þ2 for a Gaussian LFM pulse.

The behavior of the function Pð f Þ is that of a low-pass filter, removing thosefrequencies whose magnitude exceeds j�0jT=2. For large BT (j�0jT 2 � 1), it can beapproximated using

Pð f Þ � exp si�

4

� �P

f

�0T

� �; ð6:183Þ

giving the spectrum quoted in Table 6.3.Using Equation (6.178) the effective bandwidth is

�eff ¼�2

0T 2ðþ1

�1jPj2 df

; ð6:184Þ

which can be approximated by replacing jPð f Þj with the rectangle function to give

�eff � j�0jT : ð6:185Þ

6.2.2.2.3 Method of stationary phase

Sometimes the Fourier transform operations between time and frequency representa-tions require the calculation of integrals that cannot be evaluated analytically withoutapproximation. A powerful approximate method is described below.


Figure 6.10. Power spectrum 10 log10½Sð f Þ2 for a rectangular LFM pulse.

Consider a pulse whose analytic function is �ðtÞ and that has well-defined startand end times �T=2 and þT=2. It is convenient to introduce a related function��ðtÞ ¼ aaðtÞ eiðtÞ that is identical to �ðtÞ for the duration of the pulse but continuousthrough �T=2 and beyond, such that

�ðtÞ ¼ ��ðtÞPðt=TÞ ð6:186Þand

Mð f Þ ¼ðþT=2

�T=2

aaðtÞ eiðtÞ dt; ð6:187Þ

whereðtÞ ¼ ðtÞ � !t: ð6:188Þ

Assuming that the amplitude aðtÞ (and hence also aaðtÞ) is a slowly varying function oftime in the sense that its value is essentially unchanged during a complete cycle of theexponential term, this integral may be evaluated approximately using the method ofstationary phase (see Appendix A). The result is

Mð f Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2�

j00ðtoÞj

saaðtoÞ eiðtoÞPð f Þ; ð6:189Þ

where to is the instant when the phase is stationary with respect to time, such that

0ðtoÞ � ! ¼ 0: ð6:190Þ

The function Pð f Þ is given by Equation (6.179), with

u� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij 00ðtoÞj

�

r�T

2� to

� �ð6:191Þ

ands ¼ sgn 00ðtoÞ: ð6:192Þ

The instant of stationary phase to varies with frequency, and hence so also do u�and s.

Equation (6.189) is completely general and may be used for any slowly varyingpulse. For an LFM pulse, the phase term in Equation (6.187) is

LFM ¼ ��0t2 � !t: ð6:193Þ

Differentiating Equation (6.193) and setting the result to zero yields

toð f Þ ¼ f

�0

: ð6:194Þ

Using Equation (6.189) for the spectrum it follows from Equation (6.175) that

SLFMð f Þ ¼ aa½toð f Þ a0

Pð f ÞPð0Þ

��; ð6:195Þ

wherea0 � að0Þ ¼ a½toð0Þ : ð6:196Þ

If the bandwidth is sufficiently large, the function jPð f Þj may be approximated by a


rectangle of width j�0jT and centered on f ¼ 0. Substituting for to from Equation(6.194) then gives

SLFMð f Þ � að f =�0Þa0

: ð6:197Þ

A desired spectrum Sð f Þ can be synthesized by rearranging Equation (6.197) in theform

aLFMðtÞ ¼ a0Sð�0tÞ ðjtj < T=2Þ: ð6:198ÞThe result for various spectra is listed in Table 6.6 in the column headed ‘‘LFM’’.(The column headed ‘‘HFM’’ is discussed in Section 6.2.2.3.3).

6.2.2.3 HFM spectra

The spectrum associated with HFM modulation provides a more challenging integralthan the LFM case, with a formal solution in terms of the incomplete gammafunction (Kroszczynski, 1969). Although exact, the complexity of this analyticalsolution complicates its interpretation. An approximation is applied below thatdescribes the essential behavior of the HFM spectrum, without losing the relativesimplicity of the LFM case. Specifically, Equation (6.189) is used to derive thestationary phase result for an HFM pulse. The first and second derivatives of theHFM phase (from Table 6.5) are

0ðtÞ ¼ 2��0t

1 � �0t=f0ð6:199Þ

and

00ðtÞ ¼ 2��0

ð1 � �0t=f0Þ2: ð6:200Þ

Notice the singularity in the instantaneous frequency ( 0=2�) at time f0=�0. The needto avoid this singularity places an upper limit on the pulse duration of

T <2f0j�0j

: ð6:201Þ


Table 6.6. Summary of amplitude envelopes required to synthesize simple power spectra.

Desired spectrum Required amplitude envelope aðtÞ=a0

Sð f Þ LFM HFM

Pf

Df

� �P

�0t

Df

� �1

1 � �0t=f0P

�0t=Df

1 � �0t=f0

� �

exp � f

Df

� �2

� �exp � �0t

Df

� �2

� �1

1 � �0t=f0exp � �0t=f0

1 � �0t=f0

� �2� �

cos2 �f

Df

� �P

f

Df

� �cos2 �

�0t

Df

� �P

�0t

Df

� �1

1 � �0t=f0cos2 �

�0t=Df

1 � �0t=f0

� �P

�0t=Df

1 � �0t=f0

� �

The instant of stationary phase is

to ¼ f

f þ f0

f0�0

; ð6:202Þ

from which it follows that

ðtoÞ ¼2�f 2

0

�0

loge

f þ f0f0

� f

f þ f0

� �ð6:203Þ

and

00ðtoÞ ¼2��0

ð1 � �0to=f0Þ2¼ 2��0

f þ f0f0

� �2

: ð6:204Þ

The frequency spread is the change in instantaneous frequency during the timeinterval defined by

� �eff2< t < þ �eff

2: ð6:205Þ

For an HFM pulse, the instantaneous frequency at times ��eff=2 is

f� ¼ �f0�

1 � �; ð6:206Þ

where

� ¼ �0�eff2f0

: ð6:207Þ

Therefore the change in frequency in this time interval (assuming �eff to be less than2f0=j�0j) is

fþ � f� ¼ 2f0�

1 � �2; ð6:208Þ

the magnitude of which is the frequency spread

B ¼ 2f0j�j

1 � �2: ð6:209Þ

The spectrum is19

Mð f Þ � f0f þ f0

ffiffiffiffiffiffiffi1

j�0j

saðtoÞ eioð f ÞPð f Þ; ð6:210Þ

where oð f Þ is defined as the phase at the moment of stationary phase

oð f Þ � ðtoÞ; ð6:211Þsuch that

oð f Þ ¼ 2�f0�0

f0 loge

f þ f0f0

� f

� �: ð6:212Þ


19 A similar result, except with jPð f Þj approximated by a rectangle function, is derived by

Kroszczynski (1969, Eq. (32a)).

The function Pð f Þ is given by Equation (6.179), with

u� ¼ffiffiffiffiffiffiffiffiffiffi2j�0j

p� f þ f0

f0

T

2� f

�0

� �ð6:213Þ

ands ¼ sgn �0: ð6:214Þ

6.2.2.3.1 Gaussian envelope

Application of the stationary phase method for a Gaussian envelope gives the result


21=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffij�0j�eff

p exp �� t2o

� 2eff

!exp ioð f Þ þ si

�

4

h i; ð6:215Þ

where to is the instant of stationary phase, given by Equation (6.202).

6.2.2.3.2 Rectangular envelope

For a rectangular envelope of duration T , the instantaneous frequency runs from f�at �T=2 to fþ at þT=2, and Equation (6.210) becomes


eioð f Þffiffiffiffiffiffiffiffiffiffiffij�0jT

p Pð f Þ; ð6:216Þ

where oð f Þ is given by Equation (6.212). The variables u� and s, needed for Pð f Þthrough Equation (6.179), are given by Equations (6.213) and (6.214).

If the time bandwidth product is large enough, the Fresnel integrals may beapproximated as step functions, in which case jPð f Þj is a rectangle function (seeEquation 6.183). In other words

jPð f Þj � Pf � fc

B

� �; ð6:217Þ

where B and fc are the frequency spread (Equation 6.209) and center frequency

fc �f� þ fþ

2¼ f0

�2

1 � �2; ð6:218Þ

respectively, with � from Equation (6.207). While the change in frequency during thepulse (Equation 6.208) can be positive or negative depending on the sign of �0, for anHFM pulse the center frequency fc is always positive.

The effective bandwidth can be calculated from Equation (6.159) as

�eff ¼�2

0T 2

f 40

ðþ1

�1jPð f Þj4ð f þ f0Þ�4 df

: ð6:219Þ

Use of Equation (6.217) simplifies the evaluation of �eff to give

�eff �12�2

f 20jð fþ þ f0Þ�3 � ð f� þ f0Þ�3j : ð6:220Þ


The start and end frequencies are given by Equation (6.206), so that

f� þ f0 ¼ �f�=� ð6:221Þand hence

�eff �j�0jT

1 þ 13�2: ð6:222Þ

6.2.2.3.3 Synthesis of HFM envelopes

The procedure used previously for the LFM case (Section 6.2.2.2.3) can be appliedalso to HFM modulation, resulting in the equations

HFM ¼ 2��0t

1 � �0t=f0� !t ð6:223Þ

and

toð f Þ ¼ f

f þ f0

f0�0

: ð6:224Þ

The spectral and envelope amplitudes are

SHFMð f Þ � f0f þ f0

a½toð f Þ a0

; ð6:225Þ

and

aHFMðtÞ ¼ a0

1 � �0t=f0S

�0t

1 � �0t=f0

� �: ð6:226Þ

The third column of Table 6.6 (labeled ‘‘HFM’’) is derived using Equation (6.226).

6.2.2.4 Hybrid spectra

A phase modulation suggested by Rosenbach and Ziegenbein (1993) is

ðtÞ2�b

¼ � t þ T=2

2�þ ðt þ T=2Þ�þ1

ð�þ 1ÞT � ð�T=2 < t < þT=2Þ; ð6:227Þ

where � is a constant to be chosen in the interval ½0; 1 , with the extremes � ¼ 0 and 1corresponding to the CW and LFM cases, respectively. The idea is that � can be tunedwithin this interval to explore the properties of modulations that are intermediatebetween CW and LFM, in order to obtain both range and frequency resolution with asingle pulse. The time origin is chosen to coincide with the moment of zero instanta-neous frequency. The pulse duration is T and the constant b is a measure of itsbandwidth.

The first two time derivatives of the phase are

0ðtÞ2�b

¼ � 1

2�þ ðt þ T=2Þ�

T � ð6:228Þand

00ðtÞ2�b

¼ �T��

ðt þ T=2Þ1�� : ð6:229Þ


The instant of stationary phase is given by

toT

¼ f

bþ 1

2�

� �1=�

� 1

2ð6:230Þ

and hence

Mð f Þ � ðB�Þ�1=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTð f =b þ 1=2�Þð1��Þ=�

qaðtoÞ eioð f ÞPð f Þ; ð6:231Þ

where

oð f Þ2�T

¼ f

2� �

1 þ �

f

bþ 1

2�

� �ð1þ�Þ=�b: ð6:232Þ

The function Pð f Þ depends on u� and s through Equation (6.179). These are

u� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij00ðtoÞj

�

r�T

2� to

� �ð6:233Þ

and

s � sgn 00ðtoÞ ¼ sgn b; ð6:234Þwhere

00ðtoÞ ¼2�b

T

�

ð f =b þ 1=2�Þð1��Þ=�: ð6:235Þ

For the case of a rectangular envelope, the true and effective pulse durations areequal. Thus, from Equation (6.228), the instantaneous frequency finst runs from

f� ¼ �b=2� ð6:236Þat time �T=2 to

fþ ¼ f� þ b ð6:237Þ

at time þT=2. Hence, the frequency spread and effective bandwidth are

B ¼ jbj ð6:238Þand

�eff ¼ �ð2 � �ÞB: ð6:239Þ

The LFM limit (Equation 6.227, with � ¼ 1) is an optimum pulse in the sense that,for fixed B, �eff has a maximum of

�eff ¼ B: ð6:240Þ

It is shown in Section 6.2.5 that, for a fixed pulse duration, the SNR is proportionalto bandwidth. Thus, maximizing the SNR requires setting � to 1, which meansthat Doppler resolution can only be bought at the expense of a reduced SNR. Acompromise value of � ¼ 0:5 is suggested by Norrmann and Ziegenbein (1995).


6.2.3 Autocorrelation and cross-correlation functions and the matched filter

6.2.3.1 Autocorrelation function

The autocorrelation function for the envelope �ðtÞ is

Að�Þ �ðþ1

�1�ðtÞ��ðt � �Þ dt: ð6:241Þ

A useful property of the autocorrelation function is that it forms a Fourier transformpair with the power spectrum.20 Thus,

jMð f Þj2 ¼ I½Að�Þ ð6:242Þand

Að�Þ ¼ I�1½jMð f Þj2 : ð6:243Þ

Autocorrelation functions for simple (CW and LFM) pulses are listed in Table 6.7.For more complicated pulses, evaluation of Equation (6.241) becomes surprisinglydifficult, and an alternative is to use Equation (6.243) instead, with the stationaryphase approximation for Mð f Þ. In particular, using the approximation of Equation(6.217) in Equation (6.189) gives

jMð f Þj2 � 2�

j00ðtoÞjjaðtoÞj2P

f � fcB

� �: ð6:244Þ

To understand the nature of this approximation, consider its application to the LFMcase, for which use of Equation (6.244) yields

ALFMð�Þ �sincð��0TÞ rectangular

exp � �

2�2

0�2eff�

2� �

Gaussian.

(ð6:245Þ

Comparison with the exact results of Table 6.7 suggests that the approximation


Table 6.7. Autocorrelation functions for CW and LFM pulses.

Envelope Autocorrelation function

CW LFM

Gaussian (Burdic, 1984) exp ��

2

� 2

� 2eff

� �exp � �

2

� 2

� 2eff

ð1 þ �20�

4effÞ

� �

RectangularT � j� j

T½Hð� þ TÞ � Hð� � TÞ T � j� j

Tsinc½��0�ðT � j� jÞ

(Russo and Bartberger, 1965)

20 Hence the alternative name ‘‘autospectral density’’ for the power spectrum jMð f Þj2 (see

Section 6.2.2.2.1).

requires that T be large compared with j� j, implying a large bandwidth. Specifically,given that the width of Að�Þ is of order 1=ðj�0jTÞ, this requirement amounts to

j�0jT 2 � 1: ð6:246ÞApplying the same method to a rectangular HFM pulse gives

AHFMð�Þ � 2�f 20

�0Texpð�2�if0�Þ

ð!þ

!�

ei!�

!2d!; ð6:247Þ

where!� ¼ 2�ð f� þ f0Þ: ð6:248Þ

This can be written (cf. Lin, 1988)

AHFMð�Þ � �if0�

�expð�2�if0�Þ E1ð�i!��Þ�E1ð�i!þ�Þþi

ei!þ�

!þ�� ei!��

!��

!" #; ð6:249Þ

where the definition of the exponential integral E1ðzÞ (see Appendix A) is such that,for an imaginary argument, the following relationship is satisifed

E1ð�iaÞ � E1ð�ibÞ ¼ðb

a

eiu

udu: ð6:250Þ

6.2.3.2 Cross-correlation and the matched filter

Given two functions of time f ðtÞ and gðtÞ, their cross-correlation function is given(following Burdic, 1984—see also Appendix A) by

Cf gð�Þ �ðþ1

�1f ðtÞg�ðt � �Þ dt: ð6:251Þ

The cross-correlation function is a measure of how similar the functions f and g are toeach other. If they are similar in shape (though perhaps shifted by time t 0) the cross-correlation function has a maximum at � ¼ t 0 and not otherwise. If the two functionsare identical, the result is the autocorrelation function

C��ð�Þ ¼ Að�Þ: ð6:252ÞThe cross-correlation operation is of importance for sonar processing because it isused to separate a target echo from random noise in active sonar returns. This is doneby cross-correlating the received signal with a delayed replica of the transmittedpulse. The expectation is that the echo will resemble the transmitted pulse, resultingin a peak in the cross-correlation output, whereas the noise will produce no suchpeak. This process is known as matched filtering and the processor in which it isimplemented is called a matched filter. The success of a matched filter relies onreceiving an undistorted echo.

6.2.3.3 Doppler processing

A common source of echo distortion, and fortunately one that is straightforward tocorrect for, is a Doppler shift relative to the transmitted pulse. A Doppler-shifted


echo, if cross-correlated with the original undistorted pulse, will suffer a reduction inthe peak of the matched filter output, resulting in a loss in the filter’s ability todiscriminate between signal and noise. However, if the echo is correlated insteadwith a Doppler-shifted replica with the same offset in frequency, the full performanceis recovered. This process of Doppler-shifting the replica is described below,following Russo and Bartberger (1965).

The main effect of target or sonar motion21 is a shift (denoted �D) in the carrierfrequency of

�D ¼ ð� � 1Þf0; ð6:253Þwhere

� ¼ c � V

c þ Vð6:254Þ

and V is the relative target velocity, defined as the rate at which the distance betweenthe target and sonar increases with time.

The Doppler factor � describes the compression or stretching of the time axis dueto the Doppler effect. If V=c is small then � may be approximated as

� � 1 � 2V

cð6:255Þ

and hence

�D � � 2V

cf0: ð6:256Þ

In general, the target Doppler is a priori unknown, so Doppler processing involves abank of replica echoes covering a range of guessed Doppler factors, say f�ng. Eachreplica can be written

yreplicaðtÞ ¼ � 1=2n �ð�ntÞ expði!0�ntÞ: ð6:257Þ

The Doppler factor stretches not just the carrier but also the envelope function �ðtÞ.Because of this, to understand the impact of a Doppler shift on a cross-correlationreceiver it is necessary to consider the full waveform yðtÞ given by Equation (6.148).

In general, the Doppler-shifted echo can be written

yechoðtÞ ¼ �1=2�ð�tÞ expði!0�tÞ: ð6:258Þ

The cross-correlation function of the echo and replica is

ð�; �; �nÞ �ðþ1

�1yreplicaðtÞy�

echoðt � �Þ dt: ð6:259Þ

Substituting Equation (6.257) and Equation (6.258) in the right-hand side gives

ð�; �; �nÞ ¼ �1=2n �1=2 ei!0��

ðþ1

�1�ð�ntÞ��½�ðt � �Þ ei!0ð�n��Þt dt: ð6:260Þ


21 If the sonar transmitter and receiver move in unison, and assuming the water to be

stationary, the Doppler shift is determined only by the relative motion between target and

sonar.

To understand the effect of the mismatch it is sufficient to consider the special case�n ¼ 1. Thus, the Doppler autocorrelation function (DACF) can be defined as

�ð�; �Þ � e�i!0� ð�; �; 1Þ ¼ �1=2

ðþ1

�1�ðtÞ��½�ðt � �Þ ei!0ð1��Þðt��Þ dt: ð6:261Þ

The normalization of �ðtÞ ensures that �ð�; �Þ has a peak value when � ¼ 0 and � ¼ 1of

�ð0; 1Þ ¼ 1: ð6:262Þ

If the Doppler shift is small (� � 1), the DACF may be approximated by putting� ¼ 1 everywhere except in the final (exponential) term in the integrand. The result,known as the ‘‘narrowband approximation’’, is

�NBð�; �DÞ ¼ e2�i�D�

ðþ1

�1�ðtÞ��ðt � �Þ e�2�i�Dt dt: ð6:263Þ

Examples of narrowband DACFs for an LFM-modulated pulse are:

�NBð�; �DÞ ¼ e�i��DT � j� j

Tsinc½�ð�D � �0�ÞðT � j� jÞ (LFM, rect.) ð6:264Þ

for a rectangular envelope, and

�NBð�; �DÞ ¼ e�i��D exp � �

2� 2eff

½� 2 þ ð�D � �0�Þ2� 4eff

(LFM, Gauss) ð6:265Þ

for a Gaussian one. In both cases the DACF for CW modulation is obtained in thelimit of small �0. For example, in the case of a Gaussian this is

�NBð�; �DÞ ¼ e�i��D exp � �

2� 2eff

ð� 2 þ �2D�

4effÞ

� �(CW, Gauss): ð6:266Þ

The applicability of the NB approximation is limited to low Doppler shift. If the lowDoppler requirement is not met, the full BB DACF is needed, which for a rectangularenvelope gets surprisingly complicated (Russo and Bartberger, 1965). For theGaussian case the BB DACF (with LFM modulation) is

�ð�; �Þ ¼ 2�

�

� �1=2

e2�i�D� exp � �

�� 2eff

½��2�� 2 � ð��2� � i�D�2effÞ2

; ð6:267Þ

where � and �, both complex variables, are given by

� ¼ �� þ �2� ð6:268Þand

� ¼ 1 þ i�0�2eff : ð6:269Þ

For zero Doppler shift, such that

� ¼ 1 ð6:270Þand

� ¼ 2; ð6:271Þ


Equation (6.267) simplifies to

�ð�; 1Þ ¼ exp � �� 2

2� 2eff

j�j2 !

; ð6:272Þ

as given by Table 6.7.As an example, consider the BB DACF for a CW pulse, derived by putting

� ¼ 1 ð6:273Þand

� ¼ �2 þ 1 ð6:274Þ

in Equation (6.267). The result is

�CWð�; �Þ ¼ 2�

1 þ �2

� �1=2

exp�

ð1 þ �2Þ��2

D�2eff � �2 �

2

� 2eff

þ 2i�D�

!" #: ð6:275Þ

The idea of BB processing for a CW pulse sounds like a contradiction, and is ofinterest only for fast targets, moving at a relative speed (i.e., range rate) comparablewith that of sound. This is because only a very fast–moving object would cause adifference between the BB and NB calculations for this case. Thus, for a CW pulse,the requirement is more one of low Doppler than of narrow band. The NBapproximation of Equation (6.266) follows if � � 1.

The DACF is an important result. From it can be obtained the autocorrelationfunction (the zero Doppler case)

Að�Þ ¼ �ð�; 1Þ; ð6:276Þ

the power spectrum (Equation 6.242) and the ambiguity function (see Equation6.277).

6.2.4 Ambiguity function

The ambiguity function is defined as the squared magnitude of the Dopplerautocorrelation function22

Xð�; �Þ � j�ð�; �Þj2: ð6:277Þ

It provides a convenient way of representing the resolution properties of the pulse indelay Doppler space. For simplicity, attention is limited in the following to the so-called NB approximation

XNBð�; �DÞ � j�NBð�; �DÞj2: ð6:278Þ

The NB ambiguity volume is defined as the integral of XNB over all time and frequency

VNB ¼ðþ1

�1

ðþ1

�1XNBð�; �DÞ d� d�D ð6:279Þ


22 Alternative definitions of the term ‘‘ambiguity function’’ are �ð�; �Þ and j�ð�; �Þj.

and this integral is equal to unity

VNB ¼ 1: ð6:280Þ

The BB version of Equation (6.279) is

VBB ¼ðþ1

�1

ðþ1

�1Xð�; �Þ d� d�D; ð6:281Þ

which satisfies the inequality (Rihaczek, 1967; Sibul and Titlebaum, 1981)

VBB � 1: ð6:282Þ

Henceforth the subscripts NB and D are omitted, but they are implied wherever thenotation Xð�; �Þ is used.

6.2.4.1 CW pulse

To illustrate the time and Doppler resolution properties of a CW pulse, consider theambiguity function for a Gaussian envelope. This case is chosen because it has theparticularly simple form

Xð�; �Þ ¼ exp½��ð�2 þ �2Þ ; ð6:283Þ

where � and � are dimensionless time delay and frequency (Doppler) shift variablesdefined by

� ¼ �=�eff ð6:284Þand

� ¼ ��eff : ð6:285Þ

A graph of the ambiguity function Xð�; �Þ, known as an ambiguity surface, is shownin Figure 6.11, converted to decibels. The width in delay and Doppler of this surfaceindicates the resolution of the pulse. In the �–� plane, a locus of equal ambiguity is acircle. In absolute time and frequency co-ordinates (say in the �–� plane), this circlebecomes an ellipse, and the ellipse whose ambiguity is expð��Þ is known as theambiguity ellipse.23 This ellipse has width �eff in the delay axis and 1=�eff in Doppler,and area �. Thus, a long CW pulse is well suited for discriminating between differentfrequencies (i.e., target Doppler) but poorly able to measure target range (delay time).It is said that such a pulse has high ‘‘Doppler resolution’’ and low ‘‘range resolution’’.By convention the Doppler axis is often converted to speed V . If the target andplatform speeds are small compared with the speed of sound, the Doppler anddimensionless frequency are related through the expression

V � � c

2f0�eff�: ð6:286Þ


23 The choice of expð��Þ to define the ambiguity ellipse, from Burdic (1984), is not universal

(see Russo and Bartberger, 1965 for other possibilities) but it is a convenient one for a Gaussian

envelope.

Similarly, the time delay is converted to a range offset R. If the sound pathsare assumed to be confined to the horizontal plane, the range variable R anddimensionless time � are related according to

R � c�eff2

�: ð6:287Þ

The effect on the ambiguity surface of changing the pulse duration is shown inFigure 6.12.

In R–V co-ordinates, the shape of the ellipse, though not its area, depends on thevalue of the pulse duration �eff . Specifically, the ellipse is described by the equation

R2

� 2eff

þ ð f0�effÞ2V 2 ¼ c2

4: ð6:288Þ

In R–V space, the width of the ellipse (its semi-axis length) is c�eff=2 in the rangedirection and �=ð2�effÞ in Doppler. The area of the ellipse is �c2=4f0, which for acenter frequency of 1 kHz is about 1750 m2/s. Thus, to achieve a Doppler resolutionof 1m/s or better the best achievable range resolution is 1750 m, whereas a rangeresolution of 100 m would imply a Doppler resolution no better that 17.5 m/s.


Figure 6.11. Generic ambiguity surface for Gaussian CW pulse, 10 log10 X plotted vs. � ¼ ��effand � ¼ �=�eff . Lines of constant ambiguity are circles in these co-ordinates.


Figure 6.12. Ambiguity surfaces 10 log10 X plotted vs. V � �ðc=2f0�eff Þ� and R � ðc�eff=2Þ�for Gaussian CW pulses with duration 0.5 s (upper) and 2.0 s (lower). Increasing the pulse

duration increases the Doppler resolution at the expense of range resolution.

6.2.4.2 LFM pulse

For an LFM pulse, still for a Gaussian envelope, the (NB) ambiguity function is24

Xð�; �Þ ¼ expf��½ �2 þ ð�� 0�2eff �Þ2 g: ð6:289Þ

This is a function of the dimensionless time and frequency variables as before, plusthe frequency rate in the form �0�

2eff . The effect of changing the frequency rate is

illustrated by Figure 6.13. Increasing the magnitude of this parameter turns the circleinto an increasingly elongated ellipse in �–� space. As the ellipse elongates, it rotatesat the same time, starting at 45 deg with its major axis along the line �=�eff ¼ ��eff ,aligning itself eventually with the Doppler axis (� ¼ 0), meaning that high-bandwidthpulses have high range resolution and low Doppler resolution. The graph showsresults for �0 � 0. For negative �0, the ellipse rotates in the same manner but inthe opposite direction.

For sufficiently large values of j�0j� 2eff , eventually an ambiguity surface

reminiscent of that of a short CW pulse is recovered (cf. Figure 6.12). Thus, rangeresolution is achieved at the expense of Doppler, so what is the advantage? Theanswer is an improved SNR associated with the high bandwidth. For a pulse offixed duration, the higher the bandwidth the greater the control over the shape of thewaveform, which in turn gives greater discrimination over a random background, asexplained in Section 6.2.5.

The ambiguity ellipse (from Equation 6.289) can be written

�2�2 � 2 sgnð�0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

p�� þ �2 ¼ 1; ð6:290Þ

where � is the time bandwidth product

� � �eff�eff : ð6:291Þ

Equation (6.290) describes an ellipse whose axes are rotated through some angle relative to the � and � axes. The rotation angle can be found by first defining rotatedcoordinates (�; �) such that

�

�

� �¼ cos

�sin

sin

cos

� ��

�

� �: ð6:292Þ

The ellipse is then, for Cartesian co-ordinates rotated at an arbitrary angle

A�2 þ B�� þ C�2 ¼ 1; ð6:293Þwhere

A ¼ �2 cos2 þ sin2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

psin 2; ð6:294Þ

B ¼ ð1 � �2Þ sin 2� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

pcos 2; ð6:295Þ

and

C ¼ �2 sin2 þ cos2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

psin 2: ð6:296Þ


24 The (BB) DACF for a rectangular envelope is given by Russo and Bartberger (1965). See also

Lin (1988).

The natural co-ordinates for an ellipse are those aligned with the axes of that ellipse.The rotation angle associated with this natural co-ordinate system is found byrequiring the cross term to vanish (i.e., B ¼ 0 in Equation 6.293) so that

tan 2 ¼ �2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

p : ð6:297Þ

With this rotation angle, the ellipse is given by

�2 tan2 þ � 2=tan2 ¼ 1: ð6:298ÞThe area of this ellipse is equal to �, irrespective of the rotation angle .

6.2.4.3 HFM pulse

The Doppler autocorrelation function for an HFM pulse25 is calculated by Lin(1988). An important property of HFM processing is that it is robust to smallDoppler shifts, making it easier to maintain performance against a moving target.


Figure 6.13. Generic ambiguity surfaces for Gaussian LFM pulse. Increasing the frequency

rate for a fixed pulse duration increases range resolution at the expense of Doppler resolution.

(The frequency rate increases anticlockwise from the upper left panel.)

25 For a rectangular envelope.

This is because the effect of a small shift in frequency can be approximated by a timedelay. The price paid for this robustness is a reduced Doppler resolution and a smallerror in range estimation.

6.2.5 Matched filter gain for perfect replica

The gain due to a matched filter is equal to the SNR at the output of the filter dividedby that at the input, expressed either as a ratio or in decibels. This gain is a measure ofthe filter’s ability to reject noise that does not resemble the transmitted waveform.In order to quantify this gain, the signal and background are given both before andafter the matched filter in Table 6.8 for a rectangular pulse of duration T .

The gain in SNR is

MG � 10 log10 G; ð6:299Þwhere

G ¼ T

�T: ð6:300Þ

If samples are taken at the Nyquist rate for bandwidth B (such that �T ¼ 1=B), thegain is then

G ¼ BT : ð6:301Þ

There are some practical considerations that place an upper limit on the achievablevalue of the BT product as follows:

— The bandwidth of the transducer limits the maximum achievable value of B.— Many sonars are not able to receive sound during the time they are transmitting,

resulting in a blind period that lasts at least as long as the transmitted pulse.The duration of such a blind period would limit the maximum achievable valueof T , depending on the required (minimum) detection range.


Table 6.8. Derivation of matched filter gain for pulse duration T and sample interval �T . The

function f ðtÞ has unit amplitude and mean square value 12.

xðtÞ Input Output Gain¼ output/

(squared magnitude inputðx2 dt of cross-correlation

peak) jCð�0Þj2

Signal sðtÞ af ðt � �0Þa2

2T

aT

2

� �2 T

2

Random noise (or nðtÞ n2T n2T�T

2

�T

2reverberation)

SNRa2

2n2

a2

2n2

T

�T

T

�T

One way of understanding the origin of the matched filter gain is by means of thefollowing thought experiment, involving a reversal of the propagation and matchedfilter operations. Imagine that, instead of an FM pulse, a sonar were to transmit theautocorrelation function of that pulse, with its amplitude increased by a factor

ffiffiffiffiffiffiffiBT

p

with respect to the original pulse, in order to maintain the same total transmittedenergy.26 The noise level is unaffected by this change because the original pulse and itsautocorrelation function have the same bandwidth. The reverberation level wouldalso be unchanged because the reduced scattering area caused by the shorter durationprecisely balances the increased intensity of the transmitted pulse. Thus, the SNRincreases by a factor BT , entirely due to the increased signal intensity. In practice,the correlation filter is invariably applied after reception in order to minimize theamplitude of the transmitted pulse for a given signal energy.

6.2.6 Matched filter gain for imperfect replica (coherence loss)

However large the value of BT , the theoretical gain is achieved only if the echo is anidentical replica of the transmitted pulse, or departs from it in a known or predictableway (a Doppler shift is an example of a predictable departure). Unpredictabledepartures result in a reduction in the processing gain known as coherence loss.Possible causes of coherence loss include:

— multipaths (e.g., due to target highlights or multiple boundary reflections);— short-term fluctuations in the environment, causing changes in the propagation

conditions on a timescale shorter than the duration of the pulse and therebydistorting its shape;

— scattering from a rough boundary;— changes to the pulse shape due to dispersion (frequency-dependent sound speed

or attenuation, frequency-dependent target spectrum, or frequency-dependentpropagation loss)

Coherence loss can be illustrated by means of a simple example involving two scaledreplicas of the transmitted pulse, identical in every respect except for their amplitudesand arrival times. If the replicas arrive with delay times �1; �2 and amplitudes a; b, thesignal can be written

sðtÞ ¼ af ðt � �1Þ þ bf ðt � �2Þ: ð6:302Þ

The precise form of both input (s2) and output (jCð�0Þj2) depend in general on thetime separation. Results are presented in Table 6.9 for the case of small separationcompared with the duration of the original pulse (T) and large compared with that ofthe compressed one (1=B). Coherence loss CL for this situation (two scaled replicas in


26 In the thought experiment, the reversal of the order of propagation and cross-correlation

means that the cross-correlation output becomes the autocorrelation function of the actual

transmitted pulse.

random noise) may be estimated as

CL � 10 log10

T

�T� 10 log10 GSNR; ð6:303Þ

where GSNR is the SNR gain from Table 6.9. Hence, adopting the convention thatjaj � jbj

CL ¼ 10 log10

a2 þ b2

a2: ð6:304Þ

In this situation, the worst case degradation, which arises when the replicas haveequal amplitude, is 3 dB. More generally, the worst case coherence loss for Nidentically shaped replicas that are closely spaced in time (but resolved aftercompression) is 10 log10 N.

6.2.7 Array gain and total processing gain (active sonar)

A modern sonar processing chain is likely to incorporate both beamforming andmatched filtering. If the output SNR (after both processes) is Rout, the combinedprocessing gain is

PG � 10 log10

Rout

Rhp

¼ AG þ MG; ð6:305Þ

where Rhp is the input SNR at the hydrophone.As both operations are linear ones, the output of the combined beamformer plus

matched filter processing does not depend on the order in which the individualoperations are carried out. Assuming arbitrarily that the beamformer comes firstthe array gain is

AG ¼ 10 log10

Rarr

Rhp

; ð6:306Þ


Table 6.9. Effect of multipath on matched filter gain.

sðtÞ Input Outputa Gain

s2 jCð�0Þj2 GSNR

Signal af ðt � �1Þ þ bf ðt � �2Þa2 þ b2

2

aT

2

� �2 T 2

2

a2

a2 þ b2

Noise nðtÞ n2 2 �Tn2T �T

2

SNRa2 þ b2

2n2

a2

2 �Tn2T

T

�T

a2

a2 þ b2

a Squared magnitude of cross-correlation peak.

where Rarr is the SNR at the beamformer output, which in turn is the input SNR forthe matched filter. Thus, the matched filter gain is

MG ¼ 10 log10

Rout

Rarr

: ð6:307Þ

For active sonar the array gain (and hence also the total processing gain) depends onwhether the background is dominated by noise or reverberation, which depends onthe processing applied and the distance to the target. A general rule is that abeamformer is more effective (has a larger AG) against ambient noise than againstreverberation, because the noise tends to originate in directions other than that of thetarget. An exception to this rule is the use of a monostatic sonar with a horizontalreceiving array and an omni-directional transmitter, for which the gain is comparablefor both noise and reverberation.

6.3 REFERENCES

Barger, J. E. (1997) Sonar systems, in M. J. Crocker (Ed.), Encyclopedia of Acoustics (pp. 559–

579), Wiley, New York.

Burdic, W. S. (1984) Underwater Acoustic Systems Analysis, Prentice-Hall, Englewood Cliffs

NJ.

Cheston, T. C. and Frank, J. (1990) Phased array radar antennas, in M. Skolnik (Ed.), Radar

Handbook (Second Edition, pp. 7.1–7.82), McGraw-Hill, New York.


Farnett, E. C. and Stevens, G. H. (1990) Pulse compression radar, in M. Skolnik (Ed.), Radar

Handbook (Second Edition, pp. 10.1–10.39), McGraw-Hill, New York.

Harris, F. J. (1978) On the use of windows for harmonic analysis with the discrete Fourier

transform, Proc. IEEE, 66(1) 51–83.

Kroszczynski, J. J. (1969) Pulse compression by means of linear-period modulation, Proc.

IEEE, 57(7), 1260–1266.

Lin, Zhen-biao (1988) Wideband ambiguity function of broadband signals, J. Acoust. Soc.

Am., 83, 2108–2116.

McDonough, R. N. and Whalen, A. D. (1995) Detection of Signals in Noise (Second Edition),

Academic Press, San Diego.

Norrmann, J. and Ziegenbein, J. (1995) Investigation of the ambiguity function of a special

kind of sonar signals, Proc. IOA, 17(8), 259–268.

Nuttall, A. H. (1981) Some windows with very good sidelobe behavior, IEEE Transactions on

Acoustics, Speech, and Signal Processing, ASSSP-29(1).

Proakis, J. G. (1995) Digital Communications (Third Edition), McGraw-Hill, Boston.

Rihaczek, A. W. (1967) Delay-Doppler ambiguity function for wideband signals, IEEE

Transactions on Aerospace and Electronic Systems, AES-3(4), 705–711.

Rosenbach, K. and Ziegenbein, J. (1993) About the effective Doppler sensitivity of

certain nonlinear chirp signals (NLFM), paper presented at the LFAS Symposium,

SACLANTCEN, May 1993, p. H/24.

Russo, D. M. and Bartberger, C. L. (1965) Ambiguity diagram for linear FM sonar, J. Acoust.

Soc. Am., 38, 183–190.


Sibul, L. H. and Titlebaum, E. L. (1981) Volume properties of the wideband ambiguity

function, IEEE Transactions on Aerospace and Electronic Systems, AES-17(1), 83–87.

Skolnik, M. (Ed.) (1990) Radar Handbook (Second Edition), McGraw-Hill, New York.

Tucker, D. G. and Gazey, B. K. (1966) Applied Underwater Acoustics, Pergamon, Oxford.

Woodward, P. M. (1964) Probability and Information Theory with Applications to Radar

(Second Edition), Pergamon Press, Oxford.


7

Statistical detection theory

While the individual man is an insoluble puzzle, in the aggregate he becomesa mathematical certainty. You can, for example, never foretellwhat any one man will be up to, but you can say with precision

what an average number will be up to. Individuals vary, butpercentages remain constant. So says the statistician.

Arthur Conan Doyle (1890)1

Natural statistical fluctuations in both signal and noise mean that it is not possible tostate with certainty what particular signal-to-noise ratio (SNR) will result in asuccessful detection. Instead, it is necessary to consider the likelihood of an eventoccurring in percentage terms. Thus, in this chapter we deal in the currencies ofprobabilities of detection and of false alarm.

The first problem considered, in Section 7.1, is the probability of detecting asingle pulse of known shape as a function of the SNR and the probability of falsealarm. In Section 7.2 the results are generalized to the reception of multiple pulseswhose shape is still known, but whose initial phase varies randomly from one pulse tothe next. Much of detection theory applied to sonar was developed originally forradar, and the material presented in Sections 7.1 and 7.2 is largely based on a bookentitled Radar Detection (DiFranco and Rubin, 1968). The language used mightsometimes convey the impression that the application is primarily for active sonar,but this is not intended. Some derivations can be found, where not provided here,in the various source references—primarily DiFranco and Rubin (1968) andMcDonough and Whalen (1995).2 Some readers will prefer to skip these relatively

1 These words, from The Sign of Four, are spoken by the fictional character Sherlock Holmes,

misquoting Winwood Reade.2 See also Rice (1948) and Kay (1998).

mathematical sections initially and jump instead to Section 7.3, which explains howto apply the main results to passive and active sonar, before reading the relevantsections in more detail.

The methods and results of Sections 7.1–7.3 can be said to describe a singleobservation or ‘‘look’’. If a given observation is repeated under the same conditions,the probability of detection for the second observation is unchanged, but the infor-mation from the two observations can be combined, at least in principle, in such away as to improve the overall performance. The effect on detection performance ofcombining information from multiple observations is the subject of Section 7.4.

7.1 SINGLE KNOWN PULSE IN GAUSSIAN NOISE,

COHERENT PROCESSING

Consider a narrowband signal pulse with amplitude AS at the receiver and durationDt, such that the signal pressure (or voltage) can be represented by the function

sðtÞ ¼ AS sinð!tþ �Þ; 0 < t < Dt: ð7:1Þ

Adding random noise nðtÞ, the total received signal plus noise becomes

rSþNðtÞ � sðtÞ þ nðtÞ ¼ AS sinð!tþ �Þ þ nðtÞ; 0 < t < Dt: ð7:2Þ

It is assumed that the continuous function rSþNðtÞ is sampled at discrete times ti andthat the individual noise samples nðtiÞ follow a Gaussian distribution. The term �,equal to the phase at initial time t ¼ 0, is assumed to be constant for all sampleswithin the pulse. Its value is taken from a random uniform distribution in ½0; 2��. Thesignal amplitude (AS) is also assumed constant within the pulse, with the value of thisconstant assumed either fixed (Section 7.1.2.1) or taken from one of three randomdistributions specified in Sections 7.1.2.2 to 7.1.2.4. The special case with no signal(AS ¼ 0) is considered first, in Section 7.1.1, in which the false alarm probability forGaussian noise is derived. In Section 7.1.2, detection probability is given as a functionof SNR for a sinusoidal signal with various amplitude distributions. The case of amore general, but still known pulse shape is considered in Section 7.1.4.

7.1.1 False alarm probability for Gaussian-distributed noise

False alarm probability is defined as the probability that a given amplitude thresholdis exceeded in the absence of any signal. Thus, the distribution of interest is that of thenoise term

rNðtÞ ¼ nðtÞ ð7:3Þ

alone. The assumption of a (zero-mean) Gaussian distribution for nðtÞmeans that theprobability density function (pdf) of the noise amplitude distribution, after coherent

312 Statistical detection theory [Ch. 7

processing,3 is (McDonough and Whalen, 1995)

fNðAÞ ¼A

�2exp � A2

2�2

!¼fRayleighðA=�Þ

�ðA > 0Þ; ð7:4Þ

where � is the standard deviation of the noise samples; and fRayleighðvÞ is thenormalized Rayleigh pdf

fRayleighðvÞ � v expð�v2=2Þ ðv > 0Þ: ð7:5Þ

Threshold crossings caused by noise are (by definition) false alarms. If the chosenamplitude threshold is AT, the false alarm probability pfa is therefore

4

pfa ¼ð1AT

fNðAÞ dA: ð7:6Þ


pfa ¼ð1AT=�

fRayleighðvÞ dv ¼ exp � A2T

2�2

!ð7:7Þ

or, equivalently,

AT=� ¼ ð�2 loge pfaÞ1=2: ð7:8Þ

7.1.2 Detection probability for signal with random phase

In this section, four possible amplitude distributions are considered for the signal,always with Gaussian noise, so that Equation (7.8) gives the corresponding pfa for allcases. The first of the four signal distributions represents an artificial situation with anon-fluctuating signal amplitude (referred to here as a Dirac distribution). Theremaining three, all for a fluctuating signal amplitude, are the Rayleigh and Rice(or Rician)5 distributions and the so-called one-dominant-plus-Rayleigh distribution.As explained in Section 7.1.2.3, the Dirac and Rayleigh distributions are special casesof the Rice distribution.

7.1 Single known pulse in Gaussian noise, coherent processing 313]Sec. 7.1

3 For example, the Fourier amplitude in a frequency band of interest.4 An important property of pdfs is that the integral of the pdf f ðvÞ (with respect to v) over agiven range of some physical observable A, must equal the integral of the pdf f ðuÞ with respectto a related quantity u over the same range of A. This is because the integrals represent the

probability of a certain event occurring, which is independent of the variable of integration.

Allowing the range of integration to vanish it follows also that f ðvÞ dv ¼ f ðuÞ du, and hence

that f ðuÞ is not the same function as f ðvÞ; in other words, the functional form of a pdf depends

on its argument.5 The terms ‘‘Rice’’ and ‘‘Rician’’ are used interchangeably.

7.1.2.1 Signal with non-fluctuating amplitude (Dirac distribution)

7.1.2.1.1 Marcum Q-function

If the signal amplitudeAS does not fluctuate (i.e., takes a constant value for all pulses,say, equal to a), the amplitude distribution fSðAÞ is non-zero only when A is identicalto a. Because fSðAÞ is a probability distribution, its area (integrated over all ampli-tudes A) must be equal to unity. The function that satisfies these two properties is theDirac delta function

fSðAÞ ¼ ðA� aÞ: ð7:9Þ

The significance of this distribution is that, if a measurement is made of the signalamplitude A, the probability of finding the value a is unity, and the probability offinding any other value is zero. When this non-fluctuating sine wave signal is added toGaussian noise, the resulting SþN has a Rician pdf (Rice, 1948; McDonough andWhalen, 1995):

fSþNðAÞ ¼A

�2exp � Rþ A2

2�2

!" #I0

ffiffiffiffiffiffi2R

p A

�

� �; ð7:10Þ

where I0ðxÞ is a zeroth-order modified Bessel function of the first kind (Appendix A).The probability of detection is given by

pd ¼ð1AT

fSþNðAÞ dA; ð7:11Þ

and hence

pd ¼ Q1


p;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 loge pfa

p �; ð7:12Þ

where the signal-to-noise ratio R is related to the signal amplitude a and noisestandard deviation �, according to

R ¼ a2

2�2ð7:13Þ

and Q1 is the Marcum Q-function defined as (Appendix A)

Q1ð�; �Þ �ð1�

fRiceðv; �Þ dv; ð7:14Þ

where fRice is the normalized Rice pdf

fRiceðv; �Þ � v exp � v2 þ �2

2

!I0ð�vÞ: ð7:15Þ

The functionQ1 is a special case of the generalizedMarcum function introduced laterin this chapter and denoted QM .

6


6 Specifically, the case M ¼ 1.

The mean square amplitude of the distribution described by Equation (7.10) is

hA2i ¼ 2�2ð1þ RÞ: ð7:16Þ

Using Equation (7.12), pd can be calculated as a function of R and pfa. A graph ofthis function is known as a receiver operating characteristic (ROC) curve. Theinformation is plotted in Figure 7.1 in the form of R vs. pfa for fixed pd as shownin Figure 7.1 (solid lines). This form of the ROC curve is selected because it leadsdirectly to an important term in the sonar equation, namely the detection threshold.This is the value of the SNR, expressed in decibels, that results in a 50% detectionprobability (see Chapter 3 and Section 7.3.3). Robertson’s Fig. 2 shows ROC curvesfor the same situation as Figure 7.1 here (solid lines), except in the form pd vs. pfa, forfixed SNR.

7.1.2.1.2 Albersheim approximation

It is desirable to obtain an explicit expression for R as a function of pd and pfa. Theappearance of R in the argument of the Marcum function in Equation (7.12) makes itdifficult to manipulate, but an alternative, approximate solution in the desired formdue to (Albersheim, 1981) is

R Aþ 0:12ABþ 1:7B; ð7:17Þ


Figure 7.1. ROC curves in the form 10 log10ðRÞ vs. pfa for non-fluctuating amplitude signal inRayleigh noise.

where

A ¼ loge0:62

pfað7:18Þ

and

B ¼ logepd

1� pd: ð7:19Þ

Albersheim’s result is shown in Figure 7.1 (dashed lines). It can be seen that Equation(7.17) is accurate to within ca. �0.3 dB for

10�12 < pfa < 10�3 ð7:20Þand

0:3 < pd < 0:9: ð7:21Þ

Notice the very small values of pfa considered in Figure 7.1. This is necessary tocompensate for the large number of false alarm opportunities arising in some applica-tions.7 In order to keep the total number of false alarms manageable, a very low falsealarm probability is needed for each opportunity. For typical orders of magnitude, seethe worked examples of Chapter 3.

7.1.2.1.3 Limit of large SNR

In the limit of large SNR, the Bessel function of Equation (7.15) may beapproximated by the asymptotic expression (valid for large z) (Appendix A)

I0ðzÞ �ezffiffiffiffiffiffiffiffi2�z

p ; ð7:22Þ

so that

fRiceðv; �Þ �ffiffiffiffiffiffiffiffiffiv

2��

rexp �ðv� �Þ2

2

" #: ð7:23Þ

If the parameter �, which is equal toffiffiffiffiffiffi2R

p, is sufficiently large, Equation (7.23)

approximates to a Gaussian distribution centered on �. In this approximation itfollows that

pd Fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 loge pfa

p�


p �; ð7:24Þ

where FðxÞ is the right-tail probability associated with a Gaussian pdf (see AppendixA), that is,

FðxÞ � 1ffiffiffiffiffiffi2�

pð1x

exp � u2

2

!du: ð7:25Þ

The form of Equation (7.24) is reminiscent of similar results presented in Chapter 2


7 A passive narrowband system that processes (say) 100 beams and 1000 frequencies, using a

coherent integration time of 1 s has 105 detection opportunities (and therefore also 105 false

alarm opportunities) every second. The same result holds for an active FM sonar with 100

beams and 1000 range cells and a pulse repetition rate of 1/s. In either case, to achieve a false

alarm rate of one per hour would require a false alarm probability of less than 10�8.

for a Gaussian signal in a Gaussian background. The differences stem from the factthat here the background (the noise amplitude) has a Rayleigh distribution and not aGaussian one.

Although the limit of large SNR is rarely an important one in its own right, thesimple form of the resulting equations provides a simple test of the more complicatedMarcum function in this limit.

7.1.2.2 Signal with Rayleigh fading

The term ‘‘fading’’ is used to indicate fluctuations in signal amplitude, and Rayleighfading means that these fluctuations are random in nature, with individual amplitudevalues taken from a Rayleigh distribution. Only slow Rayleigh fading is consideredhere, meaning that fluctuations occur between, but not during individual pulses. Thecorresponding pdf of the signal amplitude is (McDonough and Whalen, 1995)

fSðAÞ ¼A

a2exp � A2

2a2

!ðA � 0Þ; ð7:26Þ

so that the expectation values of A and A2 are

hAi ¼ffiffiffi�

2

ra ð7:27Þ

and

hA2i ¼ 2a2: ð7:28ÞMore generally, the expectation value of the nth power of A (i.e., its nth moment) canbe written

hAni � anð10

xnþ1 e�x2=2 dx; ð7:29Þ

and hence (see Appendix A)

hAni ¼ 2n=2Gðn=2þ 1Þan: ð7:30ÞThe parameter a is the modal (i.e., most probable) amplitude in the sense that it is thevalue of A that maximizes fSðAÞ.

The Rayleigh distribution is plotted as a solid blue line in Figure 7.2. Also shownare the non-fluctuating distribution used in Section 7.1.2.1 (i.e., a Dirac delta functionat A ¼ 1) and the one-dominant-plus-Rayleigh distribution of Section 7.1.2.4 (dottedline). The parameter a is chosen in each case to ensure that the mean squareamplitude is equal to unity. The fourth curve (dashed line) is a Rician distributionwhose parameters are chosen to match the first two moments of the one-dominant-plus-Rayleigh (1DþR) distribution (see Section 7.1.2.5).

The distribution of signal plus noise is obtained from Equations (7.4) and (7.26)by the addition of variance

fSþNðAÞ ¼A=�2

1þ Rexp �A

2=2�2

1þ R

!; ð7:31Þ


where �2 is the noise variance; and R is now the expected SNR

R � hA2i2�2

¼ a2

�2: ð7:32Þ

The detection probability is then found by applying Equation (7.11)

pd ¼ exp �A2T=2�

2

1þ R

!; ð7:33Þ

or, equivalently,

pd ¼ p1=ð1þRÞfa ; ð7:34Þ

with pfa from Equation (7.8). Figure 7.3 shows a graph of 10 log10ðRÞ vs. pfa,calculated using Equation (7.34).

7.1.2.3 Signal with Rician fading

Consider a fluctuating signal comprising Gaussian-distributed fluctuationssuperimposed on an otherwise stable sinusoidal component of amplitude aS. Mathe-matically this situation is no different from adding a non-fluctuating signal toGaussian noise, which means that the results of Section 7.1.2.1, leading to a Riciandistribution for signal plus noise, apply here to the signal alone. In other words, the


Figure 7.2. Rayleigh, one-dominant-plus-Rayleigh (1DþR), Dirac, and Rice probability

distribution functions, given by Equations (7.26), (7.50), (7.9), and (7.36), respectively. The

parameter values are chosen in each case to satisfy hA2i ¼ 1.

amplitude of a sinusoid with superimposed Gaussian fluctuations follows a Riciandistribution. This property is known as Rician fading.

The total signal power is the sum of the coherent and incoherent contributions.Thus, the ratio of signal power to noise power (i.e., the usual SNR) for a signal withRician fading is

R ¼ a2S þ 2�2S2�2N

; ð7:35Þ

where �2S and �2N are variances of the signal fluctuations and noise background,respectively.

By an exact mathematical analogy with Equation (7.10), the signal amplitude hasthe distribution

fSðAÞ ¼A

�2Sexp � RS þ

A2

2�2S

!" #I0

ffiffiffiffiffiffiffiffi2RS

p A

�S

� �ðA � 0Þ; ð7:36Þ

where RS is the ratio of coherent-to-incoherent signal power

RS �a2S2�2S

: ð7:37Þ


Figure 7.3. ROC curves in the form 10 log10ðRÞ vs. pfa for Rayleigh-fading signal in Rayleigh

noise, calculated using Equation (7.34).

The mean square amplitude of this distribution is (see Table 7.1 for other moments)

hA2i ¼ 2�2Sð1þ RSÞ: ð7:38Þ

Addition of Gaussian noise to this distribution results in (Jelalian, 1992)

fSþNðAÞ ¼A

�2SþNexp � S þ A2

2�2SþN

!" #I0

ffiffiffiffiffiffi2S

p A

�SþN

� �; ð7:39Þ

where S is the ratio of coherent signal power to incoherent signal plus noise power

S ¼ a2S2�2SþN

ð7:40Þ

and

�2SþN ¼ �2S þ �2N: ð7:41Þ

Continuing further the analogy with Section 7.1.2.1 and changing the variable ofintegration to

¼ A

�SþN; ð7:42Þ

Equation (7.11) becomes

pd ¼ �SþN

ð1AT=�SþN

�SþNexp � 2 þ a2S=�

2SþN

2

!I0

aS�SþN

� �d : ð7:43Þ

It is convenient at this point to introduce the concept of equivalent false alarmprobability, denoted qfa and given by

log qfa �log pfa

1þ �2S=�2N

; ð7:44Þ

where pfa is the true false alarm probability (Equation 7.7). By analogy with Equation


Table 7.1. Comparison table: moments of probability distribution functions. The notation inðxÞdenotes a scaled version of the modified Bessel function InðxÞ as defined in Equation (7.65). The

notation 1F1ð�; �; xÞ denotes a hypergeometric function (Appendix A).

Dirac Rayleigh Rice 1DþR

(Section 7.1.2.1) (Section 7.1.2.2) (Section 7.1.2.3) (Section 7.1.2.4)

hAi a

ffiffiffi�

2

ra

ffiffiffi�

2

r�S ð1þ RSÞi0

RS

2

� �þ RSi1

RS

2

� � � ffiffiffiffiffiffi3�

8

ra

hA2i a2 2a2 2�2Sð1þ RSÞ 43a2

hAni2hA2in 1 ½Gðn=2þ 1Þ�2 ½Gðn=2þ 1Þ 1F1 ð�n=2; 1;�RSÞ�2

ð1þ RSÞn2�n½Gðn=2þ 2Þ�2

(7.12), it then follows that the detection probability is

pd ¼ Q1

ffiffiffiffiffiffi2S

p;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 loge qfa

p �: ð7:45Þ

A graph showing S vs. qfa is shown in Figure 7.4. Apart from the change to the axislabels, this graph is identical to the curves of Figure 7.1 labeled ‘‘Marcum’’. TheAlbersheim approximation could also be used, but is omitted for clarity.

The parameter S (see Equation 7.40) can be thought of as an equivalent SNR:

S ¼ R� �2S=�2N

1þ �2S=�2N

; ð7:46Þ

where R is the true SNR (Equation 7.35). The intended use of these curves is tocalculate a detection threshold (DT), for which it is necessary to obtain a value of Rgiven pfa and pd. The procedure for doing so is as follows:

— calculate qfa using Equation (7.44);— read off 10 log10 S from Figure 7.4 for this value of qfa, at the desired detection

probability;


Figure 7.4. ROC curves in the form 10 log10ðSÞ vs. qfa for fluctuating amplitude (Rician fading)signal in Rayleigh noise. The equivalent signal-to-noise ratio S and equivalent false alarm

probability qfa are related to R and pfa through Equations (7.46) and (7.44) (compare Figure

7.1).

— calculate R by rearranging Equation (7.46) as

R ¼ ð1þ �2S=�2NÞS þ �2S=�

2N: ð7:47Þ

The detection threshold is then 10 log10 R.As an example, consider the case pd ¼ 0:9, pfa ¼ 10�8, and �2S=�

2N ¼ 3. Following

the above procedure gives qfa ¼ 10�2 and 10 log10 S ¼ 9.4 dB from Figure 7.4. UsingEquation (7.47) it then follows that DT90 � 10 log10 R90 ¼ 15.8 dB. This compareswith DT90 ¼ 14.2 dB in the absence of fluctuations from Figure 7.1, implying aperformance degradation of about 1.6 dB (meaning that for a given SNR thesefluctuations reduce the signal excess by 1.6 dB). By contrast, for pd ¼ 0.1, the samefluctuations result in an enhancement (a decrease in the detection threshold DT10) of10.4� 8.8¼ 1.6 dB. For intermediate values of pd (close to 0.5), the effect of thefluctuations is small.

The Rician distribution contains both the Dirac (i.e., non-fluctuating) andRayleigh amplitude distributions as special cases in the limit of large and smallRS, respectively. When the fluctuations are small, it is useful to explore the behaviorof the distribution before it collapses to a constant value. In this situation the signalamplitude is described approximately by a distribution of the form (using Equation7.23 with � ¼

ffiffiffiffiffiffiffiffi2RS

pand ¼ A=�S)

fSðAÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA

2�aS�2S

sexp �ðA� aSÞ2

2�2S

" #ðA > 0Þ; ð7:48Þ

which for large aS=�S approximates to a Gaussian distribution.Figure 7.5 shows Rice distributions for RS between 0.3 and 100 as marked. For

values of RS exceeding 10 or so, the distribution approximates to a Gaussian of width�S. Also shown are Rayleigh and Dirac distributions corresponding to the limits ofsmall and large RS, respectively. As previously, the parameters are chosen to ensure amean square amplitude hA2i ¼ 1, implying a variance of

�2S ¼1

2ð1þ RSÞ: ð7:49Þ

7.1.2.4 Signal with one-dominant-plus-Rayleigh distribution

An alternative signal amplitude distribution, known as the one-dominant-plus-Rayleigh (1DþR) distribution (DiFranco and Rubin, 1968, p. 313), is describedby the pdf

fSðAÞ ¼9A3

2a4exp � 3A2

2a2

!ðA � 0Þ; ð7:50Þ

where a is the modal (most probable) amplitude, related to the mean and mean squareamplitudes by

hAi ¼ffiffiffiffiffiffi3�

8

ra ð7:51Þ


and

hA2i ¼ 43a2: ð7:52Þ

The 1DþR distribution is plotted as a dotted line in Figure 7.2. In common withRice, its form is intermediate between that of a non-fluctuating amplitude (Diracdistribution) and the Rayleigh distribution. The main benefit of 1DþR is that its usesimplifies the analysis of fluctuating signals compared with the more cumbersomeRician statistics (de Theije et al., 2008). If higher order moments of A are required,these can be calculated using

hAni � 9an

2

ð10

xnþ3 e�3x2=2 dx; ð7:53Þ

so that (see Appendix A)

hAni ¼ ð2=3Þn=2Gðn=2þ 2Þan: ð7:54Þ

The disadvantage is that there is no shape parameter equivalent to � and hence nopossibility of tuning 1DþR to match a desired ratio of coherent-to-incoherent-signalpower.


Figure 7.5. Rice distributions with various values of the coherent-to-incoherent-power ratioRS

between 0.3 and 100 as marked; Rayleigh and Dirac probability distribution functions from

Figure 7.2 are also shown for comparison.

For 1DþR, the probability of detection, from Equation (7.11), is (DiFranco andRubin, 1968, Eq. (9.5-8))

pd ¼ 1þ R

ð1þ R=2Þ2A2

T

4�2

" #exp �ð1 þ R=2Þ�1 A

2T

2�2

" #; ð7:55Þ

where the signal-to-noise ratio is

R ¼ 2

3

a2

�2: ð7:56Þ

Using Equation (7.8) relating the threshold AT to the probability of false alarm, thedetection probability can be written

pd ¼ 1� R=2

ð1þ R=2Þ2loge pfa

�p1=ð1þR=2Þfa : ð7:57Þ

Given a value of the signal-to-noise ratio R (and false alarm probability pfa), it isstraightforward to calculate detection probability pd using Equation (7.57). Thereverse operation, from pd to R (a necessary one to evaluate the detection threshold),can be achieved graphically (Figure 7.6), or using one of the approximate methodsderived below.

The first step towards an (approximate) explicit solution is to rearrange Equation(7.57) in the form

R ¼ log p2fa

log pd � log 1� Rð1þ R=2Þ�2 logeffiffiffiffiffiffipfa

p� �� 2: ð7:58Þ

This is an implicit equation, because R appears on both left-hand and right-hand


Figure 7.6. ROC curves in

the form 10 log10ðRÞ vs.log10ð pfaÞ for 1DþR

signal in Rayleigh noise;

coherent processing.

sides. However, the right-hand side is relatively insensitive to the value of R, whichsuggests an iterative solution in the form

Riþ1 ¼log p2fa

log pd � log 1� Rið1þ Ri=2Þ�2 logeffiffiffiffiffiffipfa

p� �� 2; ð7:59Þ

so that the result of the first iteration is

R1 ¼2 log pfa

log pd � log 1�R0ð1þ R0=2Þ�2 logeffiffiffiffiffiffipfa

p� �� 2: ð7:60Þ

The iteration can be initialized using as a seed the detection threshold for Rayleighstatistics, that is,

R0 ¼log pfalog pd

� 1: ð7:61Þ

With this seed, Equation (7.59) converges (to�0.1 dB) after about three iterations. Atthe expense of some accuracy, a simpler solution is possible by choosing to stop afteronly the first iteration. Applying a small (empirical) correction term then gives

DT 10 log102 log pfa

log pd � log 1� R0ð1þ R0=2Þ�2 logeffiffiffiffiffiffipfa

p� �� 2

( )þ 0:03

1� 10pd1� pd

:

ð7:62Þ

An even simpler approach that works well for pd ¼ 0.5 is based on the observationthat DTRayleigh is consistently higher than DT1DþR, and that the difference is approxi-mately independent of pfa. For the special case pd ¼ 1

2, the difference is 0.8 dB, which

means that a useful approximation to DT50 (accurate to within �0.1 dB for pfa in therange 10�12 to 10�4) for 1DþR statistics is

DT50 10 log10 �log2ð2pfaÞ½ � � 0:8 dB: ð7:63Þ

Alternative approximations for the detection threshold, valid over a wide range of pdand pfa values are described by Shnidman (2002) and Barton (2005).

7.1.2.5 Summary table

Table 7.1 shows the mean and mean square amplitude for each of the fourdistributions considered, as well as a general (normalized) expression for the nthmoment. The Rice and 1DþR distributions are both intermediate between thenon-fluctuating case and the completely random Rayleigh case. If desired, the freeparameter in Rice can be adjusted to match some feature of 1DþR. For example,matching their mean amplitudes results in the condition

ð1þ RSÞi0RS

2

� �þ RSi1

RS

2

� �¼

ffiffiffi3

p

2

a

�S; ð7:64Þ


where the function inðxÞ is defined in terms of the nth-order modified Bessel functionof the first kind InðxÞ as

inðxÞ � e�xInðxÞ: ð7:65Þ

Selecting hA2i ¼ 1 as before implies that

a2 ¼ 3=4 ð7:66Þand

�2S ¼1

2ð1þ RSÞ; ð7:67Þ

so the condition on RS becomes

hAi2 ¼ �

4ð1þ RSÞð1þ RSÞi0

RS

2

� �þ RSi1

RS

2

� � �2

¼ 9�

32: ð7:68Þ

The value of RS that satisfies this condition is (de Theije et al., 2008)

RS 2:805: ð7:69Þ

7.1.3 Detection threshold

There is an important difference between the amplitude threshold AT that appears insome of the above equations (e.g., Equation 7.33) and the detection threshold DTintroduced in Chapter 3. The former is the SþN amplitude above which an operatordecision changes from ‘‘no target present’’ to ‘‘target present’’, while the latter is theSNR threshold above which the detection probability exceeds 50%. To reduce therisk of confusion, the precise relationship between them is described below.8

Let the detection probability be written in the form

pd ¼ f ðR;AT=�Þ: ð7:70Þ

At the SNR threshold corresponding to a 50% detection probability ( pd ¼ 12,

R ¼ R50) this becomes12¼ f ðR50;AT=�Þ: ð7:71Þ

For any given choice of amplitude threshold AT, this equation can be solved for R50.Converting to decibels gives the detection threshold corresponding to a 50%detection probability:

DT50 ¼ 10 log10 R50ðATÞ: ð7:72Þ

This general method can be applied to any one of the distributions considered above.As an example, consider the case of Rayleigh fluctuations, for which Equation (7.33)provides a simple equation relating the detection probability pd to the signal-to-noiseratio R and amplitude threshold AT. Substituting pd ¼ 1

2and rearranging for the


8 Abraham (2010) refers to the amplitude threshold as the ‘‘detector threshold’’.

associated SNR gives (for Rayleigh statistics)

DT � 10 log10 R50ðATÞ ¼ 10 log10A2

T

ð2 loge 2Þ�2� 1

!: ð7:73Þ

7.1.4 Application to other waveforms

So far in this section, a narrowband signal has been assumed for simplicity. For other(known) waveforms, a replica correlator (cross-correlation of the received pulse withthe transmitted one) can replace the Fourier transform, and the ‘‘amplitude’’ param-eter A is then re-interpreted as the correlator output. In this way, all results of thissection apply unaltered and the same ROC curves can be used, provided the pdf of Ais known or can be estimated. The analysis requires that the shape of the waveformbe fully known, but not the start time. See, for example, DiFranco and Rubin (1968,Ch. 9) or McDonough and Whalen (1995, Secs. 7.1 to 7.3).

7.2 MULTIPLE KNOWN PULSES IN GAUSSIAN NOISE,

INCOHERENT PROCESSING

Consider a sequence of pulses of the kind described in Section 7.1, all of equalduration. The phase term � is assumed to take an unknown constant value withineach pulse, and to vary randomly from one pulse to another. As the relative phasesare unknown, the pulses cannot be combined coherently, but instead one can sum thetotal energy over all M pulses

E �XMi¼1

A2i ; ð7:74Þ

declaring a detection if E exceeds some threshold ET. This type of detector is knownas an energy detector or square law detector. (Each individual pulse is processedcoherently; ‘‘incoherent’’ processing refers to the way the pulses are combined.)

Both fluctuating and non-fluctuating signal amplitudes are considered, aspreviously for coherent processing, and with the same amplitude distributions. Incombination with incoherent processing over multiple pulses, Rayleigh and 1DþRsignal fluctuations are known as the Swerling II and Swerling IV fluctuation models,respectively (DiFranco and Rubin, 1968; Levanon, 1988). The corresponding casewith a non-fluctuating signal is sometimes known as Swerling 0.

For radar applications, a distinction is made between ‘‘pulse-to-pulse’’fluctuations and slower ‘‘scan-to-scan’’ fluctuations, involving changes from onepulse train to the next, but not between successive pulses within a single pulse train.Such scan-to-scan fluctuations are described by the Swerling I and Swerling III

7.2 Multiple known pulses in Gaussian noise, incoherent processing 327]Sec. 7.2

models.9 The scan-to-scan cases are less relevant to sonar and not consideredhere.

7.2.1 False alarm probability for Rayleigh-distributed noise amplitude

The sum of squares ofM Rayleigh-distributed amplitudes results in a chi-squared (or‘‘�2’’) distribution. If there are M independent noise samples, the resulting �2

distribution has 2M degrees of freedom. This can be written (McDonough andWhalen, 1995, p. 295)10

fNðEÞ ¼1

2�2E

2�2

� �M�1 expð�E=2�2Þ

ðM � 1Þ! ; ð7:75Þ

where �2 is the variance of the original Gaussian noise distribution. It follows that, ifthe energy threshold is ET, the false alarm probability is

pfa ¼ð1ET

fNðEÞ dE: ð7:76Þ

Making the substitution

v ¼ E

�2ð7:77Þ

results in

pfa ¼ð1ET=�

2

f�20ðvÞ dv; ð7:78Þ

where f�20ðvÞ is the dimensionless ‘‘�2’’ distribution with 2M degrees of freedom

f�20ðvÞ ¼ 1

2

ðv=2ÞM�1 e�v=2

ðM � 1Þ! : ð7:79Þ

Equation (7.78) can be written (DiFranco and Rubin, 1968, p. 347)

pfa ¼GðM;ET=2�

2ÞðM � 1Þ! ; ð7:80Þ

where Gða; xÞ is the upper incomplete gamma function (Appendix A)

Gða;xÞ �ð1x

ta�1 e�t dt: ð7:81Þ

For fixed M, the energy threshold ET controls pfa in the same way as does AT inSection 7.1. If M increases, ET must also be increased to avoid a correspondingincrease in pfa.


9 See DiFranco and Rubin (1968, p. 390) and McDonough and Whalen (1995, p. 306) for

Swerling I; and DiFranco and Rubin (1968, p. 410) for Swerling III.10 For the special case M ¼ 2, this simplifies to the so-called ‘‘one dominant plus Rayleigh’’

(1DþR) distribution for the variable A ¼ E1=2.

For the special case M ¼ 1

pfa ¼ G 1;ET

2�2

� �¼ exp � ET

2�2

� �; ð7:82Þ

or, equivalently,

�2 loge pfa ¼ET

�2: ð7:83Þ

For sufficiently large M, the following limit is reached (DiFranco and Rubin, 1968,p. 367)

pfa FET=�

2 � 2M

2ffiffiffiffiffiM

p !

; ð7:84Þ

where FðxÞ is the right-tailed probability function (see Appendix A). An equivalentexpression for pfa is obtained by defining an (RMS) amplitude threshold AT as

AT �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiET=M

p; ð7:85Þ

so that

pfa FffiffiffiffiffiM

p A2T

2�2� 1

!" #: ð7:86Þ

7.2.2 Detection probability for incoherently processed pulse train

7.2.2.1 Signal with non-fluctuating amplitude

7.2.2.1.1 General case

If all M individual pulses in the pulse train have the same amplitude a, the SþNenergy has a pdf of the form (McDonough and Whalen, 1995)11

fSþNðEÞ ¼1

2�2E

�2�2

� �ðM�1Þ=2exp � �2

2þ E

2�2

!" #IM�1

�E 1=2

�

!; ð7:87Þ

where

�2 ¼ 2MR ð7:88Þand R is the power SNR

R ¼ a2

2�2: ð7:89Þ

For sufficiently high SNR, E is just the sum over signal energies, namelyMa2, but thepresence of noise usually complicates this simple picture.

This pdf is a non-central �2 density with 2M degrees of freedom, such that

pd ¼ð1ET

fSþNðEÞ dE ¼ð1ET=�

2

f�21ðv; �Þ dv; ð7:90Þ


11 Where InðxÞ is an nth-order modified Bessel function of the first kind (Appendix A).

where

f�21ðv; �Þ ¼ 1

2

v�2

� �ðM�1Þ=2exp � vþ �2

2

!IM�1ð�v1=2Þ: ð7:91Þ

It follows that (McDonough and Whalen, 1995, Eq. 8.25)

pd ¼ QM �;E

1=2T

�

!; ð7:92Þ

where QM is the generalized Marcum function

QMð�; �Þ � �

ð1�

x

�

�Mexp � x

2 þ �2

2

!IM�1ð�xÞ dx: ð7:93Þ

Equation (7.92) can also be written in terms of the RMS amplitude threshold

pd ¼ QM �;ffiffiffiffiffiM

p AT

�

� �: ð7:94Þ

The previously defined Marcum Q-function (Q1) is a special case of the generalizedMarcum function. The Marcum Q-function has been evaluated and plotted for aselection of values for the integerM by different authors (Robertson, 1967; DiFrancoand Rubin, 1968).

Albersheim’s approximation. It is useful to be able to express R explicitly interms of M, pd, and pfa, but Equation (7.92) does not lend itself easily to this end.A cumbersome solution is to plot pdðR; pfa;MÞ for all combinations of interest(see, e.g., Robertson, 1967) and read the value of R for the desired combinationof pd; pfa;M from the graph. A simple solution is provided by the followingapproximation due to (Albersheim, 1981)12

10 log10ðRffiffiffiffiffiM

pÞ 10xðMÞ log10ðAþ 0:12AB þ 1:7BÞ; ð7:95Þ

whereA and B are given by Equation (7.18) and Equation (7.19), respectively, and thefunction xðMÞ, plotted in Figure 7.7, is defined as

xðMÞ � 0:62þ 0:456ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM þ 0:44

p : ð7:96Þ

For largeM, the accuracy of Equation (7.95) is approximately within �0.5 dB for pd


12 The precise equation proposed by Albersheim is

10 log10ðRffiffiffiffiffiM

pÞ 6:2þ 4:54ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M þ 0:44p

� �log10ðAþ 0:12ABþ 1:7BÞ:

Equation (7.95) (with Equation 7.96) is identical to this except that the constant 4.54 is replaced

by 4.56 in order to match Equation (7.17) exactly for M ¼ 1.

and pfa values satisfying the inequalities

10�12 < pfa < 10�2 ð7:97Þand

0:3 < pd < 0:7: ð7:98ÞThe error reduces to less than �0.3 dB in the reduced range 10�7 < pfa < 10�2, stillfor large M.

The ratio R given by Equation (7.95) can be written in the form

R ðA þ 0:12AB þ 1:7BÞxðMÞffiffiffiffiffiM

p : ð7:99Þ

The form of Equation (7.99) might give the impression that R (the SNR) varies withM, but this is not the case, because the SNR is unaffected by incoherent integration.Rather, the left-hand side of this equation should be interpreted as the detectionthreshold (i.e., the SNR required to achieve a certain performance). This quantity isinversely proportional toM 1=2 for largeM. It is also a function of the detection andfalse alarm probabilities, through A and B. It is apparent from Figure 7.8 (bycomparing the magnitude of 10 log10ðM 1=2RÞ with its large-M asymptote) that,for the range of parameters considered, this M�1=2 behavior is reached to withinca. 0.6 dB whenM exceeds 100 and to within ca. 0.1 dB whenM exceeds 104. Figure7.9 shows a set of ROC curves evaluated using Equation (7.99) with pd ¼ 0.5, forMbetween 1 and 1024.


Figure 7.7. Graph of xðMÞ vs.M. This function is used in Equation (7.95) or Equation (7.99) to

calculate the detection threshold.

In order to assess its accuracy, Albersheim’s approximation is evaluated forvarious pd and pfa and the results presented in Table 7.2. Where possible a compar-ison with Robertson’s original curves (which are computed, without approximation,for a non-fluctuating signal in a Rayleigh background) is made, in order to obtain anestimate of the error involved. This error estimate is given in brackets. For theexample highlighted by gray shading (M ¼ 32, pd ¼ 0.1, pfa ¼ 10�7), the value ofDTþ 5 log10 M is 6.2 dB, which means that the Albersheim approximation gives

ðDT10ÞAlbersheim 6:2� 5 log10 M ¼ �1:3 dB: ð7:100Þ

The correction in brackets provides an improved estimate of Robertson’s originalvalue of:

DT10 ðDT10ÞAlbersheim � ð�0:4Þ ¼ �0:9 dB:

ROC curves. The intended use of Equation (7.92) is for calculation of detectionprobability pd, given the signal-to-noise ratio R. Figure 7.10 shows ROC curvesevaluated in this way for the case M ¼ 30, reproduced from DiFranco and Rubin(1968). This graph, and all subsequent ones from Radar Detection, show 10 log10ð2RÞvs. pd, for fixed values of pfa between 0:693� 10�10 and 0:693� 10�1. For further


Figure 7.8. ROC curves (Albersheim approximation) in the form 10 log10ðRÞ þ 10 log10 M1=2

vs. M for pd ¼ 0.5 and various pfa between 10�12 and 10�4 for a non-fluctuating amplitude

signal in Rayleigh noise.

examples (2 �M � 3000) see DiFranco and Rubin (1968, pp. 350–358). Graphs ofthe form pd vs. pfa for various R, designed for ease of interpolation for arbitrarycombinations of pd, pfa, and SNR, are given by Robertson (1967) for M equal tointeger powers of 2 between 1 and 8,192.

It is desirable to be able to calculate R directly from pd and pfa. One convenientapproximation for doing so is that due to Albersheim, described above. Otherapproximate methods, valid for a non-fluctuating signal over a wide range of valuesof M, pd, and pfa, are described by Shnidman (2002) and Hmam (2005).

Further ROC curves in the form 10 log10 R vs. pfa, evaluated using Albersheim’sapproximation, are shown in Figure 7.11 for values of M between 1 and 300 asmarked. The benefit of incoherent integration is to average out the fluctuations,making it feasible to detect a signal with a low SNR. Figure 7.11 shows that forM exceeding 100, DT can be negative even for a false alarm probability as low as10�12.

7.2.2.1.2 Special case M ¼ 1

For the special case M ¼ 1, Equation (7.92) reduces to

pd ¼ Q1ð�;E 1=2T =�Þ; ð7:101Þ

which is equivalent to Equation (7.12).


Figure 7.9. ROC curves (Albersheim approximation) in the form 10 log10ðRÞ þ 10 log10 M1=2

vs. pfa for pd ¼ 0.5 and variousM between 1 and 1024 for a non-fluctuating amplitude signal in

Rayleigh noise; the dashed line is the limit for M ! 1.

Table 7.2. DTþ 5 log10 M vs. M and pfa for three different pd values, evaluated using

Albersheim’s approximation (Equation 7.95). See text for interpretation of error values in

brackets. Empty columns indicate regions outside the validity range of Equation (7.95); missing

error values indicate that the point is outside the range of coverage of Robertson’s curves. The

meaning of the shaded box in the table for pd ¼ 0.1 is explained in the text surrounding

Equation (7.100).

Detection probability pd ¼ 0:1

M DTþ 5 log10 M DTþ 5 log10 M DTþ 5 log10 M DTþ 5 log10 M DTþ 5 log10 M

(dB) (dB) (dB) (dB) (dB)

ð pfa ¼ 10�1Þ ð pfa ¼ 10�3Þ ð pfa ¼ 10�5Þ ð pfa ¼ 10�7Þ ð pfa ¼ 10�9Þ

1 6.4 (�1.2) 8.9 10.5

2 5.9 (�0.8) 8.1 9.6

4 5.4 (�0.7) 7.5 8.8

8 5.0 (�0.7) 6.9 (�0.4) 8.1

32 4.5 (�0.8) 6.2 (�0.4) 7.3

256 4.2 (�0.9) 5.8 (�0.6) 6.8

8192 4.0 (�1.0) 5.6 (�0.6) 6.6 (�0.4)Detection probability pd ¼ 0:5




1 2.6 8.1 (0.0) 10.4 (0.0) 11.9 13.1

2 2.4 7.4 (0.1) 9.5 (0.1) 10.9 11.9

4 2.2 6.8 (0.1) 8.7 (0.1) 10.0 10.9

8 2.0 6.3 (0.1) 8.1 (0.2) 9.3 (�0.1) 10.1

32 1.8 5.7 (0.0) 7.3 (0.0) 8.4 (0.0) 9.1

256 1.7 5.2 (0.0) 6.8 (0.0) 7.7 (0.0) 8.5

8192 1.6 5.1 (�0.1) 6.5 (0.1) 7.5 (0.2) 8.2 (0.2)

Detection probability pd ¼ 0:9




1 7.8 (0.5) 10.7 (�0.1) 12.5 (0.0) 13.7 14.7

2 7.1 (0.4) 9.8 (0.0) 11.4 (0.0) 12.5 13.4

4 6.5 (0.4) 9.0 (0.0) 10.4 (�0.1) 11.5 12.3

8 6.1 (0.4) 8.3 (�0.1) 9.7 (�0.1) 10.7 (�0.1) 11.4

32 5.5 (0.2) 7.5 (0.0) 8.7 (�0.1) 9.6 (0.0) 10.3

256 5.1 (0.2) 7.0 (0.0) 8.1 (0.0) 8.9 (0.0) 9.5

8192 4.9 (0.3) 6.7 (0.0) 7.8 (0.1) 8.6 (0.2) 9.2 (0.3)

7.2.2.1.3 Limit of large M

In the large-M limit, Equation (7.92) may be approximated by

pd F’fa �

ffiffiffiffiffiM

pRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ 2Rp

� �; ð7:102Þ

where

’fa � F�1ð pfaÞ: ð7:103Þ

Adopting a similar shorthand for the analogous function of the detection probability

’d � F�1ð pdÞ; ð7:104Þ

Equations (7.84) and (7.102) can be written, respectively,

pfa Fð’faÞ ð7:105Þand

pd Fð’dÞ; ð7:106Þ

7.2 Multiple known pulses in Gaussian noise, incoherent processing 335

Figure 7.10. ROC curves in the form 10 log10ð2RÞ vs. pd for various pfa for a non-fluctuating

amplitude signal in Rayleigh noise; example of incoherent addition of M samples, with M ¼ 30.

The pfa values are given by 0.693/n0, where n 0 is between 10 and 1010 as labeled (reprinted from

DiFranco and Rubin, 1968, Scitech, Raleigh, NC).#

where

’fa ¼ET=�

2 � 2M

2ffiffiffiffiffiM

p ð7:107Þ

and

’d ¼ ’fa �ffiffiffiffiffiM

pRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 2Rp : ð7:108Þ

It is useful to obtain an expression for R as an explicit function of ’d and ’fa, as thissimplifies the calculation of ROC curves. To this end, Equation (7.108) can be recastas a quadratic equation in R

MR2 � 2ð’2d þ

ffiffiffiffiffiM

p’faÞRþ ’2

fa � ’2d ¼ 0; ð7:109Þ

whose solution is

R ¼’2d þ

ffiffiffiffiffiM

p’fa � ’d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM þ 2

ffiffiffiffiffiM

p’fa þ ’2

d

qM

: ð7:110Þ

Specifically, for the default situation with pd ¼ 0:5 ð’d ¼ 0Þ, this gives, withoutfurther approximation

R50 ¼’faffiffiffiffiffiM

p : ð7:111Þ


Figure 7.11. ROC curves in the form 10 log10ðRÞ vs. pfa for pd ¼ 0.3, 0.5, and 0.7 as marked,

for a non-fluctuating signal amplitude.

Alternatively, for arbitrary pd, a useful approximation is obtained for largeM in theform (disregarding the unphysical root)

ffiffiffiffiffiM

pR ’fa � ’d

1þ ’d=ffiffiffiffiffiM

p þO1

M

� �: ð7:112Þ

Defining R0 as the asymptotic form of R for large M, that is,

R0 �’fa � ’dffiffiffiffiffi

Mp ; ð7:113Þ

it follows that

R R0


p : ð7:114Þ

The approximation

R R0 ð7:115Þ

is sometimes used for largeM, as this simplifies calculation of the detection threshold.This approximation is compared in Figure 7.12 with the large-M limit of Alber-sheim’s approximation (from Equation 7.99 with B ¼ 0 and A given by Equation7.18):13

Rð pfaÞ 1ffiffiffiffiffiM

p loge0:62

pfa

� �0:62

: ð7:116Þ

An important question is: how large must M be for R0 to be used as anapproximation to R? The answer depends on the desired accuracy. According toDiFranco and Rubin (1968), for values of M exceeding 100, Equation (7.115) isaccurate to within 1 dB (and hence so too are the solid curves of Figure 7.12), andthis is supported by Figure 7.8. Greater accuracy can be achieved (still forM > 100)using Equation (7.114).14

Returning to the case of large M (say M > 100), the use of Equation (7.114) isillustrated below using a graphical method. The ratio R=R0 (calculated using Equa-tion 7.114) is plotted vs.M in Figure 7.13. This graph may be used to obtain R for anarbitrary combination of M, pd, and pfa, assuming that M is large, in the followingmanner. Given pd, the first step is to read off a value of R=R0 from Figure 7.13 for thedesired value ofM, and R0

ffiffiffiffiffiffiffiffiffiffiM=2

pfrom Figure 7.12 (solid curve) for the desired value

of pfa These two factors are multiplied first together and then further byffiffiffiffiffiffiffiffiffiffi2=M

pto

obtain the result for R

R ¼ffiffiffiffiffi2

M

r� R

R0

�ffiffiffiffiffiM

pR0ffiffiffi2

p : ð7:117Þ

The second factor is a function of R and pd only, independent of pfa.As an example, consider the case of 200 pulses combined incoherently and a

desired detection probability of 90%, with a false alarm probability of 10�6. Reading


13 The use of B ¼ 0 implies that pd ¼ 12.

14 For smaller values of M, Equation (7.92) or Equation (7.95) may be used.

appropriate values from Figures 7.13 and 7.12 gives

DT90 ¼ �9:6þ 10 log10ðR0

ffiffiffiffiffiffiffiffiffiffiM=2

pÞ

and

10 log10 R0

ffiffiffiffiffiM

2

r !¼ 6:3:

The detection threshold is therefore DT90 �3:3 dB.Figure 7.12 permits assessment of the accuracy of Albersheim’s approximation in

the limit of largeM. It shows that the error made in this limit is less than 0.5 dB for0:3 � pd � 0:7 and 10�12 � pfa � 10�2,15 and Figure 7.1 confirms that this is also thecase forM ¼ 1. Further, there is no suggestion from the bracketed errors in Table 7.2that accuracy deteriorates for intermediate values ofM, so the error seems likely to beless than 0.5 dB across the entire range of M, at least for pd ¼ 0.5.16

In the limit of very largeM, such that the right-hand side of Equation (7.114) isindependent ofM (and Equation 7.115 holds), Equation (7.102) becomes (switching


Figure 7.12. ROC curves in the form 10 log10ðffiffiffiffiffiffiffiffiffiffiM=2

pRÞ vs. pfa for fixed pd values for a

broadband signal in Rayleigh noise: calculated with Equation (7.115) (i.e., the large-M

approximation) and Equation (7.116) (Albersheim’s approximation, also for large M).

15 The Albersheim approximation is only plotted in places where this error is less than 0.5 dB.16 Greater accuracy is achieved for pd close to 0.5 and pfa in the range to 10�5 � pfa � 10�2.

here to erfc notation to facilitate comparison with Chapter 2)

pd 1


ffiffiffiffiffiM

2

rR

!; ð7:118Þ

where (from Equation 7.86)

pfa 1

2erfc

ffiffiffiffiffiM

2

rA2

T

2�21� 1

!" #; ð7:119Þ

and �1 is the standard deviation of the original Gaussian noise before any averaging(i.e., for a single pulse, the special caseM ¼ 1), hitherto denoted �. Equation (7.118)is identical in form to the corresponding expression for Gaussian statistics fromChapter 2, namely:

pd ¼1


xSffiffiffi2

p�M

�ð7:120Þ

and

pfa ¼1

2erfc

xT � xNffiffiffi2

p�M

: ð7:121Þ


Figure 7.13. Supplementary ROC curves in the form 10 log10ðR=R0Þ vs.M for fixed pd values

for a broadband non-fluctuating signal in Rayleigh noise: large-M approximation (Equation

7.114).

The equivalence indicates that Gaussian statistics apply for largeM (the central limittheorem at work). The mean and standard deviation of the noise and signal plusnoise distributions can be determined by inspection. Specifically, equating thearguments of the erfc functions for pd and pfa gives (equating the right-hand sidesof Equations 7.118 and 7.120)

xSffiffiffi2

p�M

¼ffiffiffiffiffiM

2

rR ð7:122Þ

and (from Equations 7.119 and 7.121)

xT � xNffiffiffi2

p�M

¼ffiffiffiffiffiM

2

rA2

T

2�21� 1

!; ð7:123Þ

respectively.Rearranging the first of these for �M=xS gives

�MxS

¼ 1ffiffiffiffiffiM

pR: ð7:124Þ

The second gives (dividing through by xS)

xTxS

¼ A2T

2�21R; ð7:125Þ

whereR ¼ xS=xN: ð7:126Þ

Thus, if M is sufficiently large, the noise standard deviation after incoherentprocessing of M pulses is inversely proportional to M 1=2 (Equation 7.124). Thismakes the point that the gain from incoherent processing arises from a reductionin the detection threshold (for a fixed value of pd) and not from an increase in signal-to-noise ratio.

7.2.2.2 Signal with Rayleigh amplitude distribution (Swerling II)


Now consider random fluctuations in signal amplitude between successive pulses,with individual amplitude values taken from a Rayleigh distribution (Equation 7.26).For radar applications this is known as the ‘‘Swerling II’’ model. The resultingdetection probability is (DiFranco and Rubin, 1968, p. 404)

pd ¼ð1ET

fSþNðEÞ dE ¼ 1

GðMÞG M;ET=2�

2

1þ R

!; ð7:127Þ

where R is the mean signal-to-noise ratio, equal to a2=�2.The intended use of Equation (7.127) is for calculation of detection probability

pd, given the signal-to-noise ratio R. Figure 7.14 shows ROC curves evaluated in thisway for the case M ¼ 30. For additional cases (2 �M � 3000) see DiFranco andRubin (1968, pp. 395–403). To calculate R from pd a different approach is needed.


Convenient approximate methods for doing so, valid for Swerling II statistics over awide range of values ofM, pd, and pfa, are given by Shnidman (2002), Hmam (2005),and Barton (2005).


For the special case M ¼ 1, it follows from Equation (7.127) that

pd ¼ 1� � 1;ET

2�2ð1þ RÞ

� �¼ exp � ET

2�2ð1þ RÞ

�; ð7:128Þ

resulting in Equation (7.34), as for a single coherently processed pulse.

7.2.2.2.3 Limit of large M

If M � 1, Equation (7.127) becomes

pd F’fa �

ffiffiffiffiffiM

pR

1þR

� �: ð7:129Þ


Figure 7.14. ROC curves in the form 10 log10ð2RÞ vs. pd for various pfa for a Rayleigh signal inRayleigh noise; example of incoherent addition ofM samples, withM ¼ 30. The pfa values are

given by 0.693/n0, where n 0 is between 10 and 1010 as labeled (reprinted from DiFranco and

Rubin, 1968, Scitech, Raleigh, NC).#

Without further approximation this can be rearranged as

R ¼ R0


p ; ð7:130Þ

where R0 is given by Equation (7.113). Equation (7.130) has the same form asEquation (7.114), implying that Figures 7.12 and 7.13 are applicable to this case.The 50% threshold (R50) is given by Equation (7.111).

7.2.2.3 Signal with one-dominant-plus-Rayleigh amplitude distribution(Swerling IV)


The 1DþR distribution (see Section 7.1.2.4) is given by Equation (7.50). For radarapplications this case is known as the ‘‘Swerling IV’’ model. The correspondingdetection probability is given by

pd ¼ð1ET

fSþNðEÞ dE: ð7:131Þ

The result is (DiFranco and Rubin, 1968, p. 427)

pd ¼ 1� M!

ð1þ R=2ÞMXMk¼0

ðR=2Þk

k! ðM � kÞ! ðM þ k � 1Þ! � M þ k;ET=2�

2

1þ R=2

!ð7:132Þ

where

R ¼ 2a2

3�2ð7:133Þ

and � is the lower incomplete gamma function (Appendix A)

�ða; xÞ �ðx0

ta�1 e�t dt: ð7:134Þ

The intended use of Equation (7.132) is for calculation of detection probability pd,given the signal-to-noise ratio R. Figure 7.15 shows ROC curves evaluated in thisway (with Equation 7.80 for pfa) for the case M ¼ 30. For additional cases(2 �M � 3000) see DiFranco and Rubin (1968, pp. 428–436). To calculate R frompd a different approach is needed. Convenient approximate methods for doing so,valid for Swerling IV statistics over a wide range of values ofM, pd, and pfa, are givenby Shnidman (2002) and Barton (2005).


For the special case M ¼ 1, Equation (7.132) simplifies to Equation (7.57), and thecorresponding ROC curves are as for a single coherently processed pulse (Figure 7.6).


7.2.2.3.3 Limit of large MIf M � 1, Equation (7.132) simplifies to (DiFranco and Rubin, 1968, p. 439)

pd F’fa �

ffiffiffiffiffiM

pRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 2Rþ 2R2p� �

: ð7:135Þ

This can be rearranged as a quadratic equation in R

ðM � 2’2dÞR2 � 2ð

ffiffiffiffiffiM

p’fa þ ’2

dÞRþ ’2fa � ’2

d ¼ 0; ð7:136Þ

whose solution, disregarding the unphysical root, is

ffiffiffiffiffiM

pR ¼

’fa � ’dð1þ ’fa=ffiffiffiffiffiM

pÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ð’2

fa � ’2dÞ=ð

ffiffiffiffiffiM

pþ ’faÞ2

qþ ’2

d=ffiffiffiffiffiM

p

1� 2’2d=M

: ð7:137Þ


Figure 7.15. ROC curves in the form 10 log10ð2RÞ vs. pd for various pfa for a 1DþR signal in

Rayleigh noise; example of incoherent addition ofM samples, withM ¼ 30. The pfa values are

given by 0.693/n0, where n 0 is between 10 and 1010 as labeled (reprinted from DiFranco and

Rubin, 1968, Scitech, Raleigh, NC).#

For pd ¼ 0.5, Equation (7.137) simplifies—without further approximation—toEquation (7.111). For arbitrary pd it can be written17ffiffiffiffiffi

Mp

R ’fa � ’d


p þ O1

M

� �; ð7:138Þ

which has the same form as Equation (7.114), implying that Figures 7.12 and 7.13 areapplicable to this case. Thus, Figure 7.13 (in combination with Figure 7.12) may beused for 1DþR and largeM. ROC curves from DiFranco and Rubin (1968) can beused for smaller values ofM between 2 and 100. If greater accuracy is required thanobtained in this way, Equation (7.132) can be used.

7.3 APPLICATION TO SONAR

Previously in this chapter, ROC relationships were derived for two different types ofprocessing. In order to apply the results of Chapter 7 to the four sonar typesconsidered in Chapter 3, it is first necessary to map processing types onto sonartypes. In each case the SNR must be calculated in some appropriate bandwidth thatdepends on signal processing. For incoherent processing there is an additionalparameter M, equal to the number of pulses added incoherently. The four casesof Chapter 3 are considered separately below. A fifth type of processing, applicableto active sonar, involving the transmission of a broadband frequency-modulated(FM) pulse and replica correlation (i.e., convolution of the received echo with areplica of the transmitted waveform) of the received signal, is also considered.

7.3.1 Active sonar

For active sonar the interpretation is straightforward. A ‘‘pulse’’ is just that, thewaveform transmitted by the sonar, and received at some later time, usually by thesame sonar. The main possibilities are summarized in Table 7.3.

This chapter contains many different equations for detection probability andmany different corresponding ROC curves. There is no simple prescription fordetermining which of these to use for any given problem. However, once the statisticsare known, it is relatively straightforward to evaluate the corresponding detectionprobability (see Table 7.4) to an appropriate degree of accuracy.

7.3.2 Passive sonar

For passive sonar, the concept of a pulse requires some explanation. In this context,by ‘‘pulse’’ is meant a sinusoidal or otherwise known time series, of a certainduration, received by the sonar. Specifically for a narrowband (NB) CW sonar,


17 The approximation R ¼ R0


p , with R0 from Equation (7.113), holds for all three

distributions considered (Swerling II, Swerling IV, and the non-fluctuating case).

the input to the detector is a continuous sine wave. The processing is coherent, so thetheory of Section 7.1 applies. The signal is integrated over a time DtNB. Thus, thepulse is the sinusoidal wave, the pulse duration is the coherent integration time DtNB,and the bandwidth is the reciprocal of this time B ¼ 1=DtNB.

For broadband passive sonar, the appropriate assumption is that nothing isknown about the received signal except that it is within the frequency band of thereceiver. The processing is incoherent and the theory of Section 7.2 is applicable.Because the form of the signal is unknown, the ‘‘pulse’’ is a single time sample and the‘‘pulse duration’’ is the sampling interval (or Nyquist interval �N if larger). Assumingthat the signal is sampled at the Nyquist rate, the sampling interval is

t ¼ �N ¼ 1

2B: ð7:139Þ

The number of pulses to be added incoherently is then

M ¼ DtBB�N

¼ 2B DtBB; ð7:140Þ

7.3 Application to sonar 345]Sec. 7.3

Table 7.3. Application of the detection theory results of Section 7.1 to active sonar cosine wave

(CW) and frequency-modulated (FM) pulses.

Processing Signal-to-noise ratio Amplitude

ðRÞ ðAÞ

CW pulse with Doppler SNR in Doppler Spectral amplitude

processing processing band (after FFT)

CW pulse with energy SNR in total Rx Square root of total

detector bandwidth signal energy

FM pulse with matched filter SNR after matched filter Matched filter output

Table 7.4. Equations for the detection probability for different

signal amplitude distributions. In all cases the noise amplitude

is assumed to have a Rayleigh distribution, leading to Equation

(7.7) for the false alarm probability of a single pulse or Equation

(7.80) for multiple pulses.

Distribution Single pulse Multiple pulses

Dirac Equation (7.12) Equation (7.94)

Rayleigh Equation (7.34) Equation (7.127)

Rice Equation (7.45)

1DþR Equation (7.57) Equation (7.132)

where DtBB is the incoherent integration time. The situation for passive sonar issummarized in Table 7.5. The same choice of distribution applies here as for activesonar (see Table 7.4).

7.3.3 Decision strategies and the detection threshold

To make use of the available evidence (i.e., the received S þN waveform) the sonaroperator must interpret the information and make a decision, usually in the form ofeither ‘‘target present’’ or ‘‘target absent’’. Decision strategies are discussed by severalauthors (Selin, 1965; Helstrom, 1968; Kay, 1998; Lehmann and Romano, 2005).Ideally, one would like to consider the impact of this decision on subsequent events.For example, an archeologist detecting a sunken wreck might deploy a submersible toinvestigate further, whereas a ship detecting an attacking torpedo would initiateimmediate countermeasures. In both cases there is a cost associated with actionand a (possibly greater) cost associated with inaction. A powerful method to analyzesuch situations using Bayesian probabilities is described by Selin (1965, p. 11).Applications of this approach are rare, due to the difficulty in quantifying thenecessary costs and a priori probabilities in a non-trivial manner (Helstrom, 1968).

A pragmatic and widely adopted solution to this problem is to assign a maximumacceptable false alarm rate, and hence false alarm probability pfa. Once made, thischoice determines a maximum permissible value for the amplitude threshold, fromwhich an SNR threshold corresponding to a particular detection probability (i.e., thedetection threshold, DT) can be derived using the procedure outlined in Section 7.1.3.

In principle, the equations of the present chapter can be used to calculate pd(given SNR and pfa), without the need to first calculate the detection threshold atwhich a particular value of pd is achieved. However, it is common practice to quantifysonar performance in terms of signal excess (the amount by which SNR exceeds DT),which is possible only if DT is known. Doing so makes it easy to make lateradjustments in uncertain parameters such as target strength or source level, or tocarry out sensitivity studies. Estimation of DT is straightforward from the graphspresented in this chapter of 10 log10 R vs. pfa.


Table 7.5. Application of detection theory results to narrowband (NB) and broadband (BB)

passive sonar.

Processing Section Signal-to-noise ratio Pulse duration

ðRÞ

NB (coherent processing) 7.1 SNR in NB processing Coherent integration

band, B ¼ 1=DtNB DtNB

BB (incoherent processing) 7.2 SNR in BB bandwidth Sampling interval,

assumed equal to the

Nyquist interval �N(Equation 7.139)

The presentation of ROC curves for three different signal distributions begs animportant question; namely, which of them to use for any given sonar problem. Thenon-fluctuating signal corresponds to a very stable target and stable propagationconditions, a combination that is unlikely to be encountered in practice except in atightly controlled experiment. At the other extreme is the Rayleigh distribution,which describes a signal with noiselike fluctuations. The third distribution considered(1DþR) represents a signal with intermediate statistics. Table 7.6 quantifies theeffect of varying pd and pfa for each of the three distributions. The differences inDT between them are relatively small (ca. 2 dB) for pd < 50%, but significantly largerfor pd > 50%. The largest differences shown in the table (about 8 dB) are betweenthe Rayleigh and Dirac (non-fluctuating) amplitude distributions, and arise forpd ¼ 90%. All values in the table are for a single pulse. Averaging has the effectof reducing the differences.

In general, the effect of any fluctuations is to increase DT (relative to that for anon-fluctuating signal) when pd is high, and to decrease it when pd is low. The 1DþRdistribution gives a result that is intermediate between the detection thresholdscalculated using Dirac and Rayleigh distributions, as can be expected from theintermediate nature of the distribution itself (see Figure 7.2).

For some applications it is convenient to use a simple approximation to thedetection threshold. For this purpose Equation (7.62) is suitable. An alternative,for pd ¼ 0.5 only, is Equation (7.63). The accuracy of both approximations isexamined in Table 7.7, showing that the error incurred by their use is 0.2 dB orless in the range 0:1 < pd < 0:9 and 10�12 < pfa < 10�4. Alternative approximationsfor 1DþR (Swerling IV) statistics are given by Shnidman (2002) and Barton(2005).

7.3 Application to sonar 347]Sec. 7.3

Table 7.6. Detection threshold 10 log10 R vs. pd for various statistics, with

pfa and pd values as stated. Values are obtained from Figures 7.1, 7.3, and

7.6.

Detection threshold

(dB)

pfa pd ¼ 10 % pd ¼ 50 % pd ¼ 90 %

Constant 6.1 9.4 11.7

10�4 1DþR 5.2 10.1 15.6

Rayleigh 4.8 10.9 19.4

Constant 10.4 12.5 14.2

10�8 1DþR 9.0 13.3 18.5

Rayleigh 8.5 14.1 22.4

Constant 12.8 14.3 15.7

10�12 1DþR 11.1 15.1 20.2

Rayleigh 10.4 15.9 24.2

All of the results of this chapter make the assumption of Gaussian statistics forthe noise, leading to Rayleigh-distributed amplitudes, which once averaged follow a�2 distribution. If the background is due to a large number of independent contribu-tions, this assumption is a reasonable one. However, in some situations the passivesonar background originates from a relatively small number of discrete sources, suchas individual ships, thus distorting the statistics. For active sonar it is often the casethat the background is dominated by reverberation rather than ambient noise, andthe reverberation can sometimes be resolved into contributions from discrete scat-terers such as individual rocks or shipwrecks, with the same effect. In either case,Gaussian statistics do not provide a good description of the background (Abraham,2003; Nielsen et al., 2008).18 Accurate approximations for the detection threshold innoise described by Weibull and K distributions,19 based on those of Hmam (2005),are derived by Abraham (2010) for both fluctuating and non-fluctuating signalmodels.

7.4 MULTIPLE LOOKS

7.4.1 Introduction

So far, the focus of this chapter has been on detection probability associated with asingle ‘‘look’’20 of a sonar—typically a single pulse for an active sonar, or for passivesonar the result of coherent or incoherent integration over a number of successive


Table 7.7. Detection threshold 10 log10 R vs. pd for a 1DþR

amplitude distribution for the same pfa and pd values as Table

7.6. Up to three values are given for each pd–pfa combination:

the first is obtained from Figure 7.6; the second [in square

brackets] uses the approximate Equation (7.62); the third

(in round brackets, for DT50 only) uses the alternative

approximation Equation (7.63).

Detection threshold for 1DþR statistics

(dB)

pfa pd ¼ 10 % pd ¼ 50 % pd ¼ 90 %

10�4 5.2 [5.2] 10.1 [10.1] (10.1) 15.6 [15.5]

10�8 9.0 [9.0] 13.3 [13.3] (13.3) 18.5 [18.5]

10�12 11.1 [10.9] 15.1 [15.1] (15.1) 20.2 [20.3]

18 Detection in noise with non-Gaussian statistics is considered by Kassam (1987) and Kay

(1998).19 A convenient summary of these and other related distributions is given by Jackson and

Richardson (2007).20 That is, a single threshold comparison with a binary outcome.

time samples. More generally one can think of a number of successive independentlooks, some of which might result in threshold crossings for a given target and othersnot. An important question is how best to combine the information from multipledetection opportunities (and multiple threshold comparisons) in such a way as tomaximize the overall detection probability (for a fixed false alarm rate). This questionis the subject of this section. The process of combining information from multipledetection opportunities in this way is referred to below as ‘‘fusion’’.

As a simple example, consider a situation involving two nearly simultaneous (butindependent—see Weston, 1989) looks on a sonar screen, or a single simultaneouslook on each of two identical sonars, in nearly identical positions and orientations.For any given target the expectation value of the SNR is the same for each of the twolooks. Similarly, the detection and false alarm probabilities, denoted D and Frespectively, are unchanged from look to look. (The symbols pd and pfa are reservedfor the corresponding probabilities after fusion). Because there are two looks, there istwice as much information as for a single one, so intuitively one might expect animprovement in the performance. Pertinent questions are:

— How can this anticipated improvement be realized and quantified?— How does the improvement depend on the manner in which the information is

combined?

To answer these questions it is assumed, for simplicity, that within each sonar displaythere exists a signal due to one and only one target. If Rayleigh21 statistics areassumed for both signal and noise amplitudes, from Equation (7.34), F and D arerelated according to

1þ R ¼ log F

logD; ð7:141Þ

where R is the signal-to-noise ratio, constrained to the interval ½0;1� so that,according to Equation (7.141), F cannot exceed D. This is consistent with theirrespective definitions (see Chapter 2) as the probabilities of a threshold crossing,respectively, for the cases of noise only and signal plus noise.

The values of pd and pfa, the new detection and false alarm probabilities aftermerging the two displays, depend on how the available information is combined. Anequivalent SNR, denoted Req, can be defined in terms of these probabilities such that

1þ Req �log pfalog pd

: ð7:142Þ

Thus defined,Req is the SNR that would be required to achieve the same performancefrom a single look as is actually achieved by combining two independent looks.The ratio of the equivalent SNR, Req, to the true SNR, R, is a measure of the

7.4 Multiple looks 349]Sec. 7.4

21 The choice of Rayleigh statistics is made at this point for mathematical convenience, as this

choice results in simple forms for ROC relationships. Other distributions are considered in

Section 7.4.2.4.

improvement in performance and is referred to henceforth as the fusion gain. Thisparameter, denoted G, can then be written

G �Req

R¼ 1� ðlog pdÞ�1 log pfa

1� ðlogDÞ�1 log F: ð7:143Þ

The fusion gain is a useful quantitative measure of the performance of the combineddisplay. Its value is considered below for two different situations involving thecombination (fusion) of data from two pulses for the case of a single target. Theproblem for multiple targets is considered by de Theije et al. (2008), including theeffect of positioning errors.

7.4.2 AND and OR operations

Consider two different logical operations for combining the data from the two sonars,an AND operation, requiring a threshold crossing at the target position on both sonarscreens before a detection decision is made, and an OR operation, for which a singlethreshold crossing is sufficient.

7.4.2.1 AND operation for Rayleigh statistics

By definition, the probability of the target causing a threshold crossing on eachseparate image is D. If the two observations are independent, the probability ofdetection after an AND operation is

pd ¼ D2 ð7:144Þ

and the false alarm probability is

pfa ¼ F 2: ð7:145Þ


1þ R ¼ log pfalog pd

: ð7:146Þ

Thus, for this situation the false alarm probability is reduced, but so is the detectionprobability, in such a way that the true and equivalent signal-to-noise ratios areidentical. This can be written as

GAND ¼ 1; ð7:147Þ

where GAND is the fusion gain for an AND operation. This is a curious result. Itmeans that, for the assumed Rayleigh statistics, the performance of the combined(AND) system can be achieved by either one of the individual systems simply byswitching off the other one and adjusting the threshold to maintain the same falsealarm rate.


7.4.2.2 OR operation for Rayleigh statistics

For the OR operation, the detection and false alarm probabilities pd and pfa are givenby

pd ¼ 2D�D2 ð7:148Þand

pfa ¼ 2F � F 2: ð7:149Þ

The detection probability increases relative to that for a single display because thenumber of opportunities has doubled, apparently improving the performance of thesonar. However, the false alarm probability also increases and it is not immediatelyobvious whether the net gain is positive or negative. The corresponding ROC curvescan be obtained from the relationship

1þ R ¼ logð1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pfa

pÞ

logð1�ffiffiffiffiffiffiffiffiffiffiffiffiffi1� pd

pÞ : ð7:150Þ

The gain in performance (i.e., the reduction in detection threshold for fixed pd and pfa)can be calculated as

GOR ¼

log pfalog pd

� 1

logð1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pfa

pÞ


pÞ� 1

: ð7:151Þ

Figure 7.16 shows the quantity 10 log10 GOR vs. pfa for fixed values of pd. This is thegain in decibels, which is positive for the entire range of values considered, indicatingan improvement in detection performance (GOR > 1).

All of the curves are remarkably flat for small values of the false alarmprobability. The reason for this behavior can be seen by writing Equation (7.151)in the form

GOR ¼ logD

log pd

log pfa � log pdlogð1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pfa

pÞ � logD

; ð7:152Þ

which, if pfa � 1, simplifies to

GOR logD

log pd

1� ðlog pfaÞ�1 log pd1� ðlog pfaÞ�1 logð2DÞ

: ð7:153Þ

It is often the case that a sonar design requires a false alarm probability that is manyorders of magnitude less than pd, corresponding to the left half of Figure 7.16. In thissituation, Equation (7.153) may be further approximated by neglecting terms of orderðlog pfaÞ�1. In this limit GOR reaches an asymptotic value of

GOR logð1�ffiffiffiffiffiffiffiffiffiffiffiffiffi1� pd

pÞ

log pd; ð7:154Þ

independent of pfa and equal to 1.77 for pd ¼ 0.5. Converting to decibels this is a gainof 2.5 dB, consistent with Figure 7.16.



The information in Figure 7.16 can be presented alternatively in the form of gainvs. log F for fixed D values. The resulting graph is shown in Figure 7.17.

7.4.2.3 Summary table for Rayleigh statistics

The results for AND and OR operations are summarized in Table 7.8.

7.4.2.4 Simulations with Rayleigh and non-Rayleigh signal statistics

The Rayleigh distribution was chosen for the above analysis because it is particularlyamenable to algebraic manipulation. The sensitivity of the fusion gain to the choice ofpdf is considered by computing theoretical ROC curves for Rayleigh, Dirac, and1DþR distributions by means of numerical simulations.

It is convenient to start with a non-fluctuating signal. ROC curves in the form pdvs. pfa are plotted in Figure 7.18 for SNR values between 9 dB and 13 dB as marked.The solid cyan lines show theoretical ROC curves for a single pulse with SNR valuesin decibels, as labeled. For each SNR value, in addition to the solid curve, there is alsoa dashed line and a dotted one. These are the theoretical ROC curves for combining apair of pulses of the same SNR, with OR and AND fusion, respectively. ForSNR¼ 10 dB and pd ¼ 1

2, a fusion gain of 2 dB and 0.8 dB can be inferred from

the two horizontal bars labeled AND and OR, respectively.

Figure 7.16. Fusion gain 10 log10 GOR vs. log10 pfa for OR operation, with fixed values of pd as

marked (Rayleigh statistics); pd values are 0.1 (smallest gain) to 0.9 (highest gain) in steps of 0.2;

the asymptotic gain for pd ¼ 0.5 is 2.5 dB.


These calculations are repeated for 1DþR and Rayleigh signal statistics inFigures 7.19 and 7.20, respectively, and for the same SNR values. It can beseen that the gain for OR fusion increases (from 0.8 to 2.5 dB), while the gain forAND fusion decreases (from 2 to 0 dB), with increasing signal fluctuations. In

Figure 7.17. Fusion gain 10 log10 GOR vs. log10 F for OR operation, with fixed values of D as

marked (Rayleigh statistics);D values are 0.1 (smallest gain) to 0.9 (highest gain) in steps of 0.2;

the asymptotic gain for D ¼ 0.5 is 3.8 dB.

Table 7.8. ROC relationships and fusion gain for AND andOR operations for fixed SNR, with

Rayleigh statistics.

Single sonar Combined Combined

(AND) (OR)

Detection probability pd D D2 2D�D2

False alarm probability pfa F F 2 2F � F 2

1þ Rlog pfa

log pd

log pfa

log pd


pÞ


pÞ

G N/A 1

log pfa

log pd� 1


pÞ


pÞ� 1


particular, the fusion gain for the AND operation vanishes in the Rayleigh case,so the dotted curves are hidden in Figure 7.20 (they coincide with those for asingle sonar). This sensitivity of fusion gain to signal statistics means that theoptimum fusion rule depends in general on the signal amplitude distribution.However, de Theije et al. (2008) show that the AND gain is severely degradedin the presence of measurement (positioning) errors, while the OR gain is lessaffected, making the latter potentially better suited to a multistatic sonargeometry.

7.4.3 Multiple OR operations

Consider N independent looks, each with identical detection probability per lookequal toD. In principle, theN looks could correspond to simultaneous measurementsusing N sonars, but a more likely application would be to N consecutive looks anda single sonar. If the N looks are combined with multiple OR operations, theoverall probability of detection, denoted pd (the cumulative detection probability)is the chance of at least one threshold crossing out of N opportunities. In

Figure 7.18. Detection probability pd (after fusion) vs. false alarm probability pfa for non-

fluctuating signal in Rayleigh background. The gain (with SNR¼ 10 dB and pd ¼ 12) is 0.8 dB

for OR fusion and 2 dB for AND fusion.


other words22

pd ¼ 1� ð1�DÞN ð7:155Þand similarly

pfa ¼ 1� ð1� FÞN : ð7:156Þ

Assuming a Rayleigh distribution for the signal amplitude, the resulting ROCrelationship is therefore

1þ R ¼ log½1� ð1� pfaÞ1=N �log½1� ð1� pdÞ1=N �

: ð7:157Þ

The gain in performance (i.e., the reduction in detection threshold for fixed pd

Figure 7.19. Detection probability pd (after fusion) vs. false alarm probability pfa for 1DþR

signal in Rayleigh background. The gain is 1.8 dB for OR fusion (with SNR¼ 10 dB and pd ¼ 12 )

and 0.9 dB for AND fusion (approximately independent of pd).

22 More generally, if the Di values are not identical, where i is the look number:

pd ¼ 1�YNi¼1

ð1�DiÞ;

where pd is the probability that one or more threshold crossing occurs in N opportunities. The

false alarm probability pfa is similarly increased:

pfa ¼ 1�YNi¼1

ð1� FiÞ:

and pfa) is

G ¼

log pfalog pd

� 1

log½1 � ð1� pfaÞ1=N �log½1� ð1� pdÞ1=N �

� 1

: ð7:158Þ

7.4.4 ‘‘M out of N ’’ operations

A multiple OR operation amounts to a requirement that at least one thresholdcrossing is made out of N detection opportunities. Similarly, a multiple AND opera-tion (not considered explicitly) corresponds to the much more stringent requirementofN threshold crossings out of N opportunities. The former leads to high probabilityof detection, and a correspondingly high false alarm rate. The latter virtuallyeliminates false alarms (for large N) at the expense of low detection probability. Thisline of thinking suggests a middle road to be explored between these two extremes,wherebyM threshold crossings are required from N opportunities, with 1 �M � N.For a given value of N, the parameter M can be adjusted to optimize detectionperformance. This ‘‘M out of N ’’ approach23 is analyzed by Reibman and Nolte(1987), Weiner (1991), and Shnidman (1998). The general result for the probability of


Figure 7.20. Detection probability pd (after fusion) vs. false alarm probability pfa for Rayleigh

signal in Rayleigh background. The gain is 2.5 dB for OR fusion (with SNR¼ 10 dB and pd ¼ 12 )

and 0 dB for AND fusion (independent of SNR and pd).

23 Also known as ‘‘binary integration’’.

obtainingM or more threshold crossings out of N independent looks in the presenceof signal plus noise (i.e., the detection probability) is (Reibman and Nolte, 1987)

pd ¼ 1� ð1 �DÞN�MXMq¼0

N!

q!ðN � qÞ!Dqð1�DÞM�q: ð7:159Þ

Similarly, the probability of at least M threshold crossings if only noise is presentgives the false alarm probability:

pfa ¼ 1� ð1� FÞN�MXMq¼0

N!

q!ðN � qÞ!Fqð1� FÞM�q: ð7:160Þ

It follows from Equation (7.141) that the fusion gain for the ‘‘M out of N’’ case is,assuming Rayleigh statistics again,

G ¼

log pfalog pd

� 1

log F

logD� 1

: ð7:161Þ

Weiner (1991) calculates the detection threshold vs.M for fixedN for non-fluctuatingand Swerling II targets, showing that an optimum value exists forM that minimizesthe detection threshold. Shnidman (1998) calculates the optimum value ofM for non-fluctuating, Swerling II, and Swerling IV targets, and provides approximateexpressions for this optimum. These approximations, which are all valid to within10% in the range 10 � N � 500, are

M0 100:8N�0:02; ð7:162Þ

MII 100:91N�0:38 ð7:163Þand

MIV 100:873N�0:27; ð7:164Þwhere the subscript indicates the Swerling type (0 for the non-fluctuating case).Weston (1992) considers the role of prior knowledge in determining the optimumvalue of M, arguing, for example, that fewer (consecutive) threshold crossings areneeded to confirm the presence of a contact on a sonar screen if it is known in advancethat a target is in the area.

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Aerospace and Electronic Systems, 41, 1049–1052.


de Theije, P. A. M., van Moll, C. A. M., and Ainslie, M. A. (2008) The dependence of fusion

gain on signal-amplitude distributions and position errors, IEEE J. Oceanic Eng., 3(3).

DiFranco, J. V. and Rubin, W. L. (1968)Radar Detection, Prentice-Hall, Englewood Cliffs, NJ.

Helstrom, C. W. (1968) Statistical Theory of Signal Detection, Pergamon Press, Oxford.

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Jackson, D. R. and Richardson, M. D. (2007) High-Frequency Seafloor Acoustics, Springer

Verlag, New York.

Jelalian, A. V. (1992) Laser Radar Systems, Artech House, Boston.

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Prentice-Hall, Englewood Cliffs, NJ.

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Underwater Reverberation and Clutter, September 9–12, NATO Undersea Research

Center, La Spezia, Italy.

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systems, IEEE Transactions on Aerospace and Electronic Systems, AES-23(1), 24–30.

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109–157, January.

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774.

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(ARE) Seminar, Winfrith, U.K., March.


Part III

Towards Applications

8

Sources and scatterers of sound

An experiment is a question which science poses to Nature, and ameasurement is the recording of Nature’s answer.

Max Planck, Scientific Autobiography and Other Papers (1949)

The behavior of underwater sound is central to sonar performance. A theoreticaltreatment is presented in Chapter 5, but on its own that is not enough. To inspireconfidence, the theory must be supported by measurement. The purpose of thepresent chapter is to summarize relevant acoustic measurements and, where feasible,to place these in a theoretical framework.

Considered first, in Section 8.1, is the interaction of sound with ocean boundariesin the form of reflection loss and scattering strength. This is followed, in Section 8.2,by measurements pertaining to the scattering and absorption of sound due to sub-merged objects (of interest are their target strength, volume backscattering strength,and volume attenuation coefficient). In Section 8.3 a mainly empirical description ofunderwater noise sources is provided.

8.1 REFLECTION AND SCATTERING FROM OCEAN BOUNDARIES

The reflective properties of the sea surface and seabed have an important influence onlong-distance propagation, which often involves multiple interactions with either orboth boundaries, especially in shallow water (see Chapter 9). Furthermore, scatteringfrom these boundaries, resulting in reverberation at a sonar receiver, is often thelimiting factor in determining the performance of an active sonar.

Interaction of sound with the ocean’s boundaries is the subject of ongoingresearch (Pace and Blondel, 2005; Jackson and Richardson, 2007). In those situationsfor which the physics is well understood it is possible to present theoretical equations

with good predictive ability. More often, a limited theoretical understanding must besupplemented with an empirical element based on measurements, resulting in a semi-empirical approach.

It is convenient to define the parameter F as the numerical value of the frequencywhen expressed in units of kilohertz such that, if ff is the frequency in hertz,

F � ff

1,000: ð8:1Þ

8.1.1 Reflection from the sea surface

8.1.1.1 Theoretical prediction for an isotropic surface wave spectrum

8.1.1.1.1 Coherent reflection coefficient

The loss due to scattering from a rough boundary can be modeled using the coherentreflection coefficient, the squared magnitude of which (see Chapter 5) is

jRj2 ¼ 1� 4k2�2Y sin �; ð8:2Þ

where � is the grazing angle of the incident wave; k is the acoustic wavenumber; � isthe RMS boundary roughness; and the dimensionless parameter Y is given by thefollowing integral over the roughness wavenumber �

Y ¼ 2�EPT�2

2

k

� �1=2 ð

G1ð�Þ�3=2 d�; ð8:3Þ

where EPT is a constant equal to 0.3814; and G1ð�Þ is the one-dimensional roughnesswavenumber spectrum. The angle � is assumed to be small compared with ðkLÞ�1=2,where L is the correlation length of the rough surface.

For water of depth H, gravity waves satisfy the dispersion relation (Milne-Thomson, 1962)

O2 ¼ g� tanhð�HÞ; ð8:4Þ

which in deep water (�H � 1) simplifies to

O2 � g�: ð8:5Þ

For this situation, and assuming an isotropic wavenumber spectrum, thewavenumber and frequency spectra are related via (Brekhovskikh and Lysanov,2003)

G1ð�Þ ¼g1=2

4��3=2Sð ffiffiffiffiffiffig�

p Þ: ð8:6Þ

8.1.1.1.2 Pierson–Moskowitz surface wave spectrum

The Pierson–Moskowitz (PM) gravity wave frequency spectrum depends in thefollowing manner on wind speed v20 (measured at a height of 20m):

SðOÞ ¼ CPMg2O�5 exp½�BPMðg=Ov20Þ4�; ð8:7Þ

362 Sources and scatterers of sound [Ch. 8

where BPM;CPM are empirical constants (see Chapter 4); and g is the acceleration dueto gravity. Substituting Equation (8.7) in Equation (8.6) gives

G1ð�Þ ¼CPM4��4

exp ��2PM�2

!; ð8:8Þ

where

�PM ¼ffiffiffiffiffiffiffiffiffiBPM

pg=v220: ð8:9Þ

Evaluation of the integral for Y in Equation (8.3) gives

Y ¼EPTGð34ÞC

1=4PM

ðk�PMÞ1=2; ð8:10Þ

where

�2PM ¼ CPM4BPM

v420g2: ð8:11Þ

It is convenient to convert wind speed to a standard reference height of 10m. Usingthe approximate ratio (Dobson, 1981)

v10=v20 � 0:94; ð8:12Þ

the coherent reflection coefficient (see Equation 8.2) can be written in the form(Ainslie, 2005b)

�logejRPMj � 1:14F 3=2vv1010

� �3

sin �: ð8:13Þ

This simple theoretical expression is known to underestimate measured sea surfacescattering loss, by a factor of order 3 (Weston and Ching, 1989), illustrating the needfor measurements to support theory. As explained in Section 8.1.1.2.1, the discrep-ancy between measurements and rough surface-scattering theory can be attributed tothe formation of wind-generated bubbles close to the sea surface, which influence theinteraction of sound with the air–sea boundary (Norton and Novarini, 2001; Ainslie,2005b).

8.1.1.1.3 Neumann–Pierson surface wave spectrum

An alternative to the PM spectrum sometimes used in older literature is theNeumann–Pierson (NP) spectrum, given by (see Chapter 4)

SðOÞ ¼ ANPO�6 exp½�2ðg=Ov5Þ2�; ð8:14Þ

where ANP is an empirical constant. Following the same procedure as previously forthe PM spectrum results in

Y ¼ 4EPT

3ð�gk�NPÞ1=29�

2A2NP�NP

� �1=10

ð8:15Þ

8.1 Reflection and scattering from ocean boundaries 363]Sec. 8.1

and

�2NP ¼ 3ð�=2Þ1=2ANPv52g

� �5

: ð8:16Þ

Converting the wind speed to a reference height of 10m using the ratio (Dobson,1981)

v10=v5 � 1:07; ð8:17Þ

Equation (8.2) results in the following expression for reflection loss (Ainslie, 2005b)

�logejRNPj ¼ 0:74F 3=2vv1010

� �4

sin �: ð8:18Þ

As previously with Equation (8.13), this expression underestimates reflection loss.This discrepancy, which is attributed to the effects of wind-generated bubbles, isaddressed in Section 8.1.1.2.1.

8.1.1.1.4 Effect of anisotropy

Cross-wind and downwind correlation lengths for an anisotropic Pierson–Moskowitzspectrum are calculated by Fortuin (1973) (see also Fortuin and de Boer, 1971). Theresulting effect on the sea surface reflection coefficient is analyzed by Kuperman(1975).

8.1.1.2 Semi-empirical surface reflection loss models

Reflection and scattering of sound from the sea surface is the subject of ongoingresearch. At low frequency (<1 kHz) and low wind speed (<5 m/s), the surface can beregarded as perfectly flat and acoustically compliant, resulting in perfect reflectionwith a � phase change. At higher frequency, or if the sea surface is roughened bywind action, roughness scattering starts to become an issue. The presence of wind-generated bubbles near the sea surface plays an important part in this process. Inprinciple these bubbles are capable of refracting, scattering, and absorbing sound,and their precise role in surface scattering is the subject of ongoing research (Nortonand Novarini, 2001; Ainslie, 2005b; Dahl et al., 2008).

In Section 8.1.1.2.1 a low-frequency sea surface reflection loss model, based onthe measurements of Weston and Ching (1989) (henceforth abbreviated ‘‘WC89’’)and valid up to an acoustic frequency of 4 kHz, is described. The surface reflection loss(abbreviated SL) is defined as

SL � �10 log10jRj2; ð8:19Þ

where R is the plane wave amplitude reflection coefficient of the sea surface. A high-frequency model developed at the University of Washington (APL-UW, 1994),intended for use above 10 kHz, is presented in Section 8.1.1.2.2.


8.1.1.2.1 Low-frequency surface loss modelRegardless of the precise physical mechanism that gives rise to them, measured lossesare of the form (WC89)

�logejRWC89j ¼ WC89F3=2 vv10

10

� �4

; ð8:20Þ

where the parameter WC89, equal to the reflection loss in nepers at a frequency of1 kHz and wind speed 10m/s, is a constant whose value appears to depend on season.On theoretical grounds, surface loss is expected to vary with grazing angle.

The WC89 measurements involve propagation over a distance of 23 km.Variations in propagation loss were monitored over time and linked to changes inwind speed. The water was well mixed, resulting in an isothermal profile. For thegeometry of this experiment (corresponding to a surface grazing angle of 1.7 deg forthe limiting ray), measured values of WC89 are (WC89; Ainslie, 2005b)

WC89 ¼0:0677 autumn

0:132 winter�spring:

�ð8:21Þ

Equation (8.20) is applicable to frequencies up to 4 kHz and wind speeds up to 13m/s.Its functional form implies that the quantity f �3=2 logjRj is a straight line if plottedagainst the fourth power of wind speed, and this behavior is illustrated by Figure 8.1.The WC89 measurements (indicated by the symbols) do indeed follow approximatestraight lines, although the gradient of the best fit straight line is different for each ofthe two data sets; hence the two different values of WC89 given in Equation (8.21).The assumption that the loss is proportional to angle would imply a loss in nepers perradian of

WC89 ¼ F 3=2vv1010

� �4

�2:3 autumn

4:5 winter�spring3:4 average.

8<: ð8:22Þ

The average value quoted is the arithmetic mean of the other two. It exceeds thetheoretical value due to surface scattering for the NP spectrum (Equation 8.18) by afactor of 4.6.

Notice the similarity in functional form between Equations (8.13) and (8.20).Both are proportional to the product of F 3=2 and a power of v10. Ainslie (2005b)demonstrates the need for a correction to Equation (8.13) from increased compres-sibility of water close to the sea surface due to the presence of wind-generatedbubbles. This increase in compressibility causes a reduction in the near-surface soundspeed and consequently an increase in the surface grazing angle. This refraction effectcan be made explicit by writing Equation (8.13) in the form

�logejRU j � 1:14F 3=2vv1010

� �3

sin �mðv10; �Þ; ð8:23Þ

where � is the grazing angle in bubble-free water; and the angle �m is the grazing anglein bubbly water at the sea surface. If the refractive index at the sea surface is nðvÞ, it


follows from Snell’s law that

sin2 �mðv; �Þ ¼ 1�cos2 �

n2ðvÞ: ð8:24Þ

A simple relationship between the refractive index n and the gas fraction U, valid forsmall bubbles that are well below resonance, is obtained using Wood’s equation fromChapter 5. Neglecting the density of air gives

n2 � ð1 �UÞ½1þ ðBw=Ba � 1ÞU�: ð8:25Þ

Assuming further that the void fraction is small (U � 1) gives

n2ðvÞ � 1þ BwBa

UðvÞ; ð8:26Þ


Figure 8.1. Measured and predicted values of the quantity F�3=2 logeð1=jRjÞ plotted vs.

ðvv10=10Þ4. Symbols are WC89 measurements; curves are theoretical predictions. The blackand gray lines indicate calculations with and without bubbles, respectively. The difference

between the solid and dashed lines is explained in the text (reprinted with permission from

Ainslie, 2005b, American Institute of Physics).#

where Bw and Ba are, respectively, the bulk moduli of water

Bw ¼ �wc2w; ð8:27Þ

and airBa ¼ P; ð8:28Þ

where P is static pressure. The isothermal bulk modulus of air is used herefor the bubble in preference to the adiabatic modulus (Ainslie, 2005b). For theHall–Novarini (HN) bubble model (see Chapter 4), the gas fraction is given by

UðvÞ ¼ 9:29� 10�7 vv10

� �3

: ð8:29Þ

The black lines in Figure 8.1 are obtained by evaluating Equation (8.23) using the HNgas fraction. The gray lines are calculated from Equation (8.13), amounting to anassumption that the surface void fraction is zero. The difference between the solid anddashed lines is in the assumed wave height spectrum. In each case the solid line is forthe PM spectrum, and the dashed one is for the alternative NP spectrum. It is clearfrom the graphs that the difference between these two spectra is considerably less thanthe effect of the bubbles.

The proposed low-frequency (LF) surface loss algorithm is the one resulting inthe solid black line

SLLF ¼ �10 log10jRU j2; ð8:30Þ

that is, Equation (8.23) evaluated with the HN bubble model. An example is shown inFigure 8.2.

The reader’s attention is drawn to the uncertainty in the near-surface bubblepopulation, and hence also in the void fraction, for any given wind speed v (seeChapter 4). A seasonal dependence between U and v might be responsible for thevariation observed in WC89 (Ainslie, 2005b). The fourth-power dependence on windspeed means that apparently minor uncertainties either in the wind speed iself, or inthe height at which it is measured, are amplified into potentially significantuncertainties in surface loss.

8.1.1.2.2 High-frequency surface loss model

At high frequency (HF) the effects of refraction become less important relative tothose of absorption. An empirical model of sea surface reflection loss due to absorp-tion by near-surface wind-generated bubbles, incorporating effects due to bubbleabsorption, and applicable in the frequency range 10 kHz to 100 kHz is given byAPL-UW (1994). A refined version of this model (Dahl, 2004) can be written

�10 log10jRHFj2 ¼20 log10 e

sin ��SLðvv10;FÞ: ð8:31Þ

where

�SLðvv10;FÞ ¼ 10�6:45þ0:47vv10F 0:85: ð8:32Þ

Figure 8.3 shows the result of evaluating Equation (8.31) at a frequency of 20 kHz,compared with measurements from Dahl et al. (2004). The rapid rise around 8m/s to


10m/s at this frequency is explained by the onset of absorption effects due to wind-generated bubbles. While the predicted absorption continues to increase beyond10m/s, the measured losses appear to level off, possibly because some sound isreflected by the bubble cloud before being absorbed by it, as described by Dahl etal. (2004). The model of APL-UW (1994) imposes an upper limit of 15 dB to describethis effect. Dahl et al. (2008) describe an improved method for estimating bubbleabsorption loss for a given wind speed and frequency.

Unlike the low-frequency formula, Equation (8.31) does not include acontribution from the loss of coherence due to rough surface scattering. The justifica-tion for this is a subtle but important qualitative difference between low-frequencyand high-frequency propagation. The former typically takes place over long dis-tances, allowing multiple interactions with the sea surface. In this situation onlythe coherent part of the field is expected to make a significant contribution at thereceiver, because of the leakage that occurs out of the waveguide after multiplescattering.


Figure 8.2. Theoretical surface reflection loss in nepers calculated vs. angle and frequency using

Equation (8.23) for a wind speed v10 ¼ 10m/s (reprinted with permission from Ainslie, 2005b,American Institute of Physics).#

By contrast, at high frequency the propagation distances are small, and theimplicit assumption here is that only the first reflection is of interest. After a singlereflection, there is no energy loss mechanism (other than absorption) because all ofthe incident energy is reflected by the rough boundary; hence the absence of acontribution to SL from coherence loss. However, the coherence of the signal isdegraded due to rough surface scattering. Depending on the signal processing usedfor its detection this might affect the performance of the sonar.

8.1.2 Scattering from the sea surface

8.1.2.1 Theoretical prediction for Pierson–Moskowitz surface wave spectrum

8.1.2.1.1 Non-specular scattering (perturbation theory)

The general perturbation theory (PT) result for the scattering coefficient fromChapter 5, assuming an isotropic roughness spectrum, is

�PTAOð�0; �; Þ ¼ 4k4 sin2 �0 sin2 � G1ð�Þ; ð8:33Þ

where G1ð�Þ is the one-dimensional roughness spectrum. Substituting for the


Figure 8.3. Predicted surface loss (SL) vs. wind speed for a frequency of 30 kHz and grazing

angle 10 deg, using Equation (8.31) (solid line) and measured SL at the same frequency

(symbols).

Pierson–Moskowitz (PM) spectrum using Equation (8.8) gives

�PTAOð�0; �; Þ ¼CPM�

k

�

� �4

sin2 � sin2 �0 exp ��2PM�2

!; ð8:34Þ

where the constant �PM is given by

�PM ¼ffiffiffiffiffiffiffiffiffiBPM

pg

v220; ð8:35Þ

a parameter that is closely related to the reciprocal of the correlation length. Thebackscattering coefficient is obtained by equating � and �0 in Equation (8.34),together with

� ¼ 2k cos �0: ð8:36ÞThe result is

�PTAOð�Þ ¼CPM16�

tan4 � exp � �2PM4k2 cos2 �

!: ð8:37Þ

At sufficiently high frequency, the exponent in Equation (8.37) may be neglected,except for angles close to normal incidence.

8.1.2.1.2 Near-specular scattering (facet-scattering theory)

The facet-scattering formula for the near-specular scattering coefficient from Chapter5 is

�AOð�0; �; Þ ¼Rð�0Þ2

8��2ð1þ DOÞ2 exp �DO

2�2

� �; ð8:38Þ

where

DO ¼ cos2 �0 þ cos2 �� 2 cos �0 cos � cos

ðsin �0 þ sin �Þ2: ð8:39Þ

For in-plane scattering, Equation (8.38) simplifies to

�AOð�0; �; Þ ¼Rð�0Þ2

8��21

sin4 �exp � 1

2�2 tan2 �

� �; ð8:40Þ

where � is the bisector angle

� ¼12ð�0 þ �Þ ¼ �12ð�0 þ �� Þ ¼ 0

(: ð8:41Þ

The backscattering coefficient is

�AOð�Þ ¼Rð�Þ2

8��21

sin4 �exp � 1

2�2 tan2 �

� �: ð8:42Þ

The parameter �2 is the mean square slope of surface roughness, which can beestimated from wind speed using the Cox–Munk relationship from Chapter 4:

�2 ¼ ð3þ 5:12vv10Þ � 10�3: ð8:43Þ


8.1.2.2 Semi-empirical surface-scattering strength models

The surface-scattering strength (SSS) is the scattering coefficient, expressed indecibels. In other words, it is defined as

SSSð�0; �; Þ � 10 log10 �AOð�0; �; Þ dB: ð8:44Þ

Similarly, the surface backscattering strength (SBS) is the backscattering coefficient,also in decibels

SBSð�Þ � 10 log10 �AOð�Þ dB: ð8:45Þ

As in Chapter 2, the use here of a single argument (�) implies evaluation inthe backscattering direction. The scattering coefficient �AO is dimensionless, sosurface-scattering strength does not require a reference unit.

Some sea surface scattering is caused by rough interface scattering at the air–seaboundary as described above. There is in general a further contribution due tovolume scattering from wind-generated bubbles close to the boundary, the impor-tance of which increases with increasing wind speed (Ogden and Erskine, 1994).Despite the fact that the bubbles are distributed in three dimensions, their associationwith the sea surface is so strong that it is difficult to separate the effects of the bubblesfrom those of the rough surface. Consequently, these two effects are usually lumpedtogether in a single surface-scattering term. The semi-empirical models describedbelow implicitly or explicitly include contributions from both effects. The low-frequency model is valid up to 1 kHz and the high-frequency one from 12 kHzupwards. A recently developed model that bridges the gap between 1 kHz and 12 kHzis described by Gauss et al. (2006).

8.1.2.2.1 Low-frequency model

The semi-empirical model proposed by Ogden and Erskine (1994), valid in thefrequency range 50Hz to 1000Hz and wind speed 0m/s to 20m/s, is described inthis section. The model is constructed around two frequency-dependent wind speedthresholds UPT and UCH, given by

UPT ¼ 7:22 240 < ff < 1,000

21:5� 0:0595ff 50 < ff < 240

(ð8:46Þ

and

UCH ¼ 20:14� 0:0340ff þ 3:64� 10�5 ff 2 � 1:330� 10�8 ff 3: ð8:47Þ

For a low wind speed (less than the threshold UPT), Equation (8.34) may be used, sothat SBS can be written

SBSPT � 10 log10 �PTAO; ð8:48Þ


with �PTAO from Equation (8.37).1 For high wind speed (exceeding UCH) the empirical

formula of Chapman and Harris (1962) is applied:

SBSCH � 3:3�CH log10�deg30

� �� 42:4 log10 �CH þ 2:6; ð8:49Þ

where �deg is the grazing angle � expressed in degrees

�deg �180

�� ð8:50Þ

and2

�CH ¼ 158 3600

1852vv10 ff

1=3

� ��0:58: ð8:51Þ

At intermediate wind speeds, higher than UPT and lower than UCH, the Ogden–Erskine model uses linear interpolation between SBSPT and SBSCH. Thus, the finalformula for low-frequency SBS is

SBSLF ¼SBSPT U � UPT

xSBSCH þ ð1� xÞSBSPT UPT < U < UCH

SBSCH U � UCH,

8<: ð8:52Þ

where

x ¼ U �UPTUCH �UPT

: ð8:53Þ

As a practical matter, except in the expression for �PM (Equation 8.35) the intendedwind speed measurement height is 10m, with a lower limit imposed of 2.5m/s. Thus,

U ¼ maxðvv10; 2:5Þ: ð8:54ÞThe conversion to a measurement height of 20m required for �PM can be made usingEquation (8.12).

Because the Ogden–Erskine model is based on scattering measurements at graz-ing angles in the range 5 deg to 40 deg, this is the angle range in which the model maybe used with confidence. Any extrapolation outside this range should take intoaccount:

— the need for a facet-scattering term close to normal incidence;— the uncertainty in the angle dependence of the scattering coefficient close to

grazing incidence, especially for wind speeds exceeding UPT;— the possible contribution from fish close to the sea surface.

8.1.2.2.2 High-frequency model (APL)

For frequencies between 12 kHz and 70 kHz, the semi-empirical model proposed byAPL-UW (1994) may be used, for grazing angles 0.5 deg to 90 deg, with linear


1 A lower limit of 1 deg is placed on the grazing angle.2 The factor 3600/1852 arises from the conversion between knots and meters per second (see

Appendix B).

extrapolation recommended for angles less than 0.5 deg. The reported accuracy of theAPL model is �4 dB for wind speeds greater than 8m/s, and �5 dB for lower windspeeds.

The APL model explicitly considers separate contributions to the scatteringcoefficient due to rough surface scattering (�rough) and volume scattering due tobubbles (�bubble)

SBSHF ¼ 10 log10 �AO; ð8:55Þwhere

�AO ¼ �rough þ �bubble: ð8:56Þ

Expressions for these two contributions are given separately below.

Contribution from bubbles. The contribution to the scattering coefficient due tobubbles is (McDaniel, 1993; APL-UW, 1994, p. II-6; Dahl et al., 1997)3

�bubble ¼D0 sin �8��res

½1� 8jRj2 logejRj � jRj4�; ð8:57Þ

where the grazing angle � is evaluated in bubble-free water; D0 is the radiationdamping coefficient at resonance introduced in Chapter 5

D0 � 0:0137 ð8:58Þ

and �res is the total damping coefficient at resonance, which varies with frequency F inkilohertz according to

�res � 2:55� 10�2F 1=3: ð8:59Þ

The parameter jRj is the surface reflection coefficient associated with absorption bywind-generated bubbles, such that reflection loss is proportional to pathlength, whichin turn is proportional to the reciprocal of sin �

�logejRj ¼�APLðvv10;FÞ

sin �: ð8:60Þ

The constant of proportionality �APL depends on wind speed in the following manner(APL-UW, 1994; Dahl et al., 1997; Dahl, 2003)

�APLðvv10;FÞ ¼10ð�5:2577þ0:4701vv10

F

25

� �0:85

vv10 � 11

�APLð11;FÞvv1011

� �3:5

vv10 > 11.

8>>><>>>:

ð8:61Þ


3 Equation (8.57) is valid for large values of the Rayleigh parameter (Dahl, 2003).

The resulting contribution to the scattering coefficient is independent of angle ifabsorption is low and independent of wind speed if absorption is high

�bubble ¼

3D04��res

�APL �logejRj � 1

D08��res

sin � �logejRj � 1.

8>><>>: ð8:62Þ

Contribution from rough surface. The APL roughness-scattering model includescontributions from roughness on two different lengthscales. One contribution, de-noted �facet for facet scattering, is due to scattering from large-scale features at anglesclose to the specular direction, which in the backscattering case corresponds tonormal incidence. The other contribution (�ripple) is from small-scale surface ripples.Total roughness scattering is calculated using the following non-linear interpolationalgorithm

�rough ¼�facet þ ex�ripple

1þ ex 10�SLHF=10; ð8:63Þ

where the interpolation parameter is

x ¼ 0:524ð�facet � �degÞ; ð8:64Þ

and the angle �facet, explained in more detail after Equation (8.66), defines thetransition between ripple and facet scattering. The surface loss term SLHF is thehigh-frequency reflection loss associated with wind-generated bubbles (see Equation8.31).

The facet-scattering term is

�facet ¼1

8��2 sin4 �exp � 1

2�2 tan2 �

� �; ð8:65Þ

where � is the RMS roughness slope, given by

�2 ¼ 2:3� 10�3 logeð2:1vv210Þ vv10 � 10:0017 vv10 < 1.

(ð8:66Þ

The angle �facet, measured in degrees, is the angle at which 10 log10 �facet falls to a levelthat is 15 dB below its maximum value. More precisely, it is the smallest anglethat satisfies the inequality 10 log10½�facetð90�Þ=�facetð�Þ� � 15 dB (APL-UW, 1994,p. II-9).

Finally, the near-grazing (ripple) term is given by

�ripple ¼1:3� 10�5vv210 tan4 � � � 85�

0 � > 85�.

�ð8:67Þ


8.1.3 Reflection from the seabed

It is conventional to quantify the loss of acoustic energy associated with reflectionfrom the seabed using the logarithmic term bottom reflection loss, denoted BL anddefined in the same way as for surface loss:

BL � �10 log10jRj2; ð8:68Þ

where R is the plane wave amplitude reflection coefficient of the seabed.The main loss mechanism for sound incident on the seabed is through the

transmission of sound into the sediment. If there is a second reflecting layer closeto the water–seabed boundary, or if the sediment properties vary continuously withdepth, then some of the transmitted energy will subsequently be reflected or scattered.Otherwise, it continues on its downward path and no longer contributes topropagation in the ocean waveguide.

If there is no second reflection, the reflection loss depends only on the propertiesof the water–sediment boundary, and this situation is considered in Section 8.1.3.1.The effects of fine-scale layering are described for the case of an unconsolidatedsediment in Section 8.1.3.2 and for a hard solid seabed in Section 8.1.3.3, includinglayering on a depthscale of order 10m to 100m. In the former the emphasis is on highfrequency and in the latter on low frequency, although there is no clear-cut distinctionbetween the two.

8.1.3.1 Theoretical prediction for uniform unconsolidated sediment

8.1.3.1.1 Fluid sediment

Although in reality the seabed is never perfectly uniform, it may be approximated assuch when there is a single dominant reflecting boundary at the water–sedimentinterface. One is dealing with sediment properties that are averaged in depth, andthis averaging must be over a depthscale that is appropriate to the acoustic frequencyof interest—usually a few wavelengths. In Chapter 4, two distinct sets of sedimentproperties are described: near-surface properties, representative of the top few cen-timeters of sediment and intended for use at high frequency (ca. 10–100 kHz), andbulk properties for use at lower frequency (ca. 1–10 kHz).

Bottom reflection loss calculated using bulk properties is shown in Figure 8.4(blue lines, upper graph), and using near-surface properties (lower graph). Thesecalculations assume a semi-infinite uniform fluid model for the sediment with aperfectly smooth boundary, for which the Rayleigh reflection coefficient is applicable(see Chapter 5)

Rð�Þ ¼ �ð�Þ � 1�ð�Þ þ 1 ; ð8:69Þ

where

�ð�Þ ¼ wtan �

tan �sed; ð8:70Þ



Figure 8.4. Predicted seabed reflection loss vs. grazing angle for uniform unconsolidated

sediments. Upper: MF parameters (for grain size �0.5 to þ10 ); lower: HF parameters(�0.5 to þ8:5 ). Blue lines: fluid sediment; dotted red lines: solid sediment with the samecompressional speed as the fluid case, and shear speed evaluated at a depth of 10m (see

Chapter 4).

w is the density ratio

w � �sed�w

; ð8:71Þ

and � and �sed are the ray grazing angles in water and sediment, respectively.If the fractional imaginary part of the sediment wavenumber is denoted ", such

that

ksed ¼!

csedð1þ i"Þ; ð8:72Þ

and v is the sound speed ratio

v � csedcw

; ð8:73Þ

it follows using Snell’s law in the form

ksed cos �sed ¼!

cwcos �; ð8:74Þ

that �sed is a complex angle given by

�sed � arccos vcos �

1þ i"

� �: ð8:75Þ

The parameter " is related to the sediment attenuation coefficient �sed (in decibels perwavelength) according to

" ¼ loge 1040�

�sed: ð8:76Þ

The character of the Figure 8.4 reflection loss curves depends on grain size. Thecoarser (sandy) sediments exhibit a critical angle, denoted c, below which totalinternal reflection occurs, meaning that for � < c the magnitude of the reflectioncoefficient is close to unity. The critical angle is given by

c ¼ arccosð1=vÞ ðv > 1Þ: ð8:77Þ

For example, the critical angle of coarse silt (Mz ¼ 4.5), using the MF parametersfrom Chapter 4, is about 22 deg.

For grazing angles � < c, there is no transmitted wave and, in the absence ofsediment attenuation, the angle �sed is then imaginary. In this situation the reflectionloss is zero and the parameter tan �sed in Equation (8.70) is interpreted as

tan �sed ¼ iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2sed=ðk2w cos2 �Þ

q: ð8:78Þ

If some attenuation is present (i.e., if �sed > 0), the reflection loss at sub-criticalgrazing angles, though small, increases slightly with increasing angle � in proportionwith sin � such that

jRj2 � expð�2 sin �Þ ð� < cÞ; ð8:79Þor equivalently

BL � ð10 log10 eÞ2 sin �: ð8:80Þ


The parameter is a function of seabed density, sound speed, and attenuationcoefficient. Even if there is no critical angle, it is still possible to write reflectionloss in the form of Equation (8.80) for small angles, although the value of ismuch higher for clay than it is for sand. A general result that holds regardless ofseabed impedance is

¼ 2Re Zsed�wcw

; ð8:81Þ

where

Zsed ��wcwsin �w

1þ R

1� R¼ �wcwsin �w

�; ð8:82Þ

and � is the impedance ratio (defined as Zsed sin �w=ð�wcwÞ and given for a fluidsediment by Equation 8.70). The right-hand side of Equation (8.81) is understoodto be evaluated at grazing incidence.

Applying Equation (8.81) to the present example gives

0 ¼ 2wRe~vv

ð1� ~vv2Þ1=2; ð8:83Þ

where ~vv is the complex equivalent sound speed, defined as

~vv � v1þ i" ; ð8:84Þ

and the zero subscript in Equation (8.83) indicates that this expression for applies tothe case of a fluid sediment (i.e., with a zero or negligible shear speed, irrespective ofthe sound speed ratio v).

Coarse-grained sediments. If v > 1, a characteristic of coarse sediments, 0 isproportional to the imaginary part of the complex wavenumber (Weston, 1971)

0 ¼ 2w"v

ðv2 � 1Þ3=2ðv > 1Þ; ð8:85Þ

or equivalently, in terms of the critical angle c (see Equation 8.77)

0 ¼ 2w"cos2 csin3 c

ðcos c < 1Þ: ð8:86Þ

Fine-grained sediments. For fine sediments (satisfying v < 1) there is no criticalangle and Equation (8.83) simplifies to

0 � 2wv

ð1� v2Þ1=2: ð8:87Þ

If w > 1=v is also satisfied, there exists an angle of intromission at which the reflectioncoefficient passes through zero. This angle, denoted �in, is given by

sin2ð�inÞ0 ¼1� v2

v2ðw2 � 1Þ: ð8:88Þ

As before, the zero subscript indicates that this equation is for a fluid sediment.


8.1.3.1.2 Effect of a small non-zero shear speed

If a plane wave is incident on a fluid seabed at a grazing angle steeper than the criticalangle c, a compressional wave is excited in the sediment. In this situation, most ofthe incident energy is converted into this transmitted p-wave, carrying the energyaway from the water column—hence the high reflection loss. For the case of a solidseabed, there exists additionally the possibility of exciting a shear wave, whichpropagates at a grazing angle �s. If the shear speed in the sediment is less than thesound speed in water, a shear wave is always generated. It is convenient to define thecomplex sound speed ratios ~vvp and ~vvs using

~vvX � vX1þ i"X

; ð8:89Þ

where the X subscript denotes a property of either a compressional wave (if X ¼ p) ora shear wave (X ¼ s),

vX � cXcw; ð8:90Þ

and

"X ¼ loge 1040�

�X: ð8:91Þ

The corresponding impedance ratios can be defined as

�Xð�Þ ¼ wtan �

tan �X; ð8:92Þ

where�X � arccosð~vvX cos �Þ: ð8:93Þ

With these definitions, Equation (8.69) for the reflection coefficient still holds,provided that the impedance ratio � is generalized to

� ¼ �pð�Þ cos2 2�s þ �sð�Þ sin2 2�s: ð8:94Þ

The main effects on and �in of a small but non-zero (relative) shear speed vs,illustrated by the red lines in Figure 8.4, are described below.

Effect on reflection loss. An unconsolidated sediment is weakly capable ofpropagating shear waves because it has a slightly non-zero rigidity. For a non-zero sediment shear speed, Equation (8.83) becomes (Ainslie, 1992)

¼ 2wRe~vvp

ð1� ~vv2pÞ1=2ð1� 2~vv2s Þ2 þ 4~vv3s ð1� ~vv2s Þ1=2

" #: ð8:95Þ

If ~vvs has a negligible imaginary part, it follows that (Eller and Gershfeld, 1985)4

¼ 0ð1� 2v2s Þ2 þ 8wv3s ð1� v2s Þ1=2 ðvp > 1 > vsÞ: ð8:96Þ


4 A similar expression is derived by Chapman (1999), including an additional term associated

with the imaginary part of ~vvs.

For constant vs, the first term is proportional to 0, the value for a fluid, which isgiven by Equation (8.83). The lowest order effect of vs is to reduce the loss for the fluidcase by a factor of approximately (1� 4v2s ). This reduction is clearly visible in Figure8.4, especially for high-frequency (APL) parameters.

The second term represents a loss mechanism that does not exist for the fluidcase, namely the conversion of energy into sediment shear waves. This term is usuallynegligible for unconsolidated sediments, but it becomes important for harderconsolidated sediments like chalk or mudstone (see Section 8.1.3.3).

Effect on angle of intromission. The effect of shear speed on the intromissionangle can be calculated as follows. The condition for intromission is Re � ¼ 1 inEquation (8.69), with � given by Equation (8.94). This gives

sin2 �w ¼1 � v2pv2p

½1� Dð�wÞ�2

w2 � ½1� Dð�wÞ�2; ð8:97Þ

whereDð�Þ � Re½�ð�Þ � �pð�Þ�; ð8:98Þ

which can be written in the form (using the notation OðxÞ to indicate a term oforder x)

D ¼ �4v2s sinð�inÞ0 cos2ð�inÞ0 vp

w2 � 11� v2p

!1=2

þOðvsÞ" #

: ð8:99Þ

The intromission angle �in is then the value of �w that satisfies Equation (8.97). Giventhe assumption that vs is small, it follows that D must also be small. Equation (8.97)can then be written as a Maclaurin series in D

sin2 �w ¼ sin2ð�inÞ0 1�2w2

w2 � 1Dð�wÞ þOðD2Þ

" #; ð8:100Þ

where ð�inÞ0 is the intromission angle for a fluid sediment given by Equation (8.88).Expanding Equation (8.98) in powers of vs, it can be shown that

sin2 �in ¼ sin2ð�inÞ0 1þ 8v2s

w

w2 � 1cos2ð�inÞ0 þOðv3s Þ

� �: ð8:101Þ

Thus, given that w > 1 must be satisfied for the existence of ð�inÞ0, the consequence ofa small non-zero sediment shear speed is a small increase in the angle of intromission,as illustrated by Figure 8.4.

8.1.3.2 Theoretical prediction for layered unconsolidated sediment (1–100 kHz)

The somewhat arbitrary distinction made above between ‘‘high frequency’’ or ‘‘HF’’(for which near-surface properties are used) and ‘‘medium frequency’’ or ‘‘MF’’ (bulkproperties used) leads to an undesirable artifact in the predicted reflection coefficient,namely a step change in the predicted reflection loss at a frequency of 10 kHz. Forsome applications, the correct (continuous) dependence on frequency might be


required, in which case an improved model is necessary. This problem can beaddressed by constructing a layered medium whose properties vary continuouslywith depth from their near-surface values at the top of the sediment to bulk propertiesat a depth equal to the thickness of the transition region between near-surfaceproperties and bulk properties. The thickness of this transition layer is typically oforder 1 cm to 10 cm.

The layered model described below covers the approximate frequency range1 kHz to 100 kHz. Above 100 kHz the same method is applicable except that layeringon an even finer scale becomes relevant, with the near surface properties eventuallybecoming indistinguishable from those of water (Ainslie, 2005a). For lower fre-quencies (below 1 kHz), it is necessary to include large-scale features on a depthscaleof tens or even hundreds of meters (Section 8.1.3.3).

Three sediment types are considered, representing fine sand, medium silt, andcoarse clay. The properties of these three sediments, taken from Chapter 4, aresummarized in Table 8.1. Figure 8.5 shows reflection loss plotted vs. angle andfrequency for each of the three cases.

The precise sound speed profile used in the transition layer is the ‘‘linear k2’’ case,for which the wavenumber profile kðzÞ is given by

kðzÞ2 ¼ kð0Þ2ð1� 2qzÞ: ð8:102Þ

The gradient parameter q is a complex constant given by

2qh ¼ 1 � cð0Þ2

cðhÞ21þ i"ðhÞ1þ i"ð0Þ

� �2

; ð8:103Þ

where

"ðzÞ ¼ loge 1040�

�ðzÞ: ð8:104Þ

This choice of q ensures that the correct values of sound speed and attenuation areobtained at the top and bottom of the transition layer, from Table 8.1 (at z ¼ 0 and h,respectively). The complete sound speed and attenuation profiles are

cðzÞ ¼ !

Re kðzÞ ð8:105Þ


Table 8.1. Sediment properties at top and bottom of the transition layer.

Sediment Mz Sound speed ratio Density ratio Attenuation coefficient

description (c=cw) ( �=�w) � (dB/�)at z ¼ 0 (at z ¼ h) at z ¼ 0 (at z ¼ h) at z ¼ 0 (at z ¼ h)

Fine sand 2.5 1.1073 (1.1534) 1.451 (1.945) 0.85 (0.89)

Medium silt 5.5 0.9885 (1.049) 1.149 (1.601) 0.36 (0.38)

Coarse clay 8.5 0.9812 (0.9911) 1.145 (1.378) 0.08 (0.08)


Figure 8.5. Predicted seabed reflection loss vs. angle and frequency–sediment thickness product

for a layered unconsolidated sediment.Upper: fine sandMz ¼ 2:5 (�h ¼ 1.1092); lower: mediumsilt Mz ¼ 5:5 (�h ¼ 1.1840); right: coarse clay Mz ¼ 8:5 (�h ¼ 0.8741).

and

�ðzÞ ¼ 40�

loge 10

Im kðzÞRe kðzÞ : ð8:106Þ

The linear k2 profile is adopted here because it provides a simple method forpredicting the complicated frequency dependence illustrated by Figure 8.5. If theparameters were kept fixed at the values specified by Table 8.1, use of a more realisticsediment sound speed profile would alter the detailed behavior in the transitionregion, but would not influence either low-frequency or high-frequency limits.

Robins’s density profile is taken from Chapter 5, with � chosen to ensure a zerodensity gradient at z ¼ h, that is,

� ¼ ��ð0Þ tanh �h

2

� �; ð8:107Þ

so that

�ðzÞ ¼ �ð0Þ

cosh�z

2

� �� tanh �h

2

� �sinh

�z

2

� �� 2

ð8:108Þ

and

d�ðzÞdz

¼ ��ð0Þ �ðzÞ�ð0Þ

� �3=2

cosh�z

2

� �tanh

�h

2

� �� tanh �z

2

� �� : ð8:109Þ


Continuity of the density at z ¼ h is ensured by choosing �h as

�h ¼ 2 artanh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �ð0Þ

�ðhÞ

s: ð8:110Þ

At depth z > h the bulk properties are used for all parameters.In the graphs of Figure 8.5, seabed reflection loss is calculated vs. angle and

frequency using the method of Robins (1991). The reflection loss for this sedimentmodel is a function of the product fh, where f is the acoustic frequency, and not of fand h separately. This symmetry is exploited here by choosing the y-axis parameter asy ¼ log10ð ff hhÞ. The advantage of displaying the data in this way is that the graph canbe applied to combinations of frequency and layer thickness over a wide range ofparameter values.

Below fh ¼ 100m/s (i.e., for y < 2, corresponding to f < 2 kHz with a layerthickness h ¼ 5 cm), the plots tend to stabilize towards their low-frequency limits,as can be expected. Similarly, above 10 km/s ( y > 4, or f > 200 kHz with h ¼ 5 cm),they tend towards the high-frequency (APL) limit. The transition between these casesfor the sand sediment is straightforward, showing a shift in the critical angle from30 deg to 25 deg. For clay there is a similar shift, this time in the angle of intromission.For the silt case the behavior is more complex, especially close to grazing incidence,because the nature of the reflection process changes from one of total internalreflection at low frequency to almost total transmission at high frequency.

8.1.3.3 Theoretical prediction for layered solid seabed (<1 kHz)

If the frequency is sufficiently low, the layering in the uppermost meter or so ofsediment may be ignored. Instead it becomes necessary to consider deeper layering ona depthscale of 10m to 100m, depending on the acoustic frequency. It might also benecessary to consider interaction with the solid rock beneath the sediment layer.

Before proceeding to a layered model, it is useful to consider the reflectiveproperties of a half-space of solid rock. Figure 8.6 shows reflection loss vs. anglefor four representative rock types from Chapter 4. The curves for sandstone andlimestone (not shown) are similar to the one for basalt. As in the case of anunconsolidated sediment previously, the shape of each of these curves depends onthe corresponding values of vp, except that here the requirement for small vs is lifted.Consider first the case vs < 1, corresponding to the two softest rocks, chalk andmudstone. For these two cases there exists a critical angle (denoted p) atarccosð1=vpÞ, equal to 52 deg and 43 deg for chalk and mudstone, respectively.Because of the low shear speed, a shear wave is always generated in the rock,irrespective of the grazing angle in water, so that total internal reflection is notpossible. The loss of energy associated with this mechanism, proportional to v3s(see Equation 8.96), can be very high.5

For the remaining rock types, all satisfying vs > 1, there is a second critical angle,equal to arccosð1=vsÞ and denoted s. This is the critical angle for the generation of


5 See Fig. 2 from Hughes et al. (1990) for an example with a chalk sediment.

sediment s-waves. At smaller angles neither p-waves nor s-waves can be generatedand the result is total internal reflection. For the harder rocks the p and s criticalangles are clearly identifiable in Figure 8.6 due to the abrupt change in reflection lossthrough these angles. For example, the basalt s and p critical angles are 51 deg and72 deg, respectively. The critical angles for different rock types are summarized inTable 8.2.

Now consider a thin layer of unconsolidated sediment sandwiched between therock half-space below and the sea above. Two specific cases of interest are a layer of


Figure 8.6. Seabed reflection loss vs. grazing angle for rocks.

Table 8.2. p and s critical angles for representative rock parameters (see Chapter 4).

Type of rock cp=ms�1 cs=ms

�1 p=deg s=deg

Mudstone 2050 600 43 —

Chalk 2400 1000 52 —

Sandstone 4350 2550 70 54

Basalt 4750 2350 72 51

Granite 5750 3000 75 60

Limestone 5350 2400 74 52

sand over granite and a clay layer over basalt. These two combinations are repre-sentative of typical shallow-water and deep-water seabeds, respectively. Figure 8.7shows BL vs. angle and frequency for both cases, with the sediment layer treated as auniform fluid. The properties of sediment and rock layers are summarized in Table8.3. For this situation there can be up to three different critical angles: the usualcritical angle for the fluid sediment ( 2), plus p and s, the two critical angles forrock described above.

The sediment has no effect in the low-frequency limit (kh � 1) because itbecomes transparent to sound. In this situation, reflection loss is expected to be thatfor a rock half-space alone, and this is confirmed by Figure 8.7 for frequenciessatisfying fh < 100m/s. The p and s critical angles are clearly visible for both granite(upper graph) and basalt (lower graph), at the edges of the two vertical stripes. Thethickness h is that of the whole sediment layer, which in this case is the mediumsandwiched between seawater and rock.

At high frequency the opposite situation can arise. If the frequency is highenough, none of the sound reaches the rock, and reflection loss is that of an infinitesediment half-space. For the sand case the critical angle of 30 deg is clearly visible forfh exceeding 10 km/s. For clay the asymptotic limit is not quite reached, even at100 km/s, but the angle of intromission is nevertheless apparent.

At intermediate frequencies, the reflected field includes contributions from bothboundaries. The fringes visible on both graphs result from interference between theresulting multipaths.

Of particular importance to the performance of sonar is the behavior at smallangles, because it is this that determines the viability of long-range propagation. Theclay sediment exhibits a sequence of interference nulls close to grazing incidencestarting at fh � 3 km/s. This behavior, described by Hastrup (1980), is characteristicof a low sound speed sediment (satisfying c2 < cw). Hastrup’s condition for determin-ing the frequencies at which destructive interference occurs can be generalized by firstwriting the reflection coefficient in the form (from Chapter 5)

R ¼ R12 þ R23 expð2i�2hÞ1� R21R23 expð2i�2hÞ

; ð8:111Þ

where �2 is the vertical wavenumber in the sediment layer. Requiring that thenumerator of Equation (8.111) be zero, and approximating the two-layer coefficientsusing

Rij ¼�i j � 1�i j þ 1

� �expð�2�i jÞ; ð8:112Þ

the condition becomes

expð�2�12Þ þ expð2i�2h� 2�23Þ ¼ 0: ð8:113Þ

Close to grazing incidence, given the assumption of small c2, if sediment attenuationis negligible, then �12 is a real number. Assuming further that cp; cs are both largerthan cw, and neglecting rock attenuations as well, it follows that �23 is imaginary.



Figure 8.7. Predicted seabed reflection loss vs. grazing angle and frequency–sediment thickness

product for a sand sediment overlying a granite basement (upper graph) and clay over basalt

(lower). The sediment layer is treated as a uniform fluid medium.

Thus, the only way that the left-hand side of Equation (8.113) can vanish is for thephase of the second term to be an odd integer. In other words

2�2h� 2 Im �0 ¼ 2nþ 1; ð8:114Þ

where n is a non-negative integer; and �0 is the impedance ratio at the sediment–rockboundary (�23) evaluated at grazing incidence in water (�w ¼ 0)

�0 ¼ �iw c2wc22

� 1 !

1=2 1

sin p� 4 sin

2 scos4 s

1

sin s� 1

sin p

� �" #: ð8:115Þ

It is understood that �2 is also evaluated at grazing incidence.Rearranging for the frequency, Equation (8.114) can be written

fh

cw¼ 1

4�

c2wc22

� 1 !

�1=2½ð2nþ 1Þ�þ 2 Imð�0Þ� ðcw > c2Þ: ð8:116Þ

For the sand case with a granite half-space, there is a single interference null close tograzing incidence, at a frequency–thickness product close to 200m/s. This feature ischaracteristic of a seabed comprising a thin layer of sand over a hard rock basement.It is caused by a resonant evanescent wave in the sediment, which occurs at afrequency determined by (Ainslie, 2003)

fh

cw¼ 1

4� sin 2loge

�0 � 1�0 þ 1

: ð8:117Þ

The impedance ratio �0 is defined in the same way as above. For the sand case, withc2 > cw, it is more convenient to write Equation (8.115) in the form

�0 ¼ w sin 21

sin p� 4 sin

2 scos4 s

1

sin s� 1

sin p

� �" #: ð8:118Þ


Table 8.3. Defining parameters for the two cases involving a uniform fluid sediment and a hard rock

half-space. The sediment parameters are evaluated using the Bachman correlations of Chapter 4. Rock

wave speeds are from Table 8.2.

Medium fh=(m s�1) �=(kgm�3) cp=(m s�1) �p=(dB�

�1) cs=(m s�1) �s=(dB�

�1)

Water 1 1027 1490 0 0 0

Sediment

Fine sand (Mz ¼ 2:5) 30 to 105 1997 1717 0.89 0 0

Coarse clay (Mz ¼ 8.5) 30 to 105 1415 1478 0.08 0 0

Rock

Granite 1 2650 5750 0.10 3,000 0.20

Basalt 1 2550 4750 0.10 2,350 0.20

For this evanescent resonance to exist at all, the critical angle (in water) for theexcitation of shear waves in the rock layer must exceed about 17 deg (Ainslie,2003).

Figure 8.7 shows features that are characteristic of the interaction of sound witha fluid sediment layer overlying a solid half-space. Real sediments, even if unconso-lidated, have a small but non-zero shear speed, which becomes important at lowfrequency. Furthermore, the sediment layer is never perfectly uniform but has asound speed that tends to increase with increasing depth. The consequences of thisgradient are particularly important in deep water, where sediment thickness can bemeasured in hundreds or even thousands of meters. The effects of non-zero compres-sional speed gradient and non-zero shear speed are considered next, by modifying theparameters of Table 8.3 as shown in Table 8.4. Uniform sound speed and densitygradients are applied in the sediment layer of 1 s�1 and 1 kgm�4, respectively. Shearspeed also varies with depth. The value for shear attenuation corresponds to 10 dB/(mkHz), representative of typical values from Chapter 4.

The case with a sand sediment and granite substrate represents a typical shallow-water seabed and is assigned a sediment thickness of 10m. In deep water, representedby the clay–basalt combination, sediment thickness is usually much greater than this,and a value of 300m is used.

Reflection loss vs. angle and frequency for the sand–granite substrate, evaluatedusing the method of Chapman (2004) (see Chapter 5), is shown in the upper graph ofFigure 8.8. The horizontal stripes between 5Hz and 50Hz are associated with asequence of shear wave resonances in the sediment layer (Hughes et al., 1990; Ainslie,1995; Tollefsen, 1998). Similar features are also visible for clay (lower graph),although they appear at a lower frequency (0.3–3Hz) due to the greater sedimentthickness. Compared with Figure 8.7, the Hastrup resonances are shifted up infrequency, to the extent that in Figure 8.8 only one remains visible, close to200Hz. The reason for this shift is a slightly different mechanism caused by thesediment sound speed gradient. The near-grazing sound is not reflected by rock,but refracted upwards by this gradient. The resulting resonance frequencies can be


Table 8.4. Defining parameters for a layered solid medium. The shear speed profile is given by

ccsðzÞ ¼ ½79þ 41 expð�0:4MzÞ�zz0:31, with Mz equal to 2.5 or 8.5 (see Chapter 4).

Medium h=m �=(kgm�3) cp=(m s�1) �p=(dB�

�1) cs=(m s�1) �s=(dB�

�1)

Water 1 1027 1490 0 0 0

Sediment

Fine sand (Mz ¼ 2:5) 10 1997þ zz 1717þ zz 0.89 ccsðzÞ 0.01ccsðzÞCoarse clay (Mz ¼ 8.5) 300 1415þ zz 1478þ zz 0.08 ccsðzÞ 0.01ccsðzÞ

Rock

Granite 1 2650 5750 0.10 3000 0.20

Basalt 1 2550 4750 0.10 2350 0.20


Figure 8.8. Predicted seabed reflection loss vs. grazing angle and frequency for a sand sediment

of thickness 10m overlying a granite basement (upper graph) and a clay sediment of thickness

300m over basalt (lower). The sediment is treated as a layered solid medium.

determined using the expression (Ainslie and Harrison, 1989)

f ¼ 32ðnþ 1

4Þc0

vp2ð1� vpÞ

� �3=2

; ð8:119Þ

giving a value of 194Hz (for vp ¼ 0.9923 and n ¼ 0), independent of sedimentthickness.

The frequency ranges used for Figure 8.8 are chosen to correspond to the samerange of fh values used in Figure 8.7 (30m/s to 100 km/s), thus making it straightfor-ward to compare these two figures visually. For both sand–granite and clay–basaltseabeds the main effects of non-zero sediment shear speed are the horizontal lines forgrazing angles up to about 45 deg, apparent for fh < 1 km/s. The sound speedgradient is responsible for the low loss near grazing incidence for the clay–basaltcase, around 10Hz to 30Hz in Figure 8.8. The effect of the density gradient is small.

8.1.4 Scattering from the seabed

The bottom scattering strength (BSS) and the bottom backscattering strength (BBS)are defined in an analogous way to their counterparts for surface scattering, that is,

BSSð�0; �; Þ � 10 log10 �AOð�0; �; Þ dB ð8:120Þand

BBSð�Þ � 10 log10 �AOð�Þ dB: ð8:121Þ

A theoretical calculation for the scattering coefficient from a rough seabed, includingboth boundary roughness and volume scattering, is presented in Section 8.1.4.1.However, there are many additional complications due to the possible presence of:

— roughness on grossly different lengthscales, from millimeters to megameters(APL-UW, 1994);

— buried shell fragments and gravel (Goff et al., 2004; Simons et al., 2007);— hard rough boundaries exposed or beneath a layer of sediment (Essen, 1994;

Jackson and Ivakin, 1998; Gragg et al., 2001);— pockets of trapped gas (Boyle and Chotiros, 1995);— fine sediment in suspension close to the sea floor (e.g., after a storm);— demersal fish and other animals or plants resident in, on, or near the sea floor;— a sound speed gradient within the sediment layer (Mourad and Jackson, 1989);— a tidal or current shear flow.

Any of these special effects can be modeled in principle (Jackson and Richardson,2007), but knowing in advance which of them are actually needed in practice is aproblem. See, for example, the discussion in Chapman et al. (1997). A semi-empiricalapproach based partly on measurements mitigates the risk of omitting an importanteffect. Examples of empirical or semi-empirical models are described in Section8.1.4.2.


8.1.4.1 Theoretical prediction for a fluid seabed with a rough boundary and auniform distribution of embedded scatterers

In Section 8.1.4.1.1 the theoretical scattering coefficient due to seabed boundaryroughness is described. In practice, there can also be important contributions tothe scattering from volume inhomogeneities in the sediment, as described in Section8.1.4.1.2. Near the specular direction the contribution from facet scattering isimportant (Section 8.1.4.1.3).

8.1.4.1.1 Non-specular scattering from rough boundary (perturbation theory)

The scattering coefficient for the rough boundary, with roughness spectrum Wð�Þbetween water and a (fluid) seabed, can be written (Kuo, 1964)

�AOð�Þ ¼ 4k4 sin4 �jYð�Þj2Wð2k cos �Þ; ð8:122Þ

where Yð�Þ is related to the reflection coefficient as follows

Yð�Þ ¼ Rð�Þ þ 2wðw� 1Þðw tan �þ tan �sedÞ2

; ð8:123Þ

and �sed is the sediment grazing angle defined by Equation (8.75). Close to normalincidence, the correction is negligible and Yð�Þ may be approximated by Rð�Þ. Neargrazing incidence, Yð�Þ tends to

Yð�Þ ! Rð�Þ � 2wðw� 1Þsin2 c

; ð8:124Þ

where c is the sediment critical angle.Seabed roughness can be parameterized by means of a power law spectrum of the

form (Sternlicht and de Moustier, 2003)

Wð�Þ ¼ W0

100

��

� ��

; ð8:125Þ

where the constant W0, the roughness spectral density corresponding to awavenumber of 1 cm�1, is correlated with grain size according to

W0 ¼20:7 mm4 2:03846 � 0:26923Mz

1:0þ 0:076923Mz

� �2

�1 � Mz < 5:0

5:175 mm4 5:0 � Mz < 9:0

8><>: ð8:126Þ

and the exponent � is typically between 2 and 4 (APL-UW, 1994). A nominal value of3.25 is suggested by Sternlicht and de Moustier (2003). Figure 8.9 shows the theo-retical BBS associated with rough boundary scattering as a function of grazing anglefor a sand sediment, and for various frequencies between 1 kHz and 30 kHz, asmarked. The frequency dependence arises from the k4 factor in Equation (8.122)and the �� term in Equation (8.125). Combined, these give a frequency dependenceof k4�� . Comparison with measurements from Jackson and Briggs (1992) at afrequency of 35 kHz is shown in Figure 8.10. At low frequency it is expected that


scattering from sediment volume inhomogeneities, addressed in Section 8.1.4.1.2,would make a significant contribution to the total.

8.1.4.1.2 Scattering from sediment volume

Spatial fluctuations in both density and sound speed can occur within the sedimentvolume. Even if the water–sediment boundary were perfectly smooth, such fluctua-tions, called volume inhomogeneities, would scatter some of the incident sound. Suchscattering is often treated as part of the bottom scattering coefficient, as if it hadoriginated from a rough surface at the water–sediment boundary. This is because thetwo mechanisms are difficult to differentiate, similar to the situation with bubblesclose to the air–sea boundary. If the differential scattering cross-section per unitvolume of the sediment is denoted �VO, the contribution from sediment volumescattering to the scattering coefficient can be written (Stockhausen, 1963; Jacksonand Briggs, 1992)

�AOð�Þ ¼1

4v"

Imðtan �sedÞ�VO

sed

sin2 �j1� Rð�Þ2j2

cos3 �jtan �sedj2; ð8:127Þ


Figure 8.9. Predicted seabed backscattering strength for a medium sand sediment (Mz ¼ 1.5)and frequency 1 to 30 kHz, evaluated using Equation (8.122) (with Equation 8.125). The

calculation is for roughness scattering only. Lambert’s rule (Equation 8.136) with a Lambert

parameter of �25 dB is included for comparison.

where " is the fractional imaginary part of the sediment wavenumber. Equation(8.127) is valid for all angles � between 0 deg and 90 deg, regardless of the valueof the critical angle c. If � < c is satisfied—the condition for total internalreflection—it simplifies approximately to

�AOð�Þ �"

4

�VO

sedtan2 �

cos ccos �

j1 �Rð�Þ2j2

1� cos2 c

cos2 �

!3=2; ð8:128Þ

and for � > c to

�AOð�Þ �1

4

�VO

sedv2sin2 �

sin �sedj1 �Rð�Þ2j2: ð8:129Þ

Kuo’s boundary roughness model can be combined with a volume scattering term togive (Mourad and Jackson, 1989; Jackson and Briggs, 1992)

�AOð�Þ ¼ �roughð�Þ þ �volð�Þ; ð8:130Þ

where �rough and �vol are the contributions from roughness and volume scattering asgiven by Equations (8.122) and (8.127), respectively. Further refinements to thismodel are described by APL-UW (1994). For the bistatic case, see also Williamsand Jackson (1998).

The ratio �VO=sed in Equation (8.127) is a small dimensionless number of order10�2. APL-UW (1994) recommends the following values, depending on the grain


Figure 8.10. Comparison between predicted andmeasured seabed backscattering strength for a

fine sand sediment (Mz ¼ 3.0) and frequency 35 kHz. Dashed line: roughness scattering only;solid line: roughness scatteringþ volume scattering (reprinted with permission from Jackson

and Briggs, 1992, American Institute of Physics).#

size

�VO

sed¼ 20

loge 10

0:002 �1:0 � Mz < 5:5

0:001 5:5 � Mz � 9:0

�: ð8:131Þ

Figure 8.11 shows the theoretical BBS due to volume scattering as a function ofgrazing angle for a clay sediment. The prediction is independent of frequency.Comparison with measurements from Jackson and Briggs (1992) at a frequency of20 kHz is shown in Figure 8.12.

8.1.4.1.3 Near-specular scattering (facet-scattering theory)

The near-specular scattering coefficient can be calculated in the same way as for thesea surface using Equation (8.38) (for the bistatic case) or Equation (8.37) (forbackscatter). For the seabed the parameter �2 is the mean square slope of bottomroughness. Pouliquen and Lurton (1994) suggest values of roughness angle between3 deg (for mud) and 11 deg (rock), where � ¼ tan .

8.1.4.2 Empirical and semi-empirical seabed scattering strength models

Because of the difficulties associated with theoretical predictions of bottomscattering, empirical and semi-empirical models play an important role in the


Figure 8.11. Predicted seabed backscattering strength for a coarse clay sediment (Mz ¼ 8.5),evaluated using Equation (8.127) (with Equation 8.131). The calculation is for volume scatter-

ing only; Lambert’s rule (Equation 8.136) with a Lambert parameter of �25 dB is included forcomparison (reprinted with permission from Ainslie, 2007, American Institute of Physics).#

calculation of BBS for sonar performance prediction. A commonly used empiricalscattering model, known as Lambert’s rule and described in Section 8.1.4.2.1,assumes arbitrarily that incoming sound is scattered equally in all possible directions.Ellis and Crowe (1991) combine Lambert’s rule with a facet-scattering term asdescribed in Section 8.1.4.2.2. The empirical model presented in Section 8.1.4.2.3is based on the measurements of McKinney and Anderson (1964).

8.1.4.2.1 Diffuse scattering model (empirical)

Lambert’s rule If it is assumed that all incident energy is scattered diffusely (i.e.,with equal radiant intensity in all directions), a simple relationship emerges of theform (Chapman et al., 1997)

�AOð�0; �; Þ ¼ � sin �0 sin �; ð8:132Þand hence

�AOð�Þ ¼ � sin2 �: ð8:133Þ

The corresponding scattering strengths are

BSSð�0; �; Þ ¼ 10 log10 �þ 10 log10ðsin �0 sin �Þ ð8:134Þand

BBSð�Þ ¼ 10 log10 �þ 10 log10ðsin2 �Þ: ð8:135Þ

The sin � angle dependence originates from the increasing scattering area as �


Figure 8.12. Comparison between predicted andmeasured seabed backscattering strength for a

medium silt sediment (Mz ¼ 5.6) and frequency 20 kHz. Dashed line: roughness scattering only;solid line: roughness scatteringþ volume scattering (reprinted with permission from Jackson

and Briggs, 1992, American Institute of Physics).#

approaches zero. (The scattering coefficient is proportional to the ratio of scatteredradiant intensity to scattering area).

This relationship is often referred to as ‘‘Lambert’s law’’, but it is moreappropriate to call it ‘‘Lambert’s rule’’ because, while it has a simple physical inter-pretation, it has no firm foundation in scattering physics. Measurements of seabedbackscattering strength are often reported in terms of the coefficient �, known as‘‘Lambert’s parameter’’ or ‘‘Lambert’s constant’’.

Little guidance can be given regarding a suitable choice for Lambert’s parameter.It is known to be higher for rock and gravel bottoms than for unconsolidatedsediments (see Table 8.5), but the uncertainty is large. For example, Boyle andChotiros (1995) find no clear relationship between grain size and scattering strengthfor unconsolidated sediments, except for the case of specially prepared laboratorysand.

Lambert’s parameter for rock or gravel. Reported values of 10 log10 � for rock orgravel vary between about �19 dB (Gensane, 1989) and �2 dB (Urick, 1954), bothfor frequencies in the range 10 kHz to 40 kHz. Thus, a typical value is �11 dB, with alarge uncertainty of �8 dB. If Lambert’s rule is deemed to apply for all angles, thelaw of energy conservation requires that the value of � should not exceed 1=� (Urick,1983, p. 278). This means that if values larger than this (10 log10 � > �5.0 dB) occurin a limited angle range, they must be offset by lower values at other angles, implyinga departure from Lambert’s rule.

Lambert’s parameter for unconsolidated sediments. An important factor indetermining � for unconsolidated sediments, at least at high frequency, is thepercentage of gravel or shell. For gravel-free (less than about 5% gravel)


Table 8.5. Measurements of the Lambert parameter �.

Seabed 10 log10 � Frequency Reference

(dB) (kHz)

Unconsolidated sediments �20 to �15 8–40 Gensane (1989)

(sand, silt, or clay) �30 to �22 95 Goff et al. (2004)

�28 to �16 100 Simons et al. (2007)

�35 to �15 Unspecified Boyle and Chotiros (1995)

Gravel �19 8–40 Gensane (1989)

�18 to �16 95 Goff et al. (2004)

�9 to �7 100 Simons et al. (2007)

�10 to �3 Unspecified Boyle and Chotiros (1995)

Rock �4 to �2 10–60 Urick (1954)

�11 to �8 30–300 McKinney and Anderson (1964)

unconsolidated sediment, a nominal value for 10 log10 � of �25 dB is suggested, withan uncertainty of about �10 dB.

For medium sand (i.e., for sediment grain sizes in the range 250–500 mm),Greenlaw et al. (2004) show that the parameter 10 log10 �sand increases approximatelylinearly with log(frequency) between 10 kHz and 400 kHz. A linear fit to their Fig. 6 inthis frequency range gives

�sand ¼ 10�4F

5

� �1:47

; ð8:136Þ

or equivalently, in decibels

10 log10 �sand ¼ �40þ 14:7 log10ðF=5Þ: ð8:137Þ

A similar dependence of the scattering strength of sand on frequency (increase of15 dB per decade between 20 kHz and 300 kHz) is observed by Williams et al. (2002).

Equation (8.137) is consistent with a power law roughness spectrum (Equation8.125) with � ¼ 2.53. Above 400 kHz, Greenlaw’s measurements of the Lambertparameter 10 log10 �sand level off, reaching a maximum of about �9 dB at 700 kHz,before starting to decrease with increasing frequency at higher frequencies. At theother end of the spectrum, little variation is observed with frequency below about10 kHz (Urick, 1983, p. 274).

Effect of gravel (100 kHz). The presence of gravel or shell exceeding aproportion of about 0.05 is known to increase the Lambert parameter by severaldecibels at high frequency (Goff et al., 2004; Simons et al., 2007). The effect of gravelat a frequency of 100 kHz can be estimated using the empirical regression equationdue to Simons et al. (2007), derived from data for d50 between 30 mm and 500 mm,gravel fraction g up to 0.7, and shell fraction s up to 0.05,

10 log10 � ¼ ð9:5� 3:5Þð1 � g� sÞd50=ð1mmÞ

þ ð19:2� 2:1Þgþ ð173� 45Þs� 22:1� 0:9: ð8:138Þ

For medium sand the value predicted by Equation (8.138) is �18.5 dB, with anuncertainty of about �3 dB, which is consistent with �20.9 dB for a frequency of100 kHz from Equation (8.137).

8.1.4.2.2 Ellis–Crowe (semi-empirical) scattering strength model

Ellis and Crowe (1991) suggest a combination of Lambert’s rule with the facet-scattering term (see Sections 8.1.2.1.2 and 8.1.4.1.3), that is,

�AOð�0; �; Þ ¼ � sin �0 sin �þ �ð1þ DOÞ2 exp �DO2�2

� �; ð8:139Þ

where

DO ¼ cos2 �0 þ cos2 �� 2 cos �0 cos � cos

ðsin �0 þ sin �Þ2: ð8:140Þ


The parameter � is referred to by Ellis and Crowe (1991) as the facet strength. It canbe related to the seabed reflection coefficient R using

� ¼ Rð�0Þ2

8��2: ð8:141Þ

For in-plane scattering, Equation (8.139) simplifies to

�AOð�0; �; Þ ¼ � sin �0 sin �þ1

8��2Rð�0Þ2

sin4 �exp � 1

2�2 tan2 �

� �; ð8:142Þ

where

� ¼12ð�0 þ �Þ ¼ �12ð�0 þ �� Þ ¼ 0

(; ð8:143Þ

and for the backscattering case

�AOð�Þ ¼ � sin2 �þ 1

8��2R2ð�Þsin4 �

exp � 1

2�2 tan2 �

� �: ð8:144Þ

8.1.4.2.3 McKinney–Anderson (empirical)

McKinney and Anderson (1964) presents measurements of BBS vs. angle for differentbottom types and frequencies between 12.5 kHz and 290 kHz. A widely usedempirical relationship that is understood to fit McKinney–Anderson6 measurementsis (Jenserud et al., 2001)

�AO1:196

¼ 10�4:5 þ ðsin �þ 0:19ÞB cos16 �2:53F 3:2�0:8B102:8B�12GðB; �Þ; ð8:145Þ

where

GðB; �Þ ¼1 0 < �deg < 40

1þ 125 exp �2:64ðB� 1:75Þ2 � 50

B tan2 �

� �40 < �deg < 90

8<: ; ð8:146Þ

B ¼1 mud2 sand3 gravel4 rock

8><>: ð8:147Þ

and �deg is the grazing angle in degrees, given by Equation (8.50).

8.2 TARGET STRENGTH, VOLUME BACKSCATTERING STRENGTH,

AND VOLUME ATTENUATION COEFFICIENT

This section deals with scattering either from individual objects that are small incomparison with a sonar beam so that it makes sense to treat them as point-like

8.2 Target strength, volume backscattering strength 399]Sec. 8.2

6 The author is unaware of any published comparison demonstrating such a fit.

scatterers, or from large aggregations of objects occupying a large volume that, fromthe point of view of the sonar, is effectively infinite. The former are categorized bytheir target strength (TS) and the latter by their volume backscattering strength(VBS).7 Also considered is the attenuation due to distributed scatterers.

TS is an important term in the active sonar equation. It directly controls the echolevel and hence the signal-to-noise ratio. VBS can be thought of as the TS of a unitvolume of a scatterer that is extended in three dimensions. Depending on the applica-tion, such a scatterer can either be the source of a masking background, or of thesignal to be detected.

8.2.1 Target strength of point-like scatterers

The scattering properties of both natural and man-made objects are considered in thissection. The objects concerned are assumed to be point-like in the sense that they aresmall by comparison with the footprint of a typical sonar beam. Such objects can becharacterized by their target strength TS, which is related to the backscattering cross-section (BSX) (denoted �back) by8

TS ¼ 10 log10�back

4�dB re m2: ð8:148Þ

The BSX is a measure of the power scattered by an object per unit solid angle power(its radiant intensity). This is a far-field concept so any direct measurement of TSmust take place in the far field of the object (Morse and Ingard, 1968). BSX hasdimensions of area and can be thought of as the apparent physical size of an object, insquare meters, as perceived by an incident plane wave.

For simple shapes, values of �back can be estimated using the results of Chapter 5.For more complicated shapes it is necessary to resort to measurements. TS measure-ments of natural and man-made objects are presented here. Where appropriate,comparison is made with theoretical expectation. The BSX of fish shoals (and hencetheir target strength through Equation 8.148) can be estimated from that of anindividual fish as described in Chapter 5.

8.2.1.1 Marine organisms with a gas enclosure

The most important single factor in determining the likely TS of a marine animal isthe presence or absence of a gas enclosure. This is because such an enclosure greatlyenhances the acoustic scattering strength. Important examples of animals with a gasenclosure are bladdered fish (Section 8.2.1.1.1), marine mammals (Section 8.2.1.1.2),and human divers (Section 8.2.1.1.3).


7 Volume scattering strength is the volumic scattering cross-section, expressed in decibels.8 The 4� denominator is omitted by some authors, who incorporate it instead in the definitionof �back (see Chapter 5 for details.) This alternative definition, denoted �backalt , can be converted

to TS using TS ¼ 10 log10 �backalt . The value of TS is unaffected because the 4� factors cancel out.

8.2.1.1.1 Bladdered fish

A summary of available measurements of the TS of individual (bladdered) fish isprovided in Table 8.6. All of these data are for high-frequency measurements in thesense that the product of bladder radius and acoustic wavenumber k is larger thanunity. That is, kaS > 1, where aS is an equivalent radius—that of a sphere of surfacearea equal to Sbladder—defined as

aS �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSbladder4�

r; ð8:149Þ

and Sbladder is given by Equation (8.172). Measurement frequencies are between38 kHz and 420 kHz, and fish lengths range from 4 cm to about 1m.

These measurements can be compared with the theoretical TS using the high-frequency limit of �back from Chapter 5. For bladdered fish this is (neglecting thedamping term, which vanishes at high frequency)

TS ¼ 10 log10Sbladder4�

; ð8:150Þ

and hence

TS ¼ 10 log10 L2 � 26:4 dB re m2: ð8:151Þ

In fact, the derivation of Equation (8.151) requires kaS � 1, which is not compatiblewith the actual measurement frequencies of Table 8.6, and the true high-frequency TScan be up to 6 dB lower. Allowing for this uncertainty, the theoretical high-frequencytarget strength can be written

TS� 10 log10 L2 ¼ �29:4� 3:0 dB: ð8:152Þ

Remarkably (though fortuitously), this revised value is within 0.2 dB of the averageover all four TS measurements for gadoids and clupeoids from Table 8.6. Accordingto theory this expression is dominated by the bladder term, while the flesh termcontributes only 0.1 dB. The observed dependence on bladder type (there is a


Table 8.6. Target strength measurements for bladdered fish.

Species Fish length Measurement TS� 10 log10 LL2 Reference

L=m frequency

(kHz)

Gadoids 0.1–1.0 38 �27.5 Foote (1997)

(physoclists) 0.04–1.05 38–420 �27.1� 1.7 MacLennan and Simmonds (1992)a

Clupeoids 0.1–0.3 38 �31.9 Foote (1997)

(physostomes) 0.06–0.34 30–120 �31.8� 1.2 MacLennan and Simmonds (1992)b

aUnweighted average and standard deviation of 11 different species: blue whiting, cisco, cod, great silver smelt,haddock, Norway pout, Pacific whiting, redfish, saithe, sockeye salmon, and walleye pollock.bUnweighted average of 2 different species: herring and sprat.

difference of about 5 dB in TS between physoclists and physostomes) supports thiscase, at least for the physoclists. Nevertheless, there is evidence that for some speciesthe bladder is less significant. Unusually low TS values, excluded from Table 8.6, arereported for orange roughy (McClatchie et al., 1999), capelin (Halldorson andReynisson, 1983), yellowfish tuna (Bertrand and Josse, 2000), and horse mackerel(Axelsen et al., 2003), all for a frequency of 38 kHz. The dimensionless parameterTS� 10 log10 L2 on the left-hand side of Equation (8.152) is referred to henceforth asreduced target strength.

Also excluded from the table are lanternfish. These are known to possess a gas-filled bladder (Yasuma et al., 2003), and are believed to make an important con-tribution to the deep scattering layer (Section 8.2.3.1). At the time of writing, noreliable TS measurements for an individual lanternfish are known to the author.However, the modeling results of Yasuma et al. (2003) suggest a reduced TS at38 kHz of �29 dB for the Californian headlight fish and significantly lower value(about �46 dB) for the bigfin and Japanese lanternfishes.

For some applications, an aspect average might not be suitable (e.g., if the aspectdependence of the TS is required). A computer model that takes into account thethree-dimensional shape of fish flesh as well as the air-filled bladder is described byAu et al. (2004). For accurate work it might be necessary to allow for change inbladder volume due, for example, to changes in pressure with depth according toBoyle’s law (Feuillade and Nero, 1998; Nero et al., 2004).


Target strength measurements, summarized in Table 8.7, have been made for thebottlenose dolphin and four species of whale, namely the gray, humpback, northernright, and sperm whales. Humpback and gray whales exhibit high values of reducedtarget strength, consistent with scattering from a large gas cavity, presumably theirlungs. In general, it is likely that flesh also contributes to the total, especially at highfrequency, for which sound reaching the lungs might be attenuated (Miller andPotter, 2001).

The table includes measurements of aspect-averaged TS for the bottlenosedolphin (Au, 1996), northern right whale (Miller and Potter, 2001), and humpbackwhale (Love, 1973). Taking the average in decibels of these three (aspect-averaged)values gives the following estimate for the reduced target strength of mammalsgenerally

TS� 10 log10 L2 ¼ �25:1 dB: ð8:153ÞRecent measurements by Lucifredi and Stein (2007) for the gray whale, which areincluded in the table but not in the average, are significantly higher than predicted byEquation (8.153).

8.2.1.1.3 Human divers

As for marine mammals, the TS of a human diver includes contributions from flesh(including skeleton) and lungs. In addition, the diver could be accompanied byparaphernalia such as a wet suit, breathing apparatus, and exhaled bubbles, all of


Table8.7.Targetstrengthmeasurementsforwhales,sortedbyanimallength.

Species

Length

Target

strength

Reducedtarget

strength

f/kHz

Reference

L=m

(dBrem2)

TS�10log10L2(dB)

Single

Averaged

aspect

overaspect

Bottlenosedolphin

2.2

�15to

�11(sideaspect)

�21.8to

�17.8

23–30

Au(1996)

(Tursiopstruncatus)

2.2

�24to

�18(sideaspect)

�30.8to

�24.8

40–79

Au(1996)

2.2

�39(tailaspect)to

�18(side)

�45.8to

�24.8

�29.7

67

Au(1996)

Graywhale

11a

�2.9(tailaspect)to

þ12.8(side)

�23.7to

�8.0

23

LucifrediandStein(2007)

(Eschrichtiusrobustus)

Humpbackwhale

9–14

�4.8to+7.2

�27.5to

�15.5

�19.1

10–20

Love(1973)

(Megaptera

novaeangliae)

15

þ4.0(sideaspect)

�19.5

86

MillerandPotter(2001)

Northernrightwhale

8–15

�12.4to

�1.0

�31.8to

�23.7

�26.6

86

MillerandPotter(2001)

(Eubalaenaglacialis)

Spermwhale

12a

�8.8(unknownaspect)

�30.4

1Dunn(1969)

(Physetermacrocephalus)

aAssumedvalue.

which complicate theoretical estimates and could make a significant contribution tothe total. Hollett et al. (2006) report TS at 100 kHz between�25 dB and�20 dB rem2

for the ‘‘diver’s body, suit, tanks’’. The measurements by the same authors of the TSof exhaled bubbles (for a single exhalation), also at a frequency of 100 kHz, are about7 dB higher than this (i.e.,�18 to�13 dB rem2), and similar to the value suggested byUrick (1983, p. 324), although Urick does not state a measurement frequency.

8.2.1.2 Miscellaneous marine organisms, mostly without a gas enclosure

For the case of a convex object without a gas enclosure ensonified at random aspect,the BSX depends on surface area, so the shape becomes an important consideration.Many marine animals, including fish, are elongated in a well-defined direction, andsuch animals are considered first (in Section 8.2.1.2.1). This is followed in Section8.2.1.2.2 by a discussion of animals with more complex shapes. A comprehensivediscussion of scattering from zooplankton can be found in Lavery et al. (2007).

8.2.1.2.1 Animals with a pronounced elongated shape

Target strength measurements for euphausiids and bladder-less fish are shown inTable 8.8 for frequencies between 38 kHz and 2MHz and animal lengths 2 cm to35 cm.


Table 8.8. Target strength measurements for euphausiids and bladder-less fish. Early measurements (before

1980) are excluded.

Species Animal Measurement TS� 10 log10 LL2 Reference

length frequency

L=cm (kHz)

Krill 2.8–4.3 38–420 �48.0a MacLennan and Simmonds (1992, Table 6.4)

(Euphausia Pauly and Penrose (1998, Table I)

superba) Lawson et al. (2006)

Krill 1.9–2.1 420 �43.5 Simmonds and MacLennan (2005)

(Euphausia

pacifica)

Unspecified 3.0 1000–2000 �45 Griffiths et al. (2002)

fish species

Sandeel 11–14 38 �53.7 MacLennan and Simmonds (1992, Table 6.4)

(Ammodytes

spp.)

Mackerel 31–35 38–120 �46.9a MacLennan and Simmonds (1992)

(Scomber

scombrus)

a Unweighted average over more than one data set.

The high-frequency TS for bladder-less fish is determined by the surface area offish flesh (Chapter 5)

TS ¼ 10 log10Sfish16�

jRðHFÞj2� �

; ð8:154Þ

where

jRðHFÞj2 � 0:0045: ð8:155Þ

Surface area Sfish can be related to fish length using the empirical formula (Chapter 4)

Sfish ¼ 0:24L2: ð8:156Þ

Substituting these parameters into Equation (8.154) yields

TS� 10 log10 L2 ¼ �47 dB: ð8:157Þ

This estimate is within 4 dB of the measured average value for four out of the fivespecies included in Table 8.8.9 The exception is the sandeel, for which a low reducedTS can be expected because of its long, thin aspect ratio, giving it a smaller surfacearea than would be expected from its length alone.

Comparison can be made with the theoretical expectation for a concave reflectorof surface area S

TS ¼ 10 log10S

4�jRðHFÞj2

� �: ð8:158Þ

The surface area of a prolate spheroid is (from Chapter 4)

S ¼ 2�ab arcsin e

eþ b

a

� �; ð8:159Þ

where

e � 1� b2

a2

!1=2

: ð8:160Þ

For the TS of Antarctic krill, Simmonds and MacLennan (2005, p. 279) recommenduse of an expression that can be written:

TS� 10 log10 L2 ¼ �47þ 14:85 log10ð42LLÞ dB: ð8:161Þ

Equation (8.157) predicts TS within 3 dB of this expression for animals of lengthbetween 15mm and 40mm. (Equality occurs for a length of 24mm.)

8.2.1.2.2 Miscellaneous animals with irregular shapes

Target strength measurements for squid, gastropods, and jellyfish are summarized inthis section.


9 In the case of mackerel this agreement is to be expected, since the jRj2 value of Equation(8.155) is chosen to match measurements of the target strength of mackerel.

Squid. Benoit-Bird et al. (2008) give a number of empirical equations fitting thetarget strength of live squid (Dosidicus gigas) to its mantle length L. The average oftheir three high-frequency equations (covering the range 70–200 kHz), is

TS� 10 log10 L2 ¼ �27:6 dB; ð8:162Þ

for animals of mantle length in the range 28 cm to 72 cm. This value of �27.6 dB forreduced target strength is 19 dB higher than for bladder-less fish of length L, and thecause of this discrepancy is not known. Part of the difference can be explained by thedefinition here of L as the mantle length, thus excluding the size of the tentacles, buton its own this seems unlikely to explain the full 19 dB difference. Benoit-Bird et al.find that the cranium scatters a disproportionately large amount of sound for its size,perhaps explaining a further part of the difference. At 38 kHz, the measured targetstrength is even higher (about 6 dB greater than Equation 8.162). At this frequency,the arms are identified as important scatterers, having ‘‘a stronger effect’’ than thebeak or eyes (Benoit-Bird et al., 2008).

Earlier measurements of reduced target strength reported by Kawabata (2005)and Simmonds and MacLennan (2005, Table 7.2), for frequencies between 28 kHzand 120 kHz, are lower than those of Benoit-Bird et al. (2008). Excluding one highervalue of about �27 dB from Kajiwara et al. (1990), the unweighted average of theremaining four sets of measurements is

TS� 10 log10 L2 ¼ �36:8 dB; ð8:163Þ

about 9 dB lower than Equation (8.162). The animals involved in these earliermeasurements, with mantle lengths between 8 cm and 42 cm, were smaller than thoseused by Benoit-Bird et al. (2008).

Gastropods. Stanton et al. (1998a, Fig. 4) report TS measurements for thegastropod Limacina retroversa for frequencies in the range 370 kHz to 600 kHz.Their aspect averaged value can be written

TS� 10 log10 L2 ¼ �19:5 dB; ð8:164Þ

where L is the gastropod ‘‘length’’ (Stanton et al., 1998a), equal to 1.5mm for thisanimal. In this case the most important difference compared with the measurementsof Table 8.8 is probably not the shape, but the hardness of the gastropod’s shell. Forexample, putting RðHFÞ ¼ 0.5 (from Greene et al., 1998) into Equation (8.154),retaining Equation (8.156) for the surface area, results in the formula

TS� 10 log10 L2 ¼ �19:7 dB; ð8:165Þ

thus predicting a value for the reduced target strength that is remarkably close to themeasured value of Equation (8.164).

Although shape appears not to be the determining factor here, in othercircumstances it might be. Gastropods come in a variety of shapes, most but notall having a hard exterior shell.


Jellyfish and siphonophores. TS measurements for various species of jellyfish aresummarized in Table 8.9.

The siphonophore is a colonial invertebrate resembling a jellyfish. An importantfeature is a small gas enclosure called a pneumatophore. The target strength of thesiphonophore is dominated by scattering from the pneumatophore, which approx-imates in shape to a prolate spheroid. Measurements of siphonophore TS are sum-marized in Table 8.10. The size of the pneumatophore is characterized by thedimensions of its major and minor axes, denoted a and b, respectively. The reducedTS, based on measurements of Stanton et al. (1998a), is

TS� 10 log10ð2aÞ2 ¼ �11:0 dB: ð8:166Þ

Substituting the measured values of a and b from Stanton et al. (1998a) in Equation(8.163) yields the theoretical TS value of

TS ¼ �68:7 dB re m2: ð8:167Þ


Table 8.9. Target strength measurements for jellyfish (from Simmonds and MacLennan, 2005,

Table 7.2).

Disk diameter f /kHz TS TS�10 log10 D2

D=cm (dB rem2) (dB)

Aequorea victoria 4.2 420 �64.8 �37.3(crystal jelly)

Bolinopsis sp. 4.5 420 �80.0 �53.1

Aequorea aequorea 7.4 18–120 �68.5 to �66.3 �45.9 to �43.7

Aurelia aurita 9.5–15.5 120–200 �64.3 to �57.1 �43.9 to �39.8

Chrysaora hysoscella 26.8 18–120 �51.5 to �46.6 �40.1 to �35.2

Table 8.10. Target strength measurements for siphonophores.

TS (dB rem2) Frequency/kHz Pneumatophore size Reference

�75.0 200 a ¼ 0.3mm Trevorrow et al. (2005)

(estimated)

�69.1 120 a ¼ 0.6mm Warren et al. (2001)

(estimated)

�69.5 400–600 a ¼ 0.65mm Stanton et al. (1998a)

b ¼ 0:25 mm

Adjusting for the length 2a of the pneumatophore gives

TS� 10 log10ð4a2Þ ¼ �10:2 dB; ð8:168Þ

which is within 1 dB of Stanton’s measured value as given by Equation (8.166).

8.2.1.3 Man-made objects

Table 8.11 summarizes measurements of the TS of various surface ships, submarines,and underwater weapons made during and after the Second World War. The valuesare representative only, as the measurements are subject to high variability (Urick,1983, p. 324). TS measurements of their modern equivalents are usually classified.

An order of magnitude theoretical estimate of the TS of these and similar objectscan be obtained using the expression from Chapter 5 for a smooth convex target ofsurface area S at random aspect

TS ¼ 10 log10jRj2S16�

¼ �17:0 þ 10 log10 S þ 10 log10jRj2 dB re m2; ð8:169Þ

where R is the reflection coefficient. For example, the aspect-averaged TS of aperfectly reflecting convex object of surface area S ¼ 100m2 isþ3 dB rem2. Equation(8.169) is not applicable to an object with sharp edges, as the TS might then bedominated by diffraction from these edges.

Some modern military vessels are clad with special anechoic (literally ‘‘non-reflecting’’) materials. Their shapes might also be specially designed to deflect soundaway from the expected receiver position (as is done for stealth aircraft designed toachieve a low radar cross-section). For such objects the random aspect equation isnot applicable. As a result of the special materials or shapes used, comparablemodern vessels of similar size to their Second World War (WW2) counterparts arelikely to have a lower TS than the values quoted in Table 8.11.


Table 8.11. Second World War measurements of the target strength of man-made

objects (from Urick, 1983).

TS (dB rem2)

Beam aspect Bow aspect Intermediate aspect

Submarine 24 9 (stern) 14

Surface ship 24 14

Mine 9 �26 to þ9

Torpedo �21

8.2.2 Volume backscattering strength and attenuation coefficient of

distributed scatterers

If there are many point-like objects forming an extended ‘‘cloud’’ of scatterers, it canbe more useful to think of this cloud as a continuum instead of as a collection ofdiscrete objects. In this situation the relevant quantity is BSX per unit volume(volumic10 BSX), which when converted to decibels gives the volume backscatteringstrength (VBS):

VBS � 10 log10�backV

4�dB re m�1: ð8:170Þ

Low-frequency and high-frequency scattering effects are considered separately inSections 8.2.2.1 and 8.2.2.2, respectively. Volume attenuation is discussed in Section8.2.2.3.

8.2.2.1 Low-frequency VBS (mainly due to large fish)

The presence of large numbers of pelagic fish can result in high values of VBS.Measurements for a known or independently estimated fish population are very rare(see Love, 1993 for a notable exception). A theoretical estimate can be made usingthe expression for the volumic BSX �backV from Chapter 5, giving the result for arepresentative fish length Lgroup

VBS � 10 log10Sbladder4�L2

QfishQgroup

� �þ 10 log10ðL2groupNVÞ

� 10�

loge 10Q2group

f0ðLgroupÞf

� 1� �

2

; ð8:171Þ

where Sbladder is the bladder surface area (Chapter 4)

SbladderðLÞ � 0:0291L2 ð8:172Þand the resonance frequency is

f0ðLÞ � ð78:9 HzÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:1zzþ 1:75

p

LL: ð8:173Þ

Assuming a nominal value of Qgroup � 2, Equation (8.171) can be written

VBS � �23:3þ 10 log10ðQfishL2groupNVÞ � 54:6f0ðLgroupÞ

f� 1

� �2

dB re m�1: ð8:174Þ

Using this equation, a theoretical estimate of VBS can be made based on knowledgeof the average population density of bladdered fish. An example prediction for theNorth Sea follows in Table 8.12, based on fish population data from Chapter 4. Fishdepth used for the table is 25m, one half of the assumed average water depth of 50m.The main effect of changing this depth is in the resonance frequency, which is




proportional toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffizzþ 17:5

p. The table further assumes a total North Sea volume of

19 200 km3.The night-time average nature of these estimates is emphasized (during the day

the fish tend to aggregate in shoals). Local values can be significantly higher or lower,depending on the concentration of each species (Knijn et al., 1993). According to thistable, the main contributions to VBS are due to Norway pout, herring, and silverypout, the resonance frequencies for which are between 0.6 kHz and 2.4 kHz.

These estimates are based on survey data that at the time of writing are 20 yearsold. Thus, they are not intended as a quantitative prediction for the North Sea in the


Table 8.12. Predicted average night-time contribution to volume backscattering strength (VBS), column

strength (CS, defined in Section 8.2.3.1), and attenuation due to pelagic fish in the North Sea; only those fish

known to possess a swimbladder are included (thus, mackerel and sandeel are excluded from this table).Ntotis estimated North Sea population; VBS0 is average contribution to VBS at frequency f0; CS0 is average

contribution to CS at f0; and a0 is average contribution to að f Þ at frequency f0.

Species Ntot=109 NV= L/m VBS0/ CS0/dB a0/ f0/kHz

(Chap. 4) dam�3 (dB re m�1) (dB km�1)

Silvery pout 135.8 7.09 0.06 �63 �46 0.23 2.4

(Gadiculus

argenteus)

Norway pout 64.1 3.34 0.13 �59 �42 0.51 1.1

(Trisopterus

esmarkii)

Atlantic herring 11.3 0.59 0.24 �62 �45 0.30 0.6

(Clupea harengus)

Whiting 5.9 0.31 0.20 �66 �49 0.11 0.7

(Merlangius

merlangus)

European sprat 5.5 0.29 0.10 �72 �55 0.03 1.4

(Sprattus sprattus)

Haddock 2.2 0.11 0.30 �67 �50 0.09 0.5

(Melanogrammus

aeglefinus)

Horse mackerel 1.9 0.10 0.24 �69 �52 0.05 0.6

(Trachurus trachurus)

Pollock 0.4 0.02 0.45 �71 �54 0.04 0.3

(Pollachius virens)

Cod 0.1 0.01 0.70 �73 �56 0.02 0.2

(Gadus morhua)

early 21st century, but rather as an estimate of typical values to be expected in regionssustaining a high density of fish.

8.2.2.2 High-frequency VBS (partly due to small fish)

For frequencies of 10 kHz to 60 kHz, APL-UW (1994) provides default values forVBS summarized in Table 8.13. For the Arctic region (under the ice cap and in themarginal ice zone), a separate average value of �75 dB rem�1 is suggested.

8.2.2.3 Volume attenuation coefficient due to bubbles and bladdered fish

The attenuation of sound in pure seawater is described in Chapter 4. Here theadditional contributions due to air bubbles and bladdered fish are considered, usingresults from Chapter 5.

8.2.2.3.1 Bubbles

The equations presented here are for the extinction coefficient in units of nepers permeter. This coefficient can be converted to decibels per meter by multiplying thenumerical value by 20 log10 e. The expression for attenuation due to a cloud ofbubbles is

¼ 12

ð�eðaÞnðaÞ da; ð8:175Þ

where

�eðaÞ ¼ �backða; !Þ 1þ �therm þ �visc!a=cm

� �: ð8:176Þ

Expressions for the damping coefficients �therm and �visc for bubbles are given inChapter 5.

8.2.2.3.2 Dispersed fish with swimbladder

The equivalent expression for dispersed fish is

¼ 12

ð�eðLÞnðLÞ dL; ð8:177Þ


Table 8.13. Default advice for VBS between 10 kHz and 60 kHz for sparse,

intermediate, and dense marine life (except for the Arctic region), from

APL-UW (1994).

VBS/(dB rem�1)

Depth Sparse Intermediate Dense

Deep water 0–300m �94 �87 �79300–600m �81 �74 �66

Shallow water Any �85 �72 �62

with

�e ¼ �backbladderðL; !Þ 1þcm!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�aS3Vbladder

sð�therm þ �fleshÞ

" #: ð8:178Þ

where aS is the equivalent bladder radius given by Equation (8.149). Expressions forthe damping coefficients �therm and �flesh for fish are given in Chapter 5.

In the case of a large group of fish, the population distribution is likely to exhibita peak around some value (say Lgroup). Following Weston (1995), Equation (8.178)for the extinction cross-section can be replaced at resonance in the integrand ofEquation (8.177) by

�eðLÞ � 4�a2SL0ð f ÞQrad�ðL� L0ð f ÞÞ; ð8:179Þ

where L0ð f Þ is given by

L0 �78:9

ff

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:1zz þ 1:75

pm; ð8:180Þ

and �ðL� L0Þ is the Dirac delta function (see Appendix B).Using a Gaussian length distribution for nðLÞ gives the result (see Chapter 5)

� 2�a2SQradQgroupNV exp ��Q2group

L0ð f ÞLgroup

� 1� �

2� �

: ð8:181Þ

This expression is applicable to dispersed fish with a bladder, provided that theacoustic frequency is close to the resonance frequency of fish bladders. If the fishaggregate into shoals, their effect on attenuation is expected to be negligible. It isoften the case that shoals form during the day and disperse at night. Thus, a simplerule, if no better information is available, is to use Equation (8.181) for bladdered fishat night-time, and to assume no effect for day-time (or if there is no bladder). Table8.12 includes peak values of calculated for each species of fish, using Equation(8.181) with the average North Sea population density.

Notice the relationship between the extinction coefficient and VBS (Equation8.171):

¼ 2�QradQfish

10VBS=10; ð8:182Þ

where

Qrad ¼55:2

ð1 þ zz=17:6Þ1=2: ð8:183Þ

8.2.3 Column strength and wake strength of extended volume scatterers

8.2.3.1 Column strength and the deep scattering layer

If volume scatterers are distributed laterally in the two horizontal dimensions andconfined to a finite extent in depth, it is useful to define the column strength of the


distribution as depth-integrated volumic BSX, expressed in decibels

CS � 10 log10ð10VBS=10 dz ¼ 10 log10

ð�backV ðzÞ4�

dz dB: ð8:184Þ

Like SBS and BBS, CS is a dimensionless quantity.In deep water a region of high scattering strength, known as the deep scattering

layer, is often found at depths of a few hundred meters. This layer contains manydifferent species of myctophids (lanternfish) and euphausiids. The wide variety in thesize of the different species leads to a broadband acoustic response. Some of thespecies stay at an almost constant depth, while others follow a diurnal migrationpattern (Medwin and Clay, 1998).

Order of magnitude estimates of CS for the North Sea due to pelagic fish areincluded in Table 8.12. These values are independent of the assumed water depth andcan be compared with CS measurements in the Norwegian Sea and northeast Atlanticdue to Love (1993). Love’s data show a peak value of up to �40 dB around 2 kHz,attributed to blue whiting and redfish of lengths between 20 cm and 40 cm. The fish inLove’s survey were at a depth of around 300m. The increased pressure comparedwith the nominal depth of 25m used to compile Table 8.12 might explain the higherresonance frequency observed by Love.

Some of Love’s measurements in the northeast Atlantic exhibit a monotonicallyincreasing CS with frequency from 2.5 kHz to 5 kHz, a feature that he attributes tothe presence of lanternfish. At 5 kHz the CS is about �43 dB. The peak value isoutside the measured frequency range.

Estimates for high-frequency CS data can be derived from the VBSmeasurements summarized in Table 8.13. The resulting CS values for deep waterare between �56 dB and �41 dB, similar to the spread of values predicted for fishfrom Table 8.12, which is applicable to frequencies of order 1 kHz. Similar values areshown in Urick (1983, p. 260, Figs. 8.15 and 8.16) for frequencies 3 kHz to 20 kHz,with a marked drop-off below 3 kHz. The overall impression is that, in areas that aredensely populated with marine life, a reasonable default of CS in the frequency range3 kHz to 60 kHz is about �45 dB, albeit with a large uncertainty.

8.2.3.2 Wake strength

Ships and submarines have extended wakes containing many millions of bubbles. Thewake scatters sound, acting like an acoustic target that is confined in two dimensionsand extended in a third. Such a target may be characterized by wake strength (WS),equal to the TS of a unit length of the wake. Wake strength has dimensions of areaper unit length and is therefore expressed in units of decibels re 1 meter (dB rem).Measurements of WS dating to WW2 are listed for surface vessels in Table 8.14 andfor submarines in Table 8.15. According to Anon. (1946), WS is approximatelyindependent of frequency between 15 kHz and 60 kHz, and decreases with time ata rate of about 1 dB per minute. Modern measurements of volumic BSX of wakesof three different surface vessels at frequencies between 28 kHz and 400 kHz arepresented by Trevorrow et al. (1994).


8.3 SOURCES OF UNDERWATER SOUND

Underwater noise sources are an important consideration for sonar performancecalculations, as they determine the minimum signal level required for successfuldetection. Useful reviews of the main sources of underwater sound are providedby Richardson et al. (1995), Anon. (2003), McDonald et al. (2006).

The sounds are many and varied, and are caused, for example, by:

— breaking gravity waves, either due to wind or surf (Wilson, 1983; Deane, 2000);— non-linear interactions between gravity waves passing through one another

(known as ‘‘microseisms’’) (Longuet-Higgins, 1950; Webb, 1992);— precipitation (rain, snow, or hail) (Scrimger et al., 1987; McConnell et al., 1992;

Nystuen, 2001; Ma et al., 2005);— violent geological or meteorological activity such as lightning (Hill, 1985),

hurricanes (Bowen et al., 2003; Wilson and Makris, 2006), earthquakes orvolcano eruptions (Dietz and Sheehy, 1954; Northrop, 1974);

— other physical processes associated with the behavior of ice at the sea surface(Uscinski and Wadhams, 1999) or of gravel at the seabed (Thorne, 1986);


Table 8.14. Wake strength measurements for various WW2

surface ships (from Urick, 1983, p. 263).

Surface ships f /kHz WS/dB rem

Aircraft carrier, escort (CVE) 24 �9.4

Transport (AP) 24 �9.4

Destroyer (DD) 24 �11.4

Destroyer escort (DE) 24 �11.4

Table 8.15. Wake strength for various WW2 submarines (from Anon., 1946).

Submarines Speed/(m s�1) Deptha/m WS/(dB rem) f /kHz

USS-S23 (SS-128) 4.9 0.0 �13.4 60

3.1 27.4 �21.4 60

USS-S34 (SS-139) 4.9 0.0 �8.4 45

3.1 27.4 �18.4 45

USS ‘‘Tilefish’’ (SS-307) 4.9 0.0 �8.4 45

3.1 27.4 �15.4 45

USS-S18 (SS-123) 3.1 13.7 �28.4 45

a A depth of zero indicates that the submarine was surfaced at the time of the measurement.

— marine mammals, crustacea, and other living organisms (Kelly et al., 1985; Cato,1993);

— anthropogenic sources such as sonar, shipping, or industrial activity (Richardsonet al., 1995).

A composite graph of typical ambient noise spectrum levels is shown in Figure 8.13,illustrating the varied nature of underwater noise sources. Different parts of thespectrum tend to be dominated by different, but specific noise sources. For example,between 300Hz and 100 kHz the dominant source of noise is often wind-related,whereas at slightly lower frequency (30–300Hz) the strongest component is usuallydue to distant shipping. Rain noise, when present, tends to peak at a few kilohertz.Biological noise can be broadband, but can also contain strong spectral peaks (e.g.,around 20Hz due to blue whales and fin whales—see McDonald et al., 2006).

There is growing evidence that low-frequency sound levels in the sea (around40Hz) have increased on average by up to 3 dB per decade in the period between 1965and 2003 (Andrew et al., 2002; McDonald et al., 2006).11 This increase is attributed toa doubling in the number of commercial ships during that period (from 41 900 to89 900 ships) and a nearly four-fold increase in their gross tonnage (from 160 to 605million tonnes). A comparable increase in levels has been observed in the spectralpeaks associated with blue and fin whales (McDonald et al., 2006). In the same periodthe peak frequency of observed blue whale vocalizations dropped from 22Hz to16Hz.

There is a greater emphasis in Section 8.3 on measurements (as opposed totheory) than in the rest of this chapter, because the sound generation mechanismsare poorly understood compared with the mechanisms for reflection (Section 8.1) andscattering (Sections 8.1, 8.2). The modeling of propagation from the sound source toa receiver in the sea (i.e., to the sonar or animal listening to the sound) (Hamson,1997) is addressed in Chapter 9.

Measurements of ambient noise above 100 kHz are hampered by noise in thereceiving equipment due to thermal agitation. This type of interference, called thermalnoise, is described in Chapter 10. It is outside the scope of the present chapter becauseit is not caused by underwater sound. Also excluded here and included in Chapter 10are sonar transmissions and other intentional man-made sounds, such as thoseproduced by seismic survey or acoustic communications sources.

This and subsequent chapters contain numerical values of source level and soundpressure level in water. These levels are expressed in decibels and it can be tempting tocompare them with sound levels in air, also expressed in decibels. Such a comparisonis fraught with difficulty because of the following differences (see also Chapman andEllis, 1998):

— The reference pressure is different: the standard reference pressure in air is20 mPa, leading to a numerical difference of 26 dB in sound pressure level forthe same RMS pressure in air and water.

8.3 Sources of underwater sound 415]Sec. 8.3

11 An increase of 0.5 dB per year is reported by Ross (1974).


Figure 8.13. Typical ambient noise spectra. The x-axis covers five decades of frequency from

1Hz to 100 kHz. The y-axis is the noise spectrum level from 0dB to 140 dB re mPa2/Hz (adapted

from Wenz, 1962, American Institute of Physics, with permission).#

— The medium is different: the characteristic impedance of water is 3600 timesgreater than that of air. This means that the intensity of a plane wave in wateris 3600 times smaller than that of a plane wave in air of the same RMS pressure.Conversely, the RMS pressure of a plane wave in water is 60 times greater thanthat of a plane wave in air of the same acoustic intensity.

— The hearing sensitivity, pain thresholds, or damage thresholds are different, evenfor the same species: most species live either only in air or only in water, so it onlymakes sense to consider their hearing in that medium. For a few amphibiousspecies (mainly seals and human divers) it is known that the RMS pressure of asound that is just audible in water is higher than the RMS pressure of a soundthat is just audible in air. Little is known about injury thresholds in water(Southall et al., 2007).

— Reporting conventions are different: in air, measures of sound reported indecibels are almost always in the form of a sound level (i.e., the sound pressurelevel or SPL, weighted according to the sensitivity of human hearing in air). Inwater, measurements are usually reported without adjustment for hearing sensi-tivity. Finally, it is common practice to characterize a source of underwatersound by its source level, which is a measure of transmitted power and notreceived intensity.

Thus, it is rarely necessary, and almost always unwise, to compare sound levels in airand water. The natural human desire for comparison with known experience can besatisfied instead by invoking familiar sounds in water, such as that of rainfall or surf.See Figure 8.14 for some examples of underwater sounds and the correspondingsound pressure levels.

The remainder of this section is structured as follows. Measurements of shippingsource level spectra are summarized in Section 8.3.1, followed by a discussion of thesource spectra for distributed sources at the sea surface (Section 8.3.2) and on theseabed (Section 8.3.3). Section 8.3.2 includes a description of shipping noise as acontinuum of distant ships of given areic density.

8.3.1 Shipping source spectrum level measurements

Noise from distant ships, presumably in transit along commercial shipping lanes, isbelieved to dominate the underwater ambient noise spectrum at low frequency, fromabout 10Hz to a few hundred hertz. A prediction of the likely received sound levels ata given location requires an approximate shipping density distribution and an esti-mate of the average source level of an individual ship. The latter is the subject of thepresent section. The measurement of source level is a difficult one, often involving theunwitting participation of passing vessels of opportunity, and in order to provide ameaningful average the measurement must be repeated a number of times withdifferent ships. The following text presents measured spectra for individual vessels(Section 8.3.1.2), followed by some measurements of spectra averaged over manyships (Section 8.3.1.3). Only industrial and commercial shipping vessels are


considered. A summary of measurements for warships from WW2 is given by Urick(1983) and Collier (1997).

8.3.1.1 Conversion from far-field measurements

The source level of a ship is a measure of the amount of sound in the far field12 of thatship. Thus, any measurement of this quantity needs to be made in the far field, whileat the same time be close enough to the source to ignore propagation effects. Thesetwo conflicting requirements are irreconcilable in the case of a surface ship, becausethe far field inevitably contains a contribution from sea surface reflected sound. Twoquite different methods, described below, are in use for correcting for this separatecontribution.

The first method involves calculation of propagation loss (PL) for a pointmonopole source at some assumed, representative depth. The source level (SLmp)


Figure 8.14. Typical values of sound pressure level (left) and peak pressure level (right), in units

of dB re mPa2 and at a distance of 100m from the source (except for the sound of rainfall,

included as a reference). A typical range of hearing thresholds is also marked ( TNO,

reprinted with permission).

#

12 If a point in the radiated field is sufficiently distant from the sound radiator, the phase

difference between sound paths arriving from different parts of the radiator is determined only

by the bearing of the field point relative to the radiator and not by the distance between the field

point and the radiator. The region in which this is satisfied is known as the far field of the

radiator. The near field is where this condition is not satisified.

is then calculated from the measured SPL using

SLmp ¼ SPLþ PL: ð8:185Þ

This back-calculation is referred to in the following as the ‘‘monopole method’’because the result is a conventional monopole source level. The advantage of themonopole method is that the measurement can be made in shallow water. A dis-advantage is that the result is sensitive to the assumed value of source depth and tothe accuracy of propagation loss predictions.

The second method is a pragmatic one, involving measurements sufficiently farfrom the ship for spherical spreading to hold (i.e., not in the near field), while stillclose enough to neglect absorption (Ross, 1976; de Jong, 2009). Consider an ‘‘equiva-lent source level’’, SLeq, defined as the monopole source level that would result in thesame SPL as the combined ship and surface image at the measurement distance,assuming free space propagation conditions apart from the inevitable presence of thesea surface. That is,13

SLeqð�; Þ � SPLð�; Þ þ 10 log10 s2; ð8:186Þ

where s is the distance to the acoustic center of the ship. In general, this quantity is afunction of elevation (�) and bearing ( ) aspect angles. If measured at keel aspect (i.e.,with a hydrophone directly beneath the ship, at elevation � ¼ �=2), the result at lowfrequency is the source level of the dipole created by the ship and its surface image. Inthe following, this keel aspect value is referred to as the ‘‘dipole source level’’ at allfrequencies (even when the frequency is not low enough for a dipole to form) anddenoted SLdp:14

SLdp ¼ SLeqð� ¼ �=2Þ: ð8:187Þ

This second method is referred to in the following as the ‘‘equivalent source method’’.An approximate conversion between SLdp and SLeq at other elevation angles,

valid at frequencies low enough for the ship to behave as a point dipole, is

SLeqð�; Þ � SLdp þ 10 log10 sin2 �: ð8:188Þ

The result is independent of bearing for a true dipole, but departures from thisidealized behavior can be expected in some bearings (e.g., directly ahead of or behindthe ship—Arveson and Vendittis, 2000).

The source levels denoted SLmp and SLdp defined above can take quite differentvalues, especially at low frequency, for which the monopole and dipole source factorsare related via

Smp0 =Sdp0 � 1=ð4k2d 2Þ: ð8:189Þ


13 The equivalent source level defined in this way is not a property of the source only. It depends

also on measurement distance (e.g., through absorption) and water depth (e.g., through

reflections from the seabed).14 This quantity is independent of .

At high frequency and neglecting absorption, they differ—on average—by a factorof 2. Thus, a more general conversion, incorporating both high-frequency and low-frequency forms, is

Smp0 =Sdp0 � 12þ 1=ð4k2d 2Þ: ð8:190Þ

The advantage of the equivalent source method is that it does not require a choice tobe made for the source depth. A disadvantage is that a complete characterization ofthe radiated noise of even a simple source requires measurements at many angles.

8.3.1.2 Industrial and commercial shipping (individual ships)

Measurements of the source spectra of individual ships are presented here in Table8.16, in the form of third-octave source levels. The measurements are compiled fromRichardson et al. (1995) and Arveson and Vendittis (2000). This table is not intendedas a precise indication of expected source level of any given ship, as this depends notjust on the ship type and speed, but also on its cargo and type of activity, and thecondition of its propellers.15 Instead, it provides an indication of the spread ofpossible values to be expected. A further difficulty with the interpretation of thistable is the absence in some cases of details of the measurement method.

The measurements of Arveson and Vendittis (2000) are made using theequivalent source method, and include measurements at keel aspect (� ¼ �=2), mak-ing it possibe to infer the dipole source level as given by Equation (8.187). In theabsence of a statement to the contrary, the equivalent source method is assumed tohave been used for the measurements from Richardson et al. (1995) as well, althoughfor these the elevation angle is unknown.

The term ‘‘third-octave level’’ means that the spectral density Qf is integratedover a third-octave band (i.e., one-third of an octave in frequency) before beingconverted to decibels. Thus, the third-octave sound pressure level L1=3 is given by

L1=3 � 10 log10ð2þ1=6f02�1=6f0

Qf df dB re mPa2; ð8:191Þ

where the lower and upper limits of integration are respectively one-sixth of an octavebelow and above the center frequency f0. If the spectral density varies approximatelylinearly with frequency, Equation (8.191) may be replaced by

L1=3 � 10 log10½Df1=3Qf ð f0Þ� dB re mPa2; ð8:192Þwhere

Df1=3 ¼ ð2þ1=6 � 2�1=6Þ f0 � 0:2316f0: ð8:193Þ

If the spectral density of Equation (8.192) is scaled to a nominal 1m referencedistance, then L1=3 becomes the third-octave source level, with units dB re mPa

2 m2.


15 The type of propulsion system can be an important consideration in its own right.

A third-octave level can be converted into a mean spectrum density level Lf using

Lf � L1=3 � 10 log10 Df1=3; ð8:194Þ

where the average is in frequency, across the third-octave band. Average equivalentsource levels calculated in this way are plotted in Figure 8.15 (dotted blue curves).The red curves are explained in Section 8.3.1.3.

8.3.1.3 Commercial shipping (averaged source spectra)

Average source level spectra measured by Scrimger and Heitmeyer (1991) (hereafterabbreviated SH91) and by Wales and Heitmeyer (2002) (abbreviated WH02), bothusing the monopole method, are described below. SH91 estimates the source level


Table 8.16. Third-octave source levels of various commercial and industrial vessels, expressed in units

of dB re mPa2 m2. Entries are listed in descending order of the third-octave level at 500Hz. Measure-

ments for the data for cargo ship Overseas Harriette are taken from Arveson and Vendittis (2000)

(dipole source levels). The remainder are from Richardson et al. (1995, Table 6.9) (equivalent source

levels at unstated elevation).

Type Description RMS source level (third octave)

Center frequency 50Hz 100Hz 200Hz 500Hz 1 kHz 2 kHz

Icebreaker R Lemeur 177 183 180 180 176 179

Drillship Kulluk 174 172 176 176 168 —

Large tanker 174 177 176 172 169 166

Dredger Aquarius 170 177 177 171 — —

Modern cargo ship M/V Overseas Harriette 185 180 174 168 166 163

8.2m/s (16 kn)

Supply ship Kigoriak 162 174 170 166 164 159

Drillship Canmar Explorer II 162 162 161 162 156 148


6.2m/s (12 kn)

Tug and barge 5m/s 143 157 157 161 156 157

Dredger Beaver Mackenzie 154 167 159 158 — —


4.1m/s (8 kn)

Zodiac 5m length 128 124 148 132 132 138

spectra of 50 ships in the frequency band 70Hz to 700Hz. The vessels concerned wereapproaching or departing from the Mediterranean port of Genoa (Italy), with anaverage speed of 7.2m/s (14 kn). WH02 describes the source spectra between 30Hzand 1200Hz of 272 ships in the Mediterranean Sea and eastern Atlantic Ocean, andproposes the following mean spectrum for the monopole source level spectrum

SLmpf ¼ 230:0� 35:94 log10 ff þ 9:17 log10½1þ ð ff =340Þ2� dB re mPa2 m2=Hz: ð8:195Þ

Both SH91 and WH02 are plotted in Figure 8.16. The monopole source levelsmeasured by SH91 (dashed line) are about 6 dB to 12 dB higher than those ofWH02 (solid line). (The curves marked ‘‘Ov. Harriette’’ are explained in Section8.3.1.4.) For both SH91 and WH02 measurements, the source spectrum was inferredusing the monopole method by subtracting an estimate of propagation loss (PL) fromthe received spectrum at a distance of several kilometers from the source. Any bias inestimated PL would cause a bias in the inferred source spectrum, providing a possibleexplanation for the difference between reported source levels. From this point of viewthe WH02 data seem more reliable, because they involve a shorter measurementrange and hence perhaps less uncertainty in PL, and the average is computed overa larger number of individual ships. However, a definitive statement cannot be made


Figure 8.15. Measured equivalent source spectral density levels (SLeqf ), averaged over third-

octave bands for commercial and industrial shipping: individual ships, selected from Table 8.16

(blue lines); averaged source spectra plotted in red (solid red line is fromWales and Heitmeyer,

2002; dashed red line is from Scrimger and Heitmeyer, 1991—see Section 8.3.1.3 for

details).

from the available measurements. SH91 and WH02 spectra are also shown in Figure8.15 (red curves), where they are converted to dipole levels using Equation (8.190)assuming a monopole depth of 1.8m.

8.3.1.4 Effect of ship speed and acceleration

A thorough investigation of the radiated noise characteristics of a single cargo shipwas carried out by Arveson and Vendittis (2000) using the equivalent source method.Their measured spectra for different ship speeds are represented in Figure 8.16 by theblue and cyan curves. The Overseas Harriette measurements have been convertedfrom a dipole to monopole source level using Equation (8.190), for an assumedmonopole depth of 1.8m, and may be compared with the WH02 and SH91 monopolesource level spectra plotted in the same figure. The WH02 curve is extrapolatedabove 1200Hz (see dotted red line) using an empirically determined gradient(SLmpf ¼ constant� 23 log10 F), chosen to match the Arveson and Vendittis (2000)data at low ship speed.

Figure 8.16 illustrates the effect of ship speed on the radiated noise of anindividual ship traveling at constant velocity. The effect of turn rate is studied byTrevorrow et al. (2008). For the maximum turn rate considered of 4.5 deg/s, they


Figure 8.16. Estimated third-octave monopole source levels SLmp for the cargo ship Overseas

Harriette at various ship speeds, based on measurements from Arveson and Vendittis (2000);

WH02 (solid red line) and SH91 (dashed red line) are included for comparison; the WH02

spectrum is extrapolated to 15 kHz, agreeing in the extrapolated region (dotted red line) with the

measurements of Arveson–Vendittis for a ship speed close to 6.2m/s (12 kn).

report an increase of between 6 dB and 18 dB in third-octave bands between 160Hzand 4 kHz.

8.3.2 Distributed sources on the sea surface

The ubiquitous ocean noise caused by wind is considered next, followed by rain noise,which is itself also sensitive to local wind speed. Empirical relations providing dipolesource level as a function of wind speed and rain rate are given in Sections 8.3.2.1 and8.3.2.2. A uniform distribution of distant ships can also be regarded as a distributedsource at the sea surface, as described in Section 8.3.2.3.

8.3.2.1 Wind noise source level

The physical origin of wind noise is thought to be associated with the naturalpulsations of gas bubbles created by breaking waves or similar surface activity.Because of the close proximity of such bubbles to the sea surface, a dipole radiationpattern is usually assumed, for which it is convenient to define a parameter Kwind as

Kwind ¼ 3�c2�

W windAf ; ð8:196Þ

whereW windAf is the areic power spectral density. Thus, Kwind is the spectral density of

the areic dipole source factor. The quantity 10 log10 Kwind is known as the ‘‘dipole

source spectrum level’’ or sometimes just ‘‘dipole source level’’. The term ‘‘areicdipole source spectrum level’’ (i.e., the dipole source factor per unit area, expressedin decibels), is suggested for a sheet source, to distinguish it from the source level of asingle dipole.

It follows from Chapter 2 that the noise spectral density due to such a source, fora receiver in a uniform half-space, is given by

Qf � 32�cWAf ; ð8:197Þ

and hence, eliminating the power spectral density from Equations (8.196) and (8.197),

10 log10 Qf ¼ 10 log10ð�KwindÞ dB re mPa2 Hz�1: ð8:198Þ

8.3.2.1.1 High-frequency wind noise (APL model)

At frequencies above a few kilohertz, wind noise decreases monotonically withincreasing frequency. A useful parameterization from APL-UW (1994), intendedfor the frequency range 10 kHz to 100 kHz, can be written

KwindAPL ¼ 10

4:12vv2:24APL

F 1:59100:1�mPa2 Hz�1 ð8:199Þ

�ðDTÞ ¼0 DT < 1

0:26ðDT � 1:0Þ2 DT � 1

(ð8:200Þ


where DT is the temperature difference in degrees Celsius

DT ¼ TTair � TTwater; ð8:201Þ

vAPL is related to wind speed v10 according to

vvAPL ¼ maxðvv10; 1Þ ð8:202Þ

and F is frequency in kilohertz.At high frequency and sufficiently high wind speed (above about 30 kHz for a

wind speed of 10m/s, or above 10 kHz for 15m/s) special attention needs to be givento the absorbing effect of near-surface bubbles. While largely responsible for thegeneration of wind-related noise in the first place, if present in sufficient numbers,such bubbles also absorb some of the sound before it can escape the bubble layer. Theeffect can be modeled by computing the attenuation along each ray path as describedby APL-UW (1994). An alternative, more pragmatic approach is to cap the dipolesource level so that it does not exceed the following frequency-dependent value(obtained by inspection of Fig. 17 from APL-UW, 1994, p. II-43):

10 log10ð�KmaxÞ ¼ 79� 20 log10 F : ð8:203Þ

8.3.2.1.2 Low-frequency wind noise (Kuperman–Ferla measurements)

The behavior of the wind noise source level at frequencies of order 1 kHz and below ismore difficult to measure, and hence less well established than at higher frequency.One reason for this is masking from shipping noise. Another is that low-frequencysound can travel further, tending to complicate propagation effects, making it moredifficult to separate changes in propagation loss from those in the source level.

The measurements of Kuperman and Ferla (1985) exhibit a similar wind speeddependence to that of the APL formula, with a spectrum that flattens off atfrequencies less than 400Hz. The asymptotic low-frequency value can be approxi-mated (purposefully mimicking the wind speed dependence of Equation 8.199) by

KwindLF ¼ 10

4:12

1:5vv2:24 mPa2 Hz�1; ð8:204Þ

where the value of the constant in the denominator (1.5) is chosen to match themeasured source level at 400Hz.

The level and frequency dependence of this low-frequency wind noise are not wellestablished, with some measurements showing decreasing wind noise with decreasingfrequency below about 1 kHz (Ingenito and Wolf, 1989; Cato and Tavener, 1997).

An alternative wind noise model is proposed by Ma et al. (2005) for frequenciesin the range 1 kHz to 50 kHz. Using an approximate relationship relating soundpressure level to source level (Equation 8.198), their Eqs. (3) and (4) can be written

KwindMa ¼ ð53:91vv10 � 104:5Þ2

�

8

F

� �1:57

mPa2 Hz�1 ð1 < F < 50; vv10 > 2Þ: ð8:205Þ


8.3.2.1.3 Proposed composite wind noise model

A smooth transition between the Kuperman–Ferla and APL wind noise sourcespectrum models is obtained by using the following composite formula for the dipolesource factor

Kwind ¼ 104:12vv2:24APL

ð1:5þ F 1:59Þ100:1�mPa2 Hz�1; ð8:206Þ

which is plotted in Figure 8.17.

8.3.2.2 Rain noise source level

In the same way as for wind, rain-related noise sources—also attributed to thecreation of tiny air bubbles close to the sea surface (Leighton, 1994)—are commonlyassigned a dipole radiation pattern, with Equation (8.207) defining the dipole sourcefactor K for rain

K rain ¼ 3�c2�

W rainAf : ð8:207Þ

Examples of measurements of rain noise in the open ocean are Scrimger et al. (1989),McConnell et al. (1992), Nystuen (2001), andMa et al. (2005). In the absence of wind,the rain noise spectrum has a strong peak at a few kilohertz. With wind, the peak,though still present, is less pronounced (Scrimger et al., 1989; Ma et al., 2005). Themeasurements of Nystuen show an additional dependence on the type of rain. Thus,


Figure 8.17. Wind noise areic dipole source spectrum level vs. frequency: ‘‘composite’’¼ eval-uated using Equation (8.206); ‘‘saturated’’¼ composite model, capped using Equation (8.203).

while the most important single parameter is the rainfall rate, the rain noise spectrumalso depends on drop size distribution and wind speed. An ideal predictive modelwould take all three parameters into consideration. Different rain noise models ofvarying complexity are given by APL-UW (1994), Nystuen (2001), and Ma et al.(2005).

Nystuen (2001) gives a detailed algorithm with a claimed accuracy of �1 dB, butwith no explicit dependence on wind speed. The simpler model of APL, describedbelow, is based on the measurements of Scrimger et al. (1989) and includes adependence on wind speed, but not on drop size. The measurements of Ma et al.(2005) show a dependence on wind speed for light rain, but not for heavy rain.

In the APL model, valid between 1 kHz and 100 kHz, the dipole source level canbe written

10 log10 KrainAPL ¼ 10 log10 K rain

20 ðRrain; vv10Þ þ

�10 log10 F 1 � F < 10

49 log10 F � 59:0 10 � F < 16

0 16 � F � 24

�23 log10 F þ 31:7 24 < F � 100,

8>>>>><>>>>>:

ð8:208Þ

the value at 20 kHz is given by the following function of rain rate Rrain and wind speed

10 log10 Krain20 ðRrain;UÞ ¼ bðUÞ þ aðUÞ log10 min

Rrain1mm/h

; 10

� �ð8:209Þ

and F is the frequency in kilohertz as before. Finally, the functions aðUÞ and bðUÞ are

aðUÞ ¼25:0 U � 1:5

5:0þ 5:7ð5:0�UÞ 1:5 < U < 5:0

5:0 U � 5:0

8><>: ð8:210Þ

and

bðUÞ ¼41:6 U � 1:5

50:0� 2:4ð5:0�UÞ 1:5 < U < 5:0

50:0 U � 5:0.

8><>: ð8:211Þ

In this model the dipole source level is independent ofU in the limits of both high andlow wind speed. These limiting cases are plotted in Figure 8.18 as dashed and solidlines, respectively, for rain rates between 2mm/h and 10mm/h.

8.3.2.3 Shipping noise source level

In Section 8.3.1, individual ships were considered as discrete sources of backgroundnoise. It can be useful to think of distant shipping lanes as continuous (sheet or line)sources. In the following a group of ships is represented first by a sheet of monopolesources and then by a sheet of dipoles.


8.3.2.3.1 Monopole density

Imagine a distribution of distant ships with an areic number density N shipA . Each

individual ship can be characterized as a point (monopole) source a few metersbeneath the surface. If the (average) power spectral density of each monopole isW ship

f , the average areic spectral density due to this distribution is given by

W shipAf ¼ N ship

A W shipf ; ð8:212Þ

or, in terms of a source level in decibels

SLAf ¼ 10 log10 NshipA þ SLmpf ; ð8:213Þ

where SLmpf is the monopole source spectrum level of an individual ship

SLmpf ¼ 10 log10�c

4�W ship

f

� �: ð8:214Þ

8.3.2.3.2 Dipole density

The field radiated by each monopole source interferes with its surface reflection insuch a way as to create a dipole radiation pattern at low frequency. The spectralradiant intensity at grazing angle � due to this dipole is related to the power spectraldensity of the original monopole at depth d (i.e., the power that the monopole wouldradiate in an infinite uniform medium of the same characteristic impedance as the


Figure 8.18. Rain noise areic dipole source spectrum level vs. frequency, evaluated using

Equation (8.208) for wind speed up to 1.5m/s (solid lines) and exceeding 5.0m/s (dashed lines);

rain rates are 2mm/h (lowest) to 10mm/h (highest) in steps of 2mm/h.

true medium) in the following manner:

W dpfO ¼ k2d 2 sin2 �

��Wmp

f ðkd < �=4Þ: ð8:215Þ

In Chapter 2 a relationship is derived between the power and radiant intensity of adipole, which in spectral form can be written

W dpfO ¼ 3 sin

2 �

2�W dp

f : ð8:216Þ

Eliminating W dpfO from Equations (8.215) and (8.216), rearranging for W dp

f , andsubstituting into Equation (8.196) gives an expression for the corresponding dipolesource factor

K ship ¼ 4k2d 2N shipA 10SL

mp

f=10: ð8:217Þ

Databases containing estimates of shipping density for the main global shipping lanesare described by Hamson (1997), Etter (2003), and Anon. (2003).

8.3.3 Distributed sources on the seabed (crustacea)

There are times when sound sources located on the seabed drown out the waves,especially in habitats sustaining crustacea colonies. Some species of crustacea, andsnapping shrimp in particular, can cause very loud and persistent broadband noise.The sound is caused by many individuals clicking (or ‘‘snapping’’) in unison. Becauseof the large number of individuals involved, the net result is that of an extendedsource on the seabed.

8.3.3.1 Snapping shrimp

Snapping shrimp are a ubiquitous source of underwater sound in shallow water witha rock or coral bottom of depth less than 60m, and in warm latitudes within about35 deg latitude from the equator16 (Johnson et al., 1947; Cato and Bell, 1992). Typicalreported spectral levels at 5 kHz are 60 dB to 70 dB re mPa2/Hz (Anon., 2003), butsignificantly higher and lower values are sometimes encountered, perhaps dependingon the proximity of the hydrophone to the seabed. Shrimp noise is subject to up to8 dB diurnal variation, with highest levels occurring just after sunset and beforesunrise. Cato and Bell (1992) observed no significant seasonal variation.

The sound creation mechanism involves the creation and subsequent collapse ofa large cavitation bubble. The temperature and pressure reached inside the bubbleduring its collapse are so high17 that a flash of light is sometimes emitted (Lohse et al.,2001).


16 More precisely, within latitudes whose winter temperature does not fall below 11 �C.17 The estimated maximum temperature inside the cavitation bubble exceeds 5000K.

According to a laboratory experiment by Au and Banks (1998), a single shrimpsnap has an energy source level of between 127 dB and 135 dB re mPa2 m2 s. Alsoreported is the peak-to-peak source level SLp-p (see Box), the values of which arebetween 183 dB and 189 dB re mPa2 m2, depending on the size of the claw. Fergusonand Cleary (2001) obtain similar values from in situ measurements. Peak acousticpressures of up to 80 kPa are reported by Versluis et al. (2000) at a distance of 4 cm,making snapping shrimp one of the loudest animals in the sea.

The frequency spectrum of shrimp noise covers a very wide frequency band. Thespectral density falls off slowly from its peak at about 2 kHz, with significantcontributions remaining even up to 200 kHz, as illustrated by Figure 8.19.

8.3.3.2 Other crustaceans

Other species of crustacean known as sources of underwater sound, though less wellstudied than snapping shrimp, include mussels (APL-UW, 1994) and spiny lobsters(Latha et al., 2005; Patek et al., 2009). Sounds made by crustaceans are reviewed bySchmitz (2002).


Figure 8.19. Measured waveform and frequency spectrum of a single shrimp snap. The peak in

the spectrum at 2 kHz is due to the 400 ms delay between the precursor and the main arrival

(reprinted with permission from Au and Banks, 1998, American Institute of Physics).#

Peak-to-peak, zero-to-peak, and peak-equivalent RMS source levels

The term peak-to-peak (p-p) source level, abbreviated SLp-p, is used to mean 10times the base-10 logarithm of the squared difference between the maximum andminimum pressure in a short impulse-like wave form, measured in the far field ofthe source and scaled to a standard reference distance from the source ofrref ¼ 1m. If the far-field (and free-field) measurement distance is s0, it is commonpractice to obtain the source level by multiplying the measured pressure by a factorof s0=rref . The assumptions implied by this conversion are that spherical spreadingholds and that the waveform does not change in shape or duration. With theseassumptions, SLp-p can be written as

18

SLp-p ¼ 10 log10ðs20fmax½q0ðtÞ� �min½q0ðtÞ�g2Þ dB re mPa2 m2: ð8:218ÞFor the special case of a sinusoidal wave form, SLp-p is related to the definitionintroduced in Chapter 3, which is based on RMS pressure and denoted SLRMS herefor clarity (elsewhere it is simply SL), according to

SLp-p ¼ SLRMS þ 10 log10 8 � SLRMS þ 9:0 dB re mPa2 m2: ð8:219ÞFor this reason, peak pressures are sometimes reported as peak-equivalent RMSvalues, denoted SLpeRMS and defined as (Møhl et al., 2000)

SLpeRMS � SLp-p � 10 log10 8 � SLp-p � 9:0 dB re mPa2 m2: ð8:220ÞAlso used, especially in the context of explosive or seismic survey sources, is thezero-to-peak source level, SLz-p, which, given the same assumptions as above canbe defined as

SLz-p ¼ 10 log10½s20 maxjq0ðtÞj2� dB re mPa2 m2: ð8:221ÞConversions between these different measures of source level are discussed inChapter 10 for a selection of representative pulse shapes.

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18 There is some ambiguity in Equation (8.218) unless a time window is specified. It is normal

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9

Propagation of underwater sound

So if the physics is necessarily complicated it can pay to keep themathematics simple, giving a better chance of seeing

the wood despite the trees.

David E. Weston (1971)

No book on sonar would be complete without a chapter on underwater soundpropagation, and this is it. The subject is central to sonar performance modelingbecause all sound, whether contributing to the signal, ambient noise, or reverbera-tion, must propagate through the sea before arriving at the sonar. Thus, the scopeincludes not only propagation loss (PL), but also the effect of sound propagation onnoise level (NL), and for active sonar the reverberation level (RL) and target echolevel (EL). These terms were all introduced in Chapter 3, where they were applied tosimplified sonar problems. The present chapter adds more realism by describing theeffects of a reflecting seabed and a sound speed profile.

Since the pioneering work of Lichte (1919),1 modeling of underwater soundpropagation has increased steadily in sophistication, such that today a variety ofreliable computational models exists (oalib, www). The interested reader is referredto Jensen et al. (1994) for a thorough description of the different computationaltechniques used by these models and to Brekhovskikh and Lysanov (2003) for thetheoretical foundations of ocean acoustics. It is inevitable that some material fromthese books is duplicated here, but an attempt is made to keep such duplication to aminimum. (For example, the reader is assumed to be familiar with the basic conceptsof image theory, ray theory, and normal-mode theory—Jensen et al., 1994). Theemphasis here is placed on providing physical explanations for the effects, with simple

1 Lichte was the first scientist to recognize the effect that pressure, temperature, and salinity

gradients would have on the propagation of sound in the sea (see Chapter 1).

440 Propagation of underwater sound [Ch. 9

estimates of their magnitude where feasible, drawing heavily in so doing from theideas of Weston (see, e.g., Weston, 1960, 1979, 1980, 1994).

In the passive sonar equation there are two terms, namely PL and NL, that arestrongly affected by propagation. These two terms are considered first (Sections 9.1 to9.2). Both are relevant also to the active sonar equation, with the EL (Section 9.3)and RL (Section 9.4) terms also influenced by propagation effects. The signal-to-reverberation ratio is considered in Section 9.5.

9.1 PROPAGATION LOSS

The purpose of this section is to illustrate the influence on the propagation ofunderwater sound of important mechanisms omitted from the simpler descriptionpresented in Chapter 2. The main new effects considered are reflection of sound fromthe seabed and refraction in the water due to variations of sound speed with depth.Also relevant are horizontal gradients in sound speed and water depth. Althoughhorizontal sound speed gradients are small compared with vertical ones, their effectsbecome increasingly important at increasing distance from the source. These long-range effects (see, e.g., Jensen et al. 1994, p. 36, 323ff, 397ff) are outside the presentscope. Also excluded are time-dependent effects due to the motion of surface waves,currents, eddies, and internal waves (the Doppler shift associated with a movingtarget is described in Chapter 6).

The emphasis here is on simple analytical formulas rather than exact solutions,whether analytical or numerical. The approximate analytical solutions are notintended to replace those of numerical models, but to complement them by explainingtrends and promoting insight.

9.1.1 Effect of the seabed in isovelocity water

9.1.1.1 Deep water

Compared with the sea surface, the seabed is a poor reflector of sound whosereflection coefficient depends strongly on angle. The proportion of sound energyreflected increases from around 1% to 10% at normal incidence to 80% to 100%at angles close to grazing incidence. The influence of the reflected sound is especiallyimportant in shallow water and for sound traveling close to the horizontal direction,which becomes trapped between the sea surface and seabed.

The geometry of the problem is illustrated by Figure 9.1. The deep-waterassumption implies that, in relative terms, both source and receiver are close tothe sea surface. It then becomes convenient to collect ray paths in groups of fourwith an identical number of bottom reflections, as the individual rays in such a groupfollow very similar trajectories. Using m to denote the number of bottominteractions, the first such group (m ¼ 1) follows a V-shaped path, corresponding

9.1 Propagation loss 441]Sec. 9.1

to the left-hand picture of Figure 9.1, and the second one (m ¼ 2) a W shape (right-hand picture).2

If the pressure field is written as a sum over image contributions, the similaritybetween paths simplifies the analysis considerably. An example calculation is pre-sented in Figure 9.2 with the density of the seabed equal to 1.222 relative to that ofwater, and no change in sound speed. These values are deliberately chosen to providea weak reflection (only 1% of the incident energy is reflected for this problem), sothat the second and subsequent reflections are heavily damped, making it easier tostudy the first reflection separately from the others. The water depth is 1000m, whichwhile less deep than the main oceans is deep enough to illustrate the effects of interesthere. The upper graph of Figure 9.2 (solid line) shows PLðrÞ calculated using the fast-field program SAFARI (Schmidt, 1988; Jensen et al., 1994) for this case. PL increasessystematically with increasing range, and superimposed on this trend is a beat patternof increasing period and amplitude. This result can be understood in terms of

Figure 9.1. Diagrammatic ray paths illustrating the geometry for bottom reflections in

deep water (reprinted with permission from Ainslie, 1993, American Institute of

Physics).

#

2 The water is considered here to be sufficiently deep that only a small number of bottom-

reflected paths contributes to the total field, making the effects of the seabed easier to

understand.


Figure 9.2. Propagation loss [dB rem2] vs. range for reflecting seabed (�sed=�w ¼ 1.222) at

f ¼ 250Hz. Upper: propagation loss (SAFARI) and BL, LM components; lower: components

BL (blue curve: Equation 9.15), LM (green curve: Equation 9.1) and their sum (INSIGHT).

interference between Lloyd mirror paths (abbreviated LM) and bottom-reflectedones (abbreviated BL), plotted separately and color-coded green (LM) and blue(BL) in the lower graph, calculated using the INSIGHT model (Ainslie et al.,1996). It can be seen that the dominant contributions come from LM at short rangeand from BL at long range. The beat pattern can be understood as resulting frominterference between these. For example, its dynamic range is greatest when theseparate LM and BL contributions are equal (e.g., at 4 km or 5.2 km). The shapeof the individual LM and BL components is explained in Sections 9.1.1.1.1 and9.1.1.1.2, respectively.

9.1.1.1.1 Lloyd mirror

Neglecting attenuation and assuming a perfectly reflecting sea surface, the complexLM pressure can be written as the following sum of two images (Chapter 2)

pLMðr; zÞ ¼ffiffiffi2

ps0p0

eiks�

s�� eiksþ

sþ

!e�i!t; ð9:1Þ

where

s� ¼ s�ðr; zÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ ðz� z0Þ2

q: ð9:2Þ

If the depths z and z0 are both small compared with the range r, then

s� � rþ ðz� z0Þ2

2r; ð9:3Þ

and hence

pLMffiffiffi2

pp0s0

� �2 irsin

kzz0reiðkr�!tÞ: ð9:4Þ

Thus, the propagation factor (the squared modulus of Equation 9.4) is

FLM � 4

r2sin2

kzz0r: ð9:5Þ

At long range, the sine function can be replaced by its argument, giving an r�4

dependence on range (40 log10 r in dB), and explaining the shape of the LM curveof Figure 9.2.

9.1.1.1.2 Bottom-reflected paths

For each value of m, in addition to the obvious straight-there-and-back bottom-reflected (BL) ray path that does not interact with the sea surface, there are two othersthat reflect once from the surface and one that does so twice. These four multipathsare illustrated in the close-up of Figure 9.1. A convenient expression for the sumof images, derived by Harrison (Harrison, 1989; Ainslie and Harrison, 1990), is


reproduced below. The mth BL term can be written

pmðr; zÞffiffiffi2

pp0s0

¼RmBR

m�1S

exp ik0s��s��

þRSexp ik0s�þ

s�þþRS

exp ik0sþ�sþ�

þR2S

exp ik0sþþsþþ

� �e�i!t;

ð9:6Þ

where

s2�� ¼ ð2mH � z0 � zÞ2 þ r2 ð9:7Þand

s2þ� ¼ ð2mH þ z0 � zÞ2 þ r2: ð9:8Þ

Putting RS ¼ �1 gives (correcting a sign error in Eq. A1.5 of Ainslie and Harrison,1990)


pp0s0

¼�4ið�RBÞmcos �0r

ðcos �z cos �z0 sin �þi sin�z sin�z0 cos �Þ expðik0SÞ e�i!t;

ð9:9Þ

where

S ¼ r�� þ r�þ þ rþ� þ rþþ4

; ð9:10Þ

� ¼ k0 sin �0; ð9:11Þ

� ¼ z0z

rð9:12Þ

tan �0 ¼2mH

rð9:13Þ

and ¼ k0 cos �0: ð9:14Þ

From Equation (9.9) it follows that (Harrison, 1989)

FBL ¼ 16R2mB

cos2 �0r2

ðsin2 �z0 sin2 �z cos2 � þ cos2 �z0 cos2 �z sin2 �Þ: ð9:15Þ

A simpler version, valid for near-surface source and receiver, and more convenientfor comparison with later expressions for bottom-refracted contributions, follows byassuming � is negligible. The result is


pp0s0

� 4ð�1ÞmRmB

cos �0r

sin �z0 sin �z exp ik0r

cos �0

� �e�i!t; ð9:16Þ

and hence

FBL � 16R2mB

cos2 �0r2

sin2 �z0 sin2 �z: ð9:17Þ

The blue curve of Figure 9.2 (lower graph) is PLBL ¼ �10 log10 FBL for the casem ¼ 1. The beating pattern in the blue curves is due to interference between the fourmulti-paths. The deep nulls are places at which either �z or �z0 is an integer multiple


of , so that the right-hand side of Equation (9.16) vanishes. For higher order paths(m 2) the propagation loss exceeds 110 dB rem2 in the graph, and these paths areconsequently too weak to be visible. Finally, the uppermost line is the incoherent sumof both contributions (i.e., �10 log10ðFLM þ FBLÞ), color-coded according to thelarger of the individual propagation factors.

9.1.1.1.3 Bottom-refracted paths

If 1% of the energy is reflected in the above example, consider now the fate of theremaining 99%. If the seabed were an infinite uniform half-space with density andsound speed independent of depth, the sound would continue unimpeded on itsdownward path forever. In practice there are changes in impedance, some abruptand some gradual, that cause some of the sound to be reflected. Further the speed ofsound tends to increase with increasing depth in the seabed. This sound speed gradi-ent, typically of order 1 s�1 (Chapter 4), has an important effect on low-frequencysound because it refracts the sound upwards, eventually returning it to the water if theinitial angle is not too steep, after following a U-shaped path in the sediment asillustrated by Figure 9.3. High-frequency sound is refracted in exactly the same way,but the effects are less important due to the increased attenuation (see graphs ofreflection loss vs. angle and frequency in Chapter 8).

Notice the shadow near the middle of the ray trace (Figure 9.3), and the region tothe right of this filled by bottom-refracted (BR) paths. These two regions are sepa-rated by a line called a caustic, along which an infinite ray density is reached. Thesound field has a maximum close to this line, and a dramatically different charactereither side of it. For a numerical example (see Figure 9.4), we choose a sedimentsound speed gradient c0 ¼ 1/s and attenuation �sed ¼ 0.03 decibels per wavelength.Other parameters are as Figure 9.2. The expected range to the caustic, using Equation(9.20) below, is 4.9 km. At short range, before the caustic, there is little difference


(a) (b)

Figure 9.3. Bottom-refracted (BR) ray paths travel through the sediment and form a caustic

in the reflected (i.e., bottom-refracted) field (reprinted with permission from Ainslie, 1993,

American Institute of Physics).#


Figure 9.4. Propagation loss [dB rem2] vs. range for a reflecting and refracting seabed. Upper:

propagation loss (SAFARI) (reprinted with permission from Ainslie, 1993, American

Institute of Physics); lower: LM andBL components (fromFigure 9.2), BR (red curve: Equation

9.19—in this example, Equation 9.29 is not needed for the field through the caustic itself because

the steep caustic paths are absorbed during their transit through the sediment), and their sum

(INSIGHT). (c 0 ¼ 1/s, �sed ¼ 0.03 dB/�; other parameters as Figure 9.2.)

#

between the two graphs. Beyond this point, however, the total PL in Figure 9.4 isstrongly affected by the arrival of BR rays, contributions from which are shown in redin the lower graph of Figure 9.4. These paths completely dominate the field between5 km and 10 km, to the right of the caustic.

Ensonified region. The sum of images used for BL cannot be applied to BRpaths because of the sound speed gradient in the sediment. Suitable alternativemethods such as ray or mode theory are described by Jensen et al. (1994). Ainslie(1993) uses normal mode theory with the stationary phase approximation (AppendixA) to derive the result

pmðr; zÞ ¼ pþ þ p�; ð9:18Þwhere (for r > mrc)

p�ffiffiffi2

pp0s0

¼ 4ð�1Þm½RBð��Þ�mcos ��r

�� þ 1

�� 1

��1=2sin ��z0 sin ��z exp ið�� =4Þ e�i!t

ð9:19Þand rc is the caustic range

rc ¼ 4c0H

c0

� �1=2

: ð9:20Þ

Comparing Equation (9.19) with Equation (9.16), the main difference is the factorcontaining terms of the form ð� � 1Þ1=2. This factor quantifies the bunching up of raysin the vicinity of the caustic. At the caustic itself the denominator vanishes and thefactor becomes infinite. The true pressure field, which must be finite, then needs to becalculated a different way, as explained in the section entitled caustic and shadowregion below.

The amplitude reflection coefficient RB is given by

RBð�Þ ¼ exp � 2!"c0

Yð�Þ þ i2!

c0½Yð�Þ � sin ��

2

� � �; ð9:21Þ

where

Yð�Þ ¼ loge tan

4þ �

2

� �ð9:22Þ

and " is the fractional imaginary part of the sediment wavenumber

" ¼ �sed40 log10 e

; ð9:23Þ

with �sed in decibels per wavelength.The phase term is given by

�� ¼ �rþ 2m��H þ =4; ð9:24Þwhere

�� ¼ k0 sin �� ð9:25Þand

� ¼ k0 cos �� ð9:26Þ


are the vertical and horizontal wavenumbers; and �� is the ray grazing angle of eachof the two branches of the caustic

tan �� ¼ c0

4mc0r�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 �mr2c

q� �: ð9:27Þ

The phase of Equation (9.19) also includes the term �=4. The origin of this term isrelated to the transition of individual rays through the caustics. Each ray passesthrough one caustic per cycle, and each traversal results in a phase change of =2.The phase difference arises because the steeper ray has traversed one fewer causticthan the shallow one, with a consequent =2 lag in phase (Boyles, 1984, p. 235).

One final parameter, needed for the evaluation of Equation (9.19), is

�� ¼ c0c 0H

tan2 ��: ð9:28Þ

Caustic and shadow region. Equation (9.19) is valid in the ensonified region to

the right of the caustic in Figure 9.3, but not at the caustic itself. The ratio�� þ 1

�� 1

�� in

Equation (9.19) is a measure of ray density. At the caustic itself, this term becomesinfinite and the method used to derive Equation (9.19) breaks down. It can be shownusing second-order stationary phase theory that the field in the immediate vicinity ofthe caustic is described by an Airy function (see Appendix A). Through the causticitself, the field is given by (Ainslie, 1993)

pmffiffiffi2

pp0s0

¼ ð�1Þm½RBð�cÞ�mQ

H

2

cr

� �1=2

sin �cz0 sin �czAið��Þ eið�c�!tÞ; ð9:29Þ

where the phase term is

�c ¼ crþ 2m�cH þ =4 ð9:30Þ

and the wavenumber components are

�c ¼ k0 sin �c ð9:31Þand

c ¼ k0 cos �c; ð9:32Þ

where �c is the angle of the caustic grazing ray

tan �c ¼c0H

c0

� �1=2

: ð9:33Þ

The Airy function argument, which determines the position and width of the caustics,is

� ¼ r�mrcrc

4H cos2 �cc0

!2c0

2m

!1=3

: ð9:34Þ

This Airy function provides a smooth transition between the oscillatory field in the


ensonified region to the right of the caustic (� > 0) and the shadow to the left of it(� < 0).

Equation (9.29) is valid for ranges close to the integer multiples of the causticrange. There is an assumption in its derivation that the density is uniform everywhereand the sound speed is continuous across the water–sediment boundary. The smallstep in density for the present example is neglected on the grounds that only 1% ofthe energy is reflected, so its effect on the result is minor.

Sensitivity to seabed parameters. The sensitivity of the sound field to seabedparameters is considered next (see Figure 9.5). The first two graphs (upper row)are simple to understand: increasing the sediment attenuation (upper left graph)turns off BR paths (in red), while decreasing the density ratio �sed=�w (upper right)turns off BL (in blue) because of the decreasing reflection coefficient, while leavingthe BR contribution approximately unchanged. The second row is more subtle. Theeffect of csed=cw on BL is similar to that of �sed=�w, and it also influences the angles ofthe BR paths through Snell’s law, and hence the interference patterns associated withthese. The main effect of increasing c 0 (lower right) is to reduce the caustic range(Equation 9.20), in this example to about 4 km.

9.1.1.2 Shallow water

In the world’s oceans there is a clear distinction between the deep ocean of depth 3 kmto 5 km (most of it) and the continental shelf, of depth 20m to 200m, separated byregions of relatively steep slopes. The close proximity of the seabed in shallow watermeans that its acoustical properties play a central role in shallow-water propagation.The reason why the seabed is so important is the ability of the sea to trap soundbetween the seabed and sea surface, forming a waveguide able to carry sound overmany kilometers. Each time sound is reflected from the seabed a little energy is lost,eventually limiting low-frequency propagation in shallow water. The proportion ofenergy lost at each reflection depends on the bottom type as well as the grazing angleand acoustic frequency.3

Propagation in shallow water can also be strongly frequency-dependent and themain reason for this is the existence of a minimum propagation frequency, known asthe cut-off frequency, below which waveguide propagation is not supported. Thediscussion below assumes initially that the frequency is above this cut-off frequency,which is the subject of Section 9.1.1.2.5. A second assumption is that the field is notinfluenced by coherent interference effects due to cancellation between upward-traveling and downward-traveling paths due to phase reversal at the sea surface.This second assumption is addressed in Section 9.1.1.2.6.


3 Thus, a complicating feature of shallow-water propagation is the great variety of bottom

types encountered, with mud, sand, rock, and gravel all common, sometimes in close proximity

to one another.


Figure 9.5. Propagation loss [dB rem2] vs. range/km for a reflecting and refracting seabed:

sensitivity to �sed (upper left), �sed=�w (upper right), csed=cw (lower left), and c0 (lower right). Other

parameters as Figure 9.4 (INSIGHT).

9.1.1.2.1 Multipath propagation

An important consequence of the factor-40-or-so difference in depth between deepand shallow water is that, at any given range, there are many more ray arrivals toconsider in shallow water. For calculations in shallow water, it is helpful to expressthe total field as a sum over the energy contribution from each multipath (MP), in theform

FMP ¼ 4X1m¼1

cos2 �mr2

jRð�mÞj2m; ð9:35Þ

where

m ¼ r tan �m2H

: ð9:36Þ

The validity of the energy sum in Equation (9.35) requires the phase of the multipathsto be randomly related to one another, in such a way that there is no systematicconstructive or destructive interference. The approximation breaks down close to asmooth boundary (such as the sea surface), where pairs of rays tend to arrive withalmost identical pathlengths—their phase differing only by the phase change at thereflecting boundary. A rule of thumb for the use of Equation (9.35) is that thedistance of both source and receiver from the sea surface must exceed �=sin �, where� is the acoustic wavelength and � the grazing angle of the corresponding ray arrival.

Approximating the sum as an integral over a continuum in m and changing theintegration variable to � using

dm ¼ r

2H cos2 �d�; ð9:37Þ

it follows that

FMP � 2

rH

ð=20

jRð�Þj2m d�: ð9:38Þ

More generally, it is convenient to write F in terms of the differential propagationfactor Gð�Þ

F ¼ð=20

Gð�Þ d�; ð9:39Þ

where Gð�Þ d� is the contribution to the propagation factor from ray paths atelevation angles between � and �þ d�. For this example, it is given by

Gð�Þ � 2

rHjRð�Þj2m: ð9:40Þ

9.1.1.2.2 Spherical and cylindrical spreading regions

If the reflection coefficient is approximated as a Heaviside step function, such that

jRð�Þj �1 � < c

0 � > c,

�ð9:41Þ



FMP � 2 cH

r�1: ð9:42Þ

This 1=r behavior (10 log10 r in dB) is known as cylindrical spreading (CS) because ofthe cylindrical geometry that leads to it, as clarified below. For spherical spreading(SS), the area into which the sound spreads is a sphere of radius r

ASS ¼ 4r2: ð9:43Þ

In shallow water, sound cannot spread into an indefinitely large sphere, but is limitedinstead to a cylinder of height H and radius r, so that

ACS ¼ 2rH; ð9:44Þ

leading to the 1=r dependence in Equation (9.42).

9.1.1.2.3 Mode-stripping region

A modification to cylindrical spreading is needed for long ranges once the reflectionlosses due to multiple bottom reflections begin to accumulate. An improved approx-imation for FMP can be obtained by using a more realistic approximation for Rð�Þ,taking into account reflection losses near grazing incidence, of the form

jRð�Þj �expð��Þ � < c

0 � > c,

�ð9:45Þ

where � is the rate of increase of reflection loss with the angle in units of nepers perradian. It is referred to below as the ‘‘reflection loss gradient’’.

Substitution of Equation (9.45) into Equation (9.38) yields

FMP ¼ 2�effrH

erf1=2 c2�eff

; ð9:46Þ

where

�eff ¼H

4�r

� �1=2

: ð9:47Þ

Equation (9.46) contains the cylindrical spreading result (Equation 9.42) as a specialcase in the short-range limit ( c � �eff ). At long range ( c � �eff ) it becomes

FMP � 2�effrH

; ð9:48Þor, equivalently,

FMP �

�H

� �1=2

r�3=2: ð9:49Þ

This 15 log10 r dependence on range is known as mode stripping because it resultsfrom the gradual erosion of steep ray paths (high-order modes) after multiple bottomreflections.


The transition between Equation (9.42) and Equation (9.49) (the range at whichcylindrical-spreading and mode-stripping contributions are equal) occurs at a rangerCS given by

rCS ¼H

4� 2c: ð9:50Þ

To illustrate the transition from cylindrical spreading to mode stripping we nowconsider shallow-water propagation for two different bottom types, sand andmud, the relevant properties of which are summarized in Table 9.1. The reflectionloss gradient for sand or coarse silt is given by (see Chapter 8)

�sand ¼ 2"�sed�w

cos2 csin3 c

; ð9:51Þ

with a typical value of between 0.1Np/rad and 1.0Np/rad.For mud (clay or fine silt) there is no critical angle, so Equation (9.51) is not

appropriate. Instead the reflection coefficient for a refracting sediment can be usedfrom Section 9.1.1.1.3. For small �, Equation (9.21) implies

jRð�Þj � exp � 2!"c0

�

� �ð9:52Þ

and hence

�mud ¼2!"

c0: ð9:53Þ

According to this result, if " is a constant the reflection loss for the mud case isproportional to frequency, as indicated by the corresponding entry in Table 9.1. The


Table 9.1. Characteristic properties from Chapter 4 of medium

sand (Mz ¼ 1.5) and mud (Mz ¼ 8).a

Sand Mud

Grain size Mz 1.5 8

csed=cw 1.20 1.00

�sed=�w 2.1 1.4

�sed=(dB/�) 0.88 0.09

" ¼ �sed=40 log10 e 0.0161 0.00165

(Equation 9.23)

c0=s�1 0.0 1.0

�=Np rad�1 0.28 0.021ff

(Equation 9.51) (Equation 9.53)

a The mud sediment (Mz ¼ 8) has properties intermediate betweenthose of very fine silt and coarse clay.

precise frequency dependence of " is the subject of ongoing research (Buchanan,2006). The work of Hamilton (1980, 1987), and Kibblewhite (1989) demonstratesthat the general trend is consistent with the assumption of attenuation being propor-tional to frequency (consistent with constant ") over a frequency range of about fivedecades between 0.01 kHz and 1000 kHz. However, the existence of such a trend doesnot preclude departures from linearity across a more limited frequency range.

The accuracy of these expressions for � (Equation 9.51 for sand and Equation9.53 for mud) should not be taken too seriously. Both are approximations intended toillustrate the difference in behavior between sand and mud at low frequency in aqualitative manner. For example, Equation (9.51) tends to overestimate the reflectionloss for sand sediments at grazing angles close to c, as illustrated by Figure 9.6. Inpractice, the effect is less serious than it seems because it is the contribution fromnear-grazing angles, for which the error is small, that provides most of the energy atlong range. A more precise calculation of reflection loss can be found in Chapter 8 forsand (Mz ¼ 2.5) and mud (Mz ¼ 8.5).

The single most important parameter of Table 9.1 is the sound speed ratio. It isthis parameter that determines the presence or absence of a critical angle, its magni-tude if present, and hence in broad terms the overall reflectivity of the seabed. If thereis no critical angle, the most important parameter then becomes the ratio �sed=c

0,which determines the loss per cycle due to absorption in the sediment.

The importance of the seabed for shallow-water propagation is illustrated byFigure 9.7, which shows propagation loss vs. range and bottom reflection loss vs.angle for sand and mud sediments at a frequency of 250Hz. In all other respects


Figure 9.6. Reflection loss [dB] vs. angle for sand (1.5�) comparing the Rayleigh reflectioncoefficient (solid red) with the approximation of Equation 9.51 (dotted blue).


Figure 9.7. Propagation loss [dB rem2] vs. range (upper) and reflection loss [dB] vs. angle

(lower) for sand (thick solid lines) and mud (thin lines) in shallow water at frequency

250Hz (INSIGHT).

the two environments are identical. The reflection loss for mud is much higherthan that for sand, and this manifests itself as a correspondingly higher propagationloss.

9.1.1.2.4 Single-mode region

Up to this point the field has been described without taking into account the discretenature of the normal-mode spectrum. Pairs of plane waves traveling in the oceanwaveguide (one in the upward direction and one downward) combine to form‘‘modes’’ if their phases are aligned in such a way as to match the boundaryconditions of the waveguide (Jensen et al., 1994). The alignment only occurs forcertain preferred directions that depend on these boundary conditions. The densityof modes (i.e., the number of preferred directions per unit angle) increases withincreasing frequency such that at sufficiently high frequency the discrete nature ofthe modes may be disregarded. At low frequency, however, there are some features ofpropagation that cannot be explained without invoking individual modes, and thesingle-mode region is one such feature, as follows.

The process of mode stripping has the effect of gradually reducing the number ofmodes contributing to the field, starting by removing the highest order modes andcontinuing until only a few low-order modes remain. Eventually only one mode is leftand at this point the mode-stripping regime ends—there are no more modes to stripaway—and the single-mode regime begins. Beyond this point, the field is dominatedby the lowest order mode and can be approximated by the formula

FMP � 4�

H 2ersin2

z0He

sin2z

He

exp � ��2

4H 3e

r

!; ð9:54Þ

where He is the effective water depth, which is the depth at which a pressure releaseboundary appears to exist, a short distance beneath the true seabed (Weston, 1960,1994), given by4

He ¼ H þ �sed=�wð!=cwÞ sin c

: ð9:55Þ

The transition from Equation (9.49) to Equation (9.54) occurs when the effectiveangle (Equation 9.47) falls to a value between the propagation angle of the first andsecond modes. The propagation angle for the nth mode is approximately

�n �n

ð!=cwÞHe

: ð9:56Þ

The transition range between mode-stripping (MS) and single-mode regions can beestimated by equating �n and �eff with n ¼ 3=2 (halfway between integers 1 and 2)

rMS �k2H 3

e

9�: ð9:57Þ


4 For an extension of this concept to include the effects of sediment shear waves, see Chapman

et al. (1989).

The single-mode region is usually a feature of low-frequency propagation only, saybelow 100mHz in deep water and about 10Hz in shallow water. In very shallowwater, however, the single-mode region can be important to higher frequencies, up toabout 1 kHz for a water depth of 10m.

9.1.1.2.5 Cut-off frequency

An important feature of shallow-water propagation, mentioned at the start of Section9.1.1.2, is the existence of a waveguide cut-off frequency, below which ducted pro-pagation does not occur. The condition for a cut-on duct is that at least onepropagating mode exists. The total number of propagating modesN can be estimatedby requiring that the product of effective water depth He and wavenumber be aninteger multiple of , that is,

ð!=cwÞHe sin c ¼ N: ð9:58Þ

Thus, the requirement for at least one cut-on mode (i.e., N 1) translates to

f > fc; ð9:59Þ

where fc is the cut-off frequency

fc ¼� �sed=�w2 sin c

c

H: ð9:60Þ

An alternative form is

H

�>� �sed=�w2 sin c

: ð9:61Þ

9.1.1.2.6 Depth dependence

If the receiver is close to the sea surface, a coherent interference effect occurs betweenan upward-traveling path and the corresponding downward-traveling surfacereflected path. As the receiver approaches the surface the path difference tends tozero and, because of the phase reversal at the surface, the phase difference to . In thissituation (perfect reflection with phase reversal) the total field is proportional to thequantity

WðzÞ ¼ 2 sin2 �z: ð9:62Þ

The assumptions made in the flux derivation of Sections 9.1.1.2.1 to 9.1.2.1.3 amountto replacing this function by its average value, an approximation that works wellalmost everywhere except at the sea surface. A pragmatic version that retains thecorrect depth dependence at the surface without fussing about detail elsewhere is

WðzÞ � 1

1 þ ð2�2z2Þ�1 : ð9:63Þ

The same logic applies at the source, such that the combined dependence on both


source and receiver depth (assuming these are not coincident) is

Wðz0ÞWðzÞ � 1

1þ ð2�2z20Þ�11

1 þ ð2�2z2Þ�1 : ð9:64Þ

This behavior is known as ‘‘surface decoupling’’ because the pressure at the seasurface is decoupled from the rest of the medium.

9.1.2 Effect of a sound speed profile

An important characteristic of the world’s oceans is that the speed of sound in the seais not uniform, but varies with temperature T and salinity S, both of which vary inspace (especially with depth) and time. It also increases with increasing pressure P.The result of these variations in depth is called a sound speed profile. The sound speedprofile can have a profound influence on the propagation of underwater sound.Horizontal gradients in S and T can occasionally be important (e.g., across frontsand eddies), but horizontal gradients are usually much smaller than verticalgradients.

9.1.2.1 Deep water

The acoustic consequences of vertical gradients are considered below. For example,assuming uniform T and S, the inexorable increase of P with depth leads to a positivesound speed gradient. This isothermal behavior is characteristic of the deep ocean atdepths exceeding 2 km, as illustrated by the upper graph of Figure 9.8. This graphshows two different profiles, differing mainly in the top 75m. The two profiles arefor winter and summer conditions, and calculated using Mackenzie’s formula (seeChapter 4).

An isothermal layer can also arise close to the sea surface, where it is typicallycaused by wind mixing, and this is illustrated by the uppermost 75m of the winterprofile (see lower graph of Figure 9.8). The uniform temperature results in a mildlyincreasing sound speed with depth caused by the pressure gradient. In this situation,sound rays are refracted upwards in accordance with Snell’s law, in this case forminga near-surface waveguide known as a surface duct. Surface heating can reverse thissituation because an increase in surface temperature leads to a negative sound speedgradient and downward refraction for the summer profile (Figure 9.8, upper graph).In this situation, sound is then deflected away from the surface and an acousticshadow zone forms there. These two situations, and more complicated phenomenacaused by a combination of both upward and downward refraction, are described insubsequent sections.

More generally, any variation of the speed of sound with depth can result in asubtle but important change in direction of ray paths through refraction. Dependingon the details of the sound speed profile, regions of particularly high or low density ofray paths can form, resulting in correspondingly high or low acoustic intensity.Sound speed minima are particularly important features. Sound becomes trappedin these minima by refraction in the same way as for the surface duct. For this reason,



Figure 9.8. Upper: sound speed profile vs. depth for (19�N, 150�E) in the northwest Pacific (seeChapter 4 and Table 9.2): summer profile (red solid) and winter profile (blue dashed); lower:

zoomed sound speed profile (top 300m).


Table 9.2. Sound speed profiles for the northwest Pacific

location, as plotted in Figure 9.8 (calculated using tem-

perature and salinity profiles from Chapter 4).

Depth/m cðzÞ/m s�1

Summer Winter

0. 1543.503 1536.640

10. 1543.551 1536.664

20. 1543.583 1536.761

30. 1543.483 1536.848

50. 1542.269 1536.999

75. 1538.729 1536.931

100. 1535.677 1536.078

125. 1532.834 1533.442

150. 1530.063 1530.022

200. 1523.705 1523.186

250. 1518.034 1517.543

300. 1513.976 1513.821

400. 1504.685 1504.478

500. 1494.562 1493.563

600. 1487.147 1486.780

700. 1484.019 1483.651

800. 1482.513 1482.765

900. 1481.934 1482.070

1000. 1482.083 1482.393

1100. 1482.300 1482.410

1200. 1482.737 1483.290

1300. 1483.392 1484.274

1400. 1484.199 1484.278

1500. 1485.211 1485.479

1750. 1488.094 1488.326

2000. 1490.909 1490.945

2500. 1498.109 1498.281

3000. 1506.224 1506.069

3500. 1514.722 1514.350

4000. 1523.093 1522.983

4500. 1531.909 1531.800

5000. 1540.885 1541.012

the region around a sound speed minimum is known as a sound channel (or acousticwaveguide) and the minimum itself is the channel axis. Of particular importance is thebehavior in the top few hundred meters, where seasonal dependence is greatest. Thesound channel permits underwater sound to travel long distances, sometimes withoutcontact with the ocean boundaries. A well-known example is associated with theformation of convergence zones in deep water, caused by the monotonic increase insound speed at depths exceeding ca. 1 km. In the present example, the correspondingchannel axis (that of the deep sound channel ) occurs at a depth of about 900m. Forthe winter profile there is a second minimum at the sea surface, which is the axis of thesurface duct.5

9.1.2.1.1 Examples for the northwest Pacific Ocean

Despite the apparent similarity of winter and summer profiles (Figure 9.8), the smalldifferences between them are sufficient to cause very different acoustical behavior.For the situation considered below, the main effect of refraction in the summer case isassociated with the negative sound speed gradient between 30m and 100m depth,which has the effect of directing sound downwards towards the seabed. This down-ward refraction introduces shadow zones that are filled in by steep (bottom-reflected)paths, leading to increased propagation loss compared with the isovelocity casebecause the steeper paths experience greater reflection losses (upper graph of Figure9.9). At ranges up to about 70 km in this graph, the predicted field is dominated bypaths involving a single bottom reflection (BL1). Beyond this distance, only pathssuffering two or more bottom reflections can reach the receiver—hence the stepincrease at that point.6 The seabed parameters used are the mud values from Table9.1, which are representative of a deep-water sediment.

The winter case (Figure 9.9, lower graph) involves a surface duct in the top 50m,where a positive sound speed gradient exists, and a deep sound channel with its axis atdepth 900m (see Figure 9.8). The deep sound channel results in convergence zonesappearing at integer multiples of 65 km. The difference between the summer andwinter cases, which consistently exceeds 30 dB beyond 70 km, is caused by the smallchanges in the uppermost 100m of the sound speed profile (see Figure 9.8).

9.1.2.1.2 Surface duct (upward refraction)

If the sound speed gradient is positive (i.e., if the sound speed increases withincreasing depth), sound is refracted upwards and can become trapped near thesurface. This situation is typical of winter conditions, with an isothermal near-surfacelayer resulting from wind-driven mixing. The positive sound speed gradient is mainlydue to increasing pressure with depth. Such conditions typically result in long-range


5 The summer profile also has a minimum at the sea surface, but its consequences are minor

because the gradient and thickness of the resulting duct are too small to have a significant effect

on propagation in the circumstances considered.6 Although the sharpness of this step, as predicted here by the INSIGHT model, is artificial, its

presence, position, and approximate magnitude are real effects.


Figure 9.9. Propagation loss [dB rem2] vs. range for NWP summer (upper) and winter (lower).

The thin line, computed for isovelocity water of the same depth, is the same curve in both cases.

The acoustic frequency is 1.5 kHz (INSIGHT).

propagation, limited mainly by surface scattering, as illustrated in Figure 9.10 for thewinter profile of Figure 9.8. With the exception of the near-vertical stripes7 at 0 km,65 km, and 130 km, sound is restricted to the uppermost 50 meters, the depth at whichthe sound speed reaches a maximum. (The second and third of these stripes, theconvergence zones, as seen previously in Figure 9.9, are considered further in Section9.1.2.1.3.) At a given range (other than at convergence zones), propagation loss firstdecreases with increasing depth, reaching a minimum when the receiver depth passesthrough the source depth (30m) and then increasing again to the edge of the duct (seeFigure 9.10, upper graph, calculated by applying the simple flux concepts described inthis chapter). At any fixed depth in the duct, propagation loss tends to increase withincreasing range in accordance with cylindrical spreading. The lower graph is calcu-lated for the same case using a coherent ray-tracing method. It shows that the trendsillustrated by the upper graph are accompanied by fluctuations on a finer scale, causedby interference between the different multipaths contributing to the total field.

Weston’s flux theory. Weston has developed a powerful and elegant method foranalyzing the distribution of sound intensity with range and depth in a situation likethat of Figure 9.10. The method involves calculation of range-averaged energy flux(Weston, 1980). Denoting the cycle distance r0, the propagation factor so calculatedcan be expressed as the following integral over �ax, the ray grazing angle at the ductaxis (the sound speed minimum),

F ¼ 4

r

ð 0 2

tan �axr0 tan �s tan �r

� jRj2m d�ax; ð9:65Þ

where �s; �r are the ray angles evaluated at the source and receiver depth, respectively.The upper limit 0 is the grazing angle (measured at the duct axis) of the steepest raytrapped within the duct (i.e., the one that is horizontal at the depth h, where the soundspeed has a maximum). The lower limit 2 is the grazing angle of the shallowest raythat traverses both source and receiver depth. This ray is horizontal at the depth z2,defined as the deeper of the two,

z2 ¼ maxðz; z0Þ: ð9:66Þ

The depth z1 is similarly defined as the smaller of source and receiver depths,

z1 ¼ minðz; z0Þ: ð9:67Þ

The jRj2m scaling factor due to m surface reflections is addressed later.Equation (9.65) can be written

F ¼ 4

r

ð 0 2

tan �axr0 tan �1 tan �2

� jRj2m d�ax; ð9:68Þ


7 Although the stripes seem vertical in this graph, they appear so only because of the distortion

caused by the stretched depth axis. In reality the ray paths are close to horizontal.


Figure 9.10. Effect of upward refraction: propagation loss [dB rem2] vs. range and depth for

source depth z0 ¼ 30m at 2,000Hz. Upper: INSIGHT; lower: BELLHOP (oalib, www).

where

cos �1 ¼cðz1Þc0

cos �ax ð9:69Þ

and

cos �2 ¼cðz2Þc0

cos �ax: ð9:70Þ

In an isovelocity channel the term in square brackets of Equation (9.68) would be1=ð2hÞ, in which case the propagation factor becomes

F ¼ 2

rh

ð 00

jRj2mð�Þ d�; ð9:71Þ

consistent with Equation (9.38). For the isovelocity case, sound reflects from bothboundaries, so R is then the product of sea surface and seabed reflection coefficients.

Returning to the surface duct, the profile of interest here is a linear one, withgradient c0 > 0, for which it is convenient to introduce the radius of curvature

�ð�axÞ ¼c0

c0 cos �ax: ð9:72Þ

Absolute differences in sound speed are small across the duct, so rays trapped in theduct must be nearly horizontal. Therefore, for these rays8

�ð�axÞ � �ð0Þ ¼ c0c 0: ð9:73Þ

In order to proceed with evaluation of Equation (9.68) the cycle distance is needed,given by

r0 ¼ 2�0 tan �ax � 2�0�ax; ð9:74Þ

where �0 is shorthand for �ð0Þ, and hence

m � r

2�0�ax: ð9:75Þ

For a surface duct there are no reflections from the seabed, while the surfacereflection coefficient is assumed to take the form

jRj ¼ jRSj ¼ expð��S�axÞ: ð9:76ÞIt then follows that

jRj2m ¼ expð�2�SrÞ; ð9:77Þwhere

�S ¼�S2�0

: ð9:78Þ

The right-hand side of Equation (9.77) is independent of angle, which means that theintegrand of Equation (9.71) may be factored out of the integral for the propagationfactor.


8 For isothermal conditions (i.e., if c0 ¼ 0.016/s), the radius of curvature is approximately equal

to 90 km.

Following Weston (1980), the depth factor can be defined (allowing here forattenuation due to surface reflection losses) as

D � Frh

2 0e2�Sr; ð9:79Þ

so that

FSD ¼ 2 0rh

�Dðz0; zÞ e�2�Sr: ð9:80Þ

Substituting Equation (9.68) in Equation (9.79) and assuming small angles,9 thedepth factor is

D � 2h

0

ð 0 2

�axr0�1�2

� d�ax: ð9:81Þ

The result is (Weston, 1980, Eq. (17))

Dðz0; zÞ ¼ 02 2

Fð� I �Þ; ð9:82Þ

where Fð� I �Þ is an incomplete elliptic integral of the first kind (Appendix A)

Fð� I �Þ �ð�0

ð1� sin2 � sin2 �Þ�1=2 d�: ð9:83Þ

The arguments of Equation (9.83) are

� ¼ arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20 � 22 20 � 21

sð9:84Þ

and

� ¼ arcsin 1 2; ð9:85Þ

where the angles 0, 1, and 2 are the grazing angles at the duct axis of the rayswhose turning depths are equal to h, z1, and z2, respectively. Thus,

20 �2

�0h; ð9:86Þ

21 �2

�0z1; ð9:87Þ

and

22 �2

�0z2: ð9:88Þ

The arguments � and � can also be written in terms of depth variables, that is,

sin2 � ¼ z1z2

ð9:89Þ


9 tan � is approximated by its argument.

and

sin2 � ¼ h� z2h� z1

: ð9:90Þ

The result of evaluating Equation (9.82) is shown in the upper graph of Figure 9.11.Weston’s flux formula provides useful insight into depth dependence that is difficultto gain in other ways, although quantitative corrections to it are sometimes needed.For example, when the source and receiver depths are equal, the right-hand side ofEquation (9.82) becomes infinite.

This singularity is a remnant of the infinities associated with ray theory caustics,which are lines along which the density of ray paths becomes infinite, illustrated byFigure 9.12. (The caustics are the thick bunches of rays that, for example, intersect thesea surface at 9 km, 16 km, and so on.) Flux theory smooths out these features byaveraging in range, such that most of them disappear. However, caustics that form atthe source depth, known as cusps, are of a particularly resilient variety, and these canbe seen in Figure 9.12 at ranges of 6 km, 13 km, 20 km and so on, at a depth of 30m.The infinities associated with these cusps survive the range-averaging process, albeitweakened (Weston, 1980), manifesting themselves as the singularities at the sourcedepth in Figure 9.11.

Unless z2 (the larger of source and receiver depth) is small, the cusp singularitiesare logarithmic in nature (after range averaging) and their effects are confined to avery small region either side of the source depth. For small z2 there is an additionalenhancement caused by the rays being confined to a smaller and smaller proportionof the duct. This focusing behavior is analogous to the trapping of sound against acurved reflecting surface, known as a whispering gallery (Weston, 1979).

One solution to the infinity at the cusp is to use wave theory to calculate themaximum value of the depth factor at depth z ¼ z1 ¼ z2 as a function of frequency.The result is (Weston, 1980; Harrison, 1989)

D ! 1

2

h

2z

� �1=2

loge4z

"ðz1 ! z2Þ ð9:91Þ

where

" ¼ �0c20

2!2

!1=3

: ð9:92Þ

The parameter " can be interpreted as a sort of ‘‘ray thickness’’ into which the energyassociated with the cusp spreads. For �0 ¼ 90 km and wavelength about 1m, " isabout 20 wavelengths. Unless z=" is very large (or z=h is small), the right-hand side ofEquation (9.91) is always of order unity.

An alternative approach for dealing with the singularity is to find anapproximation to Equation (9.82) that does not exhibit one. Unless � and � are bothclose to =2, the integrand is of order 1 and the integral is of order �. Thus, a verysimple approximation is

Fð� I �Þ � �: ð9:93Þ



Figure 9.11. Depth

factor vs. receiver

depth. Upper:

evaluated using

Equation (9.82)

without

approximation for

z0 ¼ 0, h=2, and9h=10 (reprintedwith permission

from Weston, 1980,

American

Institute of Physics);

lower: evaluated

using various

approximations,

solid line (——):

Equation (9.94);

dashed line (– – –):

Equation (9.93);

dotted line (� � � � �):Equation (9.95).

#

A better approximation, derived in Appendix A, is

Fð� I �Þ �

2 sin �arcsin

2�

sin �

� �: ð9:94Þ

An alternative to Equation (9.93), motivated also by simplicity, is

Fð� I �Þ � sin �; ð9:95Þ

which results in a depth factor of the form

D � 1

2

h=z2 � 1

1� z1=h

� �1=2

: ð9:96Þ

This last expression is deceptively simple. It tends to underestimate the field wheneverz1 and z2 are equal, but otherwise captures the main features of the elliptic integral, asillustrated by the lower graph of Figure 9.11 (dotted line). For example, in the limit ofsmall z2=h, near-surface behavior is readily found to be

D � 1

2

h

z2

� �1=2

: ð9:97Þ

This depth factor tends to infinity as z2 approaches zero. The physical reason for thisis that sound rays whose turning depth is less than z2 are confined to a smaller and


Figure 9.12. Ray trace illustrating the formation of caustics and cusps up to a range of 40 km,

for a source depth of 30m, and for the same case as Figure 9.10 (BELLHOP).

smaller area (/ z2). The available energy also decreases, but at a slower rate ð/ffiffiffiffiffiz2

p Þso the resulting depth factor is proportional to 1=

ffiffiffiffiffiz2

p(ratio of energy to area),

consistent with Equation (9.97).10

Surface decoupling. A surface duct ray reflecting from the sea surface suffers a phase change, resulting in near-surface cancellation the same way as described inSection 9.1.1.2.6, except that here we are dealing with curved rays due to the soundspeed gradient. In this situation, Equation (9.62) can be written

WðzÞ ¼ 2 sin2 �ðzÞ; ð9:98Þwhere, using the WKB approximation (Boyles, 1984),

�ðzÞ ¼ðz0

�ð�Þ d�: ð9:99Þ

In the same vein, Equation (9.63) generalizes to

WðzÞ � 1

1þ ð2�2Þ�1: ð9:100Þ

If the sound speed gradient is c0, the integral of Equation (9.99) for a ray whosegrazing angle at the sea surface �0 is

�ðzÞ ¼ ð!=c0ÞfYð�0Þ �Y½�1ðzÞ� þ sin½�1ðzÞ� � sin �0g; ð9:101Þwhere �1 is the corresponding angle at depth z according to Snell’s law. The functionYð�Þ is defined in Equation (9.22).

Surface-scattering loss. Another important property of the sea surface is itswavy nature, which has the effect of scattering high-frequency sound. The roleplayed by near-surface bubbles in this process is described in Chapter 8. An em-pirical approach is adopted here, assuming linear variation of reflection loss withangle regardless of the mechanism. A suitable value of �S for use in Equation (9.78),based on the measurements of (Weston and Ching, 1989), is

�S ¼ 3:8F 3=2 v1010 m/s

� �4

Np/rad; ð9:102Þ

where F is the numerical value of the acoustic frequency when expressed in units ofkilohertz; and v10 is the wind speed at 10m height.

Volume attenuation. For frequencies f between 200Hz and 10 kHz, theattenuation coefficient for the representative conditions considered previously11


10 This behavior is known as the ‘‘whispering gallery’’ effect. The name originates from a

focusing effect associated with multiple reflections from a hard curved surface such as occurs in

churches or chambers of oval or spherical design.11 The conditions (see Chapters 2 and 4) are T ¼ 10 �C, S ¼ 35, and K ¼ 1:0.

can be approximated by

�V ¼ 0:0140F 2

F 2 þ 1:32þ 0:00102F 2 Np km�1; ð9:103Þ

or, converting to decibels,

aV ¼ 0:122F 2

F 2 þ 1:32þ 0:0088F 2 dB km�1; ð9:104Þ

where

aV=ðdB kmÞ�1 ¼ ð20 log10 eÞ�V=ðNp km�1Þ; ð9:105Þ

Duct cut-off frequency. In the same way as for a shallow-water waveguide(Section 9.1.1.2.5), there exists for a surface duct a cut-off frequency below whichwaveguide propagation is not supported. The cut-off frequency fc can be calculatedusing

fc ¼9c08

�1=20

ð2hÞ3=2; ð9:106Þ

and hence

fc � ð590 m/sÞffiffiffiffiffi�0h3

r: ð9:107Þ

Assuming a nominal radius of curvature of �0 ¼ 90 km (corresponding to isothermalconditions), the cut-off frequency is

fc � ð180 HzÞ 100 m

h

� �3=2

: ð9:108Þ

Irrespective of the sound speed gradient, the ‘‘ray thickness’’ (Equation 9.92) eval-uated at the cut-off frequency of Equation (9.106) is 43% of duct thickness.

The frequency of 2 kHz chosen for Figure 9.10 is close to the optimum for long-range propagation in this duct. The reason there is an optimum at all is that at higherfrequency the sound is scattered or absorbed, and at lower frequency (below the ductcut-off frequency, equal to 800Hz for this case) the energy leaks out by means of thetunneling effect. The decay rate due to low-frequency tunneling, in nepers per unitdistance, can be estimated using (Packman, 1990, Eq. (1))

�T ¼ 2�5=2ð�0hÞ�1=2 e��ð f Þ; ð9:109Þwhere

�ð f Þ ¼3�ð f Þ f > f1

3½� 0ð f1Þð f � f1Þ þ �ð f1Þ� f � f1,

(ð9:110Þ

�ð f Þ ¼ ½ð f =fcÞ2=3 � 1�3=2 f fc ð9:111Þand

f1 ¼ 1:15fc: ð9:112Þ

Optimum propagation frequency. Frequency dependence at a fixed receiver depthof 10m is illustrated by Figure 9.13 for two different wind speeds. Without wind (see



Figure 9.13. Propagation loss [dB rem2] vs. frequency and range for a surface duct with v10 ¼ 0

(upper) and v10 ¼ 15m/s (lower). Source and receiver depths are 30m and 10m (INSIGHT).

upper graph), the channel acts as a filter with a passband of 1 kHz to 10 kHz and apeak response close to 2 kHz (the horizontal lines at 65 and 130 km are convergencezones). The effectiveness of the channel is sensitive to wind speed, as can be seen bycomparing the upper graph of Figure 9.13 (for v ¼ 0) with the lower one (v ¼ 15 m/s).

Simple surface duct formula. The various effects described above can becombined to provide a simple formula that gives a reasonable approximation tothe behavior of Figures 9.10 and 9.13:

FSDðzÞ ¼2 0rh

Dðz0; zÞWðz0ÞWðzÞ expð�2�SDrÞ: ð9:113Þ

Attenuation comprises three components

�SD ¼ �S þ �T þ �V; ð9:114Þ

representing contributions due to surface scattering, tunneling, and volumeattenuation. They are given by Equations (9.78), (9.109), and (9.104), respectively.

An implicit assumption of Equation (9.113) is that the individual effects of cut-off, surface decoupling, and surface scattering may be combined multiplicatively, butthis is not always the case. For example, one complication arises from a non-linearvariation of surface loss with angle, as this changes the depth dependence of thepropagation factor, which then becomes a function of range.

9.1.2.1.3 Convergence zones

We now return to the convergence zone features in Figures 9.10 and 9.13, which arecharacteristic of long-range propagation in deep water. To understand them it isnecessary to consider the behavior of the entire sound speed profile (see Figure 9.8),and the impact that it has on relevant ray paths. The positive sound speed gradient inthe lower half of the ocean (2–5 km) refracts sound upwards, and ‘‘the convergencezone’’ is the name given to the region where this sound returns to the surface, in thiscase at a distance of some 65 km, as illustrated by the ray trace of Figure 9.14 (uppergraph), Rays then reflect from the sea surface (or refract from the thermocline if thesurface is warm enough) and the process repeats itself at about 130 km, 195 km, andso on. This yo-yo-like behavior can continue over hundreds or even thousands ofkilometers if the conditions are right.

Within the convergence zone region, sound pressure levels can be much higherthan in its immediate surroundings, as illustrated by the lower graph of Figure 9.14,showing propagation loss calculated from the ray paths shown in the upper graph.The same convergence zone features are clearly visible in the graphs of Figures 9.10and 9.13.

9.1.2.1.4 Lloyd mirror with downward refraction

Now consider a sound speed profile with a negative gradient instead of the positiveone considered so far. A negative gradient means that sound speed decreases withincreasing distance from the sea surface, typical of summer conditions with solar



Figure 9.14. Upper: ray trace for source depth 30m, illustrating convergence zones at the sea

surface at intervals of 65 km; lower: propagation loss [dB rem2] vs. range and depth for the same

case, with an acoustic frequency of 2 kHz (BELLHOP). The surface duct is visible as a

horizontal stripe across the top of each graph. Source depth is 30m.

heating. The negative gradient results in sound being deflected away from the surfaceas illustrated by the lower graph of Figure 9.15.

There is a parallel here with a well-studied problem in radar, namely propagationin an isovelocity medium (the atmosphere) close to a spherical boundary (the earth’ssurface). A transformation can be made to a co-ordinate system in which the earth isflat and the rays are curved due to an effective refractive index profile that has theeffect of refracting radio waves away from the flat surface representing the earth–atmosphere boundary. In the transformed co-ordinate system, acoustic and electro-magnetic problems are equivalent. Radar scientists have solved this problem (Fish-back, 1951; Freehafer, 1951) and their result is given below, in the form quoted byAinslie and Harrison (1990).

Consider the wavenumber profile

k2 ¼ !2

c201þ 2

�0z

� �; ð9:115Þ

where the radius of curvature �0 is related to the sound speed c0 and its gradient c0 at

the sea surface according to

�0 ¼ c0=jc 0j: ð9:116Þ

In this situation, the propagation factor is similar to Equation (9.5) except that thedepths z; z0 are replaced by the transformed co-ordinates �1; �2 (neglecting surfacereflection loss)

FLM ¼ 4

r2expð�2�VrÞ sin2

!�1�2c0r

; ð9:117Þ

where

�1 ¼ z1 �r212�0

ð9:118Þ

and

�2 ¼ z2 �r222�0

: ð9:119Þ

As previously (see Equations 9.66 and 9.67), depths z1 and z2 are the smaller andlarger of the source and receiver depths, respectively. The ranges r1 and r2 aredistances from either the source or receiver to the point of reflection. They satisfythe condition r2 > r1 and can be found by solving the following cubic equation for r2(see Appendix A)

2r32 � 3rr22 þ ½r2 � 2�0ðz1 þ z2Þ�r2 þ 2�0rz2 ¼ 0; ð9:120Þ

and then applying

r1 ¼ r� r2: ð9:121Þ

The accuracy of Equation (9.117) is demonstrated by Ainslie and Harrison (1990) fora frequency of 50Hz.



Figure 9.15. Effect of downward refraction (dc=dz ¼ �0:03/s) on propagation loss [dB rem2]

for LM at 900Hz: isovelocity (upper), downward refracting (lower). The source depth is 15m

(INSIGHT).

9.1.2.2 Shallow water

A characteristic feature of shallow-water propagation is that any sound that hastraveled a few kilometers horizontally is likely to have suffered several interactionswith either the sea surface or the seabed, or both. The importance of the sound speedprofile in shallow water arises largely from the way it influences these boundaryinteractions. Assuming a uniform sound speed gradient there are two possible typesof ray path: one that is steep enough to interact with both boundaries (surface–bottom multipaths); and one that is not, trapped instead by refraction in the waterbefore it reaches the high-speed boundary. In the latter case, a surface duct is formedif the sound speed gradient is positive (upward-refracting) and a bottom duct if it isnegative (downward-refracting). In the following, propagation from surface–bottommultipaths is referred to as ‘‘V-duct’’ (or ‘‘VD’’) propagation, because each cycle of aray path follows a V shape as for the isovelocity case, albeit slightly curved due torefraction. Similarly, a surface duct or bottom duct is referred to as a ‘‘U-duct’’ (or‘‘UD’’) because the ray paths follow a U shape. If wind speed is low, corresponding toa smooth sea surface, the surface is a good reflector of sound (see Chapter 8), whichmeans that conditions of upward refraction (surface duct) can lead to long-rangepropagation. This point is illustrated by Figure 9.16. Theoretical expressions followfor U-duct and V-duct behavior, starting with the V-duct.

9.1.2.2.1 Surface–bottom multipaths (‘‘V-duct’’)

For rays steep enough to reflect from both boundaries (the condition for a V-duct),the propagation factor of Equation (9.68) can be written

F ¼ð�max�min

Gð�axÞ d�ax; ð9:122Þ

where the integrand Gð�Þ is the differential propagation factor, which, neglectingvolume attenuation, is given by (see Equation 9.65)

Gð�axÞ ¼4

r

tan �axr0 tan �1 tan �2

� jRj2m; ð9:123Þ

where m is the number of ray cycles

mð�Þ ¼ r=r0ð�Þ: ð9:124Þ

It is convenient to introduce the subscripts ‘‘hi’’ and ‘‘lo’’ to denote the properties ofhigh-speed and low-speed boundaries, respectively, while ‘‘B’’ and ‘‘S’’ denote theproperties of the seabed and sea surface. Thus,

chi ¼ maxðcS; cBÞ; ð9:125Þ

clo ¼ minðcS; cBÞ; ð9:126Þ

�hi ¼ minð�S; �BÞ; ð9:127Þand

�lo ¼ maxð�S; �BÞ: ð9:128Þ



Figure 9.16. The thick solid curve shows propagation loss [dB rem2] vs. range for shallowwater

with a mud bottom for two different sound speed profiles.Upper: upward refraction (isothermal

profile); lower: downward refraction (thermocline). The thin line is a reference curve for

isovelocity water from Figure 9.7 (INSIGHT).

With this notation, the integration limits are:

�min �

ffiffiffiffiffiffiffi2H

�0

sð9:129Þ

and

�max ¼ arccosclocBcos c

� �: ð9:130Þ

This maximum angle is the sediment critical angle c,

c ¼ arccoscwcB; ð9:131Þ

corrected for refraction between the seabed and the duct axis according to Snell’s law.If the critical angle is large, �max may be approximated by

�max � c: ð9:132ÞThe main effect of refraction is to change the functional form of r0ð�Þ compared withthe isovelocity case:

r0ð�Þ ¼ 2�ð�loÞðsin �lo � sin �hiÞ; ð9:133Þwhere �ð�Þ is the radius of curvature given by Equation (9.72), simplifying for anglesclose to horizontal to

r0ð�Þ � 2�0ð�lo � �hiÞ: ð9:134ÞThe term Rð�Þ is the product of both surface and bottom reflection coefficients:

Rð�Þ ¼ RSð�SÞRBð�BÞ: ð9:135ÞEquation (9.123) can be written

Gð�Þ � 2

rHjRð�Þj2r=r0ð�Þ 2H tan �ax

r0 tan �1 tan �2

� ; ð9:136Þ

where the factor in square brackets is a dimensionless ratio of order unity.Defining the angle difference

D� � �lo � �hi; ð9:137ÞEquation (9.135) becomes

jRð�Þj ¼ expð��lo�loÞ exp½��hið�lo � D�Þ�: ð9:138ÞThe cumulative reflection coefficient is then

jRð�Þj2m � e2�hir e�2ð�hiþ�loÞr�lo=D�; ð9:139Þwhere

�hi ¼�hi2�0

; ð9:140Þ

and

�lo ¼�lo2�0

: ð9:141Þ


Using Snell’s law it can be shown that, if �lo and D� are both small

�loD�

� 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2min

�2lo

s !�1: ð9:142Þ

In principle, all of the ingredients needed for calculation of the propagation factorusing Equation (9.122) are now in place. The recipe involves substitution of YVDð�Þand jRð�Þj in Equation (9.136) (using Equation 9.139) and carrying out the integralover angle. One could stop here, but it is instructive to simplify the integral—withcare to avoid losing the baby with the proverbial bathwater—to facilitate its evalu-ation. The important point is to keep a careful track of the exponent of Equation(9.139), as small errors there will be amplified by exponentiation. The exponent cannevertheless be simplified by replacing Equation (9.142) with the approximation(Ainslie, 1992)

�loD�

� 2�2lo�2min

� 1: ð9:143Þ

For small �lo (i.e., close to �min), the right-hand sides of Equations (9.142) and (9.143)are both approximately equal to 1. For large �lo they approach the same asymptoticresult of 2�2lo=�

2min. The behavior for intermediate values is examined later.

The final step in the simplification process is to recognize that the term in squarebrackets in Equation (9.136) may be approximated by unity without incurring a largeerror.12 With these simplifications, and substituting Equation (9.139) into Equation(9.136), it follows from Equation (9.122) that

FVD �ð�max�min

2

rHe2�hir e�2ð�hiþ�loÞr�lo=D� d�ax ð9:144Þ

and hence

FVD � 2�effrH

e2ð2�hiþ�loÞr erf �max

ffiffiffiffiffiffiffiffiffi�totr

H

r� �1=2

� erfð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�hi þ �loÞr

pÞ

� ; ð9:145Þ

where

�2eff ¼H

4�totrð9:146Þ

and�tot ¼ �hi þ �lo: ð9:147Þ

At short range, Equation (9.145) simplifies to give the cylindrical spreading result

FVD � 2ð�max � �minÞrH

ð�eff � �maxÞ: ð9:148Þ


12 This approximation works best at short range, where ray angles are steep, because for this

situation the three angles �1, �2, and �ax are equal and r0ð�Þ ¼ 2H=tan �. At long range, wherethe angles are small, although a small error (a few dB) from this approximation is likely, it is

more important to keep tabs on the exponential terms in Equation (9.139). In any case, the VD

term is eventually exceeded in importance by the UD contribution (Section 9.1.2.2.2).

At longer range, mode stripping sets in, with an important correction factorcompared with the isovelocity case, equal to the term in curly brackets in Equation(9.149)

FVD � 2�effrH

ferfc½2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�hi þ �loÞr

p� e2ð2�hiþ�loÞrg ð�eff � �maxÞ: ð9:149Þ

For extremely long ranges, satisfying �eff � �min, the correction factor simplifies, andthe propagation factor can then be written13 (Ainslie, 1992)

FVD � 1ffiffiffiffiffiffiffiffiffiffiffiffi2H�0

p�tot

e�2�lor

r2; ð9:150Þ

resulting in exponential decay if �lo is non-zero.Because of the importance of accurately representing the exponent, the

approximations of Equations (9.142) and (9.143) are compared in Figure 9.17.The magnitude of the exponent is correct for both small and large angles as expected.For intermediate angles it is underestimated slightly.


Figure 9.17. Approximation to D�=� for values of �min (in radians) as stated. Cyan solid line:exact; dashed line: Equation (9.143); dotted line: Equation (9.142).

13 Equation (9.150) follows from Equation (9.149) using (see Appendix A)

erfc x � expð�x2Þ1=2x

ðx � 1Þ:

9.1.2.2.2 Surface or bottom duct propagation (‘‘U-duct’’)

Equation (9.145) describes the contribution to sound propagation from ray pathsthat are steep enough to reflect from both upper and lower boundaries. A completedescription must also include a contribution FUD due to paths that reflect from thelow-speed boundary but are not steep enough to reach the high-speed one. Such pathsform a surface duct in the upward-refracting case and a bottom duct in the down-ward-refracting case. These waveguides are referred to henceforth as ‘‘U’’-ducts todistinguish them from the ‘‘V’’-ducts involving reflections from both boundaries. TheU-duct contribution can be written (using Equation 9.113 and neglecting surfacedecoupling)14

FUDðr; zÞ ¼2 0rH

Dðh0; hÞ expð�2�lorÞ; ð9:151Þ

where h and h0 are the distances from receiver and source, respectively, to the lowsound speed boundary. Notice the resemblance to Equation (9.148), associated withtypical cylindrical spreading behavior.

9.1.2.2.3 Total (VDþUD)

The total propagation factor is the sum of Equations (9.145) and (9.151)

F ¼ FVD þ FUD: ð9:152Þ

9.2 NOISE LEVEL

Sound traveling in the sea is subject to many different propagation effects that need tobe taken into account when estimating the propagation loss term of the sonarequation. The noise arriving at the receiver has traveled through the same medium,so the same propagation effects can also be important in determining the level ofambient noise, which is considered next.

Typical sources of ambient noise are described in Chapter 8, the main ones beingassociated with shipping, wind, and precipitation. These sources are characterized interms of their areic15 source spectrum level, which, combined with an ambient noisemodel16 (a computer program designed to calculate the field from a continuous ordiscrete distribution of noise sources), enables prediction of the noise level term in thesonar equation.

9.2 Noise level 483]Sec. 9.2

14 The angle 0 is the same as �min used previously for V-duct. A change in notation is

appropriate because here it is not the minimum angle, but the maximum one.15 Following Taylor (1995), the adjectives ‘‘areic’’ and ‘‘volumic’’ are used, respectively, to

mean ‘‘per unit area’’ and ‘‘per unit volume’’.16 A review of ambient noise models available in 1997 is given by Hamson (1997). See also

Jensen et al. (1994) and Etter (2003).

9.2.1 Deep water

Figure 9.18 shows predicted noise spectra for two cases, one for a high-noise situationand the other for low noise. The high-noise case involves heavy shipping (see Table9.3) and a wind speed of 15m/s, and the low-noise case is for light shipping and2.5m/s. A third contribution to sonarnoise is thermal noise, which is thename given to random pressure fluctua-tions at the hydrophone due to thermalagitation of water molecules.

9.2.1.1 Typical spectra for wind,shipping, and thermal noise

Wind noise occupies a large part of theambient noise spectrum in the fre-quency range of interest to sonar. Thisis illustrated by the prediction of Figure9.18 showing wind noise dominatingthe spectrum roughly between 1 kHzand 100 kHz. At lower frequency, thecontribution due to (distant) shipping is


Figure 9.18. Predicted deep ocean noise spectra [dB re mPa2/Hz] (INSIGHT): shipping noise,

wind noise, and thermal noise as marked. See text for details.

Table 9.3. Nomenclature used for shipping

densities.

Shipping category Shipping density

Mm�2 ðaÞ nmi�2 ðaÞ

Very heavy 50 000 0.171

Heavy 5000 0.0171

Moderate 500 0.00171

Light 50 0.000171

Very light 5 0.0000171

a 1Mm (one megameter)¼ 1000 km; 1 nmi (onenautical mile)¼ 1.852 km (see Appendix B).

important, whereas at high frequency (above 300 kHz) it is thermal noise that dom-inates.

9.2.1.1.1 Shipping noise

At frequencies between about 10Hz and 100Hz, the noise from distant shippingdominates the spectrum. This component of background noise depends on distanceto the ships, their source levels, and the density of distant ships.17 Typical values ofareic shipping density are suggested in Table 9.3. (The author is unaware of anystandard definition for terms like ‘‘heavy’’ and ‘‘light’’ shipping). In addition toshipping density, a prediction of shipping noise requires an estimate of the averagesource level for a single ship.

9.2.1.1.2 Thermal noise

The thermal noise spectrum is included in Figure 9.18. Though not acoustic inorigin,18 it is nevertheless noise that interferes with the detection of high-frequencysound. Thermal noise increases sharply at high frequency, as described by theequation (see Chapter 10)

10 log10 QNf ¼ �14:7þ 10 log10 F

2 dB re mPa2=Hz; ð9:153Þ

where F is the frequency in kilohertz.

9.2.1.2 Effect of rain rate and wind speed

Rainfall is an important, though intermittent, source of broadband noise, centeredaround 10 kHz, as illustrated by Figure 9.19, showing the predicted sensitivity ofambient noise to rain rate for two different wind speeds. Similarly, Figure 9.20 showssensitivity to wind speed with and without rainfall. These graphs are both calculatedfor light shipping (50/Mm2), as defined by Table 9.3.

The reader will notice a difference between (say) Figure 9.20 and the correspond-ing wind and rain source spectra in Chapter 8, and this is partly because differentphysical quantities are considered in each chapter. Specifically, the above graphsshow received noise spectral density (QN

f ) at a depth of 30 m, whereas the Chapter8 graphs show the areic dipole source factor K (e.g., Kwind or K rain) which is ameasure of radiated acoustic power per unit area of the sea surface. Though con-ceptually different, these two quantities are closely related. They are easily confusedbecause they share the same dimensions and units (both are reported in dB re mPa2/Hz) and are similar in numerical value. For example, in isovelocity water (and infinitewater depth), the received noise spectrum is given by (see Chapter 2)

QNf ¼ 2E3ð2�zÞK ð9:154Þ


17 Nearby ships need to be treated not as a continuum, but as discrete entities.18 That is, the pressure fluctuations are not caused by traveling acoustic waves, but by thermal

agitation of the water molecules in direct contact with the hydrophone.


Figure 9.19. Predicted sensitivity of deep-water noise spectra [dB re mPa2/Hz] to rain rate, for a

wind speed of 2.5m/s (upper) and 7.5m/s (lower) (INSIGHT).


Figure 9.20. Predicted sensitivity of deep-water noise spectra [dB re mPa2/Hz] to wind speed.

Upper: no rain; lower: rain rate¼ 10mm/h (INSIGHT).

and hence

10 log10 QNf ¼ 10 log10 þ 10 log10 K þ 10 log10½2E3ð2�zÞ�: ð9:155Þ

If there is no attenuation, the factor 2E3 is equal to 1. The constant 10 log10 isapproximately 5 dB, so Equation (9.155) then simplifies to

10 log10 QNf � 5þ 10 log10 K : ð9:156Þ

Thus, there is a systematic 5 dB offset in these conditions.

9.2.1.3 Depth dependence of surface-generated noise

The variation of ambient noise with depth, illustrated by Figure 9.21, is considerednext. The high-frequency component of surface-generated noise decays exponentiallywith increasing depth, whereas thermal noise is independent of depth. The deptheffect due to absorption can be quantified by means of Equation (9.155) and using theapproximation (Appendix A)

E3ðxÞ �e�x

xþ 3� e�0:434x: ð9:157Þ


Figure 9.21. Predicted ambient noise spectral density level [dB re mPa2/Hz] vs. frequency and

depth (v10 ¼ 5 m/s) (INSIGHT).

providing an estimate for depth dependence at high frequency or in deep water19

10 log10 QNf � 10 log10

2K

2�zþ 3� expð�0:868�zÞ � ð20 log10 eÞ�z: ð9:158Þ

At low frequency, or in shallow water, it is necessary to consider the additionalpropagation effects due to refraction and reflection from the seabed. Equations(9.155) and (9.158) apply to a uniform distribution of (dipole) surface sources, suchas wind or rain noise, but not to thermal noise (which is described by Equation 9.153)or shipping (which is not uniformly distributed).

9.2.2 Shallow water

Ambient noise in shallow water is highly variable compared with deep water. This ispartly due to the presence of location-dependent noise sources that are absent in deepwater (e.g., surf, fauna) and partly because of the influence of the seabed, whichintroduces a dependence on water depth and bottom type. Figure 9.22 illustrates the


Figure 9.22. Predicted effect of the seabed on the ambient noise spectrum [dB re mPa2/Hz] in

isovelocity water of depth H ¼ 100m, for wind speed v10 ¼ 2.5m/s. Thick solid line: sand;

dashed and dotted line: clay (INSIGHT).

19 The requirement for the validity of Equation (9.158) is that the noise field be dominated by

direct contributions from the sea surface, and not from other paths (e.g., via the seabed).

effect of bottom type on ambient noise in the presence of rain and heavy shipping.The sand seabed (thick solid curve) has a higher critical angle, which has the effect ofenhancing distant contributions at moderate frequencies (100Hz to 10 kHz). At lowfrequencies (below the waveguide cut-off frequency) it is the clay seabed that is betterable to support long-distance propagation, due to low attenuation in the sediment(see Table 9.1). Hence the crossover close to 30Hz. The effect of refraction in thewater is illustrated by Figure 9.23 for a clay seabed. The presence of a surface ductenhances the contribution from distant shipping at 300Hz.

9.2.3 Noise maps

Given a suitable noise model and the necessary inputs, it is possible to predict maps ofthe geographical distribution of underwater sound due to natural or anthropogenicnoise sources. An example follows for dredger noise in the North Sea, from Ainslie etal. (2009). Figure 9.24 shows the predicted broadband noise distribution due todredging activity close to the Port of Rotterdam (see figure caption for details).The corresponding bathymetry is shown in Figure 9.25.


Figure 9.23. Predicted effect of the sound speed profile on the ambient noise spectrum

[dB re mPa2/Hz] for a clay seabed. Water depthH ¼ 100m and wind speed v10 ¼ 2.5m/s. Thick

solid line: c 0 ¼ 0.02/s; dashed line (from Figure 9.22): c0 ¼ 0 (INSIGHT).

9.3 SIGNAL LEVEL (ACTIVE SONAR)

In this section the effect of propagation is considered on the signal term in the activesonar equation (i.e., on the target echo level). Consider an active sonar with an omni-directional transmitter (Tx). A transmitted pulse travels through the sea until itreaches a submerged object, at which point some of the sound scattered by the objecttravels back through the sea to the sonar receiver (Rx). It is assumed that thetransmitted pulse is long enough to ensure that the duration of the received signalis equal to those transmitted and reflected. With this assumption, for a monostaticgeometry, the mean square pressure (MSP) at the receiver is

QS ¼ S0�

4FTxFRx; ð9:159Þ

where S0 is the source factor; � is the backscattering cross-section of the target(assumed independent of elevation angle); and the propagation factors are FTx(for the outward path from sonar to target) and FRx (return path from target toreceiver).

An important property of solutions to the wave equation is that the field at apoint B due to a monopole source at point A is the same as the field at A due to an

9.3 Signal level (active sonar) 491]Sec. 9.3

Figure 9.24. Prediction of broadband radiated noise level (10Hz to 10 kHz) [dB re mPa2] for a

hypothetical dredging operation involving two dredgers operating in the vicinity of Rotterdam

harbor. The assumed broadband source level of each dredger is about 188 dB re mPa2 m2

( TNO, 2009, reprinted with permission).#

‘‘identical’’ monopole source at B. This principle is usually interpreted as meaningthat, for a monostatic geometry, FTx and FRx in Equation (9.159) are equal. Thisinterpretation is correct if the medium density at A is equal to that at B, but nototherwise. The precise relationship between FTx and FRx, for the case when thedensities differ, is derived below.

9.3.1 The reciprocity principle

The reciprocity relationship relating the acoustic pressure at rB due to a point sourceat rA, denoted pðrB; rAÞ, to that at rB due to a point source at rB, can be written(Pierce, 1989)

pðrB; rAÞUB ¼ pðrA; rBÞUA; ð9:160Þ

where UA and UB are the respective source strengths of the sources at A and B. Themonopole source strength (the amplitude of volume velocity) of each source is relatedto its source factor S and frequency f according to

U ¼ � 2i

ffiffiffiffiS

p

�f: ð9:161Þ


Figure 9.25. Bathymetry used for Figure 9.24. The contours show (minus) water depth in

meters ( TNO, 2009, reprinted with permission).#

Therefore, Equation (9.161) can be written

pðrB; rAÞffiffiffiffiffiffiSA

p ¼ pðrA; rBÞffiffiffiffiffiffiSB

p �B�A: ð9:162Þ

Squaring both sides and interpreting point B as the sonar position and A as that ofthe target gives

FRx ¼ FTx�ðz0Þ�ðztgtÞ

� 2

: ð9:163Þ

9.3.2 Calculation of echo level

Substituting Equation (9.163) into Equation (9.159) gives for the received sonarsignal

QS ¼ S0�

4F 2Tx

�ðz0Þ�ðztgtÞ

� 2

; ð9:164Þ

which, converted to decibels, results in the echo level

EL ¼ SLþ TSþ 2PLTx þ 10 log10�ðz0Þ�ðztgtÞ

� 2

; ð9:165Þ

where SL is the sonar source level; and TS is the target strength, related to the targetbackscattering cross-section according to (see Chapter 8)20

TS ¼ 10 log10�

4: ð9:166Þ

Given the (sonar to target) propagation loss PLTx, Equation (9.165) can be used tocalculate EL. The density ratio is an important correction21 if the target is in adifferent medium to the sonar (e.g., if it is buried in sand). For the monostatic case,and assuming henceforth that the target is not buried (i.e., that �ðz0Þ and �ðztgtÞ areequal), it follows from Equation (9.152) that

QS ¼ S0�

4ðFVD þ FUDÞ2: ð9:167Þ

Equation (9.167) can be extended to a bistatic geometry by rewriting it in the form

QS ¼ S0�OðFVD þ FUDÞTxðFVD þ FUDÞRx; ð9:168Þ

9.3 Signal level (active sonar) 493]Sec. 9.3

20 The 4 denominator is omitted by some authors, who incorporate it instead into the

definition of �. See Chapter 5 for details.21 The precise form of this correction depends on the chosen definition of propagation loss

(PL). In early work on underwater acoustics (before about 1980), it was customary to define PL

as a ratio, in decibels, of the equivalent plane wave intensity rather thanMSP (see Appendix B).

If the early PL definition is used, the density ratio �ðz0Þ=�ðztgtÞ in Equation (9.165) is replacedby the sound speed ratio cðztgtÞ=cðz0Þ (Ainslie, 2008).

where the shorthand ‘‘Tx’’, ‘‘Rx’’ is used to indicate propagation from ‘‘transmitter(Tx) to target’’ and ‘‘target to receiver (Rx)’’, respectively; and �O is the differentialscattering cross-section evaluated at the (azimuthal) bistatic angle.22 The results ofSection 9.1 can be used to calculate FVD and FUD in Equation (9.168). Two specialcases are considered below. First, an isovelocity profile is considered, for which onlyFVD is relevant. This is followed by long-range propagation in a duct with a uniformnon-zero sound speed gradient, for which only FUD is relevant. The effects of surfacedecoupling and tunneling are neglected.

9.3.3 V-duct propagation (isovelocity case)

The term ‘‘V-duct’’ (abbreviated VD) is introduced in Section 9.1.2.2 to describepropagation in a shallow-water waveguide that is bounded by reflection from bothupper and lower boundaries, for which the appropriate propagation factor is FVDfrom Equation (9.46). Substituting this expression in Equation (9.168), with FUD ¼ 0,gives for the signal MSP23

QSðrTx; rRxÞ ¼ S0�O�totH

r�3=2Tx erf

ffiffiffiffiffiffiffiffiffirTx4rCS

r� �r�3=2Rx erf

ffiffiffiffiffiffiffiffiffiffirRx4rCS

r� �; ð9:169Þ

where rCS is the transition range between cylindrical-spreading and mode-strippingregimes, given by Equation (9.50); and �tot is given by Equation (9.147). Equation(9.169) simplifies at short and long range to

QSðrTx; rRxÞ ¼ S0�O

H�totrCS

1=rTxrRx rTx � rCS; rRx � rCS

rCS=ðrTxrRxÞ3=2 rTx � rCS; rRx � rCS.

(ð9:170Þ

There is no dependence on the depth of transmitter, receiver, or target in Equation(9.169). This is because of the assumption leading to Equation (9.46) that neithersource nor receiver are close to the sea surface.

9.3.4 U-duct propagation (linear profile)

For long-range propagation in a U-duct (a waveguide bounded on one side byreflection and on the other by refraction), the propagation factor is given by Equation(9.151), so that Equation (9.168) becomes

QSðrTx; rRxÞ ¼ S04�O

20

H 2

DTx e�2�lorTx

rTx

DRx e�2�lorRx

rRx; ð9:171Þ

where 0 is given by Equation (9.86). Dependence on depth arises through the two


22 The cross-section is assumed to be independent of elevation angle.23 An absorption factor of the form expð�2�VrÞ, though omitted here, is always implied. A cut-on duct is further assumed, with neither source nor receiver close to the sea surface.

depth factors, which are defined as

DTx � DðzTx; ztgtÞ ð9:172Þand

DRx � Dðztgt; zRxÞ; ð9:173Þ

where Dðz0; zÞ is calculated as described in Section 9.1.2.1.2.

9.4 REVERBERATION LEVEL

The physics and hence calculation of reverberation level (RL) has much in commonwith that of EL. Both are specific to active sonar, and for both there is propagation toa scattering region and then back to the sonar. The main difference is that the targetecho is assumed to originate from a single point and arrive after a well-defined delaytime, whereas reverberation originates from an extended region of scatterers, arrivingcontinuously from a short time after transmission, determined by the distance to thenearest boundary. Although there are plenty reverberation models to choose fromEtter (2003), these have not yet reached the level of maturity of one-way propagationmodels.

Reverberation modeling requires the computation and summation ofcontributions to the received pressure field of scattered paths from many differentlocations and directions. As such it is one of the most challenging applications ofpropagation theory in sonar performance modeling, and is the subject of intensiveongoing research (see, e.g., Nielsen et al., 2008).

Low-frequency reverberation typically decays to a level below the noisebackground after a few seconds or tens of seconds after transmission, dependingon the source level, although this decay is not necessarily monotonic. Of particularinterest is reverberation at the arrival time of the target echo, as it is this that mightlimit the sonar’s ability to discriminate the echo from the background.

Scattered paths that contribute to reverberation at a given time t originate from ascattering annulus at a distance rðtÞ from the source24

rðtÞ � c

2t; ð9:174Þ

whose width �r is determined by the pulse duration T

�rðtÞ � c

2T : ð9:175Þ

Consider a narrow beam radiated by the source that propagates to some distant pointon the seabed after multiple boundary reflections, at which point the sound isscattered into a wide range of angles for the return path. Let the grazing angles ofthe radiated beam be between �in and �in þ ��in, and of the scattered sound considerthe contribution to reverberation from a narrow return beam, between angles �out

9.4 Reverberation level 495]Sec. 9.4

24 This and subsequent equations assume that the sound is traveling close to the horizontal

direction.

and �out þ ��out. Applying the continuum flux approach used in Section 9.1.2.1, for amonostatic geometry the contribution �2QR to the reverberation MSP due to thesetwo beams combined can be written25

�2QRðtÞ ¼ S0AðtÞ4

GTxSð�in; �outÞGRx ��in ��out; ð9:176Þ

where S0 is the source factor; Sð�in; �outÞ is the seabed scattering coefficient; and G isthe differential propagation factor from Equation (9.123), to be expressed here as afunction of time (see Ainslie, 2007). The area of the scattering annulus is 2r �r:

AðtÞ � c2T

2t: ð9:177Þ

The reverberation due to the narrow transmitted beam at angle �in, integrating overall �out, is

�QRðtÞ ¼ ��inS04GTx

ðAðtÞSð�in; �outÞGRx d�out: ð9:178Þ

Reverberation from an omni-directional source is then found by integrating over acontinuum of such transmitted beams:

QRðtÞ ¼ S04

ðAðtÞGTxSð�in; �outÞGRx d�in d�out: ð9:179Þ

Assuming that Sð�in; �outÞ is a separable function of its two arguments such that

Sð�in; �outÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSBð�inÞSBð�outÞ

p; ð9:180Þ


QRðtÞ � S0AðtÞ4

ðGð�axÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiSBð�BÞ

pd�ax

� 2

; ð9:181Þ

where

Gð�axÞ ¼4

rðtÞtan �ax

r0 tan �1 tan �2

� jRj2rðtÞ=r0 : ð9:182Þ

The angles �ax and �B are the ray grazing angles at the channel axis (sound speedminimum) and seabed, respectively.

It is useful to introduce the dimensionless reverberation coefficient

X ¼ Hcwt

4

ðG

ffiffiffiffiffiffiffiffiffiffiffiffiSBð�Þ

pd�; ð9:183Þ

defined in such a way that (for an omni-directional transmitter)26

QR ¼ S02T

H 2tX 2; ð9:184Þ


25 Assuming, as for the echo, no change in medium density between sonar and scatterer, and

that the received pulse is not stretched or compressed in time relative to the transmitted one.26 For a directional transmitter, the angle 2 is replaced by the horizontal beamwidth of thetransmitter.

where T is the pulse duration. More generally, allowing the transmitter and receiverto be located at different depths, this becomes, using Equation (9.177)

QR ¼ S02T

H 2tXTxXRx; ð9:185Þ

where XTx and XRx are given by Equation (9.183) with the depth dependence oftransmitter and receiver, respectively. This result and subsequent ones in this sectionare for a monostatic geometry in range and a bistatic one in depth. Some simple casesare considered below. For treatment of a fully bistatic geometry, see Harrison(2005a, b).

9.4.1 Isovelocity water

For an isovelocity channel, Equation (9.182) becomes

Gð�Þ ¼ 4

Hcwt cos �jRjðcwt=4HÞ sin �: ð9:186Þ

9.4.1.1 General power law scattering coefficient

Consider a scattering coefficient of the form

Sð�in; �outÞ ¼ � sinq �in sinq �out; ð9:187Þ

where � and q are constants, such that the backscattering coefficient is

SBð�Þ � Sð�; �Þ ¼ � sin2q �: ð9:188Þ


X ¼ Hcwt�1=2

4

ðsinq �Gð�Þ d�; ð9:189Þ

and therefore, using a small-angle approximation in Equation (9.186) (Ainslie, 2007),

X ¼ �1=2

2�ð�; u 2cÞu��; ð9:190Þ

where c is the seabed critical angle, given by Equation (9.131). Here � is the lowerincomplete gamma function (Appendix A), the parameter � is given by

� ¼ qþ 1

2ð9:191Þ

and u is a dimensionless time variable

u ¼ �totcwt

2H; ð9:192Þ


where �tot is given by Equation (9.147). The reverberation MSP follows fromEquation (9.184)

QR ¼ S0cwT��tot4H 3u2�þ1

�ð�; u 2cÞ2: ð9:193Þ

For reverberation, the transition between cylindrical-spreading (CS) and mode-stripping (MS) behavior occurs roughly at time 2rCS=cw, where rCS is given byEquation (9.50). The asymptotic limits of Equation (9.193) are

QR � S0��totcwT

4�2H 3� 4�c =u 2cu � 1 (CS)

�2Gð�Þ2=u2�þ1 2cu � 1 (MS):

(ð9:194Þ

Equation (9.185) can be applied to scattering from either boundary, depending onwhether interest is in surface or bottom reverberation. Total reverberation from bothboundaries can be found by calculating QR for each one and then adding the twoseparate QR values. This statement relies on the absence of multiple scattering, and istherefore valid for a slightly rough boundary.

Similar expressions are derived by Zhou (1980), who approximates the gammafunction by

Gð�Þ � 1��: ð9:195Þ

9.4.1.2 Application to a reference problem with Lambert’s rule (RMW11)

An important special case is obtained with q ¼ 1 in Equation (9.188), as thiscorresponds to the widely used Lambert rule. For this case

SBð�Þ ¼ � sin2 �; ð9:196Þ

and � then becomes the Lambert parameter. It follows by substituting q ¼ 1 inEquation (9.193) that (Zhou, 1980, Eq. (4.1); Harrison, 2003, Eq. (28))

QRZHðtÞ ¼ S0

2�T

�2totc2wt

31� exp � �totcw

2c

2Ht

!" #2

: ð9:197Þ

Converting to decibels and including absorption explicitly, Equation (9.197) becomes

RLZHðtÞ ¼ SLE þ 10 log102�

�2totc2wt

3þ 20 log10 1� exp � �totcw

2c

2Ht

!" #� aVcwt;

ð9:198Þwhere

SLE ¼ SLþ 10 log10 T ð9:199Þ

and aV is the volume attenuation coefficient, given by Equation (9.105).Reverberation for the case in hand, namely isovelocity water in combination with

Lambert’s rule for seabed scattering, is plotted vs. delay time in Figure 9.26, for one


of the test cases from a reverberation modeling workshop (RMW) held in November2006 at the University of Texas at Austin. At the time of writing, the results of thisworkshop are available online from an ftp site (rmw, 2006). The case considered isbased on workshop problem XI and referred to here as ‘‘RMW11’’.27 The waterdepth is 100m and the frequency chosen for the present comparison is 3.5 kHz. Theseabed parameters correspond to fine sand (Mz � 2.5�, see Chapter 4). A completedescription of the seabed properties is provided in Table 9.4.

The sea surface is treated as a perfect reflector (�S ¼ 0), so that �tot ¼ �B. Thevalue of �B can be estimated using Equation (9.51), giving �B � 0.274Np/rad, whichis the value used for the Zhou–Harrison formula in Figure 9.26. Also plotted is animproved approximation from Harrison (2006).


Figure 9.26. Model comparison for problem RMW11 and frequency 3.5 kHz (SLE ¼ 0 dB re

mPa2 m2 s; aV ¼ 0.2397 dB/km; cwaV ¼ 0.3596 dB/s). Water depth is 100m and source and

receiver depths are 30m and 50m, respectively. The speed of sound in water is

cw ¼ 1500m/s. See Table 9.4 for further details. INTEGRAL: Equation (9.189) (see Ainslie,

2007 for details); ZHOU-HARRISON: Equation (9.198), with �B � 0.274Np/rad;

HARRISON 2006: improved approximation from Harrison (2006).

27 RMW11 is identical to Problem XI from the RMW except for the choice of source level,

which here is chosen to be SLE ¼ 0 dB re mPa2 m2 s. The source level used by workshop

participants for the chosen frequency of 3.5 kHz is SLE ¼ �28:72 dB re mPa2 m2 s, which

means that the reverberation level shown in Figure 9.26 is about 29 dB higher than the

corresponding workshop results (rmw, 2006). The latest workshop results available at the time

of writing are available through the Solutions.html link at rmw (2006).

9.4.2 Effect of refraction

In the presence of refraction there is a need to distinguish between two different cases,depending on whether the scattering occurs at the low-speed or high-speed boundary.Denoting the respective contributions Xlo and Xhi, these can be written

Xlo ¼Hct

4

ðGð�loÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSBð�loÞ

pd�lo ð9:200Þ

and

Xhi ¼Hct

4

ðGð�loÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSBð�hiÞ

pd�lo: ð9:201Þ

If scattering occurs from both boundaries, total reverberation can be found bycalculating the contribution from each boundary separately using Equation(9.185) and adding them.

The asymmetry in the integrand makes the second integral (for Xhi) more difficultto evaluate. This asymmetry arises because the argument � inGð�Þ is the grazing angleat the channel axis (i.e., �lo) regardless of where the scattering occurs. Furtherattention is restricted to evaluation of Xlo. One justification for this is that the effectof refraction is to steer the sound towards, and hence enhance scattering from, thelow-speed boundary. Further, conditions of downward refraction are usually asso-ciated with a calm sea surface, whereas upward refraction often arises with a roughone.

9.4.2.1 V-duct propagation

For a V-duct (waveguide bounded by reflection at both boundaries), the differentialpropagation factor can be approximated by (see Section 9.1.2.2.1)

G � 2

rHe2ð2�hiþ�loÞr e�4ð�hiþ�loÞr�

2=�2min ; ð9:202Þ

where the decay rates �hi and �lo are given by Equations (9.140) and (9.141); and �minis from Equation (9.129). The reverberation coefficientXlo can therefore be calculated


Table 9.4. Seabed parameters for problems RMW11 (see Figure 9.26) and

RMW12 (Figure 9.28).

Description Symbol Value Comments

Water depth H 100m

Sediment sound speed csed 1700m/s c � 0.4900 rad

Sediment attenuation �sed 0.5 dB/� �sed=csed � 0.294 dB/(mkHz)

Density ratio �sed=�w 2

Lambert parameter � 10�2:7 10 log10 � ¼ �27 dB

as28

ðXloÞVD ¼ �1=2 eð2�hiþ�loÞcwtð�max�min

sinq �lo exp½�2ð�hi þ �loÞcwt�2lo=�2min� d�lo; ð9:203Þ

where �max is given by Equation (9.130).It is convenient to make the approximation (accurate to order �qþ3)

sinq � � �q e�q�2=6; ð9:204Þ


ðXloÞVD � �1=2

2K �eð2�hiþ�loÞct½�ð�;K�2maxÞ � �ð�;K�2minÞ�; ð9:205Þ

where

K ¼ q

6þ 2ð�hi þ �loÞcwt ð9:206Þ

and � is given by Equation (9.191).

9.4.2.2 U-duct propagation

The reverberation coefficient for propagation in a U-duct (waveguide bounded byreflection at one boundary and refraction at the other), for the outward path betweensource and scatterer (using the subscript ‘‘Tx’’ here instead of ‘‘s’’ to indicateproperties at the sonar transmitter), is

ðXloÞUD ¼ �1=2 202

ð 0 Tx

sinq �

tan � tan �Txd�: ð9:207Þ

Assuming small angles � and making the change of variable � ¼ Tx=cos u results in

ðXloÞUD � �1=2

2 qþ10 Dq e

��loct; ð9:208Þwhere

Dq ¼ DqðxÞ � cosq�1 x

ðx0

du

cosq uð9:209Þ

and

x ¼ arccos Tx 0

� �: ð9:210Þ

For q ¼ 0, Equation (9.209) gives

D0ðxÞ ¼ x=cos x; ð9:211Þand for q ¼ 1,

D1ðxÞ ¼ loge tan

4þ x

2

� �¼ artanhðsin xÞ: ð9:212Þ


28 The precise value used for cw in Equation (9.203) is not critical. Any value in the range clo to

chi will give accurate results, provided that �lo and �min are correctly calculated.


Recall that Tx (previously denoted 2; see Equation 9.68) is the grazing angle at thechannel axis (low-speed boundary) of the ray that is horizontal at the source depth. Itfollows from this that

x � arccos

ffiffiffiffiffiffiffihTxH

r; ð9:213Þ

where H is the water depth; and hTx is the distance between the source and the low-speed boundary.

For the return path from the scatterer back to the receiver, the same equationsapply except that hTx is replaced with hRx, the distance between the receiver and thelow-speed boundary.

The reverberation MSP is

ðQRloÞUD ¼ S0

T�

2H 2t 2qþ20 ðDqÞTxðDqÞRx e

�2�locwt: ð9:214Þ

For integer q ¼ N, the following recursion equation for DN is obtained usingintegration by parts:

ðN � 1ÞDNðxÞ ¼ sin xþ ðN � 2Þðcos2 xÞDN�2ðxÞ ðN 2Þ: ð9:215Þ

Similar expressions for reverberation are derived for the downward refracting case byZhou (1980), whose Eq. (11) implies use of the approximation

Dq � D1�q0 Dq

1; ð9:216Þ

which is exact for q ¼ 0 and q ¼ 1, and permits approximate interpolation betweenthese two values. By comparison, Equation (9.215) is exact for any integer N greaterthan 1.

All of the depth dependence is contained within the reverberation depth factorDqðxÞ. This function is plotted for various q in Figure 9.27. The first two (for q ¼ 0; 1)are from Equations (9.211) and (9.212). The remainder (q ¼ 2; 3) are evaluated usingEquation (9.215). Thus, for example,

D2ðxÞ ¼ sin x ð9:217Þand

2D3ðxÞ ¼ sin xþ ðcos2 xÞD1ðxÞ: ð9:218Þ

For q ¼ 2, the reverberation MSP varies linearly with depth, as it is proportional tosin2 x, which is

sin2 x ¼ 1� hTx=H: ð9:219Þ

The reverberation varies more quickly with depth for small values of q than for largeones (see Figure 9.27). This sensitivity of the depth factor to q makes it possible, inprinciple, to determine the angle dependence of sea surface backscattering strength bymeasurements of the depth dependence of reverberation in a surface duct.


9.4.2.3 Application to a reference problem with Lambert’s rule (RMW12)

The reverberation for a second case based on the 2006 reverberation modelingworkshop (problem XII) is considered next. The modified problem is referred tohere as ‘‘RMW12’’.29 The details are the same as for RMW11 (see Section 9.4.1.2)except for a uniform negative gradient of magnitude 0.3/s. (The sound speeddecreases linearly from 1530m/s at the sea surface to 1500m/s at the seabed.)

Reverberation vs. time for this case is shown in Figure 9.28, predicted using threedifferent methods as described in the caption, for a frequency of 3.5 kHz. For the first10 s, this calculation is dominated by the VD term, for which there is little differencein reverberation level compared with RMW11 (see Figure 9.26). After about 30 s, theUD term dominates, resulting in an exponential decay that becomes apparent at latertimes in Figure 9.28. (For example, by 90 s, the RMW12 reverberation is about 10 dBlower than for RMW11). Between 10 s and 30 s there is a transition region in whichboth UD and VD contribute significantly to the total.

Apart from this exponential decay, another difference between RMW11 andRMW12 is a series of regularly spaced features that appears in the coherent mode

Figure 9.27. Reverberation depth factor Dq for integer q between 0 and 3. Height h ¼ distance

from the low-speed boundary.

29 ‘‘RMW12’’ is identical to ‘‘Problem XII’’ from the RMW except for the choice of source

level, which here is chosen to be SLE ¼ 0 dB re mPa2 m2 s. The source level used by workshop

participants for Problem XII at the chosen frequency is SLE ¼ �28:72 dB re mPa2 m2 s, so the

present predictions are about 29 dB higher than the corresponding RMW results (rmw, 2006).


sum of Figure 9.28. For example, three peaks between 10 s and 15 s are clearly visiblein the close-up (Figure 9.29). The presence of these maxima can be explained in termsof a sequence of caustics associated with propagation from the source to the seabedand back to the receiver. This point is illustrated by Figure 9.30, which showspropagation loss vs. range and depth for this case, with a source at depth 30m,including the first five caustics intersecting the seabed at 2:4; 4:1; . . . ; 9:2 km. Thesedistances can be predicted using the simple formula

snðhÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8nðnþ 1Þ�0h

p; ð9:220Þ

where n is a positive integer; and h is the height from the seabed to either the source orthe receiver. The ray radius of curvature �0 is given by Equation (9.73). Moregenerally, Equation (9.220) translates to arrival times given by

tnðhÞ �2snðhÞcw

: ð9:221Þ

The derivation of Equation (9.220) follows. Consider a ray with launch angle(i.e., grazing angle at the source) �Tx. This ray reflects multiple times from the seabedwith a cycle distance of 2�ð�loÞ sin �lo, where �lo is the corresponding grazing angle at

Figure 9.28. Model comparison for problem RMW12 and frequency 3.5 kHz (SLE ¼ 0 re

mPa2 m2 s; aV ¼ 0.2397 dB/km). Water depth is 100m and source and receiver depths are

30m and 50m, respectively. For the values of other parameters see text and Table 9.4 (the

parameters of Figure 9.17—with �min ¼ 0:2—are chosen to match those of RMW12). MODES

(coh): coherent mode sum (Ellis, 1995); INTEGRAL: Equation (9.200); VDþUD: Equation(9.229), with �B � 0.195Np/rad.


the low-speed boundary and �ð�Þ is the radius of curvature given by Equation (9.72).The precise ranges (horizontal distances) at which the ray intersects the seabed aretherefore

rnð�TxÞ ¼ �ð�loÞ½ð2nþ 1Þ sin �lo � sin �Tx�: ð9:222Þ

The position of each caustic is determined by the condition drn=d�Tx ¼ 0, which gives

sin �lo ¼ ð2nþ 1Þ sin �Tx: ð9:223Þ

Substituting this result into Equation (9.222), and applying Snell’s law, gives

sn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4nðnþ 1Þðc2Tx=c2lo � 1Þ

q�0: ð9:224Þ

Equation (9.220) follows using the fact that clo and cTx are approximately equal.The caustic ranges and corresponding two-way travel times for the first seven

caustics are listed for each of source and receiver height in Table 9.5. For a sourcedepth of 30m (i.e., h ¼ 70m), caustics arise at the seabed at distances 2.4 km, 4.1 km,5.8 km, and so on from the source.

The shaded entries of Table 9.5 provide an explanation for why the set of causticsbetween 10 s and 15 s stand out prominently in Figure 9.29. The arrival timescorresponding to the formation of these caustics on the outward path (caustics 4to 6 for h ¼ 70m) are approximately equal to those on the return path (caustics 5 to 7

Figure 9.29. Model comparison for problem RMW12 and frequency 3.5 kHz (close-up). See

Figure 9.28 caption for details.

for h ¼ 50m). This numerical coincidence gives enhancement in both directions,resulting in a reinforcement of these three caustics.

An approximation to the total reverberation MSP for this case can be derivedfrom the results of Sections 9.4.2.1 and 9.4.2.2 using

QR ¼ S02T

H 2t½XVD þXUDðzTxÞ�½XVD þ XUDðzRxÞ�: ð9:225Þ

Putting q ¼ 1 (Lambert’s rule), �hi ¼ 0, and �lo ¼ �B in Equation (9.205) gives the VDcontribution

XVD ¼ �1=2

2Ke�Bct½expð�2HK=�0Þ � expð� 2cKÞ�; ð9:226Þ


Figure 9.30. Upper: ray trace illustrating formation of caustics and cusps (source depth¼ 30m)for a water depth of 100m and sound speed gradient of �0.3/s (BELLHOP). Lower: propaga-tion loss at 3.5 kHz, corrected for spherical spreading [dB] (the color axis runs from 16 dB

(white) to 29 dB (black)) (SCOOTER, see oalib (www)).

where

K ¼ KðtÞ ¼ 1

6þ �Bcwt

2H: ð9:227Þ

The UD equivalent is (Equation 9.208)

XUDðzÞ ¼�1=2

2 20 artanh

ffiffiffiffiffiz

H

re��Bcwt: ð9:228Þ

Converting to decibels and including the absorption term explicitly, Equation (9.225)becomes

RLðtÞ � SLE þ 10 log10� 402H 2t

� ðaB þ aVÞcwt

þ 10 log10f½ f ðtÞ þ artanhffiffiffiffiffiffiffiffiffiffiffiffiffizTx=H

p�½ f ðtÞ þ artanh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffizRx=H

p�g; ð9:229Þ

where

f ðtÞ ¼ e2�Bcwt

20KðtÞfexp½�ð2H=�0ÞKðtÞ� � exp½� 2cKðtÞ�g ð9:230Þ

and (see Equation 9.141)

aB ¼ ð10 log10 eÞ�B�0: ð9:231Þ

The coefficient �B can be evaluated using Equation (9.51) as previously for RMW11,but this approximation tends to overestimate the exponential decay term aB andhence underestimate the reverberation level. A better approximation is obtained


Table 9.5. Caustic ranges and corresponding two-way travel arrival times

for a source at depth 30m (height h from seabed 70m) (left) and receiver at

depth 50m (height h from seabed 50m) (right). See Figure 9.30.

Source zTx ¼ 30 m ðh ¼ 70 mÞ Receiver zRx ¼ 50 m ðh ¼ 50 mÞ

n Range/km Time/s Range/km Time/s

1 2.4 3.2 2.0 2.7

2 4.1 5.5 3.5 4.6

3 5.8 7.7 4.9 6.5

4 7.5 10.0 6.3 8.4

5 9.2 12.2 7.7 10.3

6 10.8 14.5 9.2 12.2

7 12.5 16.7 10.6 14.1

by matching the reflection loss at the steepest angle sustained by the bottom duct,such that

�B ¼ � 1

0logejRð 0Þj: ð9:232Þ

The curve labeled ‘‘VDþUD’’ in Figure 9.28 is calculated using Equation (9.232) for�B, giving a value of 0.195Np/rad for this parameter.

9.5 SIGNAL-TO-REVERBERATION RATIO (ACTIVE SONAR)

If the source level is sufficiently high, detection performance is determined by thedifference between the echo and reverberation levels (i.e., the signal-to-reverberationratio). This quantity is considered in the closing section of this chapter. The echo levelis given vs. range in Section 9.3, whereas reverberation in Section 9.4 is found vs.delay time. For the purpose of determining the signal-to-reverberation ratio (SRR),what matters is the reverberation arriving at the same time as the target echo, so in thefollowing the reverberation is evaluated at the echo arrival time, given by

tEðrÞ �2r

cw: ð9:233Þ

First, the isovelocity V-duct is considered (Section 9.5.1), followed by the long-rangeU-duct case (Section 9.5.2). The geometry considered is bistatic in depth and mono-static in range. These SRR calculations do not take into account the effects ofbeamforming or matched filtering and are therefore relevant to the SRR at thehydrophone, before any processing.

9.5.1 V-duct (isovelocity case)

From Equation (9.169) the signal and reverberation can be written (hereafterincluding absorption explicitly)

QSðrÞ ¼ S0�

4�totHr3ferff cð�r=HÞ1=2�g2 expð�4�VrÞ ð9:234Þ

and (see Equation 9.193)

QRðtEÞ ¼ S0cw��

4H 3uqþ2�

qþ 1

2; u 2c

� �� 2

expð�4�VrÞ; ð9:235Þ

where the variable u (see Equation 9.192) is evaluated at time tE. The SRR is therefore

QSðrÞQRðtEÞ

¼ ��

�Hcw�

erf ½ cð�r=HÞ1=2��½ðqþ 1Þ=2; u 2c �

( )2

uq�1: ð9:236Þ

The absorption cancels in Equation (9.236), which simplifies for short and long


ranges to

QSðrÞQRðtEÞ

� ��

2�Hcw�

2qþ4c

ðqþ 1Þ2u�1 2cu� 1

Gqþ 1

2

� �� 2 uq�1 2cu� 1.

8>>>>>><>>>>>>:

ð9:237Þ

The value q ¼ 1 is critical, resulting in an SRR that is independent of target range(Zhou et al., 1997; Harrison, 2003). For q > 1, in this simple model (which neglectsrefraction) the SRR actually increases with increasing target range.

9.5.2 U-duct (linear profile)

Putting r ¼ rTx ¼ rRx in Equation (9.171) gives

QSðrÞ ¼ S04�O

20

H 2DTxDRx

e�4ð�loþ�VÞr

r2: ð9:238Þ

UD reverberation is given by Equation (9.214)

QRðtEÞ ¼ S0��c

4H 2r 2qþ20 ðDqÞTxðDqÞRx e

�4ð�loþ�VÞr; ð9:239Þ

9.5 Signal-to-reverberation ratio (active sonar) 509]Sec. 9.5

Figure 9.31. SRR depth factor YðzÞ for the same three different target depths as Figure 9.11(U-duct propagation).

and therefore

QSðrÞQRðtEÞ

¼ 4�

2�c� 2q0 rYTxðzTxÞYRxðzRxÞ: ð9:240Þ

Notice the simple (1=r) range dependence. The complexity here is in the depthdependence, and it is convenient to define the SRR depth factor YðzÞ such that

YðzTxÞ ¼DTx

ðDqÞTzð9:241Þ

and

YðzRxÞ ¼DRx

ðDqÞRx: ð9:242Þ

This depth factor is plotted in Figure 9.31 for q ¼ 1; 2. This function is relevant to thelong-range U-duct problem, with no V-duct paths.

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waveguide, including shear wave effects, J. Acoust. Soc. Am., 85, 648–653.

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2804–2814.



Fishback, W. T. (1951) Methods for calculating field strength with standard refraction, in

D. E. Kerr (Ed.), Propagation of Short Radio Waves (p. 113), McGraw-Hill.

Freehafer, J. E. (1951) The linear modified-index profile, in D. E. Kerr (Ed.), Propagation of

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1340.

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and C. Carbo Fite (Eds.) (1987) Acoustics and Ocean Bottom, II: F.A.S.E. Specialized

Conference, June 18–20, Madrid (pp. 3–58), Consejo Superior de Investigaciones

Cientıficas, Madrid.

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complex environments, Applied Acoustics, 51(3), 251–287.

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ICA, Satellite Symposium on Sea Acoustics, Dubrovnik, September 4–6 (pp. 169–172).

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bathymetry and sound speed, IEEE J. Oceanic Eng., 30, 660–675.

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Acoust., 13, 317–340.

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Application to Reverberation Calculation (NURC-FR-2006-21), NATO Undersea

Research Centre, La Spezia, Italy.



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on new low-frequency data, J. Acoust. Soc. Am., 86, 716–738.

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Cientıficas, Madrid.

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31 Translated into English by Zhou in 2007 (Jixun Xhou, pers. commun., August 7 2008).

10

Transmitter and receiver characteristics

I like Wagner’s music much better than anybody’s.It’s so loud that one can talk the whole time

without people hearing you.

Bob Marley (ca. 1970)

Before returning to the sonar equation in Chapter 11, there is one remaining piece tobe fitted in the puzzle, namely the characteristics of the sonar itself. Some sonarproperties, such as transmitter power through the source level and receiver directivitythrough the array gain, are incorporated explicitly in the sonar equations. Alsoimportant are the frequency, bandwidth, and pulse duration, which affect terms likeprocessing gain, detection threshold, and propagation loss. In this chapter thoseproperties that are intrinsic to sonar systems are collected and tabulated, with par-ticular emphasis on the transmitter source level for active sonar, whether man-madeor biological. Receiver sensitivity and self-noise are also considered, represented inthe case of biological sonar by the animal’s audiogram. Finally, thresholds are givenfor possible impact on marine life in the form of behavioral effects and hearingthreshold shifts. In the case of man-made equipment, the scope is extended to includeall deliberate use of underwater sound. The directivity index of the receiving array isconsidered in Chapter 6.

The information is compiled from many different sources, including Internet sitesand commercial literature from equipment manufacturers, as acknowledged in theindividual tables. While reasonable attempts have been made to exclude erroneousvalues, the extended use of non peer–reviewed sources makes it likely that some errorsremain. In any case, the systems themselves are subject to review and modification bymanufacturers, so that some of the data are likely to be superseded within a few yearsof publication. In addition, the source level of many sound transmitters is not fixed,but depends on other parameters such as beamwidth and pulse duration. Finally,

regarding the use of sound by, or impact of sound on marine mammals and otheraquatic animals, new publications appear continually in the scientific literature,providing new data or questioning old assumptions. Whether for man-made or forbiological systems, the reader is therefore advised to check the information bycomparing it with up-to-date sources.

Transmitters are described first in Section 10.1, which comprises mainly tables ofsource level and frequency for search sonars, seismic survey sources, explosives,acoustic deterrents, and other miscellaneous sound-producing devices. This is fol-lowed in Section 10.2 by a discussion of the sensitivity properties of sonar receivers,including hearing and behavioral thresholds for biological systems.

10.1 TRANSMITTER CHARACTERISTICS

Two of the most important parameters determining the suitability of an acoustictransmitter for a given application are its frequency and its source level. These twoproperties are listed below for a number of different sonar types. Also important, forthe assessment of both sonar performance and environmental impact, are pulseduration, bandwidth, and beamwidth. Though not presented here, in many casesthese can be found in the references provided, or obtained from the manufacturer.

A useful distinction can be made between devices whose objective it is tomeasure some property of the seafloor (echo sounders, bottom profilers, and seismicsurvey sources), generally designed to direct energy down in the vertical direction,and search sonar or communications equipment, which is generally directed in ahorizontal direction. A third category, which includes acoustic deterrents and trans-ponding or communication equipment, generally uses omni-directional transmitters.

The source level of a transducer depends on the supplied voltage and itssensitivity.1 Alternatively, the source level can be written as a function of totaltransmitted power and sonar directivity. Either way, transmitter sensitivity anddirectivity are excluded from the present scope (the interested reader is referred topublications by Tucker and Gazey, 1966; Stansfield, 1991; and Blue and VanBuren, 1997). Thus, for the remainder of this chapter the terms ‘‘sensitivity’’ and‘‘directivity’’ refer exclusively to the sonar receiver.

A common misunderstanding that arises about the term source level arises fromits definition by the American National Standards Institute (ANSI) and the Inter-national Electrotechnical Commission (IEC) as the sound pressure level at thestandard reference distance (1m) from the source. Neither of these definitions men-tions the need for the measurement to be in the far field (Morse and Ingard, 1968), asrequired by the more complete descriptions of Kinsler et al. (1982) and Urick (1983),thus limiting their applicability to compact sources. The details are deferred toChapter 11, but the important point here is that in general the source level is noteven approximately equal to the sound pressure level at 1m, making the ANSI andIEC definitions at best misleading. Briefly, the role of the source level is to provide a

514 Transmitter and receiver characteristics [Ch. 10

1 The sensitivity of a sonar transmitter is the source factor divided by the mean square voltage.

prediction of the sound pressure level in the far field of the source. It is not a usefulmeasure of the field close to an array of transducers spanning several wavelengths.

10.1.1 Of man-made systems

Man-made equipment for the transmission of underwater sound is considered here.Applications of such transmissions include conventional sonar (i.e., equipment whoseprimary purpose is the detection and localization of underwater objects), acousticdeterrents, communication and positioning equipment, and oceanographic measure-ment systems such as used for seismic or bathymetric surveys, acoustic thermometry,or current measurement. It is useful to distinguish between a continuous source, whoseRMS pressure field remains approximately unchanged during transmission, and animpulsive source, which either decays without oscillation or whose amplitude changessignificantly from one cycle to the next. These are introduced in the following twosub-sections. More specialized information about sonar transducers is provided byHunt (1954) and Stansfield (1991).

10.1.1.1 Continuous sources

By a continuous source is meant one that transmits a signal whose amplitude remainsunchanged after many cycles, such that an associated RMS pressure field can bedefined unambiguously, as is the case for a tonal source or a frequency sweep ofconstant amplitude. Typical examples include echo sounders, sidescan sonar,communications equipment, and military search sonar.

10.1.1.1.1 Single-beam echo sounders

Perhaps the most widely used man-made sonar is the basic single-beam echo sounder,designed to measure the travel time (and hence distance) to an object (usually theseabed or a shoal of fish) beneath the vessel carrying the sonar. The frequency andsource level of some single-beam echo sounders are listed in Table 10.1. Based on thistable, a typical value for the (maximum) source level of a single-beam echo sounder isabout 214� 6 dB re mPa2 m2, with little dependence on frequency between 12 kHzand 200 kHz. The values quoted are maximum source levels for each sonar. Theactual source level depends on pulse duration and beamwidth. The highest sourcelevels are usually associated with narrow beams and short pulses.

10.1.1.1.2 Sidescan sonar

A sidescan sonar works on a similar principle as an echo sounder, except that thesound is projected sideways as well as vertically downwards (Blondel, 2009). Suc-cessive echoes are recorded and used to build up a high-resolution image of thesurroundings. Table 10.2 summarizes source level data for sidescan sonars. A typical(maximum) value is about 225 dB re mPa2 m2 at 100 kHz, decreasing with increasingfrequency.

10.1 Transmitter characteristics 515]Sec. 10.1

10.1.1.1.3 Multibeam echo sounders

More sophisticated echo sounders exist that are capable of beamforming the echoand thus distinguish between vertical arrival angles of seabed returns (Lurton, 2002).Source levels and related properties of such multibeam echo sounders are listed inTable 10.3.

The trend of decreasing source level with increasing frequency is similar tothat for a sidescan sonar. Figure 10.1 shows a graph of source levels for sidescanand multibeam sonars taken from Tables 10.2 and 10.3, plotted as a function offrequency. The graph shows a downward trend with increasing frequency for fre-quencies between 12 kHz and 675 kHz. The regression curve is a straight line fit inlog F

SL ¼ 259:5 � 17:43 log10 F dB re mPa2 m2; ð10:1Þ

where F is the frequency in kilohertz.

10.1.1.1.4 Sub-bottom profilers

A depth profiler or sub-bottom profiler is similar to an echo sounder, except that alower frequency is used in order to probe more deeply into the sediment. A list ofsource levels is given in Table 10.4.


Table 10.1. Summary of single-beam echo sounder source levels, sorted by frequency.

Manufacturer System Frequency/ Max. SL/ Reference Notes

kHz (dB re

mPa2 m2Þ

Submarine Fessenden ca. 1 204 Hackmann (1984) Early echo sounder,

Signal Co. oscillator included for historical

(now interest

Raytheon)

Massa TR-1073A 12 216 massa (www)

Simrad 38/200 combi w 38 208 simrad (www) Also operates at 200 kHz

Simrad HTL 430D 42 200 kongsberg (www)

Simrad EK 500 57 214 LIPI (2006)

Simrad SD 570 57 220 LIPI (2006)

Simrad 38/200 combi w 200 208 simrad (www) Also operates at 38 kHz

Biosonics DT4000 208 221 Churnside et al. (2003)

Table 10.2. Summary of sidescan sonar source levels, sorted by frequency.


kHz (dB re

mPa2 m2Þ

Racal SeaMARC SB12 12 233 Funnell (1998, p. 121)

Simrad AMS 36/120SI 35 228 Watts (2000, p. 453) Actual frequencies are

Funnell (1998, p. 124) 33.3 and 36.0 kHz; also

operates at 120 kHz

Racal SeaMARC SB50 50 200 Funnell (1998, p. 121)

Simrad AMS 60SI 57.6 227 Watts (2000)

Funnell (1998, p. 124)

Neptune 990 Tow Fish 59 227 neptune (www)

Ultra Deepscan 60 60 231 Watts (2000, p. 446)

Funnell (1998, p. 136)

Massa TR-1101 97 223 massa (www)

Innomar Sidescan ses-2000 100 220 innomar (www)

Neptune 422 Tow Fish 100 228 neptune (www) Also operates at 500 kHz

GEC Marconi Bathyscan 100 220 Funnell (1998, p. 122) Also operates at 300 kHz

Neptune 272 Tow Fish 105 234 neptune (www) Second beam operates

(dual beam) with a source level of

229 dB re mPa2 m2


(dual frequency)

Geoacoustics DSSS 114 223 geoacoustics (www) Also operates at 410 kHz

Simrad AMS 120SI 120 224 Watts (2000, p. 453)

AMS 120SP Funnell (1998, p. 124)

Simrad AMS 36/120SI 120 224 Watts (2000, p. 453) Also operates at 35 kHz

Funnell (1998, p. 124)

Benthos C3D 200 224 benthos (www) Also operates at 100 kHz

GEC Marconi Bathyscan 300 220 Funnell (1998, p. 122) Also operates at 100 kHz

Tritech SeaKing 325 208 Funnell (1998, p. 104) Also operates at 675 kHz

Geoacoustics DSSS 410 223 geoacoustics (www) Also operates at 114 kHz


(dual frequency)


Tritech SeaKing 675 208 Funnell (1998, p. 104) Also operates at 325 kHz

Table 10.3. Summary of multibeam echo sounder source levels, sorted by increasing frequency.

Manufacturer System Frequency/ Max. SL/ Reference NoteskHz (dB re

mPa2 m2ÞSimrad EM 120 12 245 Hammerstad (www)

Simrad EM 121A 12 238 Watts (2005)Funnell (1998, p. 126)

ELAC Sea Beam 2000 12 234 Watts (2000, p. 456)Funnell (1998, p. 131)

ELAC Sea Beam 3012 12 239 seabeam (www)

Thomson TSM 5265 12 235 Watts (2000, p. 441)Marconi Sonar Funnell (1998)(Thales)

Simrad EM 12 13 238 Funnell (1998, p. 125)

ELAC Sea Beam 2120 20 247 seabeam (www)

Simrad EM 300 30 241 Hammerstad (www)

ELAC Sea Beam 1050 50 234 Kvitek et al. (1999)

Simrad EM 710 85 232 Hammerstad (www) 70–100 kHz

Simrad EM 1000 95 225 Funnell (1998, p. 127)Hammerstad (www)

Simrad EM 1002 95 226 Kvitek et al. (1999)

Simrad EM 950 95 225 Kvitek et al. (1999)Funnell (1998, p. 127)


Atlas Fansweep 20 100 227 Kvitek et al. (1999) Also operates at 200 kHz

Thomson TSM 5260 100 210 Watts (2000, p. 441)Marconi Sonar Funnell (1998, p. 135)(Thales)

Triton ISIS 100 117 219 Kvitek et al. (1999) Also operates at 234 kHz

ELAC Sea Beam 1185 180 217 Kvitek et al. (1999)

ELAC Sea Beam 1180 180 220 Funnell (1998, p. 130)

Atlas Fansweep 15 200 227 Kvitek et al. (1999)

Atlas Fansweep 20 200 227 Kvitek et al. (1999) Also operates at 100 kHz

Reson Seabat 7125 200 220 Lovgren (2007)

Reson Seabat 8124 200 210 Kvitek et al. (1999)

Simrad EM 2000 200 218 Hammerstad (www)

ECHOSCAN 200 225 Kvitek et al. (1999)

Triton ISIS 100 234 219 Kvitek et al. (1999) Also operates at 117 kHz

Reson Seabat 8101 240 217 Kvitek et al. (1999)


Reson Seabat 9001 455 210 Kvitek et al. (1999)Watts (2005)

10.1.1.1.5 Fisheries sonarSome echo sounders are adapted to look for fish by tilting them away from thevertical direction, giving them increased area coverage by scanning at oblique angles.Examples are listed in Table 10.5.

10.1.1.1.6 Military search sonar

For some military applications there is a requirement to detect objects at very longranges in order to respond early to a potential threat. By contrast, some systems aredesigned to achieve high resolution, working by necessity at much higher frequencyand hence limited to short range, leading to a wide range of sonar specifications. Thefollowing tables summarize the source levels and frequency ranges for hull-mountedsonar (Table 10.6), dipping sonar (Table 10.7), towed array sonar (Table 10.8), andother sonars (Table 10.9).

Recall that the source level is a measure of the power (more precisely the radiantintensity) projected by a sonar transmitter into its far field. This point is of specialsignificance if the extent of the sonar transmitter is large compared with the acousticwavelength, such as for an array of two or more synchronized transducers (anexample from Table 10.8 is SURTASS). In this situation, the source level is notrelated in a simple way to the sound pressure level at 1m.


Figure 10.1. Maximum multibeam echo sounder and sidescan sonar source levels vs.

transmitter frequency.

10.1.1.1.7 Minesweeping sonar

Modern navies seek to reduce the threat of sea mines by a number of differentmeasures. One way is to use high-frequency search sonar, called ‘‘minehunting’’ sonar(see Tables 10.6 and 10.9), to find the mines before avoiding or deactivating them. Analternative strategy consists of transmitting a signal that reproduces the acousticsignature of a passing ship, thus neutralizing the mine by precipitating its premature


Table 10.4. Summary of depth profilers, sorted by increasing maximum source level.

Manufacturer System Center Max. SL/ Reference Notes

frequency/ (dB re

kHz mPa2 m2Þ



Geoacoustics geochirp 7 205 geoacoustics (www) 0.5–13 kHz

Ultra Deepscan 60 10 212 Watts (2000, p. 446) 7.5–12.5 kHz

Funnell (1998, p. 136)

Geoacoustics T135 transducer 5.5 214 geoacoustics (www) 3–7 kHz

Geoacoustics Array of 16 T135 5.5 225 geoacoustics (www) 3–7 kHz

transducers

Geoacoustics geopulse 7 227 geoacoustics (www) 2–12 kHz

Innomar compact 7 236 innomar (www) 2–12 kHz

Table 10.5. Summary of fisheries sonar source levels, sorted by maximum frequency.


kHz (dB re

mPa2 m2Þ

Simrad SX90 20–30 219 simrad (www)

Simrad SP70 26 222 kongsberg (www)

Simrad SP90 26 223 kongsberg (www)

Simrad SH80 116 210 kongsberg (www)

Simrad SH80 110–122 210 simrad (www)

Simrad ES60 single beam 12–200 >206 simrad (www) 4 kW at 38 kHz

Simrad ES60 split beam 18–200 >206 simrad (www) 4 kW at 38 kHz


Table 10.6. Summary of hull-mounted search sonars, sorted by maximum frequency.

Model Frequency/ Source level/ Reference

kHz (dB re mPa2 m2)

USN AN/SQS-53C 2.6, 3.3, 3.5 235 Anon. (2001)

Anon. (2003, p. 66)

Kuperman and Roux (2007,

Table 5.3)

AGISC (2005)

USN AN/SQS-56/DE 1160 6.7, 7.5, 8.4 218–232 Watts (2005, p. 127ff)

DE 1167 HM 7.5, 12.0 227 Watts (2005, p. 161)

See also DE 1167 VDS

(Table 10.8)

Improved DE 1160 3.75, 5.0, 7.5, 232 (7.5 kHz) Watts (2005, p. 127ff)

12 238 (3.75 kHz)

SS 2450 24 213 Watts (2005, p. 148)

CTS-24 ASW OMNI sonar 24 223 Watts (2005, p. 135)

CMAS-36/39 mine detection 36, 39 223 Watts (2005, p. 437)

and avoidance sonar

CTS-36/39 OMNI sonar 36, 39 223 Watts (2005, p. 135)

SS 9500 mine detection and 95 220 Watts (2005)

avoidance sonar

Table 10.7. Summary of helicopter dipping sonars, sorted by maximum frequency.


kHz (dB re mPa2 m2)

Helras 1.31–1.45 218 Watts (2005)

AN/AQS-13F 9.2–10.7 216 Watts (2005, p. 193)

AN/AQS-18 9.2–10.7 216 Watts (2005, p. 194)

AN/AQS-18A 9.2–10.7 216 Watts (2005, p. 194)

HS 12 13 212 Watts (2005)

explosion. The transmitting device is called a ‘‘minesweeping’’ or ‘‘influence sweep’’sonar. An example from Watts (2000) is the Sterne 1 system of Thomson Marconi(now Thales Underwater Systems), with a source level of 160 dB re mPa2 m2 in thefrequency range 10Hz to 200Hz.


Table 10.8. Summary of active towed array sonars, sorted by maximum frequency.


kHz (dB re mPa2 m2)

USN SURTASS-LFAa 0.1–0.5 215 Anon. (2003, p. 66)

projector (single LFA Kuperman and Roux (2007)

frequency projector) AGISC (2005)

USN SURTASS-LFA 0.1–0.5 221–240 Hildebrand (2004)

projector (array of up to dosits (www)

18 LFA projectors)

LFATS (derived from 1.38 219–222 Watts (2005, p. 163)

Helras technologyb)

DE 1167 VDS 12 217 Watts (2005, p. 161)

See also DE 1167 HM sonar

(Table 10.6)

ST 2400 24 213 Watts (2005, p. 149)

a SURTASS-LFA: U.S. Navy Surveillance Towed Array System—Low Frequency Active.b See Table 10.7.

Table 10.9. Summary of miscellaneous search sonar (including coastguard and risk mitigation

sonar), sorted by maximum frequency.

Sonar type Model Frequency/ Source level/ Reference

kHz (dB re mPa2 m2)

Active sonobuoy RASSPUTIN 1.5 205 Watts (2005)

Active sonobuoy AN/SSQ-62RO 6.5–9.5 >199 Watts (2005)

Coastguard search SS105 14 230 Watts (2000, p. 99)

sonar

Marine mammal risk HFM3 30–40 220 Ellison and Stein

mitigation sonar (2001)

Minehunting ROVa SeaBat 6012 455 210 (nominal) Watts (2005)

scanning sonar

a ROV: remotely operated vehicle.

10.1.1.1.8 Acoustic deterrent devices

Sound transmitters are used or have potential for use underwater to protect:

— fish farms by deterring predators;— sensitive fauna by warning them away from fishing nets, explosions, pile driving,

or other equipment posing a potential danger;— valuable or sensitive harbor facilities by deterring malicious human divers or

trained animals.

Such acoustic deterrents can be grouped into two broad classes. The first classincludes relatively low power devices, sometimes known as ‘‘pingers’’ or ‘‘alarms’’,intended to deter mammals from approaching fishing gear, in order to prevent thembecoming entangled in the nets. A list of the source levels of these low-amplitudedeterrents is given in Table 10.10. The second class of deterrents, with a higher sourcelevel (sometimes known as ‘‘acoustic harassment devices’’), is used to clear a largerarea, either to protect the animals from exposure to loud sounds that might beanticipated (such as an explosion) or to protect fish farms from predators.The properties of some higher amplitude deterrents are listed in Table 10.11. Thesource levels in this table all exceed 190 dB re mPa2 m2, compared with up to179 dB re mPa2 m2 in Table 10.10.

10.1.1.1.9 Underwater communications systems and transponders

In the same way that sonar is used as an alternative to radar for the detection ofunderwater objects, sound provides an alternative to radio waves for the transmissionof underwater signals. The source levels of transducers used in underwater commun-ications systems (Table 10.12) and transponders (Table 10.13) are summarized below.

10.1.1.1.10 High-frequency imaging sonar

Very high frequency sonars, operating at frequencies around 1MHz, are able toproduce high-resolution images that resemble photographs. For this reason theyare sometimes known as ‘‘acoustic cameras’’. The properties of such systems arelisted in Table 10.14.

10.1.1.1.11 Research instruments (global oceanography)

Sonar technology is increasingly used for exploration and monitoring of the world’soceans. The source levels of a selection of research sonars are summarized in Table10.15, including the source used for the Heard Island Feasibility Test (HIFT), aglobal-scale test transmission carried out in 1991 from the Indian Ocean to bothPacific and Atlantic Oceans (Munk et al., 1994). Because the HIFT source spansseveral wavelengths, the source level is not related in a simple way to the soundpressure level at 1m.


Table 10.10. Summary of low-amplitude acoustic deterrents, sorted by maximum source level. BB: broadband.

Manufacturer System Frequency/ Source Reference Notes

or originator kHz level/

(dB re

mPa2 m2Þ

Gearin BB 122–125 Gearin et al. (2000) Peaks at 3 and 20 kHz

Lien 2.5 110–132 Fullilove (1994)

McPherson 3.5 110–132 Gordon & Northridge

(2002, table 2)

FMP 332 10 130–134 Gordon & Northridge

(2002, table 2)

Airmar Airmar gillnet 9.8 134 Kastelein et al. (2007)

SaveWave Endurance 5–110 134� 1.3 Kastelein et al. (2007) BB

Aquatec Aquamark 200 BB 134� 1.3 Kastelein et al. (2007) Frequency sweeps

Sub-sea

SaveWave White high impact 5–95 140� 0.6 Kastelein et al. (2007) BB

Fumunda FMDP 2000 9.6 141 Kastelein et al. (2007)

SaveWave Black high impact 33–97 143� 0.7 Kastelein et al. (2007) BB

Loughborough PICE 55 137–145 Culik et al. (2001) 3 tonals

University 83 133–138

100 95–120

Aquatec Aquamark 300 10 145 Gordon & Northridge

Sub-sea (2002, table 2)

Dukane NetMark 1000 10–12 120–146 Barlow and Cameron Discontinued

(2003)

Aquatec Aquamark 100 20–160 148� 3.7 Kastelein et al. (2007) Frequency sweeps

Sub-sea

Dukane NetMark 2000 10 130–150 Gordon & Northridge Discontinued

(2002, table 2)

Dukane NetMark 2MP 9–15 127–152 Kastelein et al. (2001) 16 tonals

Dukane NetMark XP-10 9–15 133–163 Kastelein et al. (2001) 16 tonals

Ocean DRS-8 0.6 172 Kastelein et al. (2007)

Engineering

Enterprise

TERECOS Type DSMS-4 4.9 179 Lepper et al. (2004)

10.1.1.2 Impulsive sources

10.1.1.2.1 General characteristics

The continuous sources described above are characterized by their mean squarepressure (MSP), or SLMSP. Pulses whose amplitudes vary with time, sometimesrapidly, are considered next. For such pulses it is not obvious how to define MSPbecause the result of averaging depends critically on the extent in time during whichthe averaging takes place. The following comments apply whether the source isnatural or man-made.

Peak-to-peak pressure, zero-to-peak pressure, and integrated pressure squared. Itis common to characterize impulsive sources in terms of their peak-to-peak (p-p) orzero-to-peak (z-p) pressure. For short pulses, changes in the shape of the pulse canoccur over time (e.g., due to multipath propagation in shallow water) so that care isneeded in the interpretation of reported peak values. For this reason, the sourceenergy (characterized by the energy source level SLE), which is not affected bychanges in pulse shape, is a more robust measure than p-p or z-p pressure for thecharacterization of short pulses.

Peak sound pressure values are often converted to corresponding source levels indecibels, denoted SLp-p or SLz-p (these parameters are defined in Chapter 8). Suchconversion is sometimes questioned on the grounds that the decibel should bereserved for expressing ratios of power or energy, whereas the peak sound pressure

10.1 Transmitter characteristics 525

Table 10.11. Summary of high-amplitude acoustic deterrents, sorted by maximum source level.

Manufacturer System Frequency/ Source Reference Notes

kHz level/

(dB re

mPa2 m2Þ

Simrad fishguard 15 191 Gordon & Northridge

(2002, table 1)

Airmar dB Plus II 10.3 192 Lepper et al. (2004)

Ace Aquatec Silent 10 193 Lepper et al. (2004) Multi-tone (19 frequencies

scrammer 16 194 ace (www)a from 3.3 to 20 kHz)

Ferranti- Mk. 2 seal 8–30 194 Gordon & Northridge Multi-tone

Thomson scrammer (2002, table 1)

Ferranti- Mk. 3 seal 25 194 Taylor et al. (1997)

Thomson scrammer

Ocean DRS-8 3.0 202 Kastelein et al. (2007)

Engineering

Enterprise

a Universal scrammer AA-01-048V2.

is neither of these, even when squared. The practical reality is that the decibel oftendoes get used for ratios that are not powers or energies, and this is one example. Theuse here of p-p and z-p source levels in decibels, while not intended as an endorsementof this practice, acknowledges its widespread adoption in the literature.


Table 10.12. Summary of acoustic communications systems, sorted by increasing maximum source level.

(LF: low frequency; HF: high frequency).

Manufacturer System Frequency/ Max. SL/ Reference

kHz (dB re mPa2 m2)

Sparton Corporation Mk. 84 SUS 3.3, 3.5 160 Watts (2005, p. 268)

(expendable air

to submarine

communications

device)

Orcatron Scubaphone 30 171 Watts (2000, p. 395)

Fugro UDI Subcom 3400 25 171 Funnell (1998, p. 262)

Nautronix Secure 8, 25 174 Funnell (1998, p. 264)

Hellephone

Ocean Technology Aquacom 30–35 176 Funnell (1998, p. 265)

Systems SSB-2010

Ocean Technology Aquacom 22–35 178 (Funnell 1998, p. 266)

Systems SSB-1001B

Tritech AM-300 8–16, or 16–24 184 tritech (www)

Orca Instrumentation MATS 12 10–14 185 Funnell (1998, p. 257)

Orca Instrumentation MATS 53 50–58 185 Funnell (1998, p. 257)

MARCOM Defence Type 185/ 8.4–11.3 186 Watts (2005, p. 265)

G732 Mk. II

Kongsberg Simrad SPT 319 24.5–32.5 195 ashtead (www)

L3 Communications UT 2000 1–60 196 (LF) Watts (2005, p. 257)

ELAC Nautik 188 (HF)

Massa TR-1036D 8 198 massa (www)

Massa TR-1055C 12 199 massa (www)

Harris Products AN/WQC-2A 1.45–3.10 (LF) 199 (LF) Watts (2005, p. 268)

Corporation 8.3–11.1 (HF) 198 (HF)

Kongsberg Simrad MPT 331DTRDUB 24.5–32.5 206 ashtead (www)


Table 10.13. Summary of selected acoustic transponders and alerts, sorted by

increasing maximum source level.


kHz (dB re mPa2 m2)

IXSEA RP402E 37.5� 1 157 ixsea (www)

IXSEA RP162E 37.5� 1 160 ixsea (www)

Sonardyne Type 7097 36–44 185 sonardyne (www)

InterOcean 1090ET 7.5–9 190 interocean (www)

Ore Offshore BRT6000 12 190 ore (www)

Kongsberg MST 319 ca. 30 190 kongsberg (www)



IXSEA MF range 20–30 191� 4 ixsea (www)

IXSEA LF range 8-16 192� 4 ixsea (www)


Sonardyne Type 8011 7.5–15 195 sonardyne (www)





Table 10.14. Summary of acoustic cameras.


kHz (dB re mPa2 m2)

Fugro-UDI Sonavision 2000 500 208 Funnell (1998, p. 101)

Tritech SeaKing DFS 325, 675 212 Funnell (1998, p. 103)

Tritech SeaKing DFP 580, 1200 212 Funnell (1998, p. 103)

It is convenient to define peak-to-peak (p-p) and zero-to-peak (z-p) source factorsðS0Þp-p and ðS0Þz-p in terms of the corresponding source levels as

ðS0Þp-p � 10ðSLp-p =10Þ mPa2 m2 ð10:2Þand

ðS0Þz-p � 10ðSLz-p=10Þ mPa2 m2: ð10:3Þ

Similarly, the energy source factor can be characterized in terms of the time-integrated squared pressure (a measure of total transmitted energy) in the form

ðS0ÞE � 10ðSLE=10Þ mPa2 m2 s: ð10:4Þ

Expressions for these three parameters are listed in Table 10.16 for two pulse shapes,both of which are symmetrical about the origin. The first one is a cosine wave withuniform amplitude between times�� andþ� , and zero outside this range; the other isa Gaussian-modulated cosine wave with the same maximum amplitude as theunweighted pulse. In both cases, the parameter � is the time at which the pulseamplitude decays to 1=e of its peak value. Thus, the pulse extends approximatelybetween �� and þ� in time. The two functions are plotted in Figure 10.2.

In both cases the z-p and p-p source levels are related via

SLp-p ¼ SLz-p þ 6:0: ð10:5Þ

Similarly, the ‘‘pulse energy’’ (strictly, the energy source factor, which for a pointsource is the product of time-integrated squared pressure and the square of the


Table 10.15. Summary of miscellaneous oceanographic sonar.

Application System Frequency/ Max. SL/ Reference

Hz (dB re mPa2 m2)

Global thermometry ATOCa source 75 195 Anon. (2003, p. 70)

Kuperman and Roux (2007)

Global thermometry HIFT source 57 206 Munk et al. (1994)

(single transducer)

Global thermometry HIFT source 57 221 Munk et al. (1994)

(array of five

transducers)

Oceanography RAFOSb 300–400 (sweep), 195 Hildebrand (2004, p. 9)

or 185–310 (CW)

a ATOC: Acoustic Thermometry of Ocean Cllimate (atoc, www).b RAFOS (for SOFAR, spelt backwards) is a system of ocean floats designed to collect oceanographic data. The floats rely fornavigation on a network of moored transmitters (rafos, www).

measurement distance, close to the source) is related to the p-p source level via

SLE ¼SLp-p þ 10 log10ð2�Þ � 9:0 unweighted

SLp-p þ 10 log10ð2�Þ � 11:0 Gaussian modulated.

�ð10:6Þ

Next consider a pressure pulse that is switched on at its maximum amplitude at timet ¼ 0, followed by an exponential decay with time constant � . Table 10.17 lists theproperties of two such asymmetrical pulses, the first a damped sine wave and thesecond a simple exponentially decaying pressure (see Figure 10.3). The relationshipsbetween SLE, SLz-p, and SLp-p corresponding to Table 10.17 are

SLp-p ¼SLz-p þ 6:0 damped sine

SLz-p exponential

�ð10:7Þ

and

SLE ¼SLp-p þ 10 log10 � � 12:0 damped sine

SLp-p þ 10 log10 � � 3:0 exponential.

�ð10:8Þ

RMS pressure. Source levels or received levels of transient fields are sometimesreported as RMS values. What this means is that the squared pressure has been


Table 10.16. Relationships between different source level definitions for two symmetrical wave

forms. The expressions for q0ðtÞ (acoustic pressure at distance s0) are valid for a point source in

free space.

Tx descriptor Unweighted cosine Gaussian weighted cosine

s0q0ðtÞffiffiffi2

pA cosð!tÞHð� � tÞHð� þ tÞ

ffiffiffi2

pA cosð!tÞ exp � t

2

� 2

� �

The duration 2� is assumed equal The duration (i.e., the width of the

to an odd number of half-cycles Gaussian) is chosen to ensure the

such that 2!� ¼ ð2nþ 1Þ�, where same pulse energy, in the high-

n is an integer frequency limit, as the unweighted

cosine

ðS0Þz-p 2A2 2A2

ðS0Þp-p 8A2 2A2 1� cos �1 exp � �21

!2� 2

� �� 2

Here �1 is the first non-zero solution

to the transcendental equation

2�1 þ !2� 2 tan �1 ¼ 0. The high-

frequency limit is �1 ! � and

ðS0Þp-p ! 8A2.

ðS0ÞE 2A2�

ffiffiffi�

2

rA2� 1þ exp �!2� 2

2

� ��


Figure 10.2. Unweighted (upper) and Gaussian-weighted (lower) cosine pulses from Table

10.16.

averaged over a time interval that is often left unspecified, making the reportedvalues ambiguous. Madsen (2005) discusses this problem and suggests two qualita-tively different ways of choosing an appropriate averaging time. The first and sim-plest method is based on an amplitude threshold. With this approach, pulse durationis defined as the time during which a specified threshold (e.g., �3 dB relative to themaximum amplitude) is exceeded. The MSP averaged over this time interval isdenoted XðÞ in Table 10.18, expressed as a fraction of the maximum MSPvalue,2 where is the chosen threshold, such that ¼ 1

2corresponds to the �3 dB

point. The RMS pressure calculated using a zero dB threshold (i.e., AffiffiffiffiffiffiffiffiffiffiffiffiffiffiXð1:0Þ

p), is

also known as the peak-equivalent RMS (peRMS) pressure. The peRMS pressure isthe RMS pressure of a hypothetical sine wave whose peak sound pressure is equal tothe peak sound pressure of the true pulse. In Table 10.18 and for the remainder ofthe present section it is assumed that the frequency is high enough for the sine-squared terms to be replaced by their mean value of 1

2. The first of the three cases

considered is the trivial one involving a top-hat or unweighted pulse, followed by theGaussian-weighted pulse from Table 10.16 and the exponentially modulated sinewave from Table 10.17.


Table 10.17. Relationships between different source level definitions for two asymmetrical

wave forms. The expressions for q0ðtÞ (acoustic pressure at distance s0) are valid for a point

source in free space.

Tx descriptor Exponentially damped sine Exponential

s0q0ðtÞ0 t < 0ffiffiffi2

pA sinð!tÞ exp � t

�

t 0

(0 t < 0

A exp � t�

t 0

(

ðS0Þz-p 2A2 sin2 �2 exp �2�2

!�

� �A2

where

�2 ¼ arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2� 2

1þ !2� 2

r

ðS0Þp-p ðS0Þz-p 1þ exp � �

!�

h i2A2

ðS0ÞEA2�

2

!2� 2

1þ !2� 2

A2�

2

2 For oscillatory pulses, this statement involves an implied assumption that the frequency is

high enough for the ‘‘mean square’’ operation to be over several cycles. In the case of

exponentially decaying acoustic pressure the MSP is equal to the square of instantaneous

pressure.


Figure 10.3. Exponentially damped sine (upper) and decaying exponential (lower) pulses from

Table 10.17.

Applying the concepts leading to Table 10.18, the 0 dB and �10 dB values arerelated, for the exponentially damped sinusoidal pulse, by

SL�10 dB ¼ SL0dB � 4:1 dB; ð10:9Þ

where the subscript indicates the amplitude threshold, and

SL0dB � SLpeRMS ¼ SLp-p � 9:0 dB: ð10:10Þ

In this case, the difference between SL�10 dB and SLp-p is therefore 13.1 dB.The second method described by Madsen (2005) for choosing the averaging time

is to define the duration as the time interval during which a specified proportion (say90%) of the total pulse energy arrives. Table 10.19 lists the MSP calculated in thisway, denoted YðRÞ, as a function of the chosen energy fraction R, for Gaussian-weighted and exponentially weighted pulses as well as for the trivial unweighted case.In this second table a fourth pulse shape is added (bottom row), made by combining arising exponential for t < 0 with a decaying one for t > 0.

For the exponentially damped sine wave, the MSP based on R ¼ 0 is 3.0 dBrelative to the peRMS value, which in turn is 9.0 dB less than the p-p value. Thus, forthis case

SL0% ¼ SLp-p � 12:0 dB: ð10:11Þ

Further, the MSP based on 0% and 97% energy fractions differ by 3.3 dB, whichmeans that (for the same exponential damping)

SL97% ¼ SLp-p � 15:3 dB: ð10:12Þ

It is clear from either Table 10.18 or Table 10.19 that, for any of the pulses except thetrivial unweighted one, if there is any leeway in choosing the averaging time the resultis an undesirable ambiguity in the RMS pressure of up to 6.3 dB for the examplesconsidered. Larger differences are possible if longer averaging times are used, up to a


Table 10.18. Relative MSP, denoted XðÞ, defined as the mean square pressure relative to its

maximum value (peRMS). The average is over the time window during which the local average

(mean square averaged over a small number of cycles) exceeds the threshold . The parameter x

used in the table is given by xðÞ ¼ �loge .

Pulse description XðÞ Xð1:0Þ Xð0:5Þ Xð0:1Þ[0 dB] [�3.0 dB] [�10 dB]

peRMS

Unweighted sine or cosine 1 1 1 1

(Table 10.16)

Gaussian-weighted sine 1

2

ffiffiffiffiffiffiffiffiffiffi�

xðÞ

rerf

ffiffiffiffiffiffiffiffiffiffixðÞ

p 1.000 0.810 0.565

(Table 10.16) (0 dB) (�0.9 dB) (�2.5 dB)

Exponentially damped sine 1�

xðÞ1.000 0.721 0.391

(Table 10.17) (0 dB) (�1.4 dB) (�4.1 dB)

maximum difference when the averaging time reaches the pulse repetition interval.For this reason, any quantitative statement of an RMS pressure for a transient soundis incomplete unless it is accompanied by a description of the averaging time used.

The preceding discussion begs the question of which MSP definition is used, orshould be used, in practice. The answer depends on circumstances. The choice XðÞhas the benefit of simplicity. The disadvantage is the ambiguity caused by possiblenon-monotonic behavior to either side of the chosen peak. Further, an amplitudethreshold takes no account of the energy present in the low-amplitude region, whichmight sometimes be a significant proportion of the total. Use of the energy fractionsolves both of these problems, but requires a measurement of total pulse energy,resulting in a need for more complicated processing.

Given a value of R, the definition in terms of YðRÞ is unambiguous, but suffersfrom the problem that, for lopsided pulses like the exponential one, the peak valuemight be excluded from the averaging time. This concern can be mitigated bychoosing the value of R satisfying YðRÞ ¼ R, forcing a short averaging time forthe (single-sided) exponential pulse, while requiring a longer time for the symmetricalpulses. The result of this choice, denoted YðYÞ, is shown in the last column of Table10.19, and illustrated by Figure 10.4.

10.1.1.2.2 Seismic survey sources

The term ‘‘seismic survey’’ sources refers to a general category of low-frequencysources designed to transmit sound deep into the seabed, with the objective ofdetermining its structure or layering. Compared with a search sonar designed to


Table 10.19. Relative MSP, denoted YðRÞ, defined as the mean square pressure relative to its maximum

value (peRMS). The average is over the time window during which the pulse energy accumulates to a

proportion R of the total. Thus, the notation YðRÞmeans that a proportion ð1� RÞ=2 of the total energy

arrives before the start (and the same proportion arrives after the end) of the averaging window.

Description YðRÞ Yð0:00Þ Yð0:50Þ Yð0:90Þ Yð0:97Þ YðYÞ

Unweighted 1 1 1 1 1 1

sine or cosine

Gaussian-ffiffiffi�

p

2

R

erf�1 R1.000 0.929 0.686 0.560 erf

ffiffiffi�

p

2� 0:790

weighted sine (0 dB) (�0.3 dB) (�1.6 dB) (�2.5 dB)

(�1.0 dB)

Exponentially R

loge½ð1þ RÞ=ð1� RÞ 0.500 0.455 0.306 0.232 e � 1

e þ 1� 0:462

damped sine (�3.0 dB) (�3.4 dB) (�5.1 dB) (�6.3 dB)

(�3.4 dB)

Double-sided R

jlogeð1� RÞj1.000 0.721 0.390 0.277 e � 1

e� 0:632

exponentially (0 dB) (�1.4 dB) (�4.1 dB) (�5.6 dB)

weighted sine (�2.0 dB)

transmit sound horizontally, a relatively small proportion of the acoustic energy fromseismic survey sources remains in the water column. For an overview of seismicsources see usgs (www).

The choice of frequency is a compromise between the desire for high resolution(i.e., high bandwidth and hence some energy at high frequency) and the need for lowattenuation (i.e., low frequency). The solution adopted is typically a BB signal withmost energy at order 100Hz (Caldwell and Dragoset, 2000). Some early survey workwas done with explosive sources (Hersey, 1977), but this is now rare and the high far-field pressures required are achieved instead by means of large air gun arrays, whichwork by injecting compressed air into the water. The air bubbles so formed expandrapidly, creating a high-amplitude pressure pulse in the required frequency range.The properties of air guns are summarized in Table 10.20. These are dipole sourcelevels in the sense that they are based on the combined effect of the air gun itself andits sea surface image, resulting in a dipole radiation pattern at low frequency.

Source levels are often reported in the seismic literature in the form of a sourceproduct (pressure multiplied by distance) in units of bar meters (barm). The bar is aunit of pressure equal to 100 kPa. Thus, a bar meter can be converted to the presentunits by means of the relation

1 bar m ¼ 105 Pa m ¼ 1011 mPa m; ð10:13Þ


Figure 10.4. MSP Y vs. energy fraction R. The various functions YðRÞ from Table 10.19 are

plotted. The intersections with the diagonal defineYðYÞ. The parameterYðYÞ is less sensitive tothe precise pulse shape than is YðRÞ for fixed R.

or, equivalently,

1 bar2 m2 ¼ 1022 mPa2 m2: ð10:14ÞTaking logarithms of this last equation it is seen that a z-p source product of 1 barmeter is equivalent to a z-p source level of 220 dB re mPa2 m2.

It is customary to deploy air guns in a synchronized array, thus achieving highersource levels than would be possible with a single device. As for the SURTASS andHIFT sources mentioned previously, air gun arrays span several wavelengths, and thesource level of such an array is not related in a simple way to the sound pressure levelat 1m distance.

The sound-producing mechanism of an air gun is associated with the suddenrelease of compressed air into the water. After the initial release of air, an expanding


Table 10.20. Summary of dipole source levels from air guns and air gun arrays.a Sorted by source level.

Type Description Zero–peak dipole source Reference

level SLdpz-p=

( dB re mPa2 m2)

Small gun 0.7 L (40 in3) 224 Dragoset (2000)

Small gun 0.2–1.0 L (12–60 in3) 223–229 sercel (www)

Medium gun 2.5–4.1 L (150–250 in3) 229–230 sercel (www)

Sub-array GECO 594 sub-array 235 Richardson et al. (1995)

Two-gun array 2-17L (120–1040 in3) 233–238 sercel (www)

Small array Various 242–246 Richardson et al. (1995)

Medium array Various 248–252 Richardson et al. (1995)

Large array 56L (3397 in3) array 254 Caldwell and Dragoset (2000)

Large array 66L array 255 Richardson et al. (1995)

Large array Western geophysical 255 Dragoset (2000)

37L (2250 in3)

Large array ARCO 4000 255 Richardson et al. (1995)

Large array LDEO arrayb 137L 257 ldeo (www)

(8385 in3)

Large array GSC 7900 259 Richardson et al. (1995)

a The inch (1 in) is a non-metric unit of distance, defined as 25.4mm. The cubic inch (1 in3, approximately equal to16 cm3 or 0.016L) is often used to quantify the volume of air used by an air gun. One liter (1 L)¼ 1 dm3. Similarly,pressure is sometimes reported in units of pounds-force per square inch (psi). One psi is approximately equal to6.9 kPa (see Appendix B).b Lamont Doherty Earth Observatory.

bubble forms which reaches a maximum volume and then begins to contract. Oncontraction, the bubble overshoots its equilibrium radius and continues to contractuntil a minimum volume is reached, ready to start expanding again and repeat thecycle. The pulsating bubble emits a series of so-called ‘‘bubble pulses’’ (or ‘‘bubbleoscillations’’) of successively smaller amplitudes. Much effort is put by the developersof air guns into reducing the effect of these bubble pulses, as they interfere with themain signal and degrade performance (Dragoset, 2000).

One method used to minimize the bubble pulse is to arrest the implosion of theinitial bubble by firing a second air gun a short time after the first. When used in thisway the first gun is called a ‘‘generator’’ (G) gun and the second one an ‘‘injector’’ (I)gun. The combination is called a ‘‘GI gun’’. A GI gun can operate in a so-called‘‘harmonic mode’’, meaning that equal volumes of air are fired by the separate G andI guns, or ‘‘true GI mode’’, for which the volume ratio is optimized to minimize thebubble pulse (sercel, www). Source levels for these combinations are listed in Table10.21, where they are compared with the source level of an air gun with the same totalvolume from Table 10.20.

The properties of seismic survey sources other than air guns are summarized inTable 10.22. These include water guns, sleeve exploders, sparkers, and boomers.(Explosives are not described well by a zero-to-peak pressure and are discussedseparately in Section 10.1.1.2.3.) A water gun operates in a similar manner to anair gun except that the expanding air, instead of being released directly into the water,is used to accelerate a piston into a cylinder filled with seawater. The water is firstpushed out into the sea and then returns suddenly, creating a negative pressure peakas the piston returns to its original position. A sleeve exploder (or gas gun) ignites anexplosive mixture of gases inside a rubber tube. The explosion causes the rubber tube


Table 10.21. Summary of zero-to-peak dipole source levels for generator–injector (GI) air guns.

Total Volume Volume Zero–peak Reference Notes

volume generator injector dipole source

ðGþ IÞ gun ðGÞ gun ðIÞ level SLdpz-p=

( dB re mPa2 m2)

1.47L 0.74L 0.74L 223.5 sercel (www) GI gun in harmonic mode



2.46L 0.74L 1.72L 223.5 sercel (www) GI gun in true GI mode

2.46L 2.46L N/A 228.5 sercel (www) Not a GI gun, but a single air

gun of the same total volume,

included for comparison from

Table 10.20

to expand, thus creating a pressure pulse in the water. In the case of a water gun orsleeve exploder, no air is released into the water so these systems do not suffer fromthe pulsating bubble. Finally, sparkers and boomers work by creating an electricaldischarge across two electrodes placed in water. The resulting spark creates apulsating bubble. Source levels for these systems are indicative only. The actualvalue depends on the conductivity of the seawater, and hence on its salinity andtemperature.

10.1.1.2.3 Explosives

Pulse amplitude, duration, and energy. The acoustic effects of explosives aresummarized by Weston (1960, 1962). The main feature is a short shock wavecomprising a sharp—almost instantaneous—rise in pressure followed by anexponential decay with a time constant � of a few hundred microseconds. Theexplosion leaves behind a large pulsating bubble whose successive expansions andcontractions give rise to a series of weaker, more symmetrical bubble pulses, in asimilar manner to an air gun (Weston, 1960; Cole, 1965). Only the initial shock isconsidered here.


Table 10.22. Summary of zero-to-peak source levels of seismic survey sources other

than air guns, sorted by type.

Type Description SLz-p= Reference

( dB re mPa2 m2)

Water gun 0.9L 217 Richardson et al. (1995)

Water gun 1.3L 239 Finneran et al. (2002)

Water gun array 24L 245 Richardson et al. (1995)

Sleeve exploder One sleeve 217 Richardson et al. (1995)

Boomer 500 J electric 212 Richardson et al. (1995)

discharge

Boomer (Huntec) 340 J electric 213 CCC (2000)

discharge

Minisparker 1.5 kJ electric 217 CCC (2000)

(SQUID 2000) discharge

Sparker array 100 J electric 206 geospark (2003)

(Geo-spark 200) discharge

Sparker array 1 kJ electric 225 geospark (2003)

(Geo-spark 200) discharge

If the peak pressure of the shock wave is denoted P, the initial shock can becharacterized by the expression

qðtÞ ¼0 t < 0

P exp � t�

t > 0.

(ð10:15Þ

Arons (1954) provides the following numerical values for P and � , as a function ofdistance s from the explosion:

P ¼ ð52:4 MPaÞ ss

WW 1=3

� ��1:13

ð10:16Þ

and

� ¼ ð92:5 msÞWW 1=3 ss

WW 1=3

� �0:22

; ð10:17Þ

whereW is the mass of explosive.3 These equations apply to a spherical TNT charge4

of density �exp ¼ 1520 kg/m3. The significance of the density is that it determines theradius of a sphere, given its mass. The parameterW 1=3 is proportional to the chargeradius, with a constant of proportionality equal to ð4��exp=3Þ1=3.

The repeated appearance of the ratio s=W 1=3 results from application of thesimilarity theory of Kirkwood and Bethe, as described by Cole (1965). Equations(10.16) and (10.17) are valid for values of this ratio, referred to henceforth as thescaled charge distance, in the range 0.5mkg�1=3 to 800mkg�1=3.

The amplitude P does not obey a simple spherical spreading law (i.e., theexponent of Equation 10.16 differs from minus 1). The main reason for this is thatas the pulse spreads out from the explosive source, its leading edge (the shock front)expands more rapidly than its exponential tail, resulting in a pulse duration thatincreases with increasing distance traveled. Conservation of energy demands that theamplitude of the pulse must decrease more quickly than spherical spreading tocompensate for the increased duration. Let the ‘‘pulse energy’’ be characterized bythe squared pressure integrated over all time and denote this quantity E:

E �ð10

qðtÞ2 dt: ð10:18Þ

If Equation (10.18) is applied, using Equation (10.16) for the amplitude and Equation(10.17) for the duration, the result (for TNT) is

E ¼ ð0:127 MPa2 sÞWW 1=3 ss

WW 1=3

� ��2:04

: ð10:19Þ

The exponent is close to minus 2, the value expected for spherical spreading, but thereis nevertheless a small departure. This departure is associated with the dissipation ofhigh-frequency components at the shock front.


3 WW is the numerical value of W , expressed in units of kg.4 The same value of the time constant applies also for pentolite.

Arons’s equations are based partly on results published previously by Cole(1965). Cole’s original results are listed separately in Table 10.23, as these includeexpressions for the pulse energy as well as data for explosives other than TNT. Cole(1965, p. 242) also gives expressions for the impulse (the time-integrated magnitude ofthe acoustic pressure).

The formulas quoted in Table 10.23 are applicable over at least the range ofCole’s data, which in the case of TNT covers one order of magnitude of the scaledcharge distance, between 0.5mkg�1=3 and 11mkg�1=3. The measurements of Aronsfor pentolite suggest that the region of validity could extend significantly farther thanthis, to scaled distances around 250mkg�1=3. For TNT, notice the consistencybetween the coefficient and exponent of Cole’s measurements (parameter E in Table10.23) and those derived from the amplitude and time constant of Arons’s data(Equation 10.19).

Energy source level. Seismic sources other than explosives were described (inSection 10.1.1.2.2) in terms of their zero-to-peak source level SLz-p. As explainedin Chapter 8, this parameter—and its close cousin the peak-to-peak source levelSLp-p—apply to a pulse whose shape does not vary with distance from the source.For explosive pulses, whose shape does vary with distance, a more robust character-ization can be made by means of the energy source level

SLE ¼ 10 log10½EðsÞs2 dB re mPa2 m2 s: ð10:20Þ

Because of the non-linear effects mentioned previously, Equation (10.20) provides anapparent source level that decreases slightly with increasing distance from the explo-sion (see Equation 10.19). The variation with scaled charge distance of the apparentsource level defined in this way is shown in Table 10.24 for distances up to 5000charge radii. (The adjective ‘‘specific’’ is used to indicate that these values are per unit


Table 10.23. Summary of peak pressure and pulse energy (Equation 10.18) for three types of

explosive (from Cole, 1965).

Explosive (density) P E=WW 1=3

TNT (1520 kgm�3) ð52:4 MPaÞ ss

WW 1=3

� ��1:13

ð0:126 MPa2 sÞ ss

WW 1=3

� ��2:05

Loose tetryl (930 kgm�3) ð51:0 MPaÞ ss

WW 1=3

� ��1:15

ð0:150 MPa2 sÞ ss

WW 1=3

� ��2:10

Pentolite (1600 kgm�3) ð54:6 MPaÞ ss

WW 1=3

� ��1:13

ð0:161 MPa2 sÞ ss

WW 1=3

� ��2:12

mass of explosive.) The pulse energy (for pentolite) is calculated using (Arons, 1954)5

EðsÞ ¼ ð0:152 MPa2 sÞWW 1=3 ss

WW 1=3

� ��2:06

: ð10:21Þ

Thus, the energy lost from the pressure wave as the shock front expands from 10to 5000 charge radii is 232.2� 230.8¼ 1.4 dB (see column 5). The value at distances ¼ 5000aexp provides an upper limit for the specific acoustic source level of230.8 dB re mPa2 m2 s kg�1 (column 5). Using the value at 5000 charge radii as anestimate of the source level (substituting Equation 10.21 in Equation 10.20) gives

SLE � 231þ 10 log10 WW dB re mPa2 m2 s: ð10:22Þ

A useful rule of thumb is that this value, if converted to energy in joules (see column4), is approximately equivalent to one megajoule of acoustic energy per kilogram ofexplosive.

Depth dependence. The equations presented so far are for shock waves, whoseenergy and spectrum are independent of depth. Much of the low-frequency energyfrom an explosion is carried by the first bubble pulse, and a complete characteriza-tion of the source spectrum, omitted here, needs to include this contribution, which isa function of the explosive depth. Weston (1960) gives a thorough and insightfulaccount of these effects. For more recent measurements, see Chapman (1988).


Table 10.24. Variation with range of specific pulse energy and apparent specific SLE for

pentolite using Equation (10.21). The ratio s=aexp is the number of charge radii, where aexpis the charge radius aexp ¼ ð3W=4��expÞ1=3.

Distance Scaled Scaled Apparent Apparent Notes

in charge charge specific specific specific

radii distance pulse explosion energy

energy energy source level

s=aexp sW�1=3 Es2=W4�

�cEs2=W SLE

(mkg�1=3) (MPa2 m2 (MJ kg�1) ( dB re mPa2

s kg�1) m2 s kg�1)

10 0.5 0.166 1.36 232.2 Lower-limit total

explosion energy

200 10.6 0.143 1.17 231.5 Representative value

5000 265.2 0.122 1.00 230.8 Upper-limit acoustic SLE

5 Valid for pentolite at a distance between 10 and 5000 charge radii.

10.1.2 Of marine mammals

Marine mammals are responsible for a remarkable repertoire of sounds in the sea. Aswith man-made sounds, in the following text a distinction is made between contin-uous and impulsive sources. Reviews of the sounds made by marine mammals can befound in Richardson et al. (1995) and Wartzok and Ketten (1999).

10.1.2.1 Continuous vocalizations

Marine mammals make a bewildering variety of continuous sounds (Au, 1993;Richardson et al., 1995; Wartzok and Ketten, 1999; Tyack, 1999; Anon., 2003). Onlya very brief summary of these is provided below. The diversity is illustrated here bythe assortment of often onomatopoeic descriptions of the vocalizations that appear inthe scientific literature, many suggestive of more familiar sounds made by terrestrialanimals, such as barks, bleats, roars, whinnies, and yelps.

The sounds made by baleen whales are summarized by Richardson et al. (1995) intheir Table 7.1. The highest source level quoted for a non-impulsive sound, equal to189 dB re mPa2 m2, is that for bowhead whale song in the frequency range 20Hz to500Hz. Collectively, baleen whales make a noticeable contribution to ambient noiseat low frequency (Watkins et al., 1987; Anon., 2003, pp. 28, 44). Richardson et al.(1995, Table 7.4) also lists source levels for pinnipeds6 and sirenians,7 for which thehighest value quoted (for the Weddell seal call) is 193 dB re mPa2 m2. The toothedwhales, widely known for their echolocation clicks, also produce a variety of whistles,chirps. and screams with source levels up to about 180 dB re mPa2 m2 (for whistles ofthe short-finned pilot whale—Richardson et al., 1995, Table 7.2).

10.1.2.2 Impulsive sources

Impulsive sounds made by marine mammals (including non-vocal sounds such as tailslaps) are described variously in the literature as clicks, gunshots, knocks, slaps, taps,and thumps. For example, Parks and Tyack (2005) describe loud ‘‘gunshot’’ soundswith (RMS) source level up to 192 dB re mPa2 m2, based on a 90% energy averagingtime (i.e., Yð0:9Þ in the notation of Table 10.19).

Summaries of the characteristics of echolocation clicks are given by Au (1993),Richardson et al. (1995), Rasmussen et al. (2002), and Zimmer et al. (2005). Of mostobvious relevance here are the echolocation clicks used by mammals for navigation,hunting, or inspection. Table 10.25 summarizes the most important properties ofsuch pulses for seven different species. (Short pulses considered suitable for echoloca-tion are included in the table whether or not it has been established they are used bythe animal for this purpose.) Though of short duration, the impulsive sounds can bevery loud (the highest reported peak-to-peak source level for the sperm whalePhyseter macrocephalus is higher than that of most of the man-made sonars listed


6 Seals, sea lions, and walruses.7 For example, manatees.

Table 10.25. Summary of echolocation pulse parameters for selected animals.

Speciesa Peak 10 dB 3dB �t= SLp-p SLE= Reference

freq./ band band ms (dB re (dB re

kHz width/ width/ mPa2 m2) mPa2 m2 sÞkHz kHz

Physeter 15 10 5 100 243 193 Zimmer et al. (2005)

macrocephalus Møhl et al. (2003)

(sperm whale)

Ziphius cavirostris 40 23 12 160 214 164 Zimmer et al. (2005)

(Cuvier’s beaked

whale)

Monodon monoceros 40 35 20 50 227 174 Møhl et al. (1990)

(narwhal)

Grampus griseus 49 66 27 40 220 164 Madsen et al. (2004b)

(Risso’s dolphin)

Pseudorca crassidens 40 63 35 30 220 163 Madsen et al. (2004b)

(false killer whale)

Tursiops truncatus 120 100 30 25 225 167 Zimmer et al. (2005)

(Atlantic bottlenose Au (1980)

dolphin)

Phocoena phocoena 140 14–46 6–26 100 205 151 Villadsgaard et al.

(2007)

a Thumbnail images (except narwhal) Garth Mix, GMIX Designs, reprinted with permission. Narwhal imagereprinted from Wikipedia.

#

in Tables 10.1 to 10.9), leading to speculation that the sound might sometimes be usednot just for echolocation, but also for stunning prey (Møhl et al., 2000).8

As with man-made sonar, the source level of a marine mammal is usually meas-ured at a distance greater than 1m from the source, and scaled back to the standardreference distance by correcting for propagation loss. Some pulses are very short (lessthan a millisecond) so there is scope for some distortion of the pulse due to dispersion,leading to variability in measured p-p source level. Variability can be reduced bycharacterizing such sounds by their energy source level SLE, which is independent ofthe shape of the pulse. Nevertheless, it is common for quantitative measurements ofthe strength of mammal echolocation clicks to be reported in the form of peak-to-peak levels, as summarized in Table 10.26 (high-frequency clicks) and Table 10.27(short, low-frequency pulses with otherwise similar characteristics to high-frequencyecholocation clicks). If SLp-p and �t are both known, it is possible to estimate SLE

either using Equation (10.6) or Equation (10.8), depending on the pulse shape.Echolocation clicks are characterized by their high intensity and short duration

(typically fewer than 20 cycles). Large cetaceans are also known to produce short,high-intensity pulses, albeit at a much lower frequency (below 100Hz). Regardless oftheir precise function,9 which might or might not include echolocation (Mellinger andClark, 2003), the resemblance in shape of these low-frequency pulses to their high-frequency counterparts makes it useful to characterize them in a similar way (seeTable 10.27).

The echolocation pulses of the killer whale (Orcinus orca) and harbor porpoise(Phocoena phocoena) are compared in Figure 10.5. Despite its higher center frequencyof 130 kHz, the Phocoena pulse has a significantly longer duration and smallerbandwidth, resulting in a recognizable oscillatory behavior, whereas Orcinus usesa characteristic broadband click. The Orcinus pulse is also louder (see Table 10.26).

For some species, there exists evidence that the click intensity is adjustedaccording to the distance s from the target. The observed relationships are (in unitsof dB re mPa2 m2)

SLp-p ¼ 151:59þ 19:37 log10 ss (finless porpoise, Li et al., 2006) ð10:23Þ

SLp-p ¼ 181:4þ 20 log10 ss (killer whale, Au et al., 2004, s < 25 mÞ ð10:24Þ

SLp-p ¼ 177:8þ 20 log10 ss (dusky dolphin, Au and Wursig, 2004) ð10:25Þ

and

SLp-p ¼ 192þ 16 log10 ss (white-beaked dolphin, Rasmussen et al., 2002)

ð10:26Þ

In each case, the decreasing distance approximately cancels the effect of sphericalspreading at short distances, so that, while the SNR at the sonar receiver increases as


8 Benoit-Bird et al. (2006) investigate this hypothesis, finding no evidence of stunning or

disorientation of fish by simulated odontocete echolocation clicks.9 Their probable function for blue and fin whales is described by Sirovic et al. (2007) as

‘‘communication during mating and feeding’’.

the target is approached, the sound levels at the target remain relatively stable. Such astrategy is likely to make it more difficult for the animal’s prey to discern the threatassociated with an approaching predator.

10.2 RECEIVER CHARACTERISTICS

10.2.1 Of man-made sonar

The maximum effectiveness of a simple sonar receiver is determined by its sensitivityand its self-noise voltage. In the presence of an external source of noise, thismaximum will not be reached unless the external noise can be filtered out.

10.2.1.1 Hydrophone sensitivity and non-acoustic noise

10.2.1.1.1 Sensitivity

Hydrophone sensitivity can be defined as the open-circuit voltage at the output of thehydrophone per unit of pressure in the water (Stansfield, 1991). Sensitivity is afunction of frequency, so it is convenient to define it as a ratio of the spectral densityof the output voltage (mean square voltage per unit frequency band), denoted Uf ,to that of the pressure, Qf . For the present purpose, the following definition ofsensitivity M is adopted for an omni-directional hydrophone

Mð f Þ �Uf ð f ÞQf ð f Þ

: ð10:27Þ

The usual reference values of voltage and pressure are one volt and one micropascal,so hydrophone sensitivity expressed in decibels (denoted HS) is

HSð f Þ � 10 log10Mð f Þ dB re V2=mPa2: ð10:28ÞMean square signal voltage at the hydrophone output is given by

US ¼MQS: ð10:29ÞMean square noise voltage, including electrical noise Nelec, is

UN ¼MQN þNelec: ð10:30ÞThe signal-to-noise ratio at the hydrophone output (restricting attention here to asingle frequency) is

SNRout �USf

UNf

¼QSf

QNf

1þðNelecÞfMð f ÞQN

f

" #�1

: ð10:31Þ

The mean square noise pressure QN includes all pressure fluctuations at thehydrophone, whether or not they are acoustic in origin. Acoustic sources aredescribed in Chapter 8. The main non-acoustic pressure fluctuations, consideredbriefly below, are flow noise, caused by a turbulent boundary layer, and thermalnoise, caused by thermal agitation of individual water molecules.

10.2 Receiver characteristics 545]Sec. 10.2

Table 10.26. Summary of maximum peak-to-peak source levels of high-frequency marine mammal clicks

with peak frequency exceeding 10 kHz. Sort order is by scientific name.

Species Imagesa Peakb Approx. SLp-p= Reference

frequency/ pulse (dB re

kHz duration mPa2 m2)

�t=ms

Cephalorhynchus 112–130 140 151 Au (1993, Table 7.2)

hectori

(Hector’s dolphin)

Delphinapterus 40–60 — 202 Au et al. (1985)

leucas (San Diego Bay)

(beluga whale) 100–115 50–80 225 Au (1993, Table 7.2)

Feresa attenuata 45 & 117 25 223 Madsen et al. (2004a)

(pygmy killer

whale)

Globicephala 30–60 — 180 Au (1993, Table 7.2)

melaena

(pilot whale)

Grampus griseus 49 40 220 Table 10.25

(Risso’s dolphin)

Lagenorhynchus 120 10–30 219 Rasmussen et al.

albirostris (white- (2002, table I)

beaked dolphin)

Lagenorhynchus 59 34–52 170 Rasmussen et al.

obliquidens (Pacific (2002, table I)

white-sided

dolphin)

Lagenorhynchus 80–110 — 210 Au and Wursig (2004)

obscurus

(dusky dolphin)

Lipotes vexillifer 100–120 — 156 Au (1993, Table 7.2)

(Chinese river

dolphin)

Monodon 40 29–45 227 Table 10.25

monoceros

(narwhal)c

Neophocoena 125 185 Li et al. (2006)

phocoenoides

(finless porpoise)c

a Unless otherwise stated, thumbnail images are Garth Mix, GMIX Designs. Reprinted with permission.b Further information on spectral content (e.g., bandwidth) is available from many of the source references.c Narwhal and finless porpoise images reprinted from Wikipedia.

#

Species Imagesa Peakb Approx. SLp-p= Reference

frequency/ pulse (dB re

kHz duration mPa2 m2)

�t=ms

Orcinus orca 45–80 30 218 Au et al. (2004)

(killer whale)

Phocoena 140 100 205 Table 10.25

phocoena

(harbor porpoise)

Phocoenoides 120–160 180–400 170 Au (1993, Table 7.2)

dalli

(Dall’s porpoise)

Physeter 15 100 243 Table 10.25

macrocephalus

(sperm whale)

Pseudorca 40 30 220 Table 10.25

crassidens 100–130 100–120 228 Au (1993,Table 7.2)

(false killer

whale)

Stenella attenuata 69 43 220 Rasmussen et al.

(pantropical (2002, table I)

spotted dolphin)

Stenella 70 31 222 Rasmussen et al.

longirostris (long- (2002, table I)

snouted spinner

dolphin)

Tursiops 52 50–250 170 Rasmussen et al.

truncatus (2002, table I)

(Atlantic bottle- 120 25 225 Table 1025

nose dolphin

Tursiops gilli 110–130 50–80 228 Au (1993, Table 7.2)

(Pacific

bottlenose

dolphin)

Ziphius 40 160 214 Table 10.25

cavirostris

(Cuvier’s beaked

whale)

a Unless otherwise stated, thumbnail images are Garth Mix, GMIX Designs. Reprinted with permission.b Further information on spectral content (e.g., bandwidth) is available from many of the source references.

#


Table 10.27. Summary of peRMS and p-p source levels of low-frequency marine mammal pulses with peak

frequency less than 100Hz. The sort order is by scientific name.

Species Imagesa Peak Approx. SLpeRMS= SLp-p= Reference

frequency/ pulse (dB re (dB re

Hz duration mPa2 m2) mPa2 m2)

�t=ms

Balaenoptera 25–29 1000 189� 3 198 Sirovic et al. (2007)

musculus

(blue whale)

Balaenoptera 15–28 1000 189� 4 198 Sirovic et al. (2007)

physalis

(finback whale)

Megaptera 25–80 300–400 178 187 Thompson et al.

novaeangliae (1986)

(humpback whale)

a Thumbnail images are Garth Mix, GMIX Designs, reprinted with permission.#

Figure 10.5. Comparison of echolocation pulses made by the harbor porpoise (upper) and killer

whale (lower). The killer whale pulse has a higher peak sound pressure, lower center frequency,

larger bandwidth, and shorter duration (reprinted from Au, 1993 and Au et al., 2004; porpoise

pulse originally from Kamminga and Wiersma, 1981). Porpoise and orca thumbnail images

Garth Mix, GMIX Designs, reprinted with permission.

#

Animal

Harbor porpoise (Phocoena phocoena)

Duration: 100 ms

Center frequency: 140 kHz

Source factor:

ðS0Þz-p ¼ 80 kPa2 m2

Killer whale (Orcinus orca)

Duration: 40 ms

Center frequency: 50 kHz

Source factor:

ðS0Þz-p ¼ 1600 kPa2 m2

Waveform

10.2.1.1.2 Molecular thermal noise

Molecular noise is caused by the thermal agitation of water molecules at the sensitiveface of a hydrophone. The resulting bombardment contributes to RMS pressure andis thus converted to electricity in the same way as an acoustic pressure would be.Mellen (1952) shows that the equivalent acoustic spectral density10 associated withthis signal is given by

ðQmolecÞf ¼ Atherm f2; ð10:32Þ

where the constant of proportionality is

Atherm ¼ 4�KT�wcw

¼ 0:0338 nPa2=Hz3; ð10:33Þ

where T is absolute temperature; and K is Boltzmann’s constant, equal to1.381� 10�23 J/K. The equivalent noise spectrum level is

10 log10 Qf ¼ �14:7þ 10 log10 F2 dB re mPa2=Hz: ð10:34Þ

It is clear that thermal noise increases with increasing frequency. At frequenciesbelow about 100 kHz it is usually negligible.

10.2.1.1.3 Flow noise

As with molecular noise, flow noise is caused by non-acoustic pressure fluctuations inthe immediate vicinity of the receiving hydrophone (Dowling, 1998). In the case offlow noise, these fluctuations only arise if sonar and water are in relative motion (e.g.,if the hydrophone is towed, or attached to the hull of a moving vessel). In thissituation, a hydrodynamic boundary layer forms between the moving sonar andthe stationary water at some distance away. The pressure fluctuations are causedby turbulence in this boundary layer. A characteristic feature of flow noise is that it isinversely proportional to frequency cubed (Urick, 1983).

10.2.1.2 Array directivity

Modern sonars use arrays of receiving hydrophones in order to filter out sound fromunwanted directions. Some relevant properties of such arrays are described inChapter 6.

10.2.2 Of marine mammals, amphibians, human divers, and fish

Hearing is not the only sensor available to animals living in the sea, but it is the mostimportant one for some species (Wartzok and Ketten, 1999). Audiograms of selectedaquatic animals are shown in Section 10.2.2.2, including the underwater hearingcapability of human divers.


10 The equivalent acoustic spectral density is the spectral density that would be required to

trigger the same RMS response in the hydrophone as caused by the thermal bombardment.

In principle, the thermal noise limit is the same for biological receivers as forman-made equipment (see Block, 1992). However, the sensitivity and self-noise ofbio-sonar cannot easily be measured separately. The combined effect of both deter-mines the hearing threshold (i.e., the minimum sound pressure level that can be heardby the animal, in the absence of external noise), and for a specified success rate, thevalue of which depends on the design of the experiment (Au, 1993, p. 33). Anaudiogram is a graph of hearing threshold as a function of acoustic frequency(Morfey, 2001). The measurement is a laborious one, requiring an animal speciallytrained to respond in a prescribed way to an auditory stimulus. Thus, data are limitedto a small number of species traditionally kept in captivity. In most cases, themeasurements are limited to one or two individuals per species.

10.2.2.1 The intensity of underwater sound: typical orders of magnitude

Before describing the audiograms themselves it is useful to get a feel for the orders ofmagnitude involved in typical values of underwater acoustic pressure and the asso-ciated intensity and particle velocity. Consider a plane wave whose RMS pressure(denoted pRMS) is 1.5mPa. The magnitude of the mean sound intensity of this planewave is

I ¼ p2RMS

Z¼ 1:5 pW m�2; ð10:35Þ

where Z is the characteristic impedance of seawater, equal to 1.5MPa s/m. Themagnitude of the corresponding RMS particle velocity is

uRMS ¼ pRMS

Z¼ 1 nm s�1: ð10:36Þ

These are truly microscopic values. For example, the velocity is of order 1 atomicdiameter per second, while the intensity corresponds to that of a 100-watt light bulbat a distance s of

s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

100 W

4�� 1:5 pW m�2

s� 2300 km:

This is approximately the distance, for example, between Amsterdam and Moscow,or between Los Angeles and Houston. The calculation is idealized in assuming a bulbof 100% efficiency and no attenuation of radiated light. Nevertheless, the point isthat light intensity at the receiver position is very weak and would be invisible to theunaided eye.11 This minuscule intensity of sound could nevertheless, depending on thefrequency, be audible to human divers (see Section 10.2.2.2.4) and to several speciesof dolphin and other small whales (Section 10.2.2.2.1).


11 The acoustic intensity of 1.5 pW/m2 can be compared with the optical intensity threshold of

a dark-adapted harp seal, which is about 1,000 times greater (Wartzok and Ketten, 1999).

10.2.2.2 Measured audiograms

10.2.2.2.1 Of cetaceans

Hearing threshold data are available for some odontecetes (toothed whales, dolphins,and porpoises), but at the time of writing none are known to the author for thelarger baleen whales. Based on available data, species that appear to be particularlysensitive are the harbor porpoise (Phocoena phocoena) (see Figure 10.6) and killerwhale (Orcinus orca) (Figure 10.7), with a mimumum threshold of between 30 dBand 35 dB re mPa2 measured at 15 kHz (orca) and 100 kHz (porpoise). For otherspecies, see Table 10.28 and reviews by Wartzok and Ketten (1999) and Nedwellet al. (2004).

10.2.2.2.2 Of pinnipeds (seals, sea lions, and walruses)

In general the hearing of pinnipeds in water is less sensitive than that of odontecetes.An example (for the harbor seal) is shown in Figure 10.8. For other examples seeRichardson et al. (1995), Wartzok and Ketten (1999), Kastelein et al. (2002b).

Pinnipeds are amphibious, and can hear sound in both air and water. The hearingof a given pinniped in water can be compared with the hearing of the same pinnipedin air. For example, the threshold of the northern fur seal (Callorhinus ursinus) in airis about 9 dB re (20 mPa)2, which converts to 35 dB re 1 mPa2. This means that for asound to be audible to Callorhinus in air, the mean square pressure (MSP) must


Figure 10.6. Underwater audiograms for harbor porpoise. The y-axis is the hearing threshold

in units of dB re mPa2. Sources as marked.

exceed 0.003mPa2 (103:5 mPa2), which is more than 100 times less than the MSPthreshold in water (0.4mPa2) for the same species.

Some prefer to compare thresholds of equivalent plane wave intensity12

(EPWI), defined as MSP divided by the characteristic impedance of the medium.When expressed in this way it is the air threshold that exceeds the one in water,this time by a factor of order 20. In Table 10.29 the MSP and EPWI thresholds inboth air and water are listed for five pinnipeds, including Callorhinus, and forhumans. This comparison illustrates the need to specify which of the two fieldvariables (MSP or EPWI) is being considered when expressing a quantity indecibels.

Complete audiograms measured in air and water are compared by Hemila et al.(2006) for four species of pinnipeds. Two of these four species, the northern elephantseal (Mirounga angurirostris) and common seal (Phoca vitulina) are phocids (earlessseals), and the other two, the California sea lion (Zalophus californianus) and north-


Figure 10.7. Underwater audiograms for killer whale. The y-axis is the hearing threshold in

units of dB re mPa2. Sources as marked.

12 The EPWI of a sound is the intensity of a plane-propagating wave of the same MSP. This

quantity can depart significantly from the true intensity. For example, the intensity of a

standing wave is identically zero, whereas the MSP (and hence EPWI also) is determined by the

amplitudes of individual traveling waves. A similar situation arises in an isotropic noise field.

Thus, it is misleading to abbreviate EPWI as ‘‘intensity’’ unless there is a single well-defined

field direction.

ern fur seal (Callorhinus ursinus) are otariids (eared seals). The lowest thresholds inwater are, for the most sensitive of both phocids (northern elephant seal) and otariids(northern fur seal), between 57 dB and 67 dB re mPa2 in the approximate frequencyrange 5 kHz to 30 kHz. The audiograms for nine pinniped species are reviewed byNedwell et al. (2004). For many of these, a threshold in air is also quoted.


Table 10.28. Hearing thresholds and sensitive frequency bands of selected cetaceans in order of

decreasing sensitivity.

Species Imagesa Minimum Frequency Upper

hearing of best frequency

threshold/ hearing/ limit/

(dB re mPa2) kHz kHz

Phocoena phocoena 31 20–150 200

(harbor porpoise)

(Kastelein et al., 2002)

Orcinus orca 34 15–30 120

(killer whale)

(Au, 1993)

(Szymanski et al., 1999)

Pseudorca crassidens 39 17–74 115

(false killer whale)

(Au, 1993)

Delphinapterus leucas 40 11–105 120

(beluga whale)

(Au, 1993)

Tursiops truncatus 42 15–110 150

(Atlantic bottlenose dolphin)

(Au, 1993)

Stenella coeruleoalba 42 32–120 160

(striped dolphin)

(Kastelein et al., 2003)

Tursiops gilli 47 30–80 135

(Pacific bottlenose dolphin)

(Au, 1993)

Inia geoffrensis 51 12–64 100

(Amazon river dolphin)

(Au, 1993)

a Thumbnail images are Garth Mix, GMIX Designs, reprinted with permission.#

10.2.2.2.3 Of sirenians

The audiograms for two species of manatee (Trichechus inunquis and Trichechusmanatus) are reviewed by Nedwell et al. (2004). The lowest threshold reported, forTrichechus manatus, is 50 dB re mPa2 at frequencies of 16 kHz and 18 kHz (Gerstein etal., 1999).

10.2.2.2.4 Of human divers

Underwater audiograms for human divers measured by Al-Masri (1993) and Parvinet al. (2002) are shown in Figure 10.9, where they are compared with audiogramsmeasured in air (see upper graph—notice the unconventional use, in order to facilitatequantitative comparisons, of 1 mPa as the reference pressure in air). The lowestthreshold measured, that of Al-Masri at 500Hz, is 55 dB re mPa2. The lower graphzooms in on the most sensitive region, with the mean, mode, and median ofAl-Masri’s measurements all plotted. Al-Masri (1993, pp. 139–140) also measuredthe effect of a neoprene hood on the diver’s hearing, reporting an increase inthreshold of between 9 dB (at 250Hz) and 30 dB (8 kHz). At 500Hz the differenceis about 20 dB. Fothergill et al. (2004) observe a smaller difference of about 10 dB at500Hz as well as a higher threshold at that frequency (85 dB re mPa2, without wet suithood).

Table 10.29 compares human hearing in air and water with that of amphibiousmammals. Taken at face value, the table appears to suggest that human hearingsensitivity is greater than that of even the most sensitive of the seals. In practice,


Figure 10.8. Underwater audiograms for harbor seal. The y-axis is the hearing threshold in

units of dB re mPa2. Sources as marked.

such a conclusion cannot be justified on the strength of this table because of themeasurement uncertainties and differences in measurement procedure.

10.2.2.2.5 Of fish

Fish are sensitive to sound in the frequency range 10Hz to 1000Hz (Hawkins andMyrberg, 1983; Tyack, 1999; Popper and Hastings, 2009). Audiograms for many fish


Table 10.29. MSP and EPWI hearing thresholds in air and water for four pinnipeds plus

human subjects, sorted by increasing reported threshold in water. Except for human hearing

in air, for which a standard value of 0 dB re (20 mPa)2 is used, the thresholds are from the

publications listed in the first column.

Species Imagea Threshold in water Threshold in air

MSP/ EPWI/ MSP/ EPWI/

mPa2 pWm�2 mPa2 pWm�2

Human 0.32 0.21 0.00040 0.95

(Al-Masri, 1993)

Homo sapiens

Northern elephant seal 0.48 0.32 5.4 13 000

(Hemila et al., 2006)

Mirounga angustirostris

Northern fur seal 0.61 0.40 0.0013 3.2


Callorhinus ursinus

Harbor sealb 1.9 1.3 0.027 63


Phoca vitulina

Harp seal 3.2 2.1 1.0 2400

(Richardson et al., 1995)

Phoca groenlandica

Californian sea lion 122 79 0.103 32


Zalophus californianus

a Unless otherwise stated, reprinted from Wikipedia.b Harbor seal thumbnail, Garth Mix, GMIX Designs, reprinted with permission.#


Figure 10.9. Underwater audiograms for human divers, measured by Al-Masri (1993) and

Parvin et al. (2002). The y-axis is the hearing threshold in units of dB re mPa2. The same

reference pressure of 1 mPa is used for both air and water. The horizontal line (upper

graph) is the standard reference pressure in air, expressed in decibels (i.e.,

0 dB re (20 mPa)2 � 26.0 dB re mPa2).

species are summarized by Nedwell et al. (2004). The most sensitive of these (thosespecies whose hearing threshold is less than 70 dB re mPa2) are listed in Table 10.30.

It is conventional to express hearing thresholds in terms of the minimum audiblesound pressure level. However, some species of fish are thought to respond to particlemotion rather than pressure (Hawkins and Johnstone, 1978; Hawkins and Myrberg,1983; Wysocki et al., 2009), especially for infrasound (Enger et al., 1993), and forthese species it might be more appropriate to quote the hearing threshold in terms of aparticle velocity or particle acceleration level. The lowest velocity level thresholdquoted by Hawkins and Myrberg (1983), for Limanda, is 66 dB re nm2/s2.

10.2.2.3 Discrimination against background noise

The terms critical ratio and critical bandwidth are both measures of an animal’s abilityto detect a tone in white broadband noise. Both measures have dimensions ofbandwidth and, in both cases, the smaller the value of this bandwidth the greaterthe discrimination ability. The difference between them is in the way the bandwidth ismeasured, as described by Au (1993) and summarized below.

10.2.2.3.1 Critical bandwidth

The concept of a critical bandwidth,13 denoted Bc, is a simple one. It is based on the


Table 10.30. Hearing thresholds in water for 10 species of fish (all hearing specialists) (from

reviews by Hawkins and Myrberg, 1983 and Nedwell et al., 2004).

Common name (species) Threshold of hearing Frequency of best

(sound pressure level hearing/Hz

[dB re mPa2])

Goldfish (Carassius auratus) 50 500

Squirrelfish (Holocentrus ascensionis) 50 1000

Soldier fish (Myripristis kuntee) 50 500–2000

Mexican blind cave fish (Astianax jordani) 52 1000

Carp (Cyprinus carpio) 58 500

Mexican river fish (Astianax mexicanus) 60 1000

Cubbyu (Equetus acuminatus) 64 600

Cod (Gadus morhua) 65–75 20–300

Elephant nose fish (Gnathonemus petersii) 67 500

Clown knifefish (Notopterus chitala) 67 500

13 Also known as the aural critical band (ASA, 1994).

premise that, for the purpose of filtering out broadband noise, the ear acts like apassband filter of bandwidth equal to Bc. Thus, if the noise bandwidth BN exceeds Bc,the excess can be filtered out, and not otherwise.

The following procedure is used for the measurement of Bc. For a given value ofBN, the masked audibility threshold of the ratio QS=QN

f is measured, where QS is thesignal MSP and QN

f is the noise spectral density. The value of this threshold isdenoted WBðBNÞ in the following discussion. This measurement of WB is repeatedfor successive BN values. If the premise is correct, two different regimes are expected:one for BN > Bc, in whichWB is independent of BN; the other for BN < Bc, whereWBincreases linearly with increasing BN. Thus, Bc is the value of BN at which thetransition takes place between these two regimes.

10.2.2.3.2 Critical ratio

The critical ratio, denotedWc, is an alternative measure of passband bandwidth thatis easier to determine experimentally than the critical bandwidth. It is defined as thevalue of the threshold WB for the case of white BB noise, that is,

Wc � limBN!1

WBðBNÞ: ð10:37Þ

This large BN limit ofWB is a natural by-product of the measurement of Bc. Typicalvalues ofWc at 10 kHz for odontecetes and pinnipeds are between 100Hz and 1 kHz(Richardson et al., 1995, Fig. 8.6). The critical ratio is expressed sometimes as aproportion of the passband center frequency (e.g., in octaves) and sometimes indecibels:

CR ¼ 10 log10Wc dB re Hz: ð10:38ÞAccording to Richardson et al. (1995), the critical ratio of many odontecetes andpinnipeds at frequencies between 2 kHz and 30 kHz is less than one-sixth of anoctave. The critical ratio of Tursiops exceeds its critical bandwidth by 7.5 dB (i.e.,about a factor of 6 in bandwidth) (Au, 1993).

Kastelein et al (2009b) reports measurements of the critical ratio of the harborporpoise in the frequency range 0.3 kHz to 150 kHz, comparing these with othermeasurements for the bottlenose dolphin, beluga, and false killer whale. At lowfrequency the harbor porpoise critical ratio is approximately constant, with a valueof 18 dB reHz up to 4 kHz. At higher frequency the critical ratio increases to39 dB reHz at 150 kHz, approximately following the relationship

CR ¼ 12:1þ 12:1 log10 F dB re Hz ð4 < F < 150Þ: ð10:39Þ

10.2.2.4 Hearing impairment and behavioral effects

Compared with hearing thresholds, little is known about levels of sound that mightdisturb or harm an animal in water. For example, a loud sound might impair ananimal’s hearing ability by increasing its hearing threshold. Such impairment isknown as a temporary threshold shift (TTS) if the animal eventually recovers normalhearing and a permanent threshold shift (PTS) if not.

The pulse energy E (defined by Equation 10.18), is a commonly used measure to


quantify the biological impact of sound, and in the context of its possible effect on ananimal is known as sound exposure (Southall et al., 2007). The impact of a givensound depends not only on the sound itself, but also on the hearing of the animalperceiving the sound. For this reason, sound exposure is sometimes weighted accord-ing to how strongly a particular animal senses each frequency present in the soundspectrum. Thus, it is standard practice to weight sounds in air according to thesensitivity of human hearing using a process known as ‘‘A-weighting’’.

A measure of underwater sound that is used to assess the impact on hearing is thesound exposure level, which is given by sound exposure, converted to decibels in theobvious way

LE ¼ 10 log10 E dB re mPa2 s: ð10:40Þ

An equivalent form is

LE ¼ 10 log10

ðQðtÞ dt dB re mPa2 s, ð10:41Þ

where QðtÞ is the MSP averaged on a timescale much shorter than the duration of thesound.

In some cases, a weighted sound exposure level is used, meaning that the spectraldensity is modified by multiplying by a dimensionless weighting function in fre-quency. Thus, denoting weighted variables by means of the subscript ‘‘w’’, theweighted sound exposure level is

ðLEÞw ¼ 10 log10

ðQw dt; ð10:42Þ

where Qw is the weighted MSP

Qw ¼ðQf ð f Þ10Wð f Þ=10 df ; ð10:43Þ

andWð f Þ is the weighting function, in decibels, applied to the spectral densityQf ð f Þ.No internationally accepted standard exists for the choice of weighting in water, soany weighting function applied needs to be specified. For example, the comprehensivereview paper by Southall et al. (2007) proposes a number of filter functions intendedfor applications involving exposure of marine mammals to sound. Southall’s filterfunctions take the form

Wð f Þ ¼ 20 log10

f 2high f2

ð f 2low þ f 2Þð f 2high þ f 2Þ; ð10:44Þ

differing only in the choice of low-frequency and high-frequency limits flow and fhigh.Southall et al. (2007) provide four pairs of values of these two frequencies for marinemammals, for each of four ‘‘functional hearing groups’’ (Table 10.31), with theresulting filter shapes plotted in Figure 10.10. The members of these four groupsare listed in Table 10.32. The process of applying Southall’s filters is known as ‘‘M-weighting’’.


Southall et al. (2007) provide advice on sound exposure levels—and other relatedparameters—that they consider likely to result in either hearing impairment orbehavioral response in marine mammals. Thresholds for marine mammals quotedin this section are based on Southall et al. (2007), with bold text indicating thresh-olds14 taken directly from that review. All other thresholds quoted from thispublication involve some interpretation by the present author.

The situation for fish is less well established. A review of present knowledge of theeffects of sound exposure on fish, with particular emphasis on pile-driving noise, isgiven by Popper and Hastings (2009). In June 2008 the Fisheries HydroacousticWorking Group (FHWG) reached an Agreement in Principle that states agreed limitson the levels of sound to which fish may be exposed in terms of peak pressure andsound exposure level (FHWG, 2008).

Whether for mammals or fish, readers intending to use thresholds from thissection are advised to consult more recent literature, where available.

10.2.2.4.1 Sound exposure thresholds for hearing impairment to mammals and fish

The thresholds of sound exposure level proposed by Southall et al. (2007) applicableto marine mammals are reproduced here in Table 10.33. Sounds of two different typesare considered, namely pulses and nonpulses.15 Pulses are further sub-divided into


Table 10.31. Parameters of bandpass filter used in M-

weighting (Equation 10.44) (Southall et al., 2007).

Functional hearing group flow/kHz fhigh/kHz

Low-frequency cetaceans (lf ) 0.007 22

Mid-frequency cetaceans (mf) 0.150 160

High-frequency cetaceans (hf ) 0.200 180

Pinnipeds in water (pw) 0.075 75

14 Southall et al. (2007) uses the term ‘‘criterion’’ rather than ‘‘threshold’’. The term is

interpreted here as a threshold above which an effect is considered likely for a functional

hearing group, based on the (limited) available evidence.15 ‘‘Pulses’’ are defined by Southall et al. (2007) as ‘‘brief, broadband, atonal, transients’’ such

as explosions, gunshots, sonic booms, seismic air gun pulses, and pile-driving strikes. They are

‘‘characterized by a relatively rapid rise from ambient pressure to a maximal pressure value

followed by a [possibly oscillatory] decay period’’ and ‘‘generally have an increased capacity to

induce physical injury as compared with sounds that lack these features.’’ This special use of

the word ‘‘pulse’’, which differs from that elsewhere in this chapter (the present special use

excludes, for example, many sonar transmissions), is indicated here by means of italics.

‘‘Nonpulses’’ are defined by Southall et al. (2007) as intermittent or continuous sounds that

lack the essential properties of pulses. They ‘‘can be tonal, broadband or both’’ such as

machinery noise, wind noise, communications signals, and many active sonar sources.


Figure 10.10. Underwater sound level weighting curves for three groups of cetaceans plus

pinnipeds, as proposed by Southall et al. (2007).

Table 10.32. Genera represented by the four groups considered by Southall et al. (2007), from

their Table 2.

Functional hearing group Genera represented

Low-frequency (LF) cetaceans Baleana, Caperea, Eschrictius, Megaptera, and

Balaeoptera

Mid-frequency (MF) cetaceans Steno, Sousa, Tursiops, Stenella, Delphinus,

Lagenodelphis, Lagenorynchus, Lissodelphis, Grampus,

Peponocephala, Feresa, Pseudorca, Orcinus,

Globicephala, Orcaella, Physeter, Delphinapterus,

Monodon, Ziphius, Berardius, Tasmacetus, Hyperoodon,

and Mesoplodon

High-frequency (HF) cetaceans Phocoena, Neophocoena, Phocoenoides, Platanista, Inia,

Kogia, Lipotes, Pontoporia, and Cephalorhynchus

Pinnipeds Arctocephalus, Callorhinus, Zalophus, Eumetopias,

Neophoca, Phocarctos, Otaria, Erignathus, Phoca, Pusa,

Halichoerus, Histriophoca, Pagophilus, Cystophora,

Monachus, Mirounga, Leptonychotes, Ommatophoca,

Lobodon, Hydrurga, and Odobenus

single or multiple pulses. For each case, two thresholds are quoted in the table. Thefirst (in bold type) is a threshold for the onset of PTS. The second (in brackets) is athreshold for the onset of TTS. The TTS thresholds are not taken directly fromSouthall et al. (2007), but are inferred from the information provided with theirTable 3. The notation Mlf , Mmf , etc., in the last column of Table 10.33 refers toone of the four weighting functions defined by Equation (10.44) and Table 10.31. Forexample, the entry for pinnipeds means that injury (PTS) might result from a soundexposure level LE (weighted with Mpw) in excess of 186 dB re mPa2 s, whereas a TTSmight result if LE exceeds 171 dB re mPa2 s.

The thresholds presented in Table 10.33 and other similar tables have aprovisional status as they are based on the small number of measurements thatexisted at the time of the review published by Southall et al. (2007). The findingsof two recent publications that are not included in Table 10.33 (Mooney et al., 2009;Lucke et al., 2009) are considered briefly below.

Mooney et al. (2009) examine the effect on a bottlenose dolphin of exposure tooctave band noise in the frequency range 4 kHz to 8 kHz. They find that TTS occursfor SEL thresholds between 190 dB and 198 dB re mPa2 s, depending on the durationof exposure T . The lowest SEL thresholds correspond to the longest exposuredurations and vice versa. Specifically, the following relationship is observed,applicable for duration T approximately in the range 100 s to 1800 s:

SELTTS ¼ 200:21� 6:17 log10

T

60 sdB re mPa2 s. ð10:45Þ

Lucke et al. (2009) examine the effect on a harbor porpoise of exposure to an air gunpulse. They find that TTS occurs for a single-pulse SEL of 164.3 dB re mPa2 s andabove, suggesting that this animal might be more sensitive than implied by Table10.33.


Table 10.33. Proposed thresholds of M-weighted sound exposure level [dB re 1 mPa2 s] for

permanent and (in brackets) temporary auditory threshold shift in cetaceans and pinnipeds;

exposure is integrated over a time interval of 24 h. Thresholds for permanent threshold shift

(PTS) (in bold text) are taken directly from Southall et al. (2007); thresholds for temporary

threshold shift (TTS) are inferred from the caption of Southall’s Table 3.a

Functional hearing group Single pulse Multiple pulses Nonpulses Weighting

LF cetaceans 198 (183) 198 (183) 215 (195) Mlf

MF cetaceans 198 (183) 198 (183) 215 (195) Mmf

HF cetaceans 198 (183) 198 (183) 215 (195) Mhf

Pinnipeds (in waterb) 186 (171) 186 (171) 203 (183) Mpw

a Specifically, the TTS thresholds are: PTS threshold minus 15 dB in the cases of a single pulse or multiplepulses; PTS threshold minus 20 dB in the case of nonpulses.b See Southall et al. (2007) for corresponding information relating to pinnipeds in air.

For fish, the FHWG Agreement in Principle specifies a SEL threshold for TTS of187 dB re mPa2 s if the mass of an individual exceeds 2 g.16 This threshold is reducedfor smaller fish to 183 dB re mPa2 s (FHWG, 2008). Although no weighting is men-tioned, according to Reyff (2009) these thresholds are of unweighted cumulative SELfor an exposure duration of 24 hours.

10.2.2.4.2 Peak sound pressure thresholds for hearing impairment to mammalsand fish

PTS thresholds of peak sound pressure of marine mammals are quoted by Southall etal. (2007) in decibels. These thresholds, converted here to linear pressure17 (androunded to the nearest integer value in kilopascals), are quoted in Table 10.34.As previously, TTS thresholds are inferred from the information provided withSouthall’s Table 3.

For fish, the FHWG Agreement in Principle specifies a peak sound pressurethreshold for the onset of TTS of 20 kPa (FHWG, 2008). The same threshold appliesto all fish, regardless of mass.

10.2.2.4.3 Thresholds for behavioral effects

Sound pressure level (nonpulses and multiple pulses). Behavioral effects are moredifficult to assess quantitatively than threshold shifts. Southall et al. (2007) deal withthis difficulty by establishing a qualitative severity scale of 10 points, ranging from‘‘no observable response’’ (response score 0) through various intermediate responses


Table 10.34. Proposed peak sound pressure thresholds for PTS in

cetaceans and pinnipeds for single pulses, multiple pulses, and non-

pulses; pressures are unweighted (bracketed values are thresholds for

TTS, inferred from the footnote of Southall’s Table 3).a

Peak sound pressure threshold for PTS (TTS)

LF cetaceans 316 kPa (158 kPa)

MF cetaceans 316 kPa (158 kPa)

HF cetaceans 316 kPa (158 kPa)

Pinnipeds in water 79 kPa (40 kPa)

a Specifically, the TTS thresholds for peak sound pressure are calculated fromthe PTS thresholds by dividing these by 100:3� 1.995.

16 The agreed limit is described by FHWG (2008) as an ‘‘injury’’ threshold, but Stadler and

Woodbury (2009) clarify that what is meant in this context by ‘‘injury’’ is the onset of TTS.17 This is done to avoid use of the inherently ambiguous term ‘‘peak sound pressure level’’,

which can mean either ‘‘peak (sound pressure level)’’, which would be the maximum value of

the (RMS) sound pressure level or ‘‘(peak sound pressure) level’’, meaning the peak sound

pressure expressed as a level.

(see Table 10.35) to ‘‘outright panic’’ (response score 9). Simplified dose-responserelationships based on Southall et al. (2007), applicable to marine mammals andusing (unweighted) sound pressure level as a metric, are provided in Table 10.36 forexposure to nonpulses and in Table 10.37 for multiple pulses. The reader’s attention isdrawn to the uncertainty associated with the wide spread of values presented inTables 10.36 and 10.37, and to the warning from Southall et al. (2007) that ‘‘Theavailable data on marine mammal behavioral responses to multiple pulse and non-pulse sounds are simply too variable and context-specific to justify proposing singledisturbance criteria for broad categories of taxa and sounds.’’ The purpose ofincluding the two tables here is partly to illustrate this uncertainty and partly topromulgate what little information there is available.


Table 10.35. Outline of the severity scale from Southall et al. (2007). The text is

paraphrased by the present author for brevity. For a complete description of

Southall’s severity scale, the reader is referred to the original in Table 4 of Southall

et al. (2007).

Response score Observed behavior

0 No observable response

3 Minor changes in behavior; no avoidance of sound source

6 Minor or moderate avoidance of sound

9 Outright panic; predator avoidance reaction

Table 10.36. Spread of sound pressure level (SPL) values [dB re mPa2] resulting in the specified

behavioral responses in cetaceans and pinnipeds for nonpulses (from Southall et al., 2007). The

modal group (the range of SPL values resulting in the largest number of occurrences of the

specified response) is given in brackets; the median is the same as the modal in every case except

for mid-frequency cetaceans with response score 6–9, for which the median group is shown

separately in square brackets. The severity scale on which the ‘‘response score’’ is measured is

summarised in Table 10.35.

Response score 0 to 2 Response score 3 to 5 Response score 6 to 9

LF cetaceansa 80–150 (90–100) 100–150 (100–110) 90–150 (110–120)

MF cetaceansb 100–200 (110–120) 80–130 (110–120) 90-200 (100–110)

[120–130]

HF cetaceansc 80–130 (90–100) Sparse data 80–170 (140–150)

Pinnipeds in waterd Sparse data Sparse data Sparse data

a Based on Southall et al. (2007, Table 15).b Based on Southall et al. (2007, Table 17).c Based on Southall et al. (2007, Table 19) (all entries in this table are for the harbor porpoise).d Based on Southall et al. (2007, Table 21).

Sound exposure level and peak sound pressure (single pulse). Very little is knownabout thresholds of SEL and peak pressure likely to cause a behavioral change. (Nosystematic review is available comparable to that leading to Table 10.36.) For thecase of a single pulse only, Southall et al. (2007) suggest adopting the TTS thresholdas a surrogate for a behavioral threshold on the grounds that a sound loud enough tocause TTS has the potential to cause a change in the animal’s behavior, even if theonly change is a short-term adjustment for the temporary loss of hearing.

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11

The sonar equations revisited

Were it offered to my choice, I should have no objection to a repetition of the samelife from its beginning, only asking the advantages authors have in a

second edition to correct some faults in the first.

Benjamin Franklin (ca. 1771)

11.1 INTRODUCTION

The objective of this chapter is to reverse (or at least to mitigate) the preference insome previous chapters for simplicity over realism. The sonar equations and selectedworked examples of Chapter 3 are revisited, with the aim of introducing the necessaryrealism to become not only didactic but also practical. To make this possible, use ismade of a computer model for the calculation of propagation loss, noise level, andreverberation level. Selected sonar equation terms are redefined to establish a morerigorous basis for the revised worked examples. Relevant material introduced in theintervening Chapters 4 to 10, a selection of which is considered in Chapter 11,includes

— the effect of boundaries and sound speed profile on sound propagation andreverberation (Chapters 4, 8, and 9);

— the effect of bubbles and suspensions on sound propagation and scattering(Chapter 5);

— the effect of a matched filter and coherence loss on processing gain (Chapter 6);— the effect of steering and shading on the sonar beam pattern and array gain

(Chapter 6);— departures of signal statistics from idealized Gaussian or Rayleigh distributions

(Chapter 7);

— the presence of ambient noise sources other than wind (Chapters 8 and 9);1

— typical measured values of target strength (Chapter 8);— typical properties of natural and man-made sonar transmitters and receivers

(Chapter 10).

Some generic examples of man-made sonar and other underwater acoustic sensorsare listed in Table 11.1. The wide range of frequencies involved (from a few tens ofhertz for low-frequency passive sonar to several megahertz for high-resolutionacoustic cameras or Doppler profilers), and also of water depths (1m to 10 km)and ranges (from less than a meter to thousands of kilometers) means that thereexists no single computer model that is suitable for all sonar performance problems.Hence, there will always be reason to check the output of any one model. Obtaining asecond opinion is a vital part of the process of increasing (or in some cases, decreas-ing) confidence in the model predictions. To bring this point home, the reader isencouraged to run his or her preferred performance model on the worked examplesprovided (starting with the simpler ones in Chapter 3), and compare the individualsonar equation terms with the values given here. The author is confident that differ-ences will be found, and that the magnitude of these differences will often not benegligible. Identifying and understanding such differences and their causes leads to abetter understanding of the problem and eventually to an improved capability tomodel it correctly.

Perhaps the single most unrealistic feature of the examples in Chapter 3 is theassumption of infinite water depth, and this is remedied here by considering depths inthe range 30m to 5,000m. A realistic sound speed profile is also introduced in orderto demonstrate its effect on acoustic propagation. Passive sonar is discussed first, withcoherent processing in Section 11.2, followed by incoherent processing in Section11.3. For active sonar (Section 11.4), only coherent processing is considered. Finally,chapter and book are rounded off with a brief indulgence in crystal ball gazing(Section 11.5).

11.2 PASSIVE SONAR WITH COHERENT PROCESSING:

TONAL DETECTOR

11.2.1 Sonar equation

The passive sonar equation with (narrowband) coherent processing is

SENB ¼ ðSL� PLÞ � ðNLf þ BW� AGÞ �DT: ð11:1Þ

The six individual terms on the right-hand side are described in turn below, followedby a worked example in Section 11.2.8.

574 The sonar equations revisited [Ch. 11

1 Wind was the only noise source considered for the examples in Chapter 3.

11.2.2 Source level (SL)

The source level SL is defined as

SL � 10 log10 S0 dB re mPa2 m2; ð11:2Þ

where S0 is the source factor, which was defined in Chapter 3 for a point source as2

S0 ¼ lims0!0

ðp20s20Þ ð11:3Þ

11.2 Passive sonar with coherent processing: tonal detector 575]Sec. 11.2

Table 11.1. List of applications of man-made active and passive underwater acoustic sensors,

with order-of-magnitude indication of frequency range for sound transmitters. An asterisk (*)

indicates a broadband source.

Application Examples of active sensors Examples of passive

[frequency range in kHz] sensors

Military surveillance Active sonobuoy [1–30] Bottom-mounted array

Active towed array sonar [0.3–10] Hull-mounted passive

Helicopter dipping sonar [1–30] sonar (e.g., flank array)

Hull-mounted search sonar [3–300] Intercept sonar

Passive sonobuoy

Passive towed array

Weapons and Active homing sonar [10–100] Acoustic sensing mine

countermeasures Minesweeping equipment* Passive homing sonar

Navigation Navigation echo sounder [10–100]

Obstacle avoidance sonar [30–300]

Fisheries Fisheries sonar [10–300]

Seismic survey Air gun array*

Boomer*

Sparker array*

Water gun array*

Underwater Acoustic transponders [0.3–100]

positioning and Underwater telephone and telemetry

communications [3–100]

Oceanographic Acoustic camera [300–3000] Environmental sensing

survey, weather Acoustic Doppler profiler [30–3000] equipment

observation, and Inverted echo sounder [10–100]

scientific research Multibeam echo sounder [10–1,000]

Sub-bottom profiler*

Sidescan sonar [10–1000]

2 The definition in Chapter 3 is not written formally as a limit in this way, but the limiting

process is implied.

and p0 is the RMS pressure at distance s0 from the source. This Chapter 3 definitiondoes not make physical sense for a source of finite size because the limit of small s0would take us first into the near field of the source and eventually inside the objectitself. This weakness is addressed below.

As pointed out in Chapters 8 and 10, the source level is a property of the far field3

radiated by a sound source, which means that it must be defined in terms of otherproperties of that far field. A suitable far-field property is radiant intensity J (theradiated power per unit solid angle, denoted WO in Chapter 3), which for a pointmonopole source is equal to S0=ð�cÞ. A non-monopole source will, in general, radiatesound with different intensity in different directions, making radiant intensity afunction of angle, which is indicated by writing J ¼ Jð�; �Þ, or alternativelyJ ¼ JðOÞ, where O ¼ Oð�; �Þ represents a direction defined by the elevation � andazimuth �. If J0 is the maximum value of JðOÞ as � is varied, with � fixed in thedirection towards the target, the source factor of a directional source can be definedas

S0 � �cJ0; ð11:4Þ

replacing Equation (11.3). In the case of an idealized (uniform, lossless) medium, thefar-field radiant intensity JðOÞ is a well-defined property of a distributed source. Tocomplete the definition of source level it is also necessary to consider a real medium(in this case water), for which the parameter J0 in Equation (11.4) is interpreted as theradiant intensity that the same source would produce in an ideal (uniform lossless)medium of the same characteristic impedance, and if driven with the same velocity onits active surfaces.

11.2.3 Narrowband propagation loss (PL)

For realistic situations, it is often necessary to calculate PL with the aid of a computermodel. There are many models to choose from, either in the form of a standalonepropagation model (oalib, www) or embedded within a complete sonar performancemodeling package (Etter, 2003). For an omni-directional source, PL is related to thenarrowband (NB) propagation factor FNB according to

PL ¼ �10 log10 FNB; ð11:5Þ

where FNB can be written in terms of the differential propagation factor GNB asfollows

FNB ¼ðGNBð�Þ d�: ð11:6Þ

For a directional source whose vertical beam pattern is bð�Þ, Equation (11.5)


3 The far-field nature of the source level, missing from the definitions of national (ASA, 1994)

and international (IEC, www) standards bodies, is essential for its correct use in the sonar

equation—Kinsler et al. (1982) and Urick (1983, pp. 75 and 328).

generalizes to

PL ¼ �10 log10ðbð�ÞGNBð�Þ d�

� �; ð11:7Þ

where

bð�Þ � Jð�; �arrÞJ0

; ð11:8Þ

and �arr is the bearing of the sonar receiver relative to the target, such that the beampattern bð�Þ is evaluated in the plane of propagation (i.e., the vertical plane contain-ing both target and sonar receiver). Notice the change in notation compared withChapter 6, where the same function was denoted Bð�Þ.

The narrowband propagation factor may be approximated by

FNB ¼ Fcohð f0Þ; ð11:9Þ

where f0 is the nominal center frequency of the narrowband tonal (and of theprocessing band). A better approximation is

FNBð f0Þ ¼1

B

ð f0þB=2f0�B=2

Fcohð f Þ df ; ð11:10Þ

where B is the effective processing bandwidth, defined here as the smaller of the tonalwidth �ftonal and true processing bandwidth �fFFT

B � minð�ftotal; �fFFTÞ: ð11:11Þ

For narrowband sonar, the bandwidth B is usually very small, and the approximationof Equation (11.9) is often adequate. An alternative, should greater accuracy berequired, is to evaluate Equation (11.10) using the method of Harrison and Harrison(1995), which replaces the frequency average with a proportional range average at asingle frequency4 ;5

FNBðr0; f0Þ �f0Br0

ðr0ð1þB=2f0Þr0ð1�B=2f0Þ

Fcohðr; f0Þ dr: ð11:12Þ

The main effect of this averaging, whether in frequency or in range, is to smooth overany interference nulls whose position is sensitive to the precise acoustic frequency.

An important consideration for application to coherent processing is that thepropagation model must be able to account for the phase difference between differentpropagation paths. A model that does so is sometimes called a ‘‘coherent’’ model, or,alternatively, might permit the user to select a ‘‘coherent’’ option. An incoherentmodel (i.e., one that does not take account of such phase differences) might missimportant features due to constructive or destructive interference between thedifferent paths. It is also necessary to distinguish between the incoherent addition


4 An additional Gaussian-weighting function is applied by Harrison and Harrison (1995).5 The benefit of converting an average over frequency to one over range is one of

computational efficiency. This is because, for many solution methods, the required

computation time is proportional to the number of frequencies considered.

of rays and that of modes, as qualitatively different information is lost in the twocases.

11.2.4 Noise spectrum level (NLf )

Background noise, foreground noise, and non-acoustic noise are consideredseparately below.

11.2.4.1 Background noise

Background noise is sound arriving at a sonar receiver from distant sources thatcannot be resolved by the sonar as coming from spatially distinct sources. The term‘‘background’’ is used here as an adjective to distinguish from ‘‘foreground’’ noise,described below.6

In Chapter 3, only wind noise—an example of background noise—is considered.In reality, at low frequency (below 1 kHz), wind noise is supplemented by contribu-tions from shipping, and at high frequency (above 100 kHz) by thermal noise.Occasional noise sources that sometimes contribute include precipitation (rain orsnow) and some fauna (especially crustaceans and marine mammals). See Chapter 8for a general review of sound sources and Chapter 9 for examples of predicted noiselevels.

11.2.4.2 Foreground noise

Foreground noise comprises all interfering sound other than background noise; thatis, sound from those sources that can be resolved by the sonar. Potential foregroundnoise contributions include

— acoustic noise radiated by the sonar platform, either via a direct path orindirectly, after one or more boundary reflections;

— acoustic noise radiated by vessels accompanying the sonar platform, either via adirect path or indirectly, after one or more boundary reflections;

— mechanical vibrations from the platform structure or tow cable;— interference from active sonar transmitters operating nearby, including echo

sounders;— sounds made by nearby animals such as marine mammals.

Some examples of foreground noise are localized in time as well as in space. These areknown as ‘‘transient sources’’ or sometimes just ‘‘transients’’.

11.2.4.3 Non-acoustic noise

Non-acoustic noise is sonar noise that is not caused by sound waves reaching thehydrophone. It is useful to distinguish between the following two categories of


6 Compare the alternative use of ‘‘background’’ as a noun to mean the combination of noise

and reverberation.

non-acoustic noise:

— non-acoustic pressure fluctuations at the hydrophone position, converted to anelectrical signal by the hydrophone’s transducer action;7

— voltage fluctuations in the receiving equipment other than those present in thehydrophone output.

11.2.4.4 Self-noise and ambient noise

11.2.4.4.1 Self-noise, including platform noise

Background, foreground, and non-acoustic noise together amount to a completedescription of noise sources relevant to passive sonar. The term ‘‘self-noise’’ is some-times used to mean all noise that is caused by the presence of the sonar itself or itsplatform. Thus, it is partly non-acoustic noise and partly (acoustic) foreground noise.That part of self-noise associated with the sonar platform is known as ‘‘platformnoise’’. This can be structure-borne noise if the sonar is mounted on the hull of itsplatform, or acoustic noise propagated through the sea if it is towed.

A special kind of (non-acoustic) self-noise arises in the context of animal hearing,for which self-noise manifests itself in the form of an auditory threshold. Speculationinto the nature of the noise that gives rise to this threshold is outside the presentscope, other than to mention the ubiquitous presence of thermal noise.

11.2.4.4.2 Ambient noise

All acoustic noise not included in the definition of self-noise can be considered to beambient noise.

11.2.5 Bandwidth (BW)

A perfect tonal has no width in frequency. For such a tonal, BW is the width of theprocessing band, expressed in decibels. In reality the tonal always has a finite widththat might exceed that of the processing band, in which case a proportion of the signalenergy is lost, leading to a degradation relative to the ideal. This degradation can bemodeled by defining BW in terms of the processing bandwidth �fFFT as

BW � 10 log10RhpRFFT

�fFFT

� �; ð11:13Þ

where Rhp is the SNR at the hydrophone in the processing bandwidth beforenarrowband processing; and RFFT is the SNR at the hydrophone in the processingbandwidth after narrowband processing and before beamfoming.8 In the ideal case,


7 Examples of non-acoustic pressure fluctuations are:

— flow noise (hydrodynamic noise from flow around dome or towed array);

— molecular thermal noise (usually unimportant for passive sonar; see Chapter 10).8 The assumed processing chain is a Fourier transform followed by a beamformer.

the entire tonal is included in the processing band, Rhp and RFFT are equal, andthe Chapter 3 expression for BW is then recovered. More generally, allowing for thepossibility that the tonal is over-resolved

RFFT � RhpB=�ftonal B < �ftonal

1 B > �ftonal,

�ð11:14Þ

where B is given by Equation (11.11). Thus, Equation (11.13) can be written

BW � 10 log10 maxð�fFFT; �ftonalÞ: ð11:15Þ

11.2.6 Array gain (AG) and directivity index (DI)

Array gain (AG) is the gain from spatial filtering (beamfoming). It can be defined interms of the ratio of Rarr (SNR after beamfoming) to RFFT

GA � RarrRFFT

: ð11:16Þ

The AG term, equal to 10 log10 GA, is rarely calculated in full because of thecomplications introduced (for example) by anisotropic noise (Harrison, 1996; Clark,2007) and departures from a plane wave signal. The calculation can be simplified bywriting Equation (11.1) in the form

SENB ¼ ðSL� PLÞ � ðNLf þ BW�DIÞ �DTþ ½AG �DI; ð11:17Þ

and neglecting the term in square brackets, so that

SENB � ðSL� PLÞ � ðNLf þ BW�DIÞ �DT: ð11:18Þ

The neglected term can be positive or negative and in an average sense (for a randomgeometry) can be expected to cancel. For any given geometry, a bias is likely, asillustrated by the examples of Chapter 3.

A simple calculation method for the directivity index (DI) of an unsteered linearray, derived in Chapter 6, is summarized below. It is convenient to write DI in theform

DI ¼ 10 log10 GD; ð11:19Þ

where GD is the directivity factor (or directivity gain), which can be written

GD � 1þ G0 tanh2

36G0

!; ð11:20Þ

where

G0 ¼ 2L=�: ð11:21Þ

The directivity index is plotted in Figure 11.1 as a function of G0 using Equation(11.20) (dashed line). Except when steered close to the endfire direction, the effect ofsteering on the directivity index of a line array is small (see Chapter 6).


11.2.7 Detection threshold (DT)

11.2.7.1 Calculation of DT for given pfa

If the target statistics (in the form of the distribution of signal amplitude, after allprocessing) are known, a sensible choice of distribution can be made from the optionsdescribed in Chapter 7. If nothing is known about the distribution, it makes sense tominimize the maximum error by selecting one of an intermediate character, such asone dominant plus Rayleigh (1DþR). An ROC curve for this case is shown inFigure 11.2.

If pfa is known or given, the detection threshold for this situation can beestimated using9

DT50ð pfaÞ � 10 log10 log21

2pfa

� �� 0:8 dB; ð11:22Þ

which is accurate to �0.1 dB for pfa < 10�2 with 1DþR statistics. For other signalamplitude distributions, the error incurred by assuming 1DþR anyway is no morethan 0.8 dB even for extreme situations of a stable signal (no fluctuations) on the one


Figure 11.1. Directivity index DI ¼ 10 log10 GD for an unsteered continuous line array vs. G0,

where G0 ¼ 2L=�. Solid blue line: full integral of Chapter 6; dashed red line: approximation ofEquation (11.20).

9 Unless otherwise specified, the detection threshold is that corresponding to a 50% detection

probability.

hand (column 2 of Table 11.2) or strongfluctuations (Rayleigh fading) on the other(column 3).

11.2.7.2 Estimation of pfa

In order to use Equation (11.22) it might benecessary to estimate the value of pfa fromknowledge of sonar parameters. Forexample, if the number of beams is notknown, the directivity factor GD (i.e.,10DI=10) can be used instead of Nbeams toestimate pfa. Specifically, use of the approx-imation Nbeams � GD in (see Chapter 3)

pfa ¼nfa

Df Nbeams

; ð11:23Þ

yields

pfa ¼nfa

Df GD: ð11:24Þ

In some circumstances there might be special interest in just one beam. In that case itcould be that contacts in all but that one beam are ignored, so that Nbeams ¼ 1 wouldbe used in Equation (11.23). The false alarm probability can then be increased (e.g.,by increasing the detection threshold) without affecting the false alarm rate nfa. Thebenefit of higher false alarm probability is a lower detection threshold and hencehigher detection probability for a given SNR in the chosen beam. The disadvantage isthat a signal in any other beam would be missed.


Figure 11.2. ROC curve in

the form 10 log10 R vs.

log10 pfa for 1DþRamplitude signal in

Rayleigh noise (the

relevant curve is the one

for pd ¼ 12 ).

Table 11.2. Error in DT (i.e., the value of

DTtrue �DT1DþR in decibels) incurred

by assuming 1DþR if the true signal

amplitude statistics follow a Dirac (non-

fluctuating) or Rayleigh (strongly fluctu-

ating) distribution.

pfa Dirac Rayleigh

10�4 �0.7 þ0.8

10�8 �0.8 þ0.8

10�12 �0.8 þ0.8

The same logic justifies a reduced value of Df for use in Equation (11.24) if thereis interest in only a narrow range of frequencies. A possible application is for atracking system, with prior knowledge about the expected target bearing and fre-quency obtained from a previous contact.

11.2.8 Worked example

The narrowband passive sonar worked example of Chapter 3 (henceforth abbreviatedNBp) is used here as a basis to develop a more realistic scenario. The extra realismmakes it necessary to make use of an automated software tool to carry out thecalculations, with the implied acceptance of assumptions and approximations madeby that tool. Where necessary, some accuracy is sacrificed to achieve the desiredrealism.

Perhaps the most unrealistic assumption of all made in Chapter 3 (NBp) was thatof infinite water depth. For the present example, a fine-grain unconsolidated sedi-ment10 seabed is placed at a depth of 5 km, representative of the deep basins of thePacific Ocean. Further, a winter sound speed profile, corresponding to the northwestPacific (NWP) case from Chapter 4 (see also Chapter 9), is considered, and the windspeed is increased from 5m/s to 7m/s.

Further minor changes compared with the Chapter 3 example include

— the array gain AG is approximated by the directivity index DI;— Hann shading is applied to the receiver array, resulting in a small reduction in DI;— the signal amplitude is assumed to follow a 1DþR distribution (instead of

a Rayleigh distribution), resulting in a reduction of 0.8 dB in the detectionthreshold;

— loss mechanisms due to sound absorption (chemical relaxation) and sea surfacescattering are introduced;

— a moderate shipping density of 500Mm�2 (i.e., 5� 10�4 km�2, or five ships per100 km square) is considered for ambient noise prediction.

11.2.8.1 Propagation loss and signal excess

The effect of these changes on propagation is demonstrated in Figure 11.3. The mostnoticeable changes in the first 50 km (compared with NBp) are associated with thereflection of sound from the seabed. The contributions from bottom-interacting pathsare labeled ‘‘bottom reflection’’ and ‘‘bottom refraction’’ in Figure 11.3. At greaterdistances, the sound speed profile makes itself felt through the appearance ofconvergence zones (CZs) at intervals of 65 km.


10 The parameters correspond to a grain size of 8� (see Chapter 4), which is intermediatebetween coarse clay and fine silt.

For the calculation of SNR it is useful to introduce the concepts of in-beamsignal and noise levels

IBSL � 10 log10 YS ¼ SL� PLþ SG ð11:25Þ

and

IBNL � 10 log10 YN ¼ NLf þ BW þNG: ð11:26Þ

The sonar equation can then be written

SENB ¼ IBSL� IBNL �DT: ð11:27Þ

These concepts are closely related to the levels LS and LN introduced in Chapter 3, asfollows:

IBSL ¼ LS þ SG ð11:28Þand

IBNL ¼ LN þ SG; ð11:29Þsuch that

IBSL� IBNL ¼ LS � LN: ð11:30Þ

In Chapter 3, LN was also referred to as ‘‘in-beam noise level’’. The change innomenclature is intended to emphasize the refinement in the definition of this term,which here distinguishes between the contributions from noise gain (NG) and signalgain (SG) to total array gain (AG). From the above equations, the difference between


Figure 11.3. Propagation loss [dB rem2] vs. range for NWP winter case. The frequency is

300Hz (INSIGHT—Ainslie et al., 1996).

IBNL and LN (and also between IBSL and LS) is equal to SG, which is normally asmall correction.

In addition to large corrections to PL, changes in the scenario also lead to asmaller, but still significant, change to NL. The combined effect of increasing thewind speed and the more realistic sound speed profile increases NL by about 4 dB.Hann shading reduces DI by 1.8 dB, resulting in an overall increase in in-beam noiseof about 6 dB, as illustrated by Figure 11.4. In general, wind speed might also affectpropagation loss through increased surface scattering, but on this occasion the effectis negligible. The upper graph of Figure 11.4 shows signal level as a function of targetrange, calculated using propagation loss from Figure 11.3. Also shown for compar-ison (lower graph) is the corresponding prediction for the NBp example. In bothgraphs the horizontal line is the noise level.

By definition, detection is likely ( pd >12) if the signal exceeds the noise by at least

the detection threshold, which for this case is 13 dB (see Table 11.3). Thresholdcrossings occur for ranges up to 2 km, and at the first CZ. The maximum signalexcess at the second CZ, though negative, is close to zero, meaning that the detectionprobability is close to 50%. A complete description of the scenario is provided inFigure 11.5.

11.2.8.2 What is the detection range?

Calculation of either detection probability or detection range can be simplified by firstcalculating the figure of merit (FOM). The FOM calculated using the numericalvalues from Table 11.3 is

FOMNBðzarrÞ ¼ SLþ ðAG� BWÞ �NLf ðzarrÞ �DT ¼ 73:8 dB re m2: ð11:31Þ

The range at which PL is equal to FOM, denoted r50, is known as the detection range.If the crossing is unambiguous, the concept is a useful one, but this is not always thecase. There can be several FOM crossings, making the definition of r50 unclear. Forthe present situation, the oscillatory nature of PL results in multiple crossings. Figure11.6 shows signal excess vs. range and depth for distances up to and including thefirst CZ. In this graph the gray shading indicates regions where the threshold is eitherjust exceeded or nearly exceeded (�2 dB< SE< þ2 dB). Black indicates that thethreshold is exceeded by at least 2 dB.

For a target depth of 10m, unambiguous detections occur during the first 2 km,but on the basis of Figure 11.6 it seems misleading to declare a detection range of only2 km. Further threshold crossings occur also at the first CZ (and even at the secondCZ for a slightly deeper target), but, because of the gaps between them, it would besimilarly misleading to declare a detection of range of 65 km or 130 km. A close-up ofthe first CZ is shown in Figure 11.7, illustrating the intermittent nature of CZdetections.

The purpose of examining signal excess behavior in this level of detail is to raiseand address the issue of highly variable signal excess. Rapid variations through theCZ region provide a natural laboratory in which to do so. No statement is made hereabout the accuracy in detail of Figures 11.6 and 11.7, but it is expected that other



Figure 11.4. In-beam signal and noise levels vs. range for NWP winter (upper) and Chapter 3

NBp worked example (lower). The frequency is 300Hz. Small differences in level between the

lower graph and its equivalent fromChapter 3 are due to the use here of a linear range scale and a

different computer model. The reason for including the comparison is to enable the reader to

identify changes due to the scenario alone.

sonar performance models would predict a comparable variation in signal excessthrough the CZ for the present worked example.

11.2.8.3 Alternative performance measures

Despite difficulties with an unambiguous definition of r50, it remains desirable torepresent sonar performance by means of a single variable. Can the definition ofdetection range be modified in a simple way to achieve this? This question leads to theconcepts of detection volume Vd and detection area Ad introduced below, andassociated ranges rV and rA. These measures are extremely robust and applicableto bistatic or multistatic sonar geometries.

11.2.8.3.1 Detection volume (radius of equivalent volume sphere)

Consider first the total volume of space ensonified by the sonar (i.e., the total volumewithin which a target, if present, is likely to be detected)

Vd ¼þpdðxÞ dV ; ð11:32Þ

where the integration is over all space.The simplicity of Equation (11.32) is appealing. It offers an elegant measure of

performance as a single parameter, but there are two complications. The first is thatpd pfa everywhere, which means that the integral is infinite. This is a consequence ofincluding (in the definition of ‘‘detection’’) those threshold crossings due to randomfluctuation in noise alone. Such crossings are treated as ‘‘detections’’ if a target ispresent, regardless of whether or not sound from the target is responsible for thatcrossing. Crossings that are not caused by the target can be removed in a pragmaticfashion by subtracting pfa from the integrand:

Vd �þðpd � pfaÞ dV ; ð11:33Þ

rendering the integral finite. In cylindrical co-ordinates this becomes (for water


Table 11.3. Sonar equation calculation for NWP winter.

Description Symbol Value

Source level SL 133.9 dB re mPa2 m2

Noise spectrum level NLf 64.2 dB re mPa2 Hz�1

Array gain AG 11.1 dB re 1

Detection threshold DT 13.0 dB re 1

Analysis bandwidth BW �6.0 dB reHz


OPTIONS Coherence CoherentProfile mode Value vs depthSignal Processing Narrow bandPassive Beam Pattern Horizontal steered—Cosine squaredWind Noise Wind noise modelShipping Noise Shipping noise modelRain Noise Rain noise offSelf Noise Self noise offNo. of bottom reflections 5No. of bottom refractions 5

PASSIVE PARAMETERS Analysis frequency 300. HzArray depth 30. mArray length 45. mElectronic steering angle 0. degreesHLA heading clockwise from North 90. degreesAnalysis bandwidth 0.25 HzDetection threshold (narrowband) 13. dB

SOUND SPEED PROFILE USER-DEFINED LAW:Water depth 5000 mc (m/s) z (m) dc/dz (/s) H (m)1536.64 0 2.e-3 101536.66 10 1.e-2 101536.76 20 9.e-3 101536.85 30 7.5e-3 201537 50 -2.8e-3 251536.93 75 -3.4e-2 251536.08 100 -0.1056 251533.44 125 -0.1368 251530.02 150 -0.1366 501523.19 200 -0.113 501517.54 250 -7.44e-2 501513.82 300 -9.34e-2 1001504.48 400 -0.1092 1001493.56 500 -6.78e-2 1001486.78 600 -3.13e-2 1001483.65 700 -8.8e-3 1001482.77 800 -7.e-3 1001482.07 900 3.2e-3 1001482.39 1000 2.e-4 1001482.41 1100 8.8e-3 1001483.29 1200 9.8e-3 1001484.27 1300 1.e-4 1001484.28 1400 1.2e-2 1001485.48 1500 1.14e-2 2501488.33 1750 1.044e-2 2501490.94 2000 1.468e-2 5001498.28 2500 1.558e-2 5001506.07 3000 1.656e-2 5001514.35 3500 1.726e-2 5001522.98 4000 1.764e-2 5001531.8 4500 1.842e-2 5001541.01 5000

VOLUME LOSS ALTERNATIVE LAW: Thorp

SURFACE LOSS ALTERNATIVE LAW: Marsh-Schulkin-KnealeWind speed 7. m/s

BOTTOM LOSS GEO-ACOUSTIC PARAMETERS:Sediment depth 500. mcs/cw 1.Sediment velocity gradient 1. /sMaximum sediment velocity 1900. m/sSediment density 1.4Sediment attenuation coefficient 6.e-002 dB/m/kHzSediment attenuation gradient 1.e-006 dB/m/kHz/m

TARGET PARAMETERS Target range *Target depth 10. mNarrow-band radiated noise level at 1kHz 133.9 dBRadiated noise gradient 0. dB/octaveTarget bearing clockwise from North 0. degrees

BACKGROUND PARAMETERS Ship length 100. mShip speed 5. m/sShipping density 5.e-004 /km2Wind speed 7. m/s

Figure 11.5. Input parameters for northwest Pacific (NWP) problem.

depth H)

Vd ¼ð10

ð20

ðH0

½ pdðr; z; �Þ � pfar dr d� dz: ð11:34Þ

Given the volume Vd, one can imagine an equivalent sphere of the same volume,whose radius is

rV � 3Vd4

� �1=3

: ð11:35Þ

In contrast to r50, the detection volume and hence the equivalent sphere radius rValso, are robust and unambiguous measures of performance. They are not overlysensitive to small changes in SNR or DT, and the equivalent radius rV provides auseful feel for sonar coverage in terms of distance, which readers might find moreintuitive than volume.

A good measure of performance needs to recognize the value of providing newinformation. If it is known in advance that no target of interest is present at aparticular location, there is little value to be added by a sonar confirming that knownfact. In other words, a good measure would include only the proportion of Vd inwhich elementary prior knowledge would not already rule out the possibility of targetpresence. The second complication is that the measure rV fails this test.


Figure 11.6. Signal excess [dB] vs. range and depth for NWP winter (30m sonar depth). The

frequency is 300Hz (INSIGHT).

11.2.8.3.2 Detection area (radius of equivalent area circle)

Regardless of any specific local knowledge, common sense tells an archeologistlooking for a sunken wreck that he can safely limit his search to targets satisfyingthe condition ztgt � H, whereas a pelagic fishing trawler is interested in targets whosedepths are in the range 0 < ztgt < H. The ability to detect targets that do not meetthese elementary criteria has little value and would be excluded from any goodperformance measure. Thus, an improvement to rV is obtained by weighting theintegrand according to the prior probability of target presence.

A suitable weighting factor is provided by the depth probability distributionfunctionWðzÞ (the prior probability of the target being within unit depth, given thatthere is one and only one target present), the application of which results in a measurewith dimensions of area11

Ad ¼þWðzÞ½ pdðr; z; �Þ � pfar d� dr dz: ð11:36Þ

Consider first the case of the fisherman who wishes to search for targets in the depth


Figure 11.7. Signal excess [dB] vs. range and depth for NWP winter (30m sonar depth) (close-

up of first CZ). The frequency is 300Hz (INSIGHT).

11 In general, the probability distributionWðzÞ could also depend on the bearing and range co-ordinates, leading naturally to a probability per unit area or volume, but this line of thinking is

not explored further.

range ½0;H. Assume for simplicity that the fish have no preference within this range,so that the appropriate weighting function is

WðzÞ ¼ 1

HP

z

H� 12

� �: ð11:37Þ

Assuming cylindrical symmetry, the integral becomes

Ad ¼2

H

ð10

ðH0

½ pdðr; zÞ � pfar dr dz: ð11:38Þ

In terms of a plan view, Ad is the area coverage of sonar for the assumed depthprobability distribution. This can be related to a distance by calculating the radius ofan equivalent circle of the same area:

rA � Ad

� �1=2

: ð11:39Þ

This distance is closely related to sweep width, which, if multiplied by platform speed,provides a simple estimate of the area coverage rate.

Another possibility is the depth being known precisely (the case of thearcheologist)

WðzÞ ¼ �ðz�HÞ; ð11:40Þwhere �ðxÞ is the Dirac delta function, and hence

Ad ¼ 2

ð10

½ pdðr;HÞ � pfar dr: ð11:41Þ

The incorporation of prior knowledge in rA makes it a more useful measure ofdetection performance than rV . Depth knowledge need not be sophisticated to beuseful—in the above examples it is almost trivial.

11.3 PASSIVE SONAR WITH INCOHERENT PROCESSING:

ENERGY DETECTOR


The passive sonar equation with incoherent broadband processing is

SEBB ¼ SL� PL� ðNL� PGÞ �DT: ð11:42ÞThe five individual terms on the right-hand side are described below. To avoid thecomplication of a beam pattern, an omni-directional source is assumed in this sectionfrom the beginning.12

11.3 Passive sonar with incoherent processing: energy detector 591]Sec. 11.3

12 In some situations it might be necessary to model the beam pattern of the sound source (e.g.,

of a dipole source such as a surface ship at low frequency—see Chapter 8). The effect of doing

so would be to increase PL at positions outside the main beam.

11.3.2 Source level (SL)

The broadband source level is the total source factor, integrated over the sonarbandwidth, in decibels.

SL � 10 log10

ððS0Þf df dB re mPa2 m2; ð11:43Þ

where, in terms of the spectral density of radiant intensity ðJf Þ,

ðS0Þf � �c

ðJf df : ð11:44Þ

11.3.3 Broadband propagation loss (PL)

Broadband propagation loss is averaged over frequency. Specifically, if

PL ¼ �10 log10 FBB; ð11:45Þ

then, assuming a white source spectrum (see Chapter 3)13

FBB ¼ 1

B

ð fmþB=2fm�B=2

Fcohð f Þ df ; ð11:46Þ

where B is the sonar bandwidth; and fm is its center frequency. The integration ofEquation (11.46) can be carried out more efficiently by first simplifying the integrand.One approach is to smooth the integrand by replacing Fcoh with Finc, so that

FBB � 1

B

ðFincð f Þ df ; ð11:47Þ

as used in Chapter 3. Here the ‘‘inc’’ subscript is an abbreviation for ‘‘incoherent’’ inthe ray sense of the word (which implies that the relative phase of multiple ray arrivalsis neglected) and not the mode sense (which would refer to the relative phase ofinterfering modes being neglected). Thus, the step from Equation (11.46) to Equation(11.47) neglects coherent interference effects such as cancellation between direct andsurface-reflected paths. This coherent effect is not negligible, even for incoherentbroadband processing, if the distance between the sonar (or target) and the seasurface is a few wavelengths or less at the center frequency, in which case use ofEquation (11.46) is required. If the bandwidth is small relative to the center fre-quency, the approximation of Harrison and Harrison (1995) may be used to avoid thecumbersome integration by replacing the average in frequency with one in range.

As an example of broadband propagation loss, consider the Lloyd mirrorproblem from Chapter 3, with a linear variation of attenuation with frequency of


13 A more general expression, vaid for a colored source spectrum, is

FBB �ÐFcohð f ÞW S

f dfÐW S

f df:

the form�ð f Þ ¼ �m þ �f ð f � fmÞ: ð11:48Þ

In this situation, Equation (11.47) becomes

FBB ¼ 2

r2expð�2�mrÞ sinhcð�f BrÞ: ð11:49Þ

For small B the broadband (BB) propagation factor FBB reduces to the usualexpression for spherical spreading

FBB � 2e�2�mr

r2: ð11:50Þ

More generally (still for small bandwidth), FBB may be approximated by Fincð fmÞ.Returning to Equation (11.49) and using the definition of sinhcðxÞ (see

Appendix A)

sinhcð�f BrÞ ¼expð�f BrÞ � expð��f BrÞ

2�f Br; ð11:51Þ

it follows that (for a sufficiently long-range or large bandwidth)

sinhcð�f BrÞ �expð�f BrÞ2�f Br

: ð11:52Þ

Therefore, in the same limit,

FBB �exp½�2rð�m � �f B=2Þ

�f Br3

: ð11:53Þ

11.3.4 Broadband noise level (NL)

The broadband noise level is the noise spectral density, integrated across the sonarbandwidth, and expressed in decibels

NL ¼ 10 log10

ð10NLf =10 df : ð11:54Þ

The previously introduced concepts of foreground and background noise, and ofacoustic and non-acoustic noise (see Section 11.2.4), apply more or less unchangedhere.

11.3.5 Processing gain (PG)

The array gain term (AG) from Chapter 3 is generalized here to processing gain (PG),which also includes filter gain (FG). The implied assumption of Chapter 3 is that thefilter response is flat (independent of frequency) within the sonar bandwidth. Theeffect of a departure from this assumption is considered here.


Let RBB denote the SNR after all beamforming and (temporal) filtering, whileRhp and Rarr are the SNRs at input and output of the beamformer, before filtering.

14

The respective gains due to beamforming and filtering, respectively, are

GA � RarrRhp

ð11:55Þ

and

GF � RBBRarr

: ð11:56Þ

Total processing gain is

GP �RBBRhp

¼ GAGF: ð11:57Þ

In decibels, these become

AG � 10 log10 GA ¼ 10 log10RarrRhp

; ð11:58Þ

FG � 10 log10 GF ¼ 10 log10RBBRarr

; ð11:59Þ

and

PG � 10 log10 GP ¼ AGþ FG: ð11:60Þ

11.3.5.1 Array gain (AG) and directivity index (DI)

For broadband sonar, array gain (AG) can be calculated using Equation (11.58).Because of the complexity involved in the full calculation, AG is sometimesapproximated by DI.15

11.3.5.2 Filter gain (FG)

Filter gain can be calculated using Equation (11.59). Let Hð f Þ denote the combinedtransfer function of all filters.16 The post-filter SNR RBB is then given by

RBB ¼

ðHð f ÞY S

f ð f Þ dfðHð f ÞYN

f ð f Þ df; ð11:61Þ


14 Or, strictly speaking, after a hypothetical flat response filter.15 This statement begs the question of how to define DI for a broadband sonar. One possibility

is as the (broadband) array gain, evaluated for a white plane wave signal in white isotropic

noise.16 The transfer function Hð f Þ incorporates the effects of hydrophone sensitivity, analoguefilters (such as an anti-alias filter), analogue-to-digital converter, and any digital filter. Recall

that Rarr is a hypothetical SNR assuming a flat response filter, whereas RBB is the true SNR,

after all filtering.

and hence, from the definition of GF (Equation 11.56) it follows that

GF ¼

ðHð f ÞY S

f ð f Þ dfðHð f ÞYN

f ð f Þ df

ðYN

f ð f Þ dfðY S

f ð f Þ df: ð11:62Þ

As a first approximation, it is common to assume GF � 1 (i.e., FG � 0 dB). Thisapproximation is a good one if either the filter response or the SNR spectrum is flat(i.e., either Hð f Þ or Y S

f ð f Þ=YNf ð f Þ is approximately constant). However, if the

narrowband signal-to-noise ratio varies with frequency, the broadband SNR, afterall filtering, depends on the frequency response of the true filter. In other words, if thereceiver response is not flat, it becomes necessary to account for departures from thehypothetical flat response filter considered so far.

An important special case is that of a pre-whitening filter,17 for which

Hð f Þ ¼ 1

YNf ð f Þ

ð11:63Þ

and hence

GF ¼ 1

B

ðY S

f ð f ÞYN

f ð f Þdf

ðYN

f ð f Þ dfðY S

f ð f Þ df: ð11:64Þ

11.3.5.2.1 Filter gain for a white signal spectrum

Consider a white signal spectrum and a power law frequency dependence for the noiseof the form

YNf / f x; ð11:65Þ

such that

GF ¼ 1

B2

ð fþf�

f x df

ð fþf�

f �x df ; ð11:66Þ

wheref� ¼ fm � B=2; ð11:67Þ

B is the sonar bandwidth; and fm is the center frequency. If the noise is also white(i.e., x ¼ 0), the filter has no effect and the filter gain is 0 dB.18 The special casex ¼ �1 (known as pink noise ) leads to

GF ¼ 1

Dloge

2þ D2� D

; ð11:68Þ


17 A pre-whitening filter is one that is designed to achieve white noise (i.e., noise whose power

spectral density is independent of frequency) at the filter output.18 More generally, if the signal and noise have the same spectral gradient (in decibels per

octave), the signal-to-noise ratio is independent of frequency and consequently there is no gain

possible from further filtering, irrespective of the functional form of Hð f Þ.

where D is the fractional bandwidth

D � B=fm: ð11:69Þ

For other power laws, Equation (11.66) can be written

GF ¼ ð f 1þxþ � f 1þx� Þð f 1�x� � f 1�xþ Þðx2 � 1ÞB2

: ð11:70Þ

An important special case is x ¼ �2 (‘‘red noise’’), as the high-frequency spectra forwind, shipping, and rain noise all approximate to this value (Chapter 8). For rednoise, Equation (11.70) simplifies to19

GF ¼ 12þ D2

12� 3D2: ð11:71Þ

If jxj � 2 and for a single octave, the effects are less than 1 dB, increasing to up to10 dB for five octaves, as indicated by the shaded entries in Table 11.4.

11.3.5.2.2 Filter gain for a colored signal spectrum

Consider the noise spectrum of Equation (11.65) and the signal spectrum

Y Sf / f y: ð11:72Þ

For this combination, filter gain becomes

GF ¼ 1

B

ð fþf�

f y

f xdf

ð fþf�

f x df

ð fþf�

f y df

: ð11:73Þ


Table 11.4. Filter gain vs. bandwidth in octaves for a white signal and colored noise.

No. of octaves Pink noise (or blue) Red noise (or violet)

ðx ¼ �1Þ ðx ¼ �2Þ

Noct D (Equation GF FG GF FG

11.69)

1 0.67 1.04 0.2 dB 1.17 0.7 dB

3 1.56 1.34 1.3 dB 3.04 4.8 dB

5 1.88 1.84 2.7 dB 11.0 10.4 dB

7 1.97 2.46 3.9 dB 43.0 16.3 dB

9 1.99 3.13 5.0 dB 171 22.3 dB

19 Equation (11.71) holds also for x ¼ þ2.

Provided that none of x, y, or y� x is equal to 1, it follows that

GF ¼ yþ 1ðxþ 1Þðy� xþ 1Þ

fy�xþ1þ � f y�xþ1�

B

f xþ1þ � f xþ1�

fyþ1þ � f yþ1�

: ð11:74Þ

11.3.6 Broadband detection threshold (DT)

11.3.6.1 Calculation of DT for given pfa

Apart from PG itself, the detection threshold (DT) is the sonar equation term forwhich the difference between coherent and incoherent processing most makes itselffelt. The objective of incoherent processing is not to increase SNR, but to reduce thefluctuations in both signal and noise in such a way as to facilitate detection at a lowerSNR—in other words, to reduce DT. The net result is a completely different form forDT, which varies with the sonar bandwidth B and incoherent integration time T inthe manner described below. The definition of DT is in terms of the SNR at theoutput of coherent processing:

DTð pfaÞ � 10 log10ðRBBÞ50: ð11:75Þ

A simple expression, valid for large BT and any false alarm probability, independentof signal fluctuations for BT exceeding 20 or so, is20

DTð pfaÞ � 10 log10½erfc�1ð2pfaÞ � 10 log10ffiffiffiffiffiffiffiBT

p; ð11:76Þ

with an accuracy of ca. �1 dB. A good approximation for a non-fluctuating signal(for pd ¼ 1

2), due to Albersheim, is (see Chapter 7)

DTð pfaÞ � 10x log10 loge0:62

pfa

� �� 10 log10

ffiffiffiffiffiffiffiffiffiffi2BT

p; ð11:77Þ

where the factor x is

x ¼ xðBTÞ ¼ 0:62þ 0:456ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2BT þ 0:44

p : ð11:78Þ

Equation (11.77) is valid for a stable (i.e., non-fluctuating) signal in Gaussian noise. Itholds for any BT across a wide range of pfa values. Figure 11.8 shows the quantityDTþ 5 log10ð2BTÞ, evaluated using Equation (11.77), plotted vs. BT for pfa between10�12 and 10�4. From Chapter 7 it can be seen that, for pfa in the range10�12 < pfa < 0:3, Equation (11.77) is accurate to within �0.5 dB. The main benefitof Albersheim’s approximation is its validity for small BT , at least for a non-fluctuating signal. For large BT , Equation (11.76) (which is not restricted to a stablesignal) is more accurate.

If BT is large, it is not unusual for the broadband detection threshold to becomenegative (i.e., DT< 0 dB), which means that detection is possible even when noise


20 This and subsequent expressions for the detection threshold follow by substitutingM ¼ 2BT

and pd ¼ 12 into the relevant results from Chapter 7 for incoherent combination of multiple

pings.

power exceeds that of the signal. This idea might sound counterintuitive, but whatmatters for detection of signal is the extent to which ‘‘signal plus noise’’ can bedistinguished from ‘‘noise alone’’, which depends not only on the signal-to-noiseratio but also on the magnitude of any fluctuations present. To illustrate this point,consider the extreme case for which noise is known to be precisely constant. Anydeparture from this constant, however small, can then be attributed to the presence ofa signal. Incoherent processing has the effect of smoothing out natural fluctuations inboth signal and noise, making it possible to detect a weak signal in the presence of astronger noise background. For example, consider the case BT ¼ 500 and pfa ¼ 10�4.From Figure 11.8, the value of the quantity DT þ 5 log10ð2BTÞ can be read off as6 dB, and hence

DT ¼ 6� 5 log10ð1000Þ � �9 dB:

Thus, for this example, a 50% chance of detection occurs when noise power is eighttimes greater than that of the signal.

11.3.6.2 Estimation of false alarm probability

The estimation of pfa can be simplified by assuming that the directivity gain GD is


Figure 11.8. Albersheim’s approximation for the detection threshold, in the form

DTþ 5 log10ð2BTÞ vs. BT for pd ¼ 0.5 and various pfa between 10�12 and 10�4 for a non-

fluctuating signal in Rayleigh noise.

fixed for all octaves (implying a nested array21 for broadband sonar). In this situation,Nbeams can be replaced by GD so that (from Chapter 3)22

pfa ¼nfaDt

NintervalsNbeams

ð11:79Þ

becomes

pfa ¼nfaDt

NintervalsGD: ð11:80Þ


The broadband passive sonar worked example of Chapter 3 (henceforth abbreviatedBBp) is used here as a basis for a more realistic scenario. The main changes are

— conversion to shallow water (30m water depth, isothermal sound speed profile.and a sand seabed23), with the target (communications transmitter) at a depth of20m;

— reduction of integration time from 10 s to 0.1 s;— introduction of rough surface scattering;— introduction of shipping noise associated with a moderate shipping density of

500/Mm2;— use of a vertical receiver array instead of a horizontal one.

Detection performance is estimated below for this revised acoustic communicationsintercept problem. The disruptive effect of rainfall is considered in Section 11.3.7.5.

11.3.7.1 Propagation loss

Propagation loss is plotted vs. range and depth in Figure 11.9 for the shallow-watersand (SWS) case, and compared also with the PL for the Chapter 3 BBp problem,which is plotted alongside for comparison. Losses are generally lower than for theBBp situation, due mainly to additional contributions to the signal resulting frommultiple reflections from the seabed (see Chapter 9).

11.3.7.2 Signal-to-noise ratio

The source spectrum level SLf is equal to 100.9 dB re mPa2 m2/Hz (unchanged from

BBp). The resulting signal spectrum (SLf � PLm) and noise spectrum (NLf ) are


21 A nested array is a hydrophone array with non-uniform spacing between hydrophones,

typically arranged with a high density (small spacing) close to the middle of the array and a low

density (large spacing) at each end. The spacing is chosen in such a way that two or more sub-

arrays can be made using selected hydrophones. For example, a short array of densely spaced

hydrophones (the middle ones) is used at high frequency, whereas a long array with widely

spaced hydrophones is used at low frequency.22 If the footprint �O is the same for all beams, the number of beams needed to cover 4steradians of solid angle is 4=�O, which is equal to GD (Chapter 6).23 The parameters correspond to a grain size of 1.5�, or medium sand (see Chapter 4).


Figure 11.9. Propagation loss [dB rem2] vs. range and depth for SWS (upper) and for the

Chapter 3 BBp worked example (lower) (INSIGHT).

plotted in Figure 11.10 as the thin solid and dashed curves, respectively (labeled‘‘hp’’). The two curves labeled ‘‘in-beam’’ are the in-beam signal and noise (i.e., arrayresponse), calculated as

IBSLf ¼ 10 log10 YSf ð11:81Þ

and

IBNLf ¼ 10 log10 YNf ; ð11:82Þ

where

Y Sf ¼

ðQS

fOBðOÞ dO ð11:83Þ

and

YNf ¼

ðQN

fOBðOÞ dO: ð11:84Þ

The in-beam signal and background spectrum levels (evaluated at the centerfrequency) are plotted vs. target distance in Figure 11.11 for the SWS case (thinlines) and for the original BBp problem (thick lines). As a result of reduced propaga-tion loss, both signal and noise are higher than for BBp. At long range the signalincreases by more than the noise, so the SNR increases there.


Figure 11.10. Signal and noise spectra for SWS at range 10 km for array depth 10m and target

depth 20m. ‘‘hp’’: values at the hydrophone, before beamforming; ‘‘in-beam’’: in-beam levels

(IBSLf , IBNLf ) (INSIGHT).

11.3.7.3 Detection threshold and signal excess

The value of the detection threshold (DT) increases by about 10 dB due to thereduced incoherent integration time compared with BBp. If the permissible falsealarm rate is held fixed (at one per hour), the probability of false alarm must decreaseto pfa ¼ 2:8� 10�5, which increases DT by a further 1.7 dB. The combined effectgives DT¼ �6.6 dB, calculated using Albersheim’s approximation (Equation 11.77).

Given that processing gain is the sum of array and filter gains

PG ¼ AGþ FG; ð11:85Þ

the broadband sonar equation (Equation 11.42) can be written as

SEBB ¼ IBSL� IBNL �DTþ FG; ð11:86Þ

where IBSL is given by Equation (11.25) and

IBNL ¼ NLþNG: ð11:87Þ

Equation (11.86) can be written in terms of the in-beam spectral density levels IBSLfand IBNLf

SEBB ¼ IBSLf � IBNLf �DTþ FGþ fIBSL� IBNL� ðIBSLf � IBNLf Þg;ð11:88Þ


Figure 11.11. In-beam signal and noise levels vs. range for SWS (thin lines) and BBp (thick

lines); the dip in IBSL at 0.1 km is caused by the beam pattern of the vertical line array, which

results in negative signal gain at this distance (INSIGHT).

which, neglecting the term in curly brackets, approximates to24

SEBB � IBSLf � IBNLf �DTþ FG: ð11:89Þ

The right-hand side of Equation (11.89) is plotted vs. range and intercepting sonardepth in Figure 11.12, with spectral densities evaluated at the center frequency andFG¼ 0 dB. The target (in this context the communications transmitter) is at a depthof 20m. It is apparent that the achieved performance is sensitive to the chosen sonardepth, with the optimum value in the region of 15m to 20m, and poor performanceclose to the boundaries. A good choice of depth is therefore important, illustratingone of the uses of sonar performance modeling, namely as a decision aid for sonardeployment. A summary of relevant sonar equation terms is presented in Table 11.5.For a more detailed description of the scenario see Figure 11.13.


Figure 11.12. Signal excess [dB] vs. range and depth for SWS. White indicates areas of

high detection probability ( pd > 0.5) and black means low detection probability ( pd < 0.5).

In the gray region, detection probability is close to 50% (signal excess is in the range

�1 dB< SE< þ1 dB) (INSIGHT).

24 The introduction of the term in curly brackets in Equation (11.88), only to neglect it in the

very next line, serves to make explicit the nature of the approximation leading to Equation

(11.89).


Table 11.5. Sonar equation calculation for shallow-water sand (SWS).


Source spectrum level SLf 100.9 dB re mPa2 m2 Hz�1

Noise spectrum level (center frequency) NLf 51.7 dB re mPa2 Hz�1

Array gaina AG 14.0 dB re 1

Detection threshold DT �6.6 dB re 1

a Array gain is approximated for this example by the directivity index, evaluated at the centerfrequency.

OPTIONS Coherence IncoherentProfile mode Value vs depthSignal Processing Broad bandPassive Beam Pattern Vertical steered—RectangularWind Noise Wind noise modelShipping Noise Shipping noise modelRain Noise Rain noise offSelf Noise Self noise offNo. of bottom reflections 25No. of bottom refractions 5

PASSIVE PARAMETERS Analysis frequency 3. kHzArray depth 10. mArray length 6. mElectronic steering angle 0. degreesAnalysis bandwidth 2.5e-004 kHzDetection threshold (broadband) -6.6 dB

SOUND SPEED PROFILE USER-DEFINED LAW:Water depth 30 mc (m/s) z (m) dc/dz (/s) H (m)1500 0 1.6e-2 101500.16 10 1.6e-2 101500.32 20 1.6e-2 101500.48 30

VOLUME LOSS ALTERNATIVE LAW: Thorp

SURFACE LOSS ALTERNATIVE LAW: Marsh-Schulkin-KnealeWind speed 5. m/s

BOTTOM LOSS GEO-ACOUSTIC PARAMETERS:Sediment depth 500. mcs/cw 1.2Sediment velocity gradient 1.e-006 /sMaximum sediment velocity 1900. m/sSediment density 2.1Sediment attenuation coefficient 0.489 dB/m/kHzSediment attenuation gradient 1.e-006 dB/m/kHz/m

TARGET PARAMETERS Target range *Target depth 20. mBroad-band spectrum level at 1kHz 100.9 dB/HzBroad-band spectral slope 0. dB/octaveTarget bearing clockwise from North 0. degrees

BACKGROUND PARAMETERS Ship length 100. mShip speed 5. m/sShipping density 5.e-004 /km2Wind speed 5. m/s

Figure 11.13. Input parameters for shallow-water sand (SWS).

11.3.7.4 Effect of filter gain

The effect of a non-zero filter gain is now considered. Applying Equation (11.74) tothe spectral slopes measured from Figure 11.10, the gain from noise whitening isGF � 1:12, which in decibels amounts to FG¼ 0.5 dB. Applying this correction toFigure 11.12 results in an increase in the detection range of about 1 km.

11.3.7.5 Effect of rainfall

The main effect of rainfall is to increase the level of ambient noise by an amountthat depends on frequency and rainfall rate. For a rain rate Rrain of 3mm/h, theincrease in noise level at 3 kHz is about 4 dB (see Figure 11.14). A second effect is toflatten the noise spectrum and hence reduce filter gain slightly. Figure 11.15 showssignal excess vs. range and rain rate (calculated assuming FG¼ 0 dB, irrespective ofrain rate). The detection range r50 varies from about 11 km with no rain to 5 km forRrain ¼ 10mm/h.


Figure 11.14. In-beam signal (solid curve) and noise (dashed curves) spectra [dB re mPa2/Hz]

for SWS at a range of 10 km. The rainfall rate increases from 0 to 9mm/h in steps of 3mm/h

(INSIGHT).

11.4 ACTIVE SONAR WITH COHERENT PROCESSING:

MATCHED FILTER


The processing of both continuous wave (CW) and frequency-modulated (FM)pulses is considered in this section. A CW pulse can be thought of as a special caseof an FM pulse, with a phase acceleration of zero (see Chapter 6). The active sonarequation, in its most general form, is

SE ¼ ELE � ðBLE � PGÞ �DT; ð11:90Þ

where ELE is the echo energy level; and BLE is that of the total masking background.An equivalent form is

SE ¼ EL� ðBL� PGÞ �DT; ð11:91Þ

where EL and BL are sound pressure levels of the echo and background, respectively,averaged over the echo duration; and PG is the gain in SNR due to all processing.25

The main qualitative difference in background level between active and passivesonar is the presence of reverberation in the former. Methods for the calculation of


Figure 11.15. Signal excess [dB] vs. range and rainfall rate for SWS (INSIGHT).

25 Apart from initial filtering into the processing bandwidth as described in Chapter 3.

both echo and reverberation are described in Chapter 9. Discussions of the individualterms of Equation (11.91) follow.

11.4.2 Echo level (EL), target strength (TS), and equivalent target strength (TSeq)

Echo level (EL) depends on source level (SL), propagation loss (PL), and targetstrength (TS). In the presence of multipaths (multiple propagation paths), there isusually no simple equation enabling the calculation of EL from SL, TS, and PLseparately. Instead, it is necessary, in general, to sum contributions to the meansquare pressure (MSP) of the echo, over many possible multipaths. Specifically, ifthe sonar source factor is S0 and the differential scattering cross-section of the targetis Oð�in; �outÞ, the signal MSP can be written26 as a double integral over the incidentand scattered grazing angles �in and �out

QS ¼ S0

ðbTxð�inÞGTxð�inÞ Oð�in; �outÞGRxð�outÞ d�in d�out; ð11:92Þ

where GTx and GRx are the differential propagation factors to and from the target (seeChapter 9); and bTx is the transmitter beam pattern.

It is usual to characterize the properties of the target in terms of its targetstrength. This quantity was defined in Chapter 3 for a point target in terms of ratiosof MSP, and shown there to be equal to the differential scattering cross-section,evaluated in the backscattering direction and expressed in decibels. For a finite target,the relationship between TS and the backscattering cross-section is adopted here as adefinition of target strength. Thus,

TS � 10 log10 back

4dB re m2: ð11:93Þ

The target strength of an object, as defined by Equation (11.93), is a function only ofthe properties of the object itself and of sonar frequency. Unfortunately, there is nogeneral way of converting Equation (11.92) into an equation containing only TS andthe other terms in the active sonar equation. An alternative approach is to introducethe concept of an equivalent target strength, which is defined here as

TSeq � EL� SLþ PLTx þ PLRx; ð11:94Þ

such that without approximation

EL ¼ SL� PLTx þ TSeq � PLRx: ð11:95Þ

By comparison, the equivalent target strength (TSeq), as defined above, is a function

11.4 Active sonar with coherent processing: matched filter 607]Sec. 11.4

26 It is assumed here that the receiver is in the same vertical plane as the transmitter and target.

This assumption makes it possible to restrict attention to propagation in that vertical plane.

Equation (11.92) further assumes that the duration of the received pulse is equal to that of the

transmitted one. In this situation the target strength based on energy (TSE) and the one based

on mean square pressure (TSMSP) are equal, so no further distinction is made between them.

also of the propagation environment and sonar geometry. For this reason, TSeq isoften approximated by TS in the same way as AG is often approximated by DI.

For FM sonar (and for broadband sonar in general), it might be necessary tocompute Equation (11.92) for all frequencies of interest and consider the resultingspectrum. This frequency dependence is ignored for the time being, focusing insteadon the contributions from different multipaths at a single frequency, assuming im-plicitly that beam pattern, propagation factor, and other frequency-dependentparameters do not change much over the bandwidth of interest. This weakness isaddressed in the worked example of Section 11.4.6, where frequency dependence isconsidered for a broadband sonar.

11.4.2.1 Outward propagation loss (PLTx) and sonar source level (SL)

To complete the definition of TSeq, it is necessary to define the quantities PLTxand PLRx for use in Equation (11.94). Propagation loss for the outward path isstraightforward:

PLTx � SL� Ltgt; ð11:96Þ

where SL is the sonar source level; and Ltgt is the sound pressure level that would bemeasured at the target position resulting from the transmitted sonar signal, if thetarget were not present.

To define the sonar source level, let JTxð�Þ denote the far-field radiant intensity ofthe transmitted sound as a function of elevation angle �. Source level is then the valueof this quantity evaluated on the sonar axis (i.e., its maximum, denoted here by thesubscript zero), converted to MSP and expressed in decibels. In other words

SL � 10 log10 S0; ð11:97Þwhere

S0 � �cðJTxÞ0: ð11:98Þ

Outward propagation loss can be written in terms of the projector beam patternbTxð�Þ and differential propagation factor GTxð�Þ as

PLTx ¼ �10 log10ðbTxð�ÞGTxð�Þ d�

� �; ð11:99Þ

where the transmitter beam pattern is

bTxð�Þ �JTxð�ÞðJTxÞ0

: ð11:100Þ

11.4.2.2 Return propagation loss (PLRx)

By analogy with Equations (11.96) to (11.98), a natural definition for returnpropagation loss is

PLRx � 10 log10½�cðJtgtÞ0 � EL; ð11:101Þ

where the notation ðJtgtÞ0 indicates the maximum value of the radiant intensity of the


scattered field as a function of the scattered elevation angle �out. In other words

ðJtgtÞ0 � max Jtgtð�outÞ; ð11:102Þ

where

Jtgtð�outÞ ¼S0�c

ðbTxð�inÞGTxð�inÞ Oð�in; �outÞ d�in: ð11:103Þ

With the above definitions of PLTx and PLRx, the equivalent target strength(Equation 11.94) becomes

TSeq ¼ 10 log10½�cðJtgtÞ0 � Ltgt: ð11:104Þ

11.4.2.3 Special case: separable target cross-section

A case of special interest is now considered to illustrate the calculation of propagationloss and equivalent target strength. The example chosen is for a simple target whosedifferential scattering cross-section is a separable function of incident and scatteredangles:

Oð�in; �outÞ ¼1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi backð�inÞ backð�outÞ

q: ð11:105Þ

In this situation, echo MSP is

QS ¼ S04

ðbTxð�inÞGTxð�inÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi backð�inÞ

qd�in

� � ðGRxð�outÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi backð�outÞ

qd�out

� �:

ð11:106Þ

The expression for PLTx (Equation 11.99) does not simplify for this case, but PLRx(Equation 11.101) becomes

PLRx ¼ �10 log10ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

backð�outÞ back0

sGRxð�outÞ d�out

" #; ð11:107Þ

where

back0 ¼ max backð�Þ: ð11:108Þ

Equivalent target strength is then

TSeq ¼ 10 log10

ffiffiffiffiffiffiffiffiffiffiffi back0

q4

ðbTxð�inÞGTxð�inÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi backð�inÞ

qd�inð

bTxð�inÞGTxð�inÞ d�in

2664

3775; ð11:109Þ

which simplifies for the case of an omni-directional scatterer (i.e., one for which O isindependent of angle) such that

backð�Þ ¼ 4 O; ð11:110Þ


to the true target strength

TSeq ¼ 10 log10 O ¼ TS: ð11:111Þ

11.4.3 Background level (BL)

Background level is a combination of noise and reverberation given by

BL ¼ 10 log10 QBð�Þ; ð11:112Þ

where

QBð�Þ ¼ QN þQRð�Þ: ð11:113Þ

The individual noise and reverberation terms are

QN ¼ðQN

f df ð11:114Þ

and

QRð�Þ ¼ðQR

f ð�Þ df ; ð11:115Þ

where the integration is over the processing bandwidth. In Equation (11.114), QNf is

the noise spectral density

QNf ¼ 10NLf =10 mPa2=Hz; ð11:116Þ

where NLf is the noise spectrum level (see Section 11.2.4). Similarly, in Equation(11.115), QR

f ð�Þ is the reverberation spectral density evaluated at the arrival time ofthe target echo.

11.4.4 Processing gain (PG)

Processing gain is the gain from all receiver processing except that from a nominal(flat response) bandpass filter in the sonar processing bandwidth. That is, if Rout is theoutput SNR after all processing, then

PG ¼ 10 log10RoutRin

; ð11:117Þ

where Rin is the input SNR

Rin ¼QS

QN þQRð�Þ ð11:118Þ

and the MSPs QS, QN, and QR are all defined in the processing bandwidth.


It is conventional to distinguish between the gain from beamforming (AG) andfrom time domain processing such as a matched filter (MG), so that

AG ¼ 10 log10RarrRin

; ð11:119Þ

MG ¼ 10 log10RoutRarr

; ð11:120Þ

and thereforePG ¼ AGþMG: ð11:121Þ

11.4.4.1 Array gain (AG) and directivity index (DI)

The calculation of AG in the presence of reverberation is described below. If thesignal can be approximated as a plane wave, AG can be written (see Chapter 3)

AG ¼ 10 log10QN þQRð�Þ

QNGN þQRð�ÞGR; ð11:122Þ

where QN and QR are the MSPs due to noise and reverberation, respectively,integrated over the processing bandwidth. Thus

QN ¼ðð

QNfO df dO ð11:123Þ

and

QRð�Þ ¼ðð

QRfOð�Þ df dO; ð11:124Þ

where QNfO and Q

RfO are the respective differential spectral densities (i.e., the MSP per

unit solid angle and unit bandwidth) of noise and reverberation. Noise gain is (seeChapter 6)

GN ¼

ððQN

fObRxðOÞ df dO

QN; ð11:125Þ

where bRxðOÞ is the receiver beam pattern. Similarly, reverberation gain is

GR ¼ GRð�Þ ¼

ððQR

fOð�ÞbRxðOÞ df dO

QRð�Þ : ð11:126Þ

If the noise field is isotropic, noise gain is equal to the reciprocal of the directivityfactor GD ¼ 10DI=10, where DI is the directivity index. It is common to approximateGN by 1=GD, even when the noise field is not perfectly isotropic, especially if noisedirectionality is unknown. By contrast, reverberation is usually of a strongly direc-tional nature and so may not be approximated in this way. This point is addressed inthe worked example of Section 11.4.6. The gain depends not only on the receiverbeam pattern (through DI), but also on the source level, transmitter beam pattern,and on the details of propagation to and from the object responsible for thescattering.


11.4.4.2 Matched filter gain (MG)

If the received pulse is a perfect scaled replica of the transmitted one, the gain of amatched filter for a pulse of bandwidth B and duration T is given by

MG ¼ 10 log10ðBTÞ: ð11:127Þ

If the received pulse is a less-than-perfect replica of the transmitted one, the gain willbe less than this, and the difference is known as coherence loss (CL). Thus, moregenerally (see Chapter 6)

MG ¼ 10 log10ðBTÞ � CL: ð11:128Þ

11.4.5 Detection threshold (DT)

11.4.5.1 Calculation of DT from pfa

The active sonar detection threshold depends on signal statistics in the same way asfor narrowband passive sonar, described in Section 11.2.7.1. An important qualita-tive difference can arise through background statistics. In both cases the backgroundcan be thought of as a summation of individual contributions from discrete eventssuch as breaking waves at the sea surface. If the sonar averages over a large numberof such events, a Rayleigh distribution is expected to result for the backgroundamplitude. For modern high-resolution sonar, the number might not be so large,and in extreme cases discrete noise events (known as ‘‘clutter’’) would then beresolved. The effect of clutter on background statistics is the subject of ongoingresearch (e.g., Abraham, 2003; Nielsen et al., 2008).

11.4.5.2 Estimation of pfa

A general expression for pfa for an active sonar with pulse repetition time Dt is

pfa �nfaNtot

Dt; ð11:129Þ

where Ntot is the total number of detection opportunities per transmission, the valueof which depends on the processing used. For a generic sonar capable of resolution inangle (with Nbeams sonar beams), range (Nranges range cells), and frequency (NDoppler

Doppler cells), it is given by

Ntot ¼ NbeamsNDopplerNranges: ð11:130Þ

For a CW sonar the number of range cells is

Nranges ¼Dt�t; ð11:131Þ

or, equivalently (using the CW propertyDf

NDoppler

�t � 1)

NDopplerNranges � Dt Df : ð11:132Þ


For FM the number of range and Doppler cells is

Nranges ¼Dt1=Df

¼ Df Dt ð11:133Þ

andNDoppler ¼ 1: ð11:134Þ

Thus, Equation (11.132) holds for the FM case as well as for CW.Substituting Equations (11.130) and (11.132) into Equation (11.129) gives

pfa �nfa

NbeamsDf; ð11:135Þ

where

Df ¼NDoppler�f CW

�f FM.

�ð11:136Þ

If the array directivity factor is known, the number of sonar beams can beapproximated by GD ¼ 10DI=10. The bandwidth for use in Equation (11.135) canbe estimated by knowledge of the task the sonar is required to perform. For CWit is related to the maximum Doppler speed of interest (vDoppler)

Df ¼2vDoppler

cf ; ð11:137Þ

whereas, for FM it is determined by the required range resolution ð�rÞ

Df ¼ c

2�r: ð11:138Þ

Thus,

pfa ¼nfa

2cNbeams

c2=ðvDoppler f Þ CW

4�r FM.

�ð11:139Þ

Consider a combined (CW plus FM) sonar system with common values ofNbeams ¼ 100, c ¼ 1500m/s, and a maximum acceptable false alarm rate of one perhour. For an operating frequency of 1 kHz, if the maximum Doppler speed is 10m/sand the FM range resolution is 1m, then (from Equation 11.139)

pfa ¼2:1� 10�7 CW

3:7� 10�9 FM.

�ð11:140Þ

The point of this comparison is to demonstrate that, for a given false alarm rate nfa,the false alarm probability required of the FM system is two orders of magnitudesmaller than for the CW sonar. This is a consequence of the large FM bandwidth,which, while delivering a high SNR on the one hand, results in a high false alarm rateon the other unless compensated for by a low false alarm probability.


The example chosen here is inspired by the pioneering modeling work of Au et al.(2004), who consider the task faced by a killer whale (Orcinus orca) in hunting its


prey, the chinook salmon (Oncorhynchus tshawytscha). The same basic problem isconsidered here, with the following changes:

— the detailed fish model of Au et al. (2004) is replaced here by a point scatterer,with a representative TS value for salmon from Chapter 8;

— reverberation from the sea surface, neglected by Au et al. (2004), is included inthe sources of background.

The orca is swimming close to the sea surface (shown at depth dO in Figure 11.16),looking ahead and down at the salmon. The fish, of length 80 cm, is at a depth ofdF ¼ 25 m. For the present purpose the orca is assumed to use a baffled circular platetransducer, of diameter 10 cm, for both transmission and reception. The (measured)pulse shape and spectrum of an orca echolocation pulse are shown in Figure 11.17.The pulse has a duration of approximately 30 ms. The wide range of frequenciespresent in the orca’s transmitted pulse, which extends between 20 kHz and 80 kHz,means that the broadband nature of this problem, which has so far been ignored,needs to be considered.

Questions to be addressed are:

(1) What is the detection range in the absence of any masking noise or reverberation(i.e., if limited only by hearing sensitivity)? Assume that the spectral density ofthe transmitted pulse and the target strength of the fish are independent offrequency.

(ii) What is the detection range in the presence of wind noise if wind speed (at 10mheight) is 2m/s? (Surface reverberation is negligible at this wind speed). For thepurpose of calculating DT, assume that the signal satisfies 1DþR statistics in aGaussian background and that the orca accepts a false alarm probability of0.1%.

(iii) What is the sensitivity of the detection range to increasing wind speed ifreverberation is neglected? (Consider wind speeds of 2, 6, and 10m/s).

(iv) What effect does reverberation have on detection range?


Figure 11.16. Geometry for worked

example involving killer whale (orca)

hunting salmon (adapted from Au et al.,

2004, American Institute of Physics,

reprinted with permission).

#

11.4.6.1 Part (i) maximum audibility range (no background)

If the background term is neglected, there is no need for the sonar equation in theusual sense. Instead, the requirement for audibility is that the echo level must exceedthe animal’s hearing threshold. The echo level and hearing threshold are consideredbelow.

11.4.6.1.1 Echo level (EL)

Echo level is given by

EL ¼ 10 log10 QS; ð11:141Þ

where, generalizing Equation (11.106) to the broadband case and further assumingspherical spreading (neglecting surface reflection) and an omni-directional scatterer( backð�Þ independent of angle), gives for the signal at the center of the beam (usingbTx ¼ 1)

QS ¼ð fmaxfmin

ðS0Þfexpð�2�sÞ

s2

� �2 back

4df : ð11:142Þ

where ðS0Þf is the source factor spectral density, such that

SLf ¼ 10 log10ðS0Þf : ð11:143Þ

The integral over frequency in Equation (11.142) is necessary because of thebroadband nature of the animal’s sonar. The lower and upper limits of integration


Figure 11.17.

Example

measurements of

orca pulse shapes

and power spectra

(from Au et al.,

2004, American

Institute of Physics,

reprinted with

permission).

#

arefmin ¼ fm � B=2 ð11:144Þ

andfmax ¼ fm þ B=2: ð11:145Þ

Numerical values of fm and B are 50 kHz and 60 kHz, respectively.If target and source spectra are both white, the only frequency dependence arises

through the absorption term in the propagation factor, and Equation (11.142) cantherefore be written

QS ¼ ðS0Þf BFTx back

4FRx; ð11:146Þ

where (see Equation 11.49)

FTx ¼1

s2expð�2�msÞ sinhcð�f BsÞ ð11:147Þ

and

FRx ¼1

FTx

1

Bs4

ð fmþB=2fm�B=2

expð�4�sÞ df : ð11:148Þ

Substituting Equation (11.146) into Equation (11.141) gives

EL ¼ SL� ðPLTx þ PLRxÞ þ TS; ð11:149Þwhere

SL ¼ 10 log10½ðS0Þf B; ð11:150Þ

PLTx ¼ �10 log10 FTx; ð11:151Þ

PLRx ¼ �10 log10 FRx; ð11:152Þand

TS ¼ 10 log10 back

4: ð11:153Þ

The terms on the right-hand side of Equation (11.149) are discussed below.

Source level (SL). The value quoted for the peak-to-peak source level of anorca in Chapter 10 is SLp-p ¼ 218 dB re mPa2 m2, which is close to the highest valuesmeasured (see Figure 11.18). These measurements show significant variability, with asystematic decrease in level with decreasing distance to the target. For a representa-tive value, consider the black curve in Figure 11.18, which follows the relationshipdue to Au et al. (2004) (see also Chapter 10)

SLp-p ¼ 181:4þ 20 log10 s: ð11:154Þ

Putting s ¼ 25 m (the largest distance for which a source level measurement isavailable) gives a value of 209.4 dB re mPa2 m2 for SLp-p. Before it can be used inEquation (11.149), this value needs to be converted to an RMS level. The peak-equivalent RMS value (corresponding to a hypothetical average over a full cycle atpeak amplitude) is obtained by subtracting 9 dB (see Chapter 10), and the true RMSlevel is less than this because on average the amplitude is lower than its peak value.


The difference between RMS and peRMS levels depends on the pulse shape and thechoice of averaging time, with realistic values, from the last column of Table 10.19,between 1.0 dB and 3.4 dB. Taking the average of these two values (2.2 dB) asrepresentative for the orca pulse, the resulting (RMS) source level is 11.2 dB belowthe peak–peak value:

SL ¼ 209:4� 11:2 ¼ 198:2 dB re mPa2 m2: ð11:155Þ

Propagation loss (PLTx þ PLRx). Equation (11.148) is similar to thecorresponding integral for the broadband passive example (Equation 11.47), andcan be evaluated in the same way. Assuming a linear variation of attenuation withfrequency as in Equation (11.48), the result is

FTxFRx ¼1

s4expð�4�msÞ sinhcð2�f BsÞ: ð11:156Þ

Figure 11.19 (upper graph) shows PL (calculated using spherical spreading plusabsorption) plotted at three nominal frequencies of 20 kHz, 50 kHz, and 80 kHz.For use in the broadband sonar equation, broadband PL values are necessary andthese are shown in the lower graph, calculated using Equations (11.147) and (11.156).This second graph shows that, for a target at a distance of 500m, the broadbandpropagation loss for the outward path is 2 dB lower than PLð fmÞ (i.e.,PLTx ¼ 60.5 dB rem2 at this distance). For the return path, the loss is 2 dB lowerstill, so that PLRx ¼ 58.5 dB rem2, making the sum of these two terms (still for adistance of 500m)

PLTx þ PLRx ¼ 119 dB re m4: ð11:157Þ


Figure 11.18.

Variation in orca

source level (SLp-p)

with distance from

target (reprinted

with permission

from Au et al.,

2004, American

Institute of

Physics).

#


Figure 11.19. Upper: propagation loss PLð f Þ vs. distance for f ¼ 20, 50, 80 kHz; lower: broad-

band correction PLð fmÞ � PLBB, where PLð fmÞ is the value for f ¼ 50 kHz from the upper graph

and PLBB is one of PLTx and PLRx as indicated by the legend. The distance s is the slant range,

equal to ðx2 þ ðdF � dOÞ2Þ1=2 as illustrated by Figure 11.16.

The reference unit used here (m4) is explained as follows:

— before dividing by reference units, both PL terms have dimensions of area (seeChapter 3);

— addition of logarithms implies multiplication of propagation factors and hencealso of their reference units;

— the product of two areas is a squared area, with reference unit m2 �m2 ¼ m4.

Target strength (TS). Salmon is a physoclist, which means that its bladder isclosed (see Appendix C). The target strength for physoclists can be estimated usingthe formula (see Chapter 8)

TS� 10 log10 L2 ¼ �27:1� 1:7 dB: ð11:158ÞUsing L ¼ 0:8 m for the salmon length gives TS ¼ �29.0 dB rem2.

11.4.6.1.2 Hearing threshold (HT)

Based on measurements by Hall and Johnson (1972) and Szymanski et al. (1999),Wensveen and Van Roij (2007) propose a hearing threshold (in dB re mPa2) for theorca of the form

HTð f Þ ¼445:2F�0:05401

kHz � 344:3 0:5 � FkHz < 11:3

242:9F�0:7578kHz þ 0:5643F 1:076kHz 11:3 � FkHz < 46:2

2:792F 0:7537kHz � 2:064 46:2 � FkHz � 80,

8>><>>: ð11:159Þ

where FkHz is the frequency in kilohertz

FkHz ¼f

1 kHz: ð11:160Þ

The audiogram calculated using Equation (11.159) is plotted in Figure 11.20,including the original measurements on which it is based. Using the same formula,the minimum hearing threshold, of 39.0 dB re mPa2, occurs at 22.6 kHz.

11.4.6.1.3 Maximum audibility range

Strictly speaking, what is needed for a calculation of audibility is the hearingthreshold for a broadband pulse. Because such an audiogram is not available, arough estimate is made instead using the hearing threshold at the pulse centerfrequency. The threshold using Equation (11.159) at the pulse center frequency(50 kHz) is 51.2 dB re mPa2. Armed with this information, the figure of merit (thevalue of one-way PL for which the unmasked sound is just audible) can be calculatedas (see Table 11.6)

FOMHT ¼ SLþ TS�HT2

¼ 59:0 dB re m2; ð11:161Þ

which intersects PLð fmÞ at about 400m (see Figure 11.19). This is the maximumdistance at which the echo could be heard if the frequency were precisely 50 kHz and


there were no masking noise or reverberation. The actual broadband propagationloss is slightly lower than the mid-frequency value PLð fmÞ, and the broadbandhearing threshold is also lower than at fm (Figure 11.20). Both factors favor evenlonger ranges, illustrating this animal’s remarkable potential in a quiet environ-ment.27 The real-world limitations of the orca’s detection capability caused by thepresence of background noise are considered next.


Figure 11.20. Orca audiogram due to Wensveen and Van Roij (2007) and individual hearing

threshold measurements of Hall and Johnson (1972) and Szymanski et al. (1999).

Table 11.6. Sonar equation calculation for active sonar example (orca vs. salmon)—

limited by hearing threshold.



Target strength TS �29.0 dB rem2

Hearing threshold at 50 kHz HT 51.2 dB re mPa2

Figure of merit (hearing threshold limited) FOMHT 59.0 dB rem2

27 Most of the energy is in the lower half of the frequency range, to which the animal is most

sensitive.

11.4.6.2 Part (ii) detection range for low wind speed (noise-limited)

At what distance s is the orca able to detect the salmon using its active sonar? Toanswer this question, it is necessary to consider the background against which theecho is to be detected, which for low wind is determined by ambient noise. Signalexcess is given by Equation (11.91), which, neglecting reverberation, simplifies to

SE ¼ EL � ðNL�AGÞ �DT: ð11:162Þ

The four terms on the right-hand side are now considered in turn. First, echo level(EL) is plotted in Figure 11.21 (red curve, calculated using Equation 11.141). Thecalculation of noise level (NL), plotted as a dashed blue line, is described in Section11.4.6.2.1, followed by discussions of array gain (AG) in Section 11.4.6.2.2 anddetection threshold (DT) in Section 11.4.6.2.3. Finally, signal excess and detectionrange are the subject of Section 11.4.6.2.4.

11.4.6.2.1 Noise level (NL)

In the frequency range of interest here it is appropriate to use the APL formula for thewind noise source level from Chapter 8. The noise MSP at the sea surface is obtained


Figure 11.21. Echo level (EL) vs. distance between orca and salmon. Also shown (horizontal

lines) are noise level (NL), and the quantities (NL�AG) and (NL�AGþDT), all for windspeed v10 ¼ 2 m/s. The detection range (i.e., the intersection of EL with NL �AGþDT) isapproximately 240m.

by integrating the dipole source spectrum over the sonar bandwidth and multiplyingby . Thus,28

QN ¼

ðKwindAPL df ; ð11:163Þ

where

KwindAPL ¼ �ðvÞ

F 1:59kHz

mPa2 kHz�1 ð11:164Þ

and (assuming unstable conditions, with water temperature exceeding that of air)

�ðvÞ ¼ 107:12vv2:24: ð11:165ÞEvaluating the integral gives

QN ¼ �ðvÞ0:59

ðF�0:59min � F�0:59

max Þ mPa2: ð11:166Þ

The integrated noise level 10 log10 QN is plotted in Figure 11.21 as the dashed blue

horizontal line at 75 dB re mPa2.29

11.4.6.2.2 Array gain (AG)

Consider array gain in linear form, defined as

GBB � 10AG=10; ð11:167Þ

whereAG ¼ SG�NG: ð11:168Þ

The signal is assumed to originate from a single direction (no multipaths areinvolved), which means that SG¼ 0. In this situation, array gain is determined bynoise gain alone

GBB ¼ 10�NG=10: ð11:169ÞThe (broadband) in-beam noise is

QN

GBB¼

ðKwindAPL

1

GDð f Þdf ; ð11:170Þ

where the directivity factor GDð f Þ is30

GDð f Þ ¼ ðD=�Þ2: ð11:171Þ


28 The orca is assumed sufficiently close to the surface for the effect of absorption on ambient

noise to be negligible.29 Following Au et al. (2004), it is assumed here that the receiver bandwidth is equal to that of

the transmitter pulse.30 The use of the directivity factor GDð f Þ in the integrand of Equation (11.170) instead of thenoise gain at frequency f is made for simplicity only. In the model problem, the sonar is

pointing down (away from the sea surface and hence also from the noise source), so a more

careful calculation of AG is expected to result in a value greater than DI.

It follows that

QN

GBB¼ �c2

2:59D2ðF�2:59

min � F�2:59max Þ mPa2

kHz2: ð11:172Þ

Rearranging Equation (11.172) for GBB gives

GBB ¼ 2:59

0:59

f �0:59min � f �0:59max

f �2:59min � f �2:59max

2D2

c2: ð11:173Þ

Inserting numerical values, array gain is found to be 16.5 dB.31

11.4.6.2.3 Detection threshold (DT)

Assuming 1DþR statistics for the signal, the detection threshold (DT) can beapproximated by Equation (11.22). Putting pfa ¼ 10�3 in this equation givesDT¼ 8.7 dB.

11.4.6.2.4 Signal excess (SE) and detection range

Signal excess is given by

SE ¼ EL� ðNL�AG þDTÞ: ð11:174Þ

The condition for detection is SE> 0 dB, so the detection range is the intersectionbetween EL and NL �AG þDT in Figure 11.21, which occurs at a range of ca.240m.

An alternative calculation of the detection range is by means of the figure of merit

FOM ¼ SLþ TS�NLþAG �DT2

; ð11:175Þ

which, using Table 11.7, is found to be 51.0 dB rem2. This value can be used incombination with Figure 11.19 for broadband propagation loss. In this way, thesame result for the detection range is found by equating FOMwith ðPLTx þ PLRxÞ=2.

11.4.6.3 Part (iii) detection range for high wind speed (noise-limited)

The result of Figure 11.21 for wind speed 2m/s is repeated in Figure 11.22 for windspeeds of 6m/s and 10m/s. The noise-limited ranges for the three wind speeds arestated in the figure caption.

The above calculations assume that wind is the only source of ambient noise. Auet al. (2004) point out that the increase in ambient noise due to rain has a detrimentaleffect on the orca’s detection performance. It is left as an exercise for the reader toshow this.


31 For sufficiently large bandwidth ( fmax � fmin), Equation (11.173) simplifies to

GBB � 4:4GDð fminÞ.

11.4.6.4 Part (iv) effect of reverberation (for high wind speed)

As wind speed increases, so do roughness of the sea surface and the near-surfacebubble population density, and consequently so too does surface reverberation. Thepurpose of this last exercise is to illustrate the effect of this increased reverberation onpredicted sonar performance.


Table 11.7. Sonar equation calculation for active sonar example (orca

vs. salmon)—limited by wind noise.



Target strength TS �29.0 dB rem2

Noise level NL 75.0 dB re mPa2

Array gain AG 16.5 dB re 1

Detection threshold DT 8.7 dB re 1

Figure of merit (noise-limited) FOMNL 51.0 dB rem2

Figure 11.22. Echo level (red curve) vs. distance between orca and salmon. The intersections

between echo level (EL) and the cyan horizontal lines (NL�AGþDT) are the predicteddetection ranges of 117, 149, and 241m, respectively, for v10 ¼ 10, 6, and 2m/s.

Signal excess is

SE ¼ EL� ðBL� PGþDTÞ: ð11:176Þ

Echo level is not affected by the presence of reverberation and it is assumed forsimplicity that the detection threshold is also unaffected. Of the four terms on theright-hand side, this leaves BL and PG to be determined.

11.4.6.4.1 Background level (BL)

Background level (see Equation 11.113) is

BL ¼ 10 log10½QRð�Þ þQN: ð11:177Þ

The noise term QN has already been considered (see Section 11.4.6.2). Contributionto the background from reverberation is calculated below.

As previously for echo and ambient noise, reverberation needs to be integratedover the entire frequency band of the animal’s sonar. There are several frequency-dependent effects, associated with the source spectrum ðS0Þf , propagation factor, seasurface backscattering coefficient sð f Þ, and beam pattern bTxðuÞ. Taking all of theseinto account, the contribution to reverberation from a sea surface area element dA ofazimuthal width d� is

dQRf ð�Þ ¼ ðS0Þf

expð�2�sÞs2

� �2

bTxðuÞ sð�S; f Þ dA; ð11:178Þ

where the area element is

dA ¼ sc �t

2d� ð11:179Þ

and s is the one-way pathlength corresponding to a delay time �

s ¼ c�=2: ð11:180Þ

Reverberation arriving at this delay time is caused by scattering at the sea surfacefrom paths whose elevation angle is

�S ¼ arcsindOs: ð11:181Þ

Following Au et al. (2004), the beam pattern is assumed to be that of a circular disk(Chapter 6)

bTxðuÞ ¼2J1ðuÞu

� �2

; ð11:182Þ

where

uð�Þ ¼ ðD=�Þ sin � ð11:183Þ

and � is the angle measured from the projector axis (� ¼ 0 corresponds to the centerof the beam). Approximating the beam pattern by a top-hat function of horizontal


width Fð�Þ and integrating Equation (11.178) over azimuth gives

QRf ð�Þ � Fð�S þ �FÞðS0Þf

expð�2�sÞs2

� �2

sð�S; f Þsc �t

2; ð11:184Þ

where �F is the elevation angle of a straight line between fish and orca

�F ¼ arcsindF � dO

s: ð11:185Þ

Assuming that the ray angles of interest are close to horizontal (i.e., if the distance s islarge compared with the fish depth), the angle F in Equation (11.184) may beapproximated using

Fð�S þ �FÞ2 � F0ð f Þ2 � ðdF=sÞ2; ð11:186Þ

where F0 is the full width

F0ð f Þ ¼4c

Df: ð11:187Þ

Integrating the spectral density over frequency and approximating the right-handside of Equation (11.186) by its first term only32 then gives

QRð�Þ � sc �t

2

ðF0ð f ÞðS0Þf

expð�2�sÞs2

� �2

sð�S; f Þ df : ð11:188Þ

The integrand factors are now considered in turn. First, the beamwidth F0 is given byEquation (11.187). Second, for a center frequency of fm and bandwidth B, theassumed source spectrum is flat from fm � B=2 to fm þ B=2, equal to

ðS0Þf ¼S0B

ð11:189Þ

inside this range and zero outside it. Next is the propagation factor, which varies withfrequency via the frequency-dependent attenuation �, in the same way as the echo(Section 11.4.6.1.1).

The only remaining term is the scattering coefficient s. If the orca swims close tothe surface, the grazing angle �S at the sea surface of ray paths contributing to surfacereverberation is close to zero. Because of the small angle, rough surface scattering isnegligible and consequently the surface scattering coefficient can be approximated bythe contribution from bubble scattering alone. The bubble scattering coefficientvaries strongly with wind speed when the wind speed is low and only weakly whenit is high. The high wind speed limit is a simple function of grazing angle � and


32 This approximation is consistent with the assumption that the orca is looking ahead rather

than down, so that dF=s is small. Furthermore, the high-frequency contributions to the integralare filtered out by the exponential decay in the propagation factor, so the second term needs

only to be negligible at low frequency. A necessary condition is s > ð fm � B=2ÞDdF=2c, whichis satisfied for distances exceeding 30m.

frequency f given by (Chapter 8)33

sð�; f Þ ¼ bubble � 0:214 ff �1=3 sin �: ð11:190Þ

Using this expression for the surface scattering coefficient gives

QRð�Þ ¼ S0Fð fmÞc �t

2 bubbleð fm; �SÞ

1

B

ð fþf�

expð�4�sÞs4

fmf

� �4=3

df

� �ð fþ > f�Þ:

ð11:191Þ

The limits of integration are determined on the one hand by the bandwidth, and onthe other by the condition that the right-hand side of Equation (11.186) be positive.This second condition imposes an upper bound on fþ equal to

fD ¼ 2c

D

s

dF: ð11:192Þ

The lower-frequency and upper-frequency limits are therefore

f� ¼ fm � B=2 ð11:193Þand

fþ ¼ minð fm þ B=2; fDÞ: ð11:194Þ

The frequency fD is that at which the edge of the main beam intersects the sea surfaceprecisely at a distance s from the animal. At frequencies higher than this, reverbera-tion enters only through sidelobes and is disregarded from the present calculation.

It is convenient to define a (one-way) propagation factor, weighted in frequencyaccording to the various frequency-dependent mechanisms and denoted FW, as thesquare root of the quantity in curly brackets in Equation (11.191):

FWðsÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

B

ð fþf�

expð�4�sÞs4

fmf

� �4=3

df

s: ð11:195Þ

With this definition, the expression for reverberation becomes

QR ¼ 0:214

ff1=3m

dOs� S0F0ð fmÞ

c �t

2FWðsÞ2 ð fþ > f�Þ: ð11:196Þ

The integral for FW can be simplified by assuming a linear variation of attenuationwith frequency (see Equation 11.48)

FWðsÞ2 ¼exp½�4ð�m � � 0fmÞs

s4B

ð fþf�

expð�4� 0fsÞ fmf

� �4=3

df : ð11:197Þ

Using the change of variable x ¼ 4� 0sf , this integral can be evaluated in terms of the


33 If the wind speed is low, the problem becomes noise-limited.


incomplete gamma function �ða; xÞ:34

FWðsÞ ¼exp½�2ð�m � � 0fmÞs

s2ð4�0sfmÞ1=6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifmB½�ð� 1

3; 4� 0sfþÞ � �ð� 1

3; 4� 0sf�Þ

r:

ð11:198Þ

Figure 11.23 shows the result of evaluating Equation (11.196) with FW given byEquation (11.198) and F0 from Equation (11.187). The graph shows NL (blue)and RL (cyan) separately vs. distance. The combined noise plus reverberation is alsoshown as the dashed blue line. Because reverberation is much weaker than (unpro-cessed) noise, this dashed line almost coincides with noise alone. Processing gain isconsidered next.

11.4.6.4.2 Processing gain (PG)

The gain from any hypothetical matched filter used by the orca is neglected. This isjustifiable because the duration of the transmitted pulse is close to the theoreticalminimum for the bandwidth used, so no gain is possible from further compression.Thus

PG � AG: ð11:199Þ

Figure 11.23. Background level vs. distance between orca and salmon (v10 ¼ 10 m/s).

34 From Appendix A:

�ða; xÞ �ðx0

e�tta�1 dt:


It is appropriate to use GR ¼ 1 in Equation (11.122) because reverberation arrivespredominantly in the main beam.35 Assuming for simplicity that GN � 1=GBB (seeEquation 11.173) it follows that

GAð�Þ ¼QN þQRð�Þ

QN=GBB þQRð�Þ : ð11:200Þ

Figure 11.23 shows the effect of processing on background noise. The differencebetween the curves ‘‘BL’’ and ‘‘BL� PG’’, plotted in Figure 11.24, is processing gain(in this case equal to array gain), which is a function of distance to the target. It passesa minimum at a distance of 60m, corresponding to the peak in RL at that range.

11.4.6.4.3 Signal excess and detection range

The level that must be exceeded by signal level for a detection is BL� PGþDT (thecyan curve in Figure 11.25). Thus, the detection range, which is reduced from 120min Figure 11.22 to about 70m in this case, is determined by the intersection betweenthis curve and EL (solid red line).

Figure 11.24. Array gain (equal to processing gain) vs. distance between orca and salmon

(v10 ¼ 10 m/s).

35 This is a consequence of the assumption that the same transducer is used for both

transmission and reception.

11.5 THE FUTURE OF SONAR PERFORMANCE MODELING

Nearly a century after the invention of sonar, sound provides the only practicalwindow into the sea and its contents. Much of the world’s oceans remains unex-plored, so the need for sonar will increase well into the 21st century. As Niels Bohronce said prediction is very difficult, especially of the future, but a brief look ahead intothe future of sonar and sonar performance modeling seems an appropriate way to endthis account.

11.5.1 Advances in signal processing and oceanographic modeling

The future can be expected to bring

— increasingly sophisticated sonar hardware and processing software, leading to aneed for increasingly detailed modeling of linear or non-linear pressure andparticle velocity fields;

— increasingly sophisticated knowledge and understanding of the oceanographicfeatures responsible for fluctuations in both signal and background (surfacewaves, internal waves, multiscale seabed roughness, etc.);

— increasingly powerful computing facilities, leading to an increasing ability tomodel this detail, including closer integration of sonar models with oceano-graphic databases and ocean forecasting models.


Figure 11.25. Signal and background levels vs. distance between orca and salmon

(v10 ¼ 10 m/s).

These trends point towards increasingly sophisticated sonar performance predictionmodels, capable of modeling not just fluctuations in the signal and backgroundwaveforms themselves, but also the oceanographic features that cause them, andtheir consequences for sonar processing gain and detection probability.

11.5.2 Autonomous platforms

It seems reasonable to expect an increase in the autonomy of underwater vehiclesgenerally and hence also of sonar platforms. A possible scenario involves a group ofsmall autonomous platforms working together or separately to execute a designatedtask. Specific applications for such platforms might include

— Ocean survey: In this scenario the vehicles gather data independently, exchangingfindings with any neighboring platforms within acoustic communication dis-tance, and surfacing occasionally to upload data, recharge batteries, and perhapsreceive new instructions. Such surveys would take place initially in benign con-ditions of temperature and pressure and then in increasingly hostile ones, perhapseventually to support planetary exploration.36 Less exotic applicationsmight include a seabed survey, monitoring of gas exchange processes with theatmosphere for climate modeling, and a census of protected species.

— Military reconnaissance or marine archeology: The vehicles rendezvous at apredetermined location and carry out co-ordinated, possibly covertmeasurements and upload data to a network or mother platform.

Common to these applications is the reduced scope for human intervention comparedwith the more traditional use of sonar on manned (or unmanned, remotely operated)platforms. The challenge to sonar performance modeling is to provide a robustsolution to the effective deployment and co-ordination of multiple autonomousplatforms and their sensors.

11.5.3 Environmental impact of anthropogenic sound

Many sea animals, especially marine mammals, rely on underwater sound to carryout routine tasks such as foraging, communication, and navigation, in much the sameway as humans rely on light in air. There is a growing concern that such animalsmight be adversely affected by anthropogenic sources of sound such as sonar (includ-ing echo sounders, seismic survey sources, and communications equipment), acousticdeterrents, underwater explosions (such as arising from the controlled disposal ofunexploded ordnance), and radiated sound from shipping vessels (Richardson et al.,

11.5 The future of sonar performance modeling 631]Sec. 11.5

36 The potential for acoustic remote sensing is demonstrated by Collins et al (1995, 1997) in

their analysis of the impact of Comet Shoemaker-Levy on Jupiter. Leighton and others make a

case for acoustic monitoring in suspected lakes on Titan (liquid methane/ethane lake)

(Leighton et al., 2005) and Europa (ice-covered liquid water) (Lee et al., 2003; Leighton et

al., 2008).

1995; Southall et al., 2007; Dolman et al., 2009; Popper and Hastings, 2009). Theacoustic modeling tools that have been developed for predicting sonar performance(and similar modeling tools developed for the offshore prospecting industry) are wellsuited to the task of estimating the sound pressure field and its spectrum at a givenlocation. What these models are not yet able to do is assess the impact on individualanimals exposed to the sound, on groups of animals, or on the ecosystem as a whole.The need to develop models with this capability will lead to increasing co-operationbetween biologists and sonar scientists.

11.6 REFERENCES

Abraham, D. A. (2003) Signal excess in K-distributed reverberation, IEEE J. Oceanic Eng., 28,

526–536.

Ainslie, M. A. (2004) The sonar equation and the definitions of propagation loss, J. Acoust.

Soc. Am., 115, 131–134.

Ainslie, M. A., Harrison, C. H., and Burns, P. W. (1996) Signal and reverberation prediction

for active sonar by adding acoustic components, IEE Proc.-Radar, Sonar Navig., 143(3),

190–195.

ASA (1994) American National Standard: Acoustical Terminology, ANSI S1.1-1994 (ASA 111-

1994, revision of ANSI S1.1-1960 (R1976)), Acoustical Society of America, New York.

Au, W. W. L., Ford, J. K. B., Horne J. K., and Newman Allman, K. A. (2004) Echolocation

signals of free-ranging killer whales (Orcinus orca) and modelling of foraging for chinook


Clark, C. A. (2007) Vertical directionality of midfrequency surface nois in downward-

refracting environments, IEEE J. Oceanic Eng., 32, 609–619.

Collins, M. D., McDonald, B. E., Kuperman W. A., and Siegmann, W. L. (1995) Jovian

acoustics and Comet Shoemaker–Levy 9, J. Acoust. Soc. Am., 97, 2147–2158.

Collins, M. D., McDonald, B. E., Kuperman, W. A., and Siegmann, W. L. (1997) Jovian

acoustic matched-field processing, J. Acoust. Soc. Am., 102, 2487–2493.

Dolman, S. J., Weir, C. R., and Jasny, M. (2009) Comparative review of marine mammal

guidance implemented during naval exercises, Marine Pollution Bulletin, 58, 465–477.



Hall, J. D. and Johnson, C. S. (1972) Auditory thresholds of a killer whale Orcinus orca

Linnaeus, J. Acoust. Soc. Am., 106, 1134–1141.

Harrison, C. H. (1996) Formulas for ambient noise level and coherence, J. Acoust. Soc. Am.,

99, 2055–2066.

Harrison, C. H. and Harrison, J. A. (1995) A simple relationship between frequency and range

averages for broadband sonar, J. Acoust. Soc. Am., 97, 1314–1317.

IEC (www) Electropedia, Acoustics and Electroacoustics/IEV 801 (International Electro-

technical Commission), available at http://www.electropedia.org/iev/iev.nsf (last accessed

June 23, 2009).

Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V. (1982) Fundamentals of Acoustics

(Third edition), Wiley, New York.

Lee, S., Zanolin, M., Thode, A. M., Pappalardo, R. T., and Makris, N. C. (2003) Probing

Europa’s interior with natural sound sources, Icarus, 165, 144–167.


Leighton, T. G., White, P. R., and Finfer, D. C. (2005) The sounds of seas in space, Proc.

International Conference on Underwater Acoustic Measurements: Technologies and Results,

Heraklion, Crete, Greece, June 28–July 1, 2005 (edited by J. S. Papadakis and L. Bjørnø,

pp. 833–840).

Leighton, T. G., Finfer, D. C., and White, P. R. (2008) The problems with acoustics on a small

planet, Icarus, 193(2), 649–652.

Nielsen, P. L., Harrison, C. H., and Le Gac, J. C. (2008) International Symposium on

Underwater Reverberation and Clutter, September 9–12, 2008, NATO Undersea Research

Center, La Spezia, Italy.

oalib (www) Ocean Acoustics Library, available at http://oalib.hlsresearch.com/ (last accessed

February 9, 2009).

Popper, A. N. and Hastings, M. C. (2009) The effects of anthropogenic sources of sound on

fishes, Journal of Fish Biology, 75, 455–489.

Richardson, W. J., Greene, C. R., Malme, C. I., and Thomson, D. H. (1995)Marine Mammals



Kastak, D., Ketten, D. R., Miller, J. H., Nachtigall, P. E., Richardson, W. J.,

Thomas, J. A., and Tyack, P. L. (2007) Marine mammal noise exposure criteria: Initial

scientific recommendations, Aquatic Mammals, 33(4), 411–521.

Szymanski, M. D., Bain, D. E., Kiehl, K., Pennington, S., Wong, S., and Henry, K. R. (1999)

Killer whale (Orcinus orca) hearing: Auditory brainstem response and behavioral

audiograms, J. Acoust. Soc. Am., 106, 1134–1141.

Urick, R. J. (1983) Principles of Underwater Sound, Peninsula Publishing, Los Altos, CA.

Wensveen, P. J. and Van Roij, Y. A. L. (2007) Exposure Data Analysis: 3S-2006 Trial (TNO

report TNO-DV 2007 SV291). TNO, The Hague, The Netherlands.


Appendix A

Special functions and mathematical operations

The purpose of this appendix is to define the special functions and mathematicaloperations used in the main text, and to describe their most important properties. Thematerial draws heavily from two valuable resources: the Handbook of MathematicalFunctions edited by Abramowitz and Stegun (1965) and Weisstein’s MathWorld(Weisstein, www). Unless stated otherwise, the symbols x and z denote real andcomplex variables, respectively.

A.1 DEFINITIONS AND BASIC PROPERTIES OF SPECIAL FUNCTIONS

A.1.1 Heaviside step function, sign function, and rectangle function

Three closely related functions are the Heaviside step function

HðxÞ �0 x < 0

1=2 x ¼ 0

1 x > 0,

8<: ðA:1Þ

the sign function

sgnðxÞ ��1 x < 0

0 x ¼ 0

þ1 x > 0

8<: ðA:2Þ

and the rectangle function

PðxÞ �1 jxj < 1=2

1=2 jxj ¼ 1=2

0 jxj > 1=2.

8><>: ðA:3Þ

It follows from these definitions that

sgnðxÞ ¼ 2½HðxÞ � 12 ðA:4Þ

andPðxÞ ¼ Hðx þ 1

2Þ � Hðx � 1

2Þ: ðA:5Þ

A.1.2 Sine cardinal and sinh cardinal functions

The sine cardinal, or ‘‘sinc’’, function is

sincðxÞ � sin x

x; ðA:6Þ

some integrals of which are included in Table A.1.Similarly, the sinh cardinal function is (Weisstein,2003a)

sinhcðxÞ � sinh x

x: ðA:7Þ

A.1.3 Dirac delta function

Dirac’s delta function has zero magnitude everywhere except the origin, and unitarea. It can be defined in terms of a limiting form of, for example, the rectanglefunction

�ðxÞ ¼ lim"!0

Pðx="Þ"

; ðA:8Þor the Gaussian

�ðxÞ ¼ lim"!0

exp½�ðx="Þ2ffiffiffi�

p"

: ðA:9Þ

A.1.4 Fresnel integrals

The Fresnel integrals are

CðxÞ �ðx

0

cos�

2u2

� �du ðA:10Þ

and

SðxÞ �ðx

0

sin�

2u2

� �du: ðA:11Þ

Asymptotic properties are

limx!1

CðxÞ ¼ð10

cos�

2u2

� �du ¼ 1

2ðA:12Þ

and

limx!1

SðxÞ ¼ð10

sin�

2u2

� �du ¼ 1

2: ðA:13Þ

636 Appendix A

Table A.1. Integrals of integer

powers of the sine cardinal

function (Weisstein, 2006).

N

ð10

dx sincN x

1 �=2

2 �=2

3 3�=8

4 �=3

5 115�=384

A.1.5 Error function, complementary error function, and right-tail probability

function

The error function is

erfðxÞ � 2ffiffiffi�

pðx

0

e�t2 dt: ðA:14Þ

Its limiting value for large x is

limx!1

erfðxÞ ¼ 1: ðA:15Þ

The complementary error function, plotted in Figure A.1 (cyan line of upper graph), is

erfcðxÞ � 1� erfðxÞ ¼ 2ffiffiffi�

pð1

x

e�t2 dt: ðA:16Þ

A simple approximation to erfcðxÞ, shown as ‘‘approx 1’’ in Figure A.1 and valid forlarge x, is

erfcðxÞ e�x2ffiffiffi�

px: ðA:17Þ

A slightly more accurate version (‘‘approx 2’’) is (Abramowitz and Stegun, 1965)

erfcðxÞ 2ffiffiffi�

p e�x2

x þffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ 2

p : ðA:18Þ

At the expense of a little more complication, a very accurate value can be obtainedusing the approximation

erfcðxÞ 2ffiffiffi�

p e�x2

x þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ 2� ð1� 2=�Þ21�1:2117x

p ; ðA:19Þ

shown as ‘‘approx 3’’. The fractional errors for all three approximations are alsoplotted (lower graph). For Equation (A.19), the fractional error is less than 0.1% forall x � 0. For negative arguments, the following symmetry property can be used

erfcðxÞ ¼ 2� erfcð�xÞ: ðA:20Þ

The erfc function is closely related to the right-tail probability function (Kay, 1998,p. 21), defined as

FðxÞ � 1ffiffiffiffiffiffi2�

pð1

x

exp � u2

2

!du: ðA:21Þ

The precise relationships between these two functions and their inverses are

FðxÞ ¼ 1

2erfc

xffiffiffi2

p�

ðA:22Þ

and

F�1ðxÞ ¼ffiffiffi2

perfc�1ð2xÞ: ðA:23Þ

Appendix A 637

638 Appendix A

Figure A.1. The complementary error function erfcðxÞ and approximations 1 to 3 (upper graph)

and fractional error (lower). The approximations are indicated by ‘‘approx 1’’ (Equation A.17),

‘‘approx 2’’ (Equation A.18), and ‘‘approx 3’’ (Equation A.19).

A.1.6 Exponential integrals and related functions

A.1.6.1 Definition of the exponential integral

The exponential integral of order n is (Abramowitz and Stegun, 1965)

EnðzÞ �ð11

e�zt

tndt: ðA:24Þ

The recursion relation between EnðzÞ and Enþ1ðzÞ, for positive integers n � 1, is

nEnþ1ðzÞ ¼ e�z � zEnðzÞ: ðA:25Þ

For real positive arguments, n > 0, the function is bounded by the inequality(Abramowitz and Stegun, 1965, Eq. (5.1.19))

1

x þ n< exEnðxÞ <

1

x þ n � 1: ðA:26Þ

A.1.6.2 Exponential integral of first order (imaginary argument)

An example of particular interest is the first-order exponential integral (i.e., EquationA.24 with n ¼ 1) with a purely imaginary argument

E1ðixÞ �ð11

e�ixt

tdt: ðA:27Þ

This can be written in the equivalent form

�E1ð�ixÞ ¼ þ loge x þðx

0

eiu � 1

udu � i�=2; ðA:28Þ

where is the Euler–Mascheroni constant

0:57722: ðA:29Þ

A.1.6.3 Exponential integral of third order (real argument)

The third-order exponential integral (Equation A.24 with n ¼ 3), this time with a realargument, is

E3ðxÞ �ð11

e�xt

t3dt: ðA:30Þ

This function is introduced in Chapter 2, for calculation of the radiated noise field ofan infinite sheet. An approximation to it, for all x � 0 (based on Equation A.26) is

E3ðxÞ e�x

x þ 3� e�0:434x: ðA:31Þ

For values of x in the range ½0; 2, the largest fractional error in E3ðxÞ incurred by theuse of Equation (A.31) is 2%.

Appendix A 639

A.1.6.4 Sine and cosine integral functions

The sine integral and cosine integral functions are, respectively,

SiðxÞ �ðx

0

sin u

udu ðA:32Þ

and

CiðxÞ � þ loge x þðx

0

cos u � 1

udu: ðA:33Þ

These two functions are related to the exponential integral via (Abramowitz andStegun, 1965, p. 232)

SiðxÞ ¼ �

2þ 1

2i½E1ðixÞ � E1ð�ixÞ ðA:34Þ

and

CiðxÞ ¼ � 1

2½E1ðixÞ þ E1ð�ixÞ: ðA:35Þ

It follows thatE1ð�ixÞ ¼ �CiðxÞ � i½SiðxÞ � �=2: ðA:36Þ

Asymptotic values are

limx!1

SiðxÞ ¼ð10

sin u

udu ¼ �

2ðA:37Þ

andlim

x!1CiðxÞ ¼ 0: ðA:38Þ

A.1.7 Gamma function and incomplete

gamma functions

A.1.7.1 Gamma function

A.1.7.1.1 Definition and importantvalues

The gamma function is

GðzÞ �ð10

tz�1 e�t dt; ðA:39Þ

which for real arguments satisfies theproperty

Gðx þ 1Þ ¼ xGðxÞ ðx > 0Þ: ðA:40ÞImportant values of GðxÞ are listed inTable A.2. It follows from Equation(A.39) and the result Gð1Þ ¼ 1 that,for integer n

Gðn þ 1Þ ¼ n! ðn � 1Þ: ðA:41Þ

640 Appendix A

Table A.2. Selected values of the gamma

function GðxÞ for 0 < x � 1. Values outside

this range can be calculated using

Gðx þ 1Þ ¼ xGðxÞ. All GðxÞ values in the

table are approximate except Gð1Þ. The exactvalue of Gð1=2Þ is �1=2.

x GðxÞ

1/5 4.5908

1/4 3.6256

1/3 2.6789

2/5 2.2182

1/2 1.7725

3/5 1.4892

2/3 1.3541

3/4 1.2254

4/5 1.1642

1 1

A.1.7.1.2 Approximations

Stirling’s formula can be used to estimate the value of n! for large arguments(Abramowitz and Stegun, 1965):

limn!1

n!ffiffiffiffiffiffi2�

pnnþ1=2 e�n

¼ 1: ðA:42Þ

The assumption that Equation (A.42) may be generalized to non-integer n (throughuse of Equation A.41) results in the approximation

GðxÞ GStirlingðxÞ ¼ffiffiffiffiffiffi2�

pxx�1=2 e�x; ðA:43Þ

where Equation (A.43) serves to define the function GStirlingðxÞ. A more generalversion is obtained using Stirling’s series (Weisstein, 2004a)

loge GðxÞ ¼ logeffiffiffiffiffiffi2�

pþ ðx � 1=2Þ loge x � x þ 1

12x� 1

360x3þ O

1

x5

� ; ðA:44Þ


GðxÞ ¼ GStirlingðxÞ 1þ 1

12xþ 1

288x2þ O

1

x3

� � �: ðA:45Þ

A convenient approximation is obtained by retaining the first two terms of thisexpansion

GðxÞ GStirlingðxÞ 1þ 1

Kx

� �; ðA:46Þ

with

K ¼ 12: ðA:47Þ

Alternative values of K for Equation (A.46) are now considered. Insisting thatEquation (A.46) should give the correct value of GðxÞ at x ¼ 1 (i.e., Gð1Þ ¼ 1) resultsin

K ¼ 1

e=ffiffiffiffiffiffi2�

p� 1

11:843: ðA:48Þ

When substituted in Equation (A.46), Equations (A.47) and (A.48) both give goodaccuracy for large x, but result in large errors in the region 0 < x < 1, especially at thelower end of this range. This problem can be remedied by applying Equation (A.40)for x < 1. Thus,

GðxÞ 1þ 1

Kx

� GStirlingðxÞ x � 1

1

x1þ 1

Kðx þ 1Þ

� GStirlingðx þ 1Þ 0 < x < 1.

8>>><>>>:

ðA:49Þ

Appendix A 641

In general, there is a small discontinuity through x ¼ 1, which can be removed bychoosing

K ¼ e�ffiffiffi2

pffiffiffi8

p� e

11:840: ðA:50Þ

Figure A.2 shows the gamma function with various approximations (upper graph)and the fractional error incurred by these (lower). The approximation obtained usingEquation (A.49) (with Equation A.50 for K) is not shown in the upper graph becauseit cannot be distinguished from the exact function GðxÞ on this scale. The largestfractional error incurred by use of this approximation (shown as a cyan curve in thelower graph) is about 0.01%, and occurs when x 3:5.

A.1.7.1.3 Use of the gamma function

Integrals of the form ð10

xp expð�BxqÞ dx ðA:51Þ

appear in several chapters of this book. It follows from the definition of the gammafunction (Equation A.39) that this integral can be writtenð1

0

xp expð�BxqÞ dx ¼ B�ðpþ1Þ=q

qG

p þ 1

q

� : ðA:52Þ

A.1.7.2 Incomplete gamma functions

Two incomplete gamma functions are of interest here. The first, known as the lowerincomplete gamma function, is defined as (Abramowitz and Stegun, 1965, p. 260)

ða; xÞ �ðx

0

e�t ta�1 dt: ðA:53Þ

The second is the upper incomplete gamma function (Abramowitz and Stegun, 1965;Weisstein, 2002)

Gða;xÞ �ð1

x

e�t ta�1 dt: ðA:54Þ

These two functions are complementary in the sense that their sum gives an ordinary(i.e., complete) gamma function

ða; xÞ þ Gða; xÞ ¼ GðaÞ: ðA:55ÞImportant properties include

Gða; 0Þ ¼ limx!1

ða; xÞ ¼ GðaÞ ðA:56Þ

and (Weisstein, 2002)

Gð0;xÞ ¼E1ðxÞ � i� x < 0

�E1ð�xÞ x > 0.

�ðA:57Þ

642 Appendix A

Appendix A 643

Figure A.2. Upper graph: the gamma function GðxÞ defined by Equation (A.39) and

approximations ‘‘Stirling1’’ (Equation A.43), ‘‘Stirling3’’ (Equation A.45), ‘‘K¼ 12’’ (Equa-

tion A.49þEquation A.47); lower graph: fractional error incurred by the three approximations

from the upper graph, plus a fourth approximation, labeled ‘‘K¼ 11.840’’ (Equation

A.49þEquation A.50).

The asymptotic behavior of ða; xÞ is

ða; xÞ xa=a x � 1

GðaÞ x � 1.

�ðA:58Þ

An alternative form, used in some textbooks devoted to detection theory, is Pearson’sincomplete gamma function Iðu; pÞ, defined as (Abramowitz and Stegun, 1965)

Iðu; pÞ � 1

Gðp þ 1Þ

ðuffiffiffiffiffiffipþ1

p

0

e�t tp dt: ðA:59Þ

This function is related to the lower incomplete gamma function of Equation (A.55)via

ð p þ 1; uffiffiffiffiffiffiffiffiffiffiffip þ 1

pÞ ¼ Gð p þ 1ÞIðu; pÞ: ðA:60Þ

A.1.8 Marcum Q functions

The ordinary Marcum Q function is

Qð; �Þ �ð1�

x exp �x2 þ 2

2

!I0ðxÞ dx; ðA:61Þ

where I0 is the modified Bessel function of order zero. Helstrom (1968, p. 219) definesthe generalized Marcum function as

QMð; �Þ �ð1�

xx

� �M�1

exp � x2 þ 2

2

!IM�1ðxÞ dx; ðA:62Þ

where IN is a modified Bessel function of order N.To simplify the notation and to reinforce the point that Q1ð; �Þ ¼ Qð; �Þ, the

ordinary Marcum Q function is denoted Q1ð; �Þ in Chapter 7.

A.1.9 Elliptic integrals

Elliptic integrals of the first and second kind, introduced in Chapter 9, are describedbelow. The elliptic integral of the first kind is defined as (Abramowitz and Stegun,1965, p. 589)

Fð’ IÞ �ð’0

ð1� sin2 sin2 Þ�1=2 d : ðA:63Þ

The integrand of Equation (A.63) is always greater than or equal to unity, so theintegral must be greater than or equal to ’. If sin in the integrand is approximatedby 2 =�, the integral becomes

Fð’ IÞ �

2 sin �ð’; Þ; ðA:64Þ

644 Appendix A

where

�ð’; Þ � arcsin2’

�sin

� : ðA:65Þ

The right-hand side of Equation (A.64) satisfies the inequality

’ � �

2 sin �ð’; Þ � Fð’ IÞ: ðA:66Þ

The function Fð’ IÞ has a singularity at � ¼ ¼ �=2. Use of Equation (A.64)avoids this singularity, while still providing a useful approximation away from it.

The elliptic integral of the second kind is

Eð’ IÞ �ð’0

ð1� sin2 sin2 Þþ1=2 d : ðA:67Þ

A similar approximation to that leading to Equation (A.64) gives

Eð’ IÞ �

4 sin ð� þ sin � cos �Þ; ðA:68Þ

where � ¼ �ð’; Þ is given by Equation (A.65). This approximation satisfies theinequality

’ � �

4 sin ð� þ sin � cos �Þ � Eð’ IÞ: ðA:69Þ

A.1.10 Bessel and related functions

A.1.10.1 Bessel function of the first kind

Bessel functions of the first kind are solutions to the ordinary differential equation(Abramowitz and Stegun, 1965, p. 358)

z2d2w

dz2þ z

dw

dzþ ðz2 � �2Þw ¼ 0: ðA:70Þ

The solutions to this equation, denoted J��ðzÞ, are Bessel functions (of the first kind)of order ��. The normalization (for positive integer n) is (Weisstein, 2004b)ð1

0

½JnðxÞ2 dx ¼ 1: ðA:71Þ

Related integrals are (Wolfram, www)ð10

1

xJ�ðxÞ2 dx ¼ 1

2�ðRe � > 0Þ ðA:72Þ

and (Weisstein, 2004b) ð10

J1ðxÞx

� �2

dx ¼ 4

3�: ðA:73Þ

Appendix A 645

A series expansion is (Abramowitz and Stegun, 1965, p. 360)

J�ðxÞ ¼x

2

� ��X1n¼0

ð�x2=4Þn

n! Gð� þ n þ 1Þ : ðA:74Þ

The asymptotic behavior of J�ðxÞ for small and large x is given by (Abramowitz andStegun, 1965)

J�ðxÞ

1

Gð� þ 1Þx

2

� ��x � 1ffiffiffiffiffiffi

2

�x

rcos x � ��

2� �

4

� �x � 1,

8>><>>: ðA:75Þ

valid for x > 0 and real, non-negative �.

A.1.10.2 Modified Bessel function

Modified Bessel functions of the first kind, denoted I��ðzÞ, are solutions to theordinary differential equation (Abramowitz and Stegun, 1965)

z2d2w

dz2þ z

dw

dz� ðz2 þ �2Þw ¼ 0: ðA:76Þ

They are related to J�ðzÞ according to (Abramowitz and Stegun, 1965, p. 375):

I�ðzÞ ¼expð��i=2ÞJ�ðizÞ �� < arg z � �=2

expð3��i=2ÞJ�ð�izÞ �=2 < arg z � �.

�ðA:77Þ

Other important properties include

I�nðzÞ ¼ InðzÞ; ðA:78Þ

I�ðzÞ ¼z

2

� ��X1k¼0

ðz2=4Þk

k! Gð� þ k þ 1Þ ; ðA:79Þ

and

I�ðzÞ �ezffiffiffiffiffiffiffiffi2�z

p 1� 4�2 � 1

8zþ Oðz�2Þ

" #jarg zj < �=2: ðA:80Þ

Levanon (1988) suggests the approximation

I0ðxÞ 1

6ð1þ cosh xÞ þ 1

3cosh

x

2þ cosh

ffiffiffi3

px

2

!: ðA:81Þ

The modified Bessel function is plotted in Figure A.3 (upper graph), together with theapproximation of Equation (A.81). The fractional error increases with increasingargument (lower graph). For the range 0 < x < 15 the error is less than 2%.

646 Appendix A

Appendix A 647

Figure A.3. Upper graph: the modified Bessel function I0ðxÞ and Levanon’s approximation

(Equation A.81); lower graph: fractional error incurred by use of Levanon’s approximation.

A.1.10.3 Airy functions

The second-order differential equation

d2w

dz2� z

dw

dz¼ 0 ðA:82Þ

has two independent solutions, known as Airy functions, one of which, denotedAiðzÞ, vanishes for large real values of its argument, while the other, BiðzÞ, isunbounded in this limit. They are related to the Bessel functions J�1=3 and I�1=3

via (Abramowitz and Stegun, 1965, p. 446)

AiðzÞ ¼ffiffiffiz

p

3½I�1=3ð�Þ � Iþ1=3ð�Þ ðA:83Þ

and

BiðzÞ ¼ffiffiffiz

3

r½I�1=3ð�Þ þ Iþ1=3ð�Þ ðA:84Þ

where

� ¼ 23z3=2: ðA:85Þ

Alternative expressions that are more convenient to use for negative arguments are

Aið�zÞ ¼ffiffiffiz

p

3½Jþ1=3ð�Þ þ J�1=3ð�Þ; ðA:86Þ

and

Bið�zÞ ¼ffiffiffiz

3

r½J�1=3ð�Þ � Jþ1=3ð�Þ: ðA:87Þ

The value and gradient of the Airy functions at the origin are given by

Aið0Þ ¼ Bið0Þffiffiffi3

p ¼ 3�2=3

Gð2=3Þ 0:35503 ðA:88Þ

and

�Ai0ð0Þ ¼ Bi 0ð0Þffiffiffi3

p ¼ 3�1=3

Gð1=3Þ 0:25882: ðA:89Þ

A.1.11 Hypergeometric functions

A.1.11.1 Gauss’s hypergeometric function

Gauss’s hypergeometric function (sometimes abbreviated as the ‘‘hypergeometricfunction’’) is (Weisstein, 2004c)

2F1ða; b; c; zÞ ¼GðcÞ

GðbÞGðc � bÞ

ð10

tb�1ð1� tÞc�b�1

ð1� tzÞa dt: ðA:90Þ

This function is a solution of the differential equation

zð1� zÞ d2u

dz2þ ½c � ða þ b þ 1Þz du

dz� abu ¼ 0 ðA:91Þ

648 Appendix A

that is regular at the origin, and normalized such that

2F1ða; b; c; 0Þ ¼ 1: ðA:92ÞIf jxj < 1, Equation (A.90) may be expanded as a power series:

2F1ða; b; c; xÞ ¼GðcÞ

GðaÞGðbÞX1n¼0

Gða þ nÞGðb þ nÞGðc þ nÞ xn: ðA:93Þ

Of particular interest (for Chapter 5, in connection with the bulk modulus of bubblywater) is the special case for b ¼ c � 1 ¼ a

2F1ða; a; a þ 1; zÞ ¼ a

ð10

ta�1

ð1� tzÞa dt: ðA:94Þ

A.1.11.2 Confluent hypergeometric function of the first kind

The confluent hypergeometric function of the first kind, denoted 1F1ða; b; zÞ, is(Weisstein, 2003b)

1F1ða; b; zÞ ¼GðbÞ

Gðb � aÞGðaÞ

ð10

ezt ta�1

ð1� tÞ1þa�bdt: ðA:95Þ

Of particular interest (for Chapter 7, in connection with the third and highermoments of the Rician probability distribution function) is the special case b ¼ 1

1F1ða; 1; zÞ ¼1

GðaÞGð1� aÞ

ð10

ezt ta�1

ð1 � tÞa dt: ðA:96Þ

A.2 FOURIER TRANSFORMS AND RELATED INTEGRALS

A.2.1 Forward and inverse Fourier transforms

The Fourier transform of the function f ðxÞ is written I½ f ðxÞ. The outcome of thisoperation, denoted FðkÞ, is defined as:

FðkÞ ¼ I½ f ðxÞ �ðþ1

�1f ðxÞ expð�ikxÞ dx: ðA:97Þ

The inverse Fourier transform is

f ðxÞ ¼ I�1½FðkÞ � 1

2�

ðþ1

�1FðkÞ expðþikxÞ dk: ðA:98Þ

An equivalent alternative form used in Table A.3 is

Gð f Þ ¼ I½gðtÞ �ðþ1

�1gðtÞ expð�2�iftÞ dt; ðA:99Þ

Appendix A 649

with

gðtÞ ¼ I�1½Gð f Þ ¼

ðþ1

�1Gð f Þ expðþ2�iftÞ df : ðA:100Þ

A.2.2 Cross-correlation

The cross-correlation operation between two complex functions hðtÞ and gðtÞ,denoted here by the operator s, is defined by Weisstein (wwwa) as

hðtÞsgðtÞ �ðþ1

�1h�ð��Þgðt � �Þ d�; ðA:101Þ

where h�ðtÞ denotes the complex conjugate of hðtÞ. From this definition it follows that

hðtÞsgðtÞ ¼ðþ1

�1h�ð�Þgðt þ �Þ d�: ðA:102Þ

An important result, known as the cross-correlation theorem, is (Weisstein, wwwb)

hsg ¼ I�1½H �ð f ÞGð f Þ; ðA:103Þ

650 Appendix A

Table A.3. Examples of Fourier transform pairs (based on Weisstein, 2004d).

Function gðtÞ Gð f Þ

Constant 1 �ð f Þ

Cosine cosð2�f0tÞ 12½�ð f � f0Þ þ �ð f þ f0Þ

Sine sinð2�f0tÞ1

2i½�ð f � f0Þ � �ð f þ f0Þ

Dirac delta function �ðt � t0Þ expð�2�ift0Þ

Exponential expð�2�f0jtjÞ1

�

f0

f 2 þ f 20

Complex Gaussian exp½�ða þ ibÞt2ffiffiffiffiffiffiffiffiffiffiffiffi�

a þ ib

rexp � �f 2

a þ ib

�

Shifted Heaviside step function Hðt � t0Þ1

2�ð f Þ � i

�f

� �expð�2�ift0Þ

Rectangle Pðt=TÞ T sincð�fTÞ

Symmetrical ramp ð1� jtj=TÞPðt=2TÞ T sinc2ð�fTÞ

Sine cardinal sincð�t=aÞ aPð faÞ

Reciprocal (Cauchy principal value) 1=t �i½2Hð�f Þ � 1

whereHð f Þ ¼ I½hðtÞ ðA:104Þ

andGð f Þ ¼ I½gðtÞ: ðA:105Þ

The special case with h ¼ g, known as the Wiener–Khinchin theorem, relates theautocorrelation function hsh to the Fourier transform of the power spectrum:

hsh ¼ I�1½jHð f Þj2: ðA:106Þ

An alternative definition, used in Chapter 6 (following Burdic, 1984; McDonoughand Whalen, 1995), is

ChgðtÞ �ðþ1

�1hð�Þg�ð� � tÞ d�: ðA:107Þ

The two definitions are related according to

hðtÞsgðtÞ ¼ C�hgð�tÞ: ðA:108Þ

A.2.3 Convolution

The convolution operation between functions hðtÞ and gðtÞ is denoted here by theoperator � and defined as (Weisstein, 2003c)

hðtÞ � gðtÞ �ðþ1

�1hð�Þgðt � �Þ d�: ðA:109Þ

It follows from Equations (A.101) and (A.109) that

hðtÞsgðtÞ ¼ h�ð�tÞ � gðtÞ: ðA:110ÞThe Fourier transform of the product hðtÞgðtÞ is equal to the convolution of theindividual transforms Hð f Þ and Gð f Þ (i.e., Weisstein, 2003c)

I½hðtÞgðtÞ ¼ Hð f Þ � Gð f Þ: ðA:111ÞEquation (A.111) is known as the convolution theorem. Alternative forms of thetheorem are (Weisstein, wwwc)

I½h � g ¼ FG; ðA:112Þ

I�1½HG ¼ h � g; ðA:113Þ

and

I�1½H � G ¼ hg: ðA:114Þ

A.2.4 Discrete Fourier transform

The discrete Fourier transform (DFT) of the function xðnÞ is

XðmÞ �XN�1

n¼0

xðnÞ exp �i2�m

Nn

� ; ðA:115Þ

Appendix A 651

the inverse transform of which is (Oppenheim and Schafer, 1989)

xðnÞ ¼ 1

N

XN�1

m¼0

XðmÞ exp þi2�mn

N

� ; n ¼ 0; 1; 2; . . . ;N � 1: ðA:116Þ

A common application of the DFT is for a continuous function of time, say FðtÞ, thathas been sampled at discrete time intervals

tn ¼ t0 þ n �t: ðA:117ÞIn the analysis of signals of this form, it is common to evaluate expressions of theform

Gð!Þ �XN�1

n¼0

FðtnÞ expð�i!tnÞ; tn ¼ t0 þ n �t: ðA:118Þ

The inverse transform that follows from Equation (A.116) is

FðtnÞ ¼1

N

XN�1

m¼0

Gð!mÞ expðþi!mtnÞ; n ¼ 0; 1; 2; . . . ;N � 1; ðA:119Þ

where

!m ¼ 2�

N �tm: ðA:120Þ

A.2.5 Plancherel’s theorem

The Fourier transform pair gðtÞ and Gð f Þ are related according to Plancherel’stheorem (Weisstein, wwwd)ðþ1

�1jgðtÞj2 dt ¼

ðþ1

�1jGð f Þj2 df : ðA:121Þ

Thus, jGð f Þj2 is the energy spectral density of the time series gðtÞ. The correspondingrelationship for the discrete transform pair is

XN�1

n¼0

jxðnÞj2 ¼ �f �tXN�1

n¼0

jXðmÞj2: ðA:122Þ

A.3 STATIONARY PHASE METHOD FOR EVALUATION

OF INTEGRALS

A.3.1 Stationary phase approximation

The stationary phase method is a way of approximating integrals of the form

Iða; bÞ ¼ðb

a

f ðxÞ exp½i�ðxÞ dx; ðA:123Þ

where f ðxÞ is a slowly varying function; and �ðxÞ is a phase term. It is one of a more

652 Appendix A

general class of approximations known as saddle point methods (Skudrzyk, 1971;Chapman, 2004). The basic requirement is for f ðxÞ to vary slowly compared with �,in such a way that the amplitude f does not change significantly during a period ofei�. There is also a requirement that the phase approaches a maximum or minimumeither within or close to the integration interval. If there is only one such point ofstationary phase, the integral is

Iða; bÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2�

j� 00ðx0Þj

sf ðx0ÞEsð; �Þ ei�ðx0Þ; ðA:124Þ

where x0 is the point of stationary phase such that

� 0ðx0Þ ¼ 0 ðA:125Þand

s ¼ sgn½� 00ðx0Þ: ðA:126ÞThe variables and � are related to a and b according to

¼ gðaÞ; ðA:127Þand

� ¼ gðbÞ ðA:128Þwhere

gðxÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij� 00ðx0Þj

�

rðx � x0Þ: ðA:129Þ

Finally, the function Esð; �Þ is defined as

Esð; �Þ �1ffiffiffi2

pð�

exp si�

2x2

� �dx; ðA:130Þ

which in terms of Fresnel integrals becomes

Esð; �Þ ¼Cð�Þ � CðÞ þ si½Sð�Þ � SðÞffiffiffi

2p : ðA:131Þ

If there is more than one stationary phase point, and if these are not too closetogether, their individual contributions may be added.

A.3.2 Derivation

The derivation of Equation (A.124) follows. It is convenient to write the integrationlimits as x� such that

I ¼ðxþ

x�

f ðxÞ exp½i�ðxÞ dx ðA:132Þ

and expand �ðxÞ around some point x0 (to be specified)

�ðxÞ ¼ �ðx0Þ þ � 0ðx0Þðx�x0Þ þ 12� 00ðx0Þðx�x0Þ2 þ 1

6� 000ðx0Þðx�x0Þ3 þ � � � ðA:133Þ

If �ðxÞ is a rapidly varying function, the exponential is oscillatory and the netcontribution to the integral averaged over many cycles is small. However, if the

Appendix A 653

phase slows down, the contributions can build up quickly. For this reason it is usefulto expand about points at which the first derivative vanishes (known as points of‘‘stationary phase’’). Thus, the value of x0 is chosen to ensure that �0ðx0Þ ¼ 0, andtherefore

�ðxÞ ¼ �ðx0Þ þ 12� 00ðx0Þðx � x0Þ2 þ 1

6� 000ðx0Þðx � x0Þ3 þ � � � ðA:134Þ

and

I ¼ ei�ðx0Þðxþ

x�

f ðxÞ expfi½12� 00ðx0Þðx � x0Þ2 þ 1

6� 000ðx0Þðx � x0Þ3 þ � � �g dx: ðA:135Þ

So far no approximation has been made, other than the assumptions that a point ofstationary phase exists and the function �ðxÞ may be replaced by a Taylor expansionabout that point. To proceed further, the third and higher order derivatives areassumed to make a negligible contribution to the phase in the vicinity of x0, suchthat the phase of Equation (A.135) is approximated by its first term only

I ei�ðx0Þðxþ

x�

f ðxÞ expfi½12�00ðx0Þðx � x0Þ2g dx: ðA:136Þ

If the variation in the amplitude term is assumed to be negligible in the region ofinterest, f ðx0Þ may then be factored out of the integral

I f ðx0Þ ei�ðx0Þðxþ

x�

expfi½12� 00ðx0Þðx � x0Þ2g dx: ðA:137Þ

Changing the integration variable to

u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij�00ðx0Þj

�

rðx � x0Þ; ðA:138Þ

Equation (A.137) can be written (without further approximation)

I ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2�

j� 00ðx0Þj

sf ðx0Þ ei�ðx0ÞEsðu�; uþÞ; ðA:139Þ

where

u� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij� 00ðx0Þj

�

rðx� � x0Þ ðA:140Þ

and

s ¼ sgn½� 00ðx0Þ: ðA:141ÞThus,


2�

j� 00ðx0Þj

sf ðx0ÞEsðu�; uþÞ ei�ðx0Þ; ðA:142Þ

which is equivalent to Equation (A.124).The function Esðu�; uþÞ is a linear combination of Fresnel integrals (see Equation

A.131). If the limits of integration in Equation (A.137) are extended to infinity it

654 Appendix A

becomes

limju�j!þ1

Esðu�; uþÞ ¼ eis�=4: ðA:143Þ

Therefore (in this limit)


2�

j� 00ðx0Þj

sf ðx0Þ ei½�ðx0Þþs�=4; ðA:144Þ

which is the standard stationary phase result quoted in many textbooks and is validwhen the point of stationary phase is well within the range of integration. Equation(A.142) is a generalization that retains its accuracy for situations with a stationaryphase point close to the integration limits.

A.4 SOLUTION TO QUADRATIC, CUBIC, AND QUARTIC EQUATIONS

A.4.1 Quadratic equation

Readers will be familiar with the quadratic equation

Ax2 þ Bx þ C ¼ 0 ðA:145Þand its solution in the form

x ¼ �B �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 � 4AC

p

2A: ðA:146Þ

A.4.2 Cubic equation

There are times when the solution to a third-order polynomial (a cubic equation) isneeded and this is given below. Any cubic equation can be written in the form

x3 þ Ax2 þ Bx þ C ¼ 0: ðA:147ÞThere are three solutions to Equation (A.147), given by (Archbold, 1964; Weisstein,2004e)

xn ¼ yn ¼ A

3; ðA:148Þ

where

yn ¼ bn �Q

3bn

; ðA:149Þ

bn ¼ e2�in=3 �R

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR

2

� 2

� Q

3

� 3

s !1=3

; ðA:150Þ

Q ¼ �A2

3þ B; ðA:151Þ

Appendix A 655

and

R ¼ 2A3

27� AB

3þ C: ðA:152Þ

The three solutions to Equation (A.147) are obtained using n ¼ 0, 1, and 2 (or anythree consecutive integers) in Equation (A.150). The choice of sign in Equation(A.150) is arbitrary,1 but once made it must remain the same for all three valuesof n.

A.4.3 Quartic and higher order equations

Sometimes a fourth-order polynomial (quartic equation) is encountered. The solutionto such an equation is described by Archbold (1964) and Weisstein (2004f ).

The visionary 19th-century mathematician Evariste Galois proved that nogeneral purpose formula, comparable with the algorithm given above for the solutionto the cubic equation, exists for polynomials of order 5 or higher. In doing so he alsolaid the foundations of modern group theory, all before a tragic death at the age ofjust 20. Livio (2005) gives a fascinating historical account of the events leading up tothis proof.

A.5 REFERENCES

Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions, U.S. Govern-

ment Printing Office, Washington, D.C., available at http://www.math.sfu.ca/�cbm/aands/

(last accessed March 23, 2009).

Archbold, J. W. (1964) Algebra (Third Edition), Pitman, London.

Burdic, W. S. (1984) Underwater Acoustic Systems Analysis, Prentice Hall, Englewood Cliffs,

NJ.

Chapman, C. H. (2004) Fundamentals of Seismic Wave Propagation (Appendix D: Saddle-point

Methods), Cambridge University Press, Cambridge, U.K.

Helstrom, C. W. (1998) Statistical Theory of Signal Detection, Pergamon Press, Oxford, U.K.

Kay, S. M. (1998) Fundamentals of Statistical Signal Processing: Detection Theory, Prentice

Hall, Upper Saddle River, NJ.

Levanon, N. (1988) Radar Principles, Wiley, New York.

Livio, M. (2005) The Equation that Couldn’t Be Solved: How Mathematical Genius Discovered

the Language of Symmetry, Simon & Schuster, New York.


Academic Press, San Diego, CA.

Oppenheim, A. V. and Schafer, R. W. (1989) Discrete-Time Signal Processing, Prentice Hall,

Englewood Cliffs, NJ.

Skudrzyk, E. (1971) The Foundations of Acoustics: Basic Mathematics and Basic Acoustics,

Springer Verlag, Vienna.

656 Appendix A

1 Although in theory the two roots give identical answers, any practical implementation is

subject to rounding errors. These can be reduced by choosing the larger of the two roots in

magnitude.

Weisstein, E. W. (2002) Incomplete gamma function, available at http://mathworld.wolfram.

com/IncompleteGammaFunction.html (last accessed August 28, 2008).

Weisstein, E. W. (2003a) Sinhc function, available at http://mathworld.wolfram.com/Sinhc

Function.html (last accessed August 28, 2008).

Weisstein, E. W. (2003b) Confluent hypergeometric function of the first kind, available at

http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html (last

accessed August 28, 2008).

Weisstein, E. W. (2003c) Convolution, available at http://mathworld.wolfram.com/

Convolution.html (last accessed August 28, 2008).

Weisstein, E. W. (2004a) Stirling’s series, available at http://mathworld.wolfram.com/Stirlings

Series.html (last accessed August 28, 2008).

Weisstein, E. W. (2004b) Bessel function of the first kind, available at http://mathworld.

wolfram.com/BesselFunctionoftheFirstKind.html (last accessed August 28, 2008).

Weisstein, E. W. (2004c) Hypergeometric function, available at http://mathworld.wolfram.com/

HypergeometricFunction.html (last accessed August 28, 2008).

Weisstein, E. W. (2004d) Fourier transform, available at http://mathworld.wolfram.com/

FourierTransform.html (last accessed August 28, 2008).

Weisstein, E. W. (2004e) Cubic formula, available at http://mathworld.wolfram.com/Cubic

Formula.html (last accessed August 28, 2008).

Weisstein, E. W. (2004f) Quartic equation, available at http://mathworld.wolfram.com/Quartic

Equation.html (last accessed August 28, 2008).

Weisstein, E. W. (2006) Sinc function, available at http://mathworld.wolfram.com/Sinc

Function.html (last accessed August 28, 2008).

Weisstein, E. W. (www) Wolfram MathWorld, available at http://mathworld.wolfram.com/

(last accessed April 12, 2007).

Weisstein, E. W. (wwwa) Cross-correlation, available at http://mathworld.wolfram.com/Cross-

Correlation.html (last accessed July 10, 2007).

Weisstein, E. W. (wwwb) Cross-correlation theorem, available at http://mathworld.wolfram.

com/Cross-CorrelationTheorem.html (last accessed July 10, 2007).

Weisstein, E. W. (wwwc) Convolution theorem, available at http://mathworld.wolfram.com/

ConvolutionTheorem.html (last accessed July 10, 2007).

Weisstein, E. W. (wwwd) Plancherel’s theorem, available at http://mathworld.wolfram.com/

PlancherelsTheorem.html (last accessed November 28, 2008).

Wolfram (www) Wolfram functions, available at http://functions.wolfram.com/BesselAiry

StruveFunctions/BesselJ/21/02/02/] (last accessed April 11, 2007).

Appendix A 657

Appendix B

Units and nomenclature

B.1 UNITS

B.1.1 SI units

The International System of Units (abbreviated SI, from the French Systeme Inter-nationale d’Unites) is used throughout this book (bipm, www; Taylor and Thompson,2008; Anon., 2008). For example, energy is expressed in joules (symbol J), pressure inpascals (symbol Pa), and intensity in watts per square meter (W/m2). Further,standard SI prefixes are used to denote multiples of integer powers of 1000, suchas ‘‘mega’’ for one million and ‘‘milli’’ for one thousandth, as indicated by Table B.1.Also in use are prefixes for integer powers of 10 between 10�2 and 10þ2, the mostcommon being centi for 10�2 (as in centimeter). These are listed in Table B.2.

B.1.2 Non-SI units

For mainly historical reasons, units that are not part of SI are sometimes encounteredin underwater acoustics, especially for units of distance or pressure. Some commonnon-SI units are listed in Table B.3, together with a conversion to their SI equivalents.For the definition of many other units see Rowlett (www).

B.1.3 Logarithmic units

Logarithmic units form a special category of (non-SI) units that are typically used toquantify ratios of parameters that might vary by many orders of magnitude. Specialnames are typically given to such logarithmic units to help remind us of the physicalquantity they represent. Common examples are the octave (a base-2 logarithmic unitused to quantify frequency ratios), the decibel (a base-10 logarithmic unit used to

quantify power ratios) and the neper (a base-e logarithmic unit used to quantifyamplitude ratios). These and other relevant logarithmic units are described below.

B.1.3.1 Base-10 logarithmic units

B.1.3.1.1 Bel and decibel

Relative levels. The bel is a logarithmic unit of power or energy ratio. A physicalparameter that is proportional to power or energy is referred to in the following as a

660 Appendix B

Table B.1. SI prefixes for indices equal to an integer

multiple of 3. One terajoule (1012 J) is written 1TJ.

The range of prefixes most likely to be encountered is

in the white (unshaded) region. Those least likely to be

encountered are shaded dark gray.

Prefix name Symbol Index Example

yotta- Y 24 1YJ¼ 1024 J

zetta- Z 21 1ZJ¼ 1021 J

exa- E 18 1EJ¼ 1018 J

peta- P 15 1PJ¼ 1015 J

tera- T 12 1TJ¼ 1012 J

giga- G 9 1GJ¼ 109 J

mega- M 6 1MJ¼ 106 J

kilo- k 3 1 kJ¼ 103 J

— — 0 1 J¼ 100 J

milli- m �3 1mJ¼ 10�3 J

micro- m �6 1 mJ¼ 10�6 J

nano- n �9 1 nJ¼ 10�9 J

pico- p �12 1 pJ¼ 10�12 J

femto- f �15 1 fJ¼ 10�15 J

atto- a �18 1 aJ¼ 10�18 J

zepto- z �21 1 zJ¼ 10�21 J

yocto- y �24 1 yJ¼ 10�24 J

‘‘power-like’’ quantity. The level of a power-like quantity W2 is N bels higher thanthat of W1 if (Morfey, 2001)

N ¼ log10W2

W1

: ðB:1Þ

The symbol for the bel is B.The decibel is defined as one tenth of a bel. Thus, the same two power levels differ

by M decibels if

M ¼ 10 log10W2

W1

: ðB:2Þ

The symbol for the decibel is dB. Neither the bel nor the decibel are recognized as SIunits, but use of the decibel is permitted alongside SI units by the InternationalCommittee for Weights and Measures (CIPM) and at least one national standardsbody (Taylor and Thompson, 2008).

For example, the decibel is used to express ratios of mean squared acousticpressure (MSP) of statistically stationary pressure signals pðtÞ in this way using

MMSP ¼ 10 log10hp2i2hp2i1

: ðB:3Þ

It is sometimes argued that the MSP in both the numerator and denominator ofEquation (B.3) must first be divided by the characteristic acoustic impedance, inorder to convert to the equivalent plane wave intensity (EPWI).1 In other words

MEPWI ¼ 10 log10hp2i2=ð�cÞ2hp2i1=ð�cÞ1

; ðB:4Þ

Appendix B 661

Table B.2. SI prefixes for indices equal to an integer

betweenþ3 and�3. One decijoule (10�1 J) is written 1 dJ.

Name Symbol Index Example

kilo k 3 1 kJ¼ 103 J

hecto h 2 1 hJ¼ 102 J

deca da 1 1 daJ¼ 101 J

— — 0 1 J¼ 100 J

deci d �1 1 dJ¼ 10�1 J

centi c �2 1 cJ¼ 10�2 J

milli m �3 1mJ¼ 10�3 J

1 The EPWI is the intensity of a propagating plane wave whose MSP is equal to that of the true

acoustic field.

Table B.3. Frequently encountered non-SI units (in alphabetical order).

Unit Symbol SI equivalent Notes

atmosphere See standard atmosphere

bar bar 100 kPa 1Pa¼ 1N/m2

dyne dyn 10 mN 1dyn¼ 1 g cm/s2; 1N¼ 1 kg m/s2

dyne per square centimeter dyn/cm2 0.1 Pa

erg erg 0.1 mJ 1 erg¼ 1 dyn cm; 1 J¼ 1Nm

erg per square centimeter erg/cm2 1mJ/m2

fathom 1.8288m 1 fathom¼ 6 ft (international fathom)

foot ft 304.8mm

hour h 3600 s

inch in 25.4mm 1 ft¼ 12 in. The capacity of air guns (see Chapter

10) is sometimes expressed in cubic inches

(1 in3 � 16:39 cm3)

international nautical mile nmi 1.852 km There is no internationally recognized symbol or

abbreviation for this unit. The abbreviation

‘‘nmi’’ is adopted (preferred over ‘‘nm’’ to avoid

a conflict with the SI symbol for a nanometer)

knot kn (1852/3600)m/s The knot is defined as one nautical mile per hour

� 0.5144m/s (1 nmi/h), such that 9 kn¼ 4.63m/s, exactly

liter L 1000 cm3 The uppercase ‘‘L’’ is preferred to the alternative

(lowercase) letter ‘‘l’’ to avoid possible confusion

with the number ‘‘1’’

microbar mbar 0.1 Pa 1 mbar¼ 10�6 bar

millimeter per hour mm/h 1mm/(3600 s) Used as a unit of rainfall rate

�0.2778 mm/s

MKS rayl See rayl

nautical mile See international nautical mile

poise 0.1 Pa s 1 poise¼ 1 dyn s/cm2

pound-force per square inch psi �6.895 kPa

rayl dyn s/cm2 10 Pa s/m One pascal second per meter (1 Pa s/m) is

sometimes known as an ‘‘MKS rayl’’. The rayl is

not an SI unit.

standard atmosphere 101.325 kPa Pressure under standard conditions of

temperature and pressure, denoted PSTP (see

Section 14.2.2)

yard yd 0.9144m 1 yd¼ 3 ft

where ð�cÞn is the characteristic impedance at the measurement location indicated bythe value of the subscript n. Often the impedance is the same at locations 1 and 2, inwhich case Equations (B.3) and (B.4) are equivalent. In all other cases it is importantto state which of the two is being used. Throughout this book the convention ofEquation (B.3) (MSP ratio) is adopted, partly to conform to the de facto definition ofpropagation loss used in underwater acoustics, which since 1980 omits the impedanceratio (Ainslie and Morfey, 2005) and partly to avoid the ambiguities associated withthe EPWI definition in the absence of an agreed standard reference value for theimpedance (Ainslie, 2004, 2008).

Absolute levels. It is common practice to specify absolute power levels by re-placing the denominator W1 in Equation (B.2) with an agreed standard referencevalue. Thus, a power W may be expressed as an absolute level by defining the powerlevel LW in decibels, relative to a reference value Wref , as

LW 10 log10W

Wref

: ðB:5Þ

When the decibel is used in this way, to avoid ambiguity both the reference value andthe nature of the quantity W (in this case power) must be stated. Internationallyaccepted reference values for power and energy levels are 1 pW and 1 pJ, respectively.For example, a sound source of acoustic power (one watt) has a power level of10 log10ð1=10�12Þ ¼ 120 dB re pW.2

The sound pressure level Lp is defined in terms of the MSP (Morfey, 2001)

Lp 10 log10hp2ip2ref

; ðB:6Þ

where the reference pressure pref is equal to 1 mPa, making the MSP reference valueequal to 1 mPa2. Thus, the sound pressure level of an acoustic field whose RMSpressure is one pascal (MSP¼ 1 Pa2) is 10 log10ð1=10�12Þ ¼ 120 dB re mPa2. The samequantity is often written 120 dB re mPa. The squared unit is adopted here to avoidinconsistencies that otherwise arise when this quantity is combined with other ratiosin decibels.3 For example, it seems more natural to express the spectral density level indB re mPa2/Hz than in dB re mPa/

ffiffiffiffiffiffiffiHz

p.

Other physical parameters relevant to acoustics are energy density and intensity.When expressed as levels, their standard reference values are, respectively, 1 pJ/m2

and 1 pW/m2 (Morfey, 2001). When used in a spectral density, the reference unit forfrequency is one hertz. For example, the power spectral density level has the unitdB reW/Hz.

Appendix B 663

2 Or, equivalently, 120 dB re 1 pW.3 It is p2ref and not pref that appears in the denominator of Equation (B.6).

B.1.3.1.2 pH (acidity measure)

The pH of a solution is a logarithmic measure of the reciprocal concentration ofhydrogen ions dissolved in the solution.

pH ¼ �log10½Hþ�; ðB:7Þ

where ½Hþ� denotes the molar concentration of hydrogen (Hþ) ions. The precisedefinition depends on convention. For example, it might include only the concentra-tion of free protons (the free proton scale) or might also include that of protonsassociated with other ions.

Chapter 4 mentions four different pH scales: the U.S. National Bureau ofStandards4 scale ( pHNBS), the ‘‘seawater scale’’ ( pNSWS), the ‘‘total proton scale’’( pHT), and the ‘‘free proton scale’’ ( pHF). As there is no single universally adoptedconvention, a choice is necessary between these. The NBS scale is considered un-suitable for modern use in seawater (Brewer et al., 1995; Millero, 2006). The otherthree are defined below (following Millero, 2006).

The free proton scale is given by

pHF �log10½Hþ�F; ðB:8Þ

where the notation ½X� indicates the concentration of ion X, defined as the number ofmoles of that ion per kilogram of solution. Thus, ½Hþ�F is the concentration of freehydrogen ions in units of moles per kilogram (Brewer et al., 1995).

The total proton scale is given by

pHT �log10½Hþ�T; ðB:9Þ

where ½Hþ�T includes hydrogen sulfate ions

½Hþ�T ¼ ½Hþ�F þ ½HSO�4 �: ðB:10Þ

Finally, the SWS scale, recommended by UNESCO for use in seawater (Dickson andMillero, 1987), also includes the concentration of hydrogen associated with fluorideions. Thus

pHSWS �log10½Hþ�SWS; ðB:11Þwhere

½Hþ�SWS ¼ ½Hþ�T þ ½HF�: ðB:12Þ

B.1.3.1.3 Decade

The decade is a logarithmic unit of frequency ratio. The frequency f2 is N decadeshigher than f1 if (Pierce, 1989)

N ¼ log10f2f1: ðB:13Þ

IfN is negative then it is more conventional to say that f2 is jNj decades lower than f1.

664 Appendix B

4 Now the National Institute of Standards and Technology (NIST).

B.1.3.2 Base-e logarithmic unit (neper)

The neper is a logarithmic unit of amplitude ratio. Consider a sinusoidal oscillation ofamplitude A2. The amplitude level of this oscillation is N nepers higher than that ofanother of amplitude A1 if (Morfey, 2001)

N ¼ logeA2

A1

: ðB:14Þ

The symbol for the neper is Np.A change in amplitude level of 1Np is associated with a change in power level of

20 log10 e decibels. However, it is not correct to say that 1Np is equal to 20 log10 edecibels unless the neper is redefined in terms of (the square root of ) a power ratio(Mills and Morfey, 2005).

B.1.3.3 Base-2 logarithmic units

B.1.3.3.1 Octave

The octave is a logarithmic unit of frequency ratio. The frequency f2 is N octaveshigher than f1 if (Pierce, 1989)

N ¼ log2f2f1: ðB:15Þ

If N is negative then it is more conventional to say that f2 is jNj octaves lower than f1.

B.1.3.3.2 Phi

The phi unit is a logarithmic unit of reciprocal grain diameter. A spherical sedimentgrain of diameter5 d has a grain size of N phi units if (Krumbein and Sloss, 1963)

N ¼ �log2d

dref; ðB:16Þ

where the reference diameter is

dref 1 mm: ðB:17ÞThe symbol for the phi unit is �. For example, if d ¼ 0.25mm, the grain sizeexpressed in phi units is written 2�.

B.2 NOMENCLATURE

B.2.1 Notation

A concerted effort has been made to employ a consistent notation throughout thisbook. While there is no separate list of symbols, the notation used is defined as and

Appendix B 665

5 The ‘‘diameter’’ of non-spherical grains is defined implicitly in terms of the mesh sizes of

sieves able to separate them.

where it is introduced.The following notation conventions are used:

— variable names are italic: frequency f ;— two- or three-letter abbreviations for sonar equation terms are upright and upper

case: detection threshold is DT, whereas DT would mean a product of thevariables D and T ;

— other abbreviations are also upright, though often lower case: ‘‘fa’’ in ‘‘pfa’’ is anabbreviation of ‘‘false alarm’’;

— symbols for some standard functions are upright: sin x;— non-standard function names are italic: f ðxÞ or FðkÞ;— differential operators are upright: dðsin xÞ=dx ¼ cos x;— mathematical constants are upright: e ¼ expð1Þ; i ¼

ffiffiffiffiffiffiffi�1

p; � ¼ 2 arccos(0);

— variable names with a circumflex denote the numerical value of that variablewhen expressed in the corresponding (base) unit in the SI system. For example,if the frequency f is 3 kHz, then ff is a dimensionless number equal to(3 kHz)/(1Hz)¼ 3000. Thus ff f f gHz and cc fcgm=s.

The following conventions are used for subscripts. Subscripts are used for a variety ofpurposes, indicating, for example:

(1) the medium to which the subscripted parameter corresponds: �air is the density ofair (if no medium is specified, water is usually implied);

(2) a derivative with respect to the subscript variable: Wf is the power spectraldensity (power W per unit frequency f ; i.e., dW=df ); higher order derivativesare indicated in the same way, so that the power spectral density per unit areaA isdenoted WAf , meaning d2W=dA df ;

(3) a calculation method: ‘‘inc’’ in Finc stands for ‘‘incoherent’’, indicating that thepropagation factor F is evaluated without regard for phase information;

(4) evaluation for particular conditions: the ‘‘50’’ in DT50 means that the detectionthreshold corresponds to a 50% detection probability.

B.2.2 Abbreviations and acronyms

The abbreviations and acronyms used are listed in Table B.4. Abbreviations withmultiple meanings (e.g., BL) are further qualified with an integer in brackets: BL (2),meaning ‘‘bottom reflection loss’’, is the second of three uses of the abbreviation‘‘BL’’.

B.2.3 Names of fish and marine mammals

Many animals have more than one common name, and a small number have morethan one scientific name. Where the author has found more than one name in use hehas followed Froese and Pauly (2007) for fish and Read et al. (2003) for marinemammals.

666 Appendix B

Appendix B 667

Table B.4. List of abbreviations and acronyms, and their meanings.

Abbreviation Meaning

AG array gain

ANSI American National Standards Institute

APL Applied Physics Laboratory (University of Washington)

arr array

atm atmospheric

ATOC acoustic thermometry of ocean climate

BB broadband

BBS bottom backscattering strength

BIPM Bureau International des Poids et Mesures (International Bureau of

Weights and Measures)

BL (1) background level

BL (2) bottom reflection loss

BL (3) bottom reflected (path)

BR bottom refracted (path)

BSS bottom scattering strength

BSX backscattering cross-section

BW the quantity BW ¼ 10 log10 BB, where BB is the numerical value of the

bandwidth in hertz

CIPM Comite International des Poids et Mesures (International Committee for

Weights and Measures)

coh coherent

CS column strength

CW continuous wave

dB decibel (see Section B.1.3)

deg degree (angle)

DFT discrete Fourier transform

(continued)

668 Appendix B

Table B.4 (cont.)


DI directivity index

DT detection threshold

EPWI equivalent plane wave intensity

FFT fast Fourier transform

FG filter gain

FL fork length (of fish)

FM frequency modulation

FOM figure of merit

FRF flat response filter

ft foot (see Table B.3)

ftp file transfer protocol

fwhm full width at half-maximum

GEOSECS Geochemical Ocean Sections Study

GI generator injector (air gun)

h hour (see Table B.3)

HF high frequency

HFM hyperbolic frequency modulation

HIFT Heard Island feasibility test

hp hydrophone

IEC International Electrotechnical Commission

in inch (see Table B.3)

inc incoherent

kn knot (see Table B.3)

L liter (see Table B.3)

LF low frequency

LFM linear frequency modulation

Appendix B 669


LPM linear period modulation

MKS meter kilogram second system of units (predecessor to SI)

MSP mean square (acoustic) pressure

NB narrowband

NBS National Bureau of Standards (now NIST)

NIST National Institute of Standards and Technology

NL noise level

nmi international nautical mile (see Table B.3)

Np neper (see Section B.1.3)

pdf (1) probability density function

pdf (2) portable document format

peRMS peak equivalent RMS

PG processing gain

pH logarithmic measure of acidity (see Section B.1.3)

PL propagation loss

p-p peak to peak

psi pound-force per square inch (see Table B.3)

RAFOS ‘‘SOFAR’’ spelt backwards

RL reverberation level

RMS root mean square

ROC receiver operating characteristic

Rx receiver

SBR signal to background ratio

SBS surface backscattering strength

SE signal excess

(continued)

670 Appendix B

Table B.4 (cont.)


SI Systeme Internationale d’Unites (International System of Units)

SL (1) source level

SL (2) surface reflection loss

SL (3) standard length (of fish)

SNR signal to noise ratio

SOFAR sound fixing and ranging

SPL sound pressure level

SRR signal to reverberation ratio

SSS surface scattering strength

stat static

STP standard temperature and pressure; note: at STP the temperature and

pressure are YSTP ¼ 273:15 K and PSTP ¼ 101:325 kPa (one standard

atmosphere), respectively

SWS seawater scale (of pH)

tgt target

tot total

TL total length (of fish)

TPL total path loss

TS target strength, the quantity TS ¼ 10 log10��back

4�, where ��back is the

backscattering cross-section in square meters

Tx transmitter

UNESCO United Nations Educational, Scientific and Cultural Organization

VBS volume backscattering strength

vs. versus

WMO World Meteorological Organization

WS wake strength

B.3 REFERENCES

Ainslie, M. A. (2004) The sonar equation and the definitions of propagation loss, J. Acoust.

Soc. Am., 115, 131–134.

Ainslie, M. A. (2008) The sonar equations: Definitions and units of individual terms, Acoustics

’08, Paris, June 29–July 4, 2008, pp. 119–124. This article is missing from the search index

of the CD version of the Acoustics ’08 Proceedings. The paper can be located on the CD

by means of its identification number (475), at /data/articles/2008/000475.pdf It is also

available at http://intellagence.eu.com/acoustics2008/acoustics2008/cd1 (last accessed

April 12, 2010).

Ainslie, M. A. and Morfey, C. L. (2005) ‘‘Transmission loss’’ and ‘‘propagation loss’’ in

undersea acoustics, J. Acoust. Soc. Am., 118, 603–604.

Anon. (2008) The Little Big Book of Metrology, National Physical Laboratory, Teddington,

U.K.

bipm (www) The International System of Units (SI), Bureau International des Poids et Mesures,

available at http://www.bipm.org/en/si (last accessed September 21, 2008).

Brewer, P. G., Glover, D. M., Goyet, C., and Shafer, D. K. (1995) The pH of the North

Atlantic Ocean: Improvements to the global model of sound absorption, J. Geophysical

Res., 100(C5), 8761–8776.


Dickson, A. G. and Millero, F. J. (1987) A comparison of the equilibrium constants for the

dissociation of carbonic acid in sea water media, Annex 3 of Thermodynamics of the

Carbon Dioxide System in Seawater (report by the Carbon Dioxide Sub-panel of the Joint

Panel on Oceanographic Tables and Standards, Unesco Technical Papers in Marine

Science 51, Unesco, Paris.

Froese, R. and Pauly D. (Eds.), FishBase, version (01/2007), available at http://www.fishba-

se.org/search.php (last accessed March 23, 2009).



Krumbein, W. C. and Sloss, L. L. (1963) Stratigraphy and Sedimentation (Second Edition),

Freeman, San Francisco.

Kuperman, W. A. and Roux, P. (2007) Underwater Acoustics, in T. D. Rossing (Ed.), Springer

Handbook of Acoustics (pp. 149–204), Springer Verlag, New York.

Appendix B 671


WW1 First World War

WW2 Second World War

yd yard (see Table B.3)

z-p zero to peak

Kuperman, W. A. (1997) Propagation of sound in the ocean, in M. J. Crocker (Ed.), Ency-

clopedia of Acoustics (pp. 391–408), Wiley, New York.

Millero, F. J. (2006) Chemical Oceanography (Third Edition), CRC/Taylor & Francis.

Mills, I. and Morfey, C. L. (2005). On logarithmic ratio quantities and their units, Metrologia,

42, 246–252.

Morfey, C. L. (2001) Dictionary of Acoustics, Academic Press, San Diego, CA.

Pierce, A. D. (1989) Acoustics: An Introduction to its Physical Principles and Applications,


Read, A. J., Halpin, P. N., Crowder, L. B., Hyrenbach, K. D., Best, B. D., and Freeman S. A.

(Eds.) (2003) OBIS-SEAMAP: Mapping Marine Mammals, Birds and Turtles, World

Wide Web electronic publication, available at http://seamap.env.duke.edu/species (last


Rossing, T. D. (Ed.) (2007) Springer Handbook of Acoustics, Springer Verlag, New York.

Rowlett (www) R. Rowlett, A Dictionary of Units, available at http://www.unc.edu/�rowlett/

units/ (last accessed April 2, 2007).

Taylor, B. N. and Thompson, A. (2008) The International System of Units (SI) (NIST Special

Publication 330, 2008 Edition), U.S. Department of Commerce, National Institute of

Standards & Technology.

672 Appendix B

Appendix C

Fish and their swimbladders

C.1 TABLES OF FISH AND BLADDER TYPES

The scattering properties of fish generally are sensitive to the presence or absence of agas enclosure, or ‘‘swimbladder’’. The main purpose of this appendix is to enable thereader to assess the likelihood that a particular order, family, or species of fish isequipped with such a bladder, and where a bladder is present to provide furtherinformation about its relevant properties. General rules are described in Table C.3(by order) and Table C.4 (by family). Where known to the author, information aboutfish length is also provided.

Table C.7 presents a long list of information by individual species, but despite itslength it is not a complete list. In fact it is not even close to complete. Rather, itcomprises relevant information collected by the author over a number of years.Regardless of its shortcomings, its existence at all owes itself partly to David Weston,who impressed upon the author the importance of bladdered fish in underwateracoustics, and partly to FishBase (Froese and Pauly, 2007), from which much ofthe information is gleaned.

Table C.1 describes abbreviations used to describe types of fish in terms ofwhether or not a bladder is present, and if so whether a duct is present connectingit to the gut of the fish (in which case the fish is known as a physostome) or not (aphysoclist). The shape of the bladder varies between different species.

Each time the bladder code is used, it is accompanied by a lower case suffixindicating the source of the information, and these suffixes are listed in Table C.2. Forexample, ‘‘Sw’’ means that the fish is a physostome according to Whitehead andBaxter (1989), whereas ‘‘Nb’’ means that it has no swimbladder according to Froeseand Pauly (2007).

Two more keys are presented below to aid the interpretation of the main list ofspecies in Table C.7. The first (Table C.5) describes a list of categories, referred tohere as ‘‘Yang groups’’, which describe the likely behavior of the fish. The groups are

used by Yang (1982) to describe the relative ‘‘catchability’’ of the different species forhis population estimates. The reason they are useful here is that catchability isinfluenced by the fish’s behavior which in turn affects its likely acoustical properties,its environment, or both. For example, groups B and C are demersal fish, whichmeans that their properties are easily confused with (and might be affected by) theproperties of the seabed. The other groups are pelagic. For Yang’s group C, the terms‘‘sandeels’’ and ‘‘gobies’’ are interpreted here, respectively, as Ammodytidae andGobidae.

The second key (Table C.6) defines the abbreviations used to describe the fishlength information (last column of Table C.7).

674 Appendix C

Table C.1. Bladder presence and type key used in Tables C.3, C.4, and C.7.

Bladder code Means

J Bladder missing in adults ( juveniles physoclist or physostome)

L Physoclist

M With bladder (bladder sometimes partly or completely filled with fat;

uncertain air fraction)a

N No bladder

P With bladder (physoclist or physostome)

S Physostome

a The ‘‘M’’ stands for ‘‘Myctophidae’’, a family representative of this category.

Table C.2. Reference key.

Reference code Means

b Froese and Pauly (2007)

e Egloff (2006)

f Foote (1997)

i Iversen (1967)

k Kitajima et al. (1985)

m Simmonds and MacLennan (2005)

r Bertrand et al. (1999)

w Whitehead and Baxter (1989)

Table C.3. Bladder type by order for ray-finned fishes (Actinopterygii). See Tables C.1 and C.2 for bladder and

reference codes used in the last column.

Order Families Bladder Relevant extract

present

(bladder

code)

Anguilliformes Anguillidae, Chlopsidae, Colocongridae, Yes (Sb) ‘‘Swim bladder present, duct

Congridae, Derichthyidae, usually present’’

Heterenchelyidae, Moringuidae,

Muraenesocidae, Muraenidae,

Myrocongridae, Nemichthyidae,

Nettastomatidae, Ophichthidae,

Serrivomeridae, Synaphobranchidae

Clupeiformes Chirocentridae, Clupeidae, Denticipitidae, Yes (Sw) ‘‘Clupeoids . . . are physostomesEngraulidae, Pristigasteridae with one, or often two, ducts

between the swimbladder and

the exterior: a pneumatic duct

from the stomach, and an anal

duct to the ‘cloaca’,’’ p. 300.

‘‘[Pneumatic duct] is invariably

present,’’ p. 346

Gadiformes Bregmacerotidae, Euclichthyidae, Yes (Lb) ‘‘Swim bladder without

Gadidae, Lotidae, Merluccidae, Moridae, pneumatic duct’’

Muranolepididae, Phycidae

Gadiformes Macrouridae, genus Squalogadus No (Nb) ‘‘The swim bladder is absent in

Melanomus and Squalogadus’’

Gadiformes Melanonidae No (Nb ) ‘‘The swim bladder is absent in

Melanomus and Squalogadus’’

Myctophiformes Myctophidae Yes (Mb) ‘‘Swim bladder usually present’’

Myctophiformes Neoscopelidae, genus Scopelengys No (Nb) ‘‘Swim bladder present in all

but Scopelengys’’

Myctophiformes Neoscopelidae, except Scopelengys Yes (Mb) ‘‘Swim bladder present in all

but Scopelengys’’

Notacanthiformes Halosauridae, Notacanthidae Yes (Pb) ‘‘Swim bladder present’’

Perciformes Sciaenidae Yes (Pb) ‘‘Swim bladder usually having

many branches and used as a

resonating chamber’’

Perciformes Ammodytidae No (Nb) ‘‘No swim bladder’’

Pleuronectiformes Achiridae, Achiropsettidae, Bothidae, Only in ‘‘Adults almost always without

Citharidae, Cynoglossidae, juveniles swim bladder’’

Paralichthyidae, Pleuronectidae, (Jb)

Psettodidae, Samaridae, Scophthalmidae,

Soleidae

Table C.4. Bladder type by family; see Table C.3 for details.

Family Order Bladder code

Achiridae Pleuronectiformes Jb

Achiropsettidae Pleuronectiformes Jb

Ammodytidae Perciformes Nb

Anguillidae Anguilliformes Sb

Bothidae Pleuronectiformes Jb

Bregmacerotidae Gadiformes Lb

Chirocentridae Clupeiformes Sw

Chlopsidae Anguilliformes Sb

Citharidae Pleuronectiformes Jb

Clupeidae Clupeiformes Sw

Colocongridae Anguilliformes Sb

Congridae Anguilliformes Sb

Cynoglossidae Pleuronectiformes Jb

Denticipitidae Clupeiformes Sw

Derichthyidae Anguilliformes Sb

Engraulidae Clupeiformes Sw

Euclichthyidae Gadiformes Lb

Gadidae Gadiformes Lb

Halosauridae Notacanthiformes Pb

Heterenchelyidae Anguilliformes Sb

Lotidae Gadiformes Lb

Macrouridae, genus Squalogadus Gadiformes Nb

Melanonidae Gadiformes Nb

Merluccidae Gadiformes Lb

Moridae Gadiformes Lb

Moringuidae Anguilliformes Sb

Muraenesocidae Anguilliformes Sb

Muraenidae Anguilliformes Sb

Muranolepididae Gadiformes Lb

Myctophidae Myctophiformes Mb

Myrocongridae Anguilliformes Sb

Nemichthyidae Anguilliformes Sb

Neoscopelidae, except Scopelengys Myctophiformes Mb

Neoscopelidae, genus Scopelengys Myctophiformes Nb

Nettastomatidae Anguilliformes Sb

Notacanthidae Notacanthiformes Pb

Table C.5. ‘‘Catchability’’ key

(Yang groups) used in Table

C.7.

Yang group Means

A Cod-like

B Flatfish

C Eels

D Herring-like

E Mackerel-like

Table C.6. Length key used in Table C.7.

Length code Name Description

FL Fork length Distance from tip of snout to end of middle caudal rays

(Froese and Pauly, 2007)

SL Standard length Distance from tip of snout to end of vertebral column

(roughly the start of the caudal fin) (Froese and Pauly,

2007)

TL Total length Distance from tip of snout to end of caudal fin (Froese

and Pauly, 2007)

L50 — The length at which 50% of females have reached sexual

maturity (Knijn et al., 1993)

Table C.4. (cont.)

Family Order Bladder code

Ophichthidae Anguilliformes Sb

Paralychthyidae Pleuronectiformes Jb

Pleuronectidae Pleuronectiformes Jb

Phycidae Gadiformes Lb

Pristigasteridae Clupeiformes Sw

Psettodidae Pleuronectiformes Jb

Samaridae Pleuronectiformes Jb

Sciaenidae Perciformes Pb

Scophthalmidae Pleuronectiformes Jb

Serrivomeridae Anguilliformes Sb

Soleidae Pleuronectiformes Jb

Synaphobranchidae Anguilliformes Sb

678 Appendix C

TableC.7.Fishandtheirbladders,sortedbyscientificname.Keys:forbladdercodeseeTablesC.1andC.2;forYanggroupseeTableC.5.

MaximumlengthisfromFroeseandPauly(2007)(seeTableC.6);L50isfromKnijnet

al.(1993).

Species(scientificname)

Commonname

Family

Bladder

Yang

Max.length/cm

code

group

(TL,SL,orFL);

L50/cm

Acantholabruspalloni

Scale-rayedwrasse

Labridae(wrasses)

25.0(TL)

Acipensersturio

Sturgeon

Acipenseridae(sturgeons)

500(TL)

Agonuscataphractus

Hooknose

Agonidae(poachers)

B21.0(TL)

Alosa

pseudoharengus

Alewife

Clupeidae(herrings,shads,sardines,

Sm

40.0(SL)

menhadens)

Ammodytesmarinus

Lessersand-eel

Ammodytidae(sandlances)

C25.0(TL)

Ammodytestobianus

Smallsand-eel


C20.0(SL)

Anarhichasdenticulatus

Northernwolffish

Anarhichadidae(wolf-fishes)

180(TL)

Anarhichaslupus

Wolf-fish


A150(TL)

Anarhichasminor

spottedwolffish


180(TL)

Anguilla

anguilla

Europeaneel

Anguillidae(freshwatereels)

133(TL)

Anisarchusmedius

Stouteelblenny

Stichaeidae(pricklebacks)

30.0(TL)

Anoplogaster

cornuta

Commonfangtooth

Anoplogastridae

15.2(SL)

Antimora

rostrata

Bluehake

Moridae(moridcods)

Aphanopuscarbo

Blackscabbardfish

Trichiuridae(cutlassfishes)

110(SL)

Aphia

minuta

Transparentgoby

Gobiidae(gobies)

C7.9(TL)

Arctogadusglacialis

Arcticcod

Gadidae(codsandhaddocks)

32.5(TL)

Appendix C 679Argentinasilus

Greaterargentine

Argentinidae(argentinesorherring

Lm

D70.0

smelts)

Argentinasphyraena

Argentine

Argentinidae(argentinesorherring

D

smelts)

Argyropelecushem

igymnus

Half-nakedhatchetfish

Sternoptychidae

3.9(SL)

Argyropelecusolfersii

Hatchet-fish

Sternoptychidae

ArgyrosomushololepidotusMadagascarmeagre

Sciaenidae(drumsorcroakers)

Le

200(TL)

Argyrosomusregius

Meagre


Arnoglossuslaterna

Scaldfish

Bothidae(lefteyeflounders)

B25.0(SL)

Artediellusatlanticus

Atlantichookearsculpin

Cottidae(sculpins)

15.0(SL)

Aspitrigla

cuculus

EastAtlanticredgurnard

Triglidae(sea-robins)

B50.0(TL)

Astronesthes

gem

mifer

Snaggletooth

Stomiidae(barbeleddragonfishes)

17.0(SL)

Belonebelone

Garpike

Belonidae(needlefishes)

Benthodesmuselongatus

Elongatefrostfish


100.0(TL)

Benthosemafibulatum

Spinycheeklanternfish

Myctophidae(lanternfishes)

10.0

Benthosemaglaciale

Glacierlanternfish


10.3(SL)

Benthosemapanamense

Lampfish


5.5

Benthosemapterotum

Skinnycheeklanternfish


7.0

Benthosemasuborbitale

Smallfinlanternfish


3.9(SL)

Beryxdecadactylus

Alfonsino

Berycidae(alfonsinos)

(continued)

680 Appendix C

TableC.7(cont.)


Commonname

Family

Bladder

Yang

Max.length/cm

code

group

(TL,SL,orFL);

L50/cm

Bolinichthysdistofax


9.0(SL)

Bolinichthysindicus

Lanternfish


4.5(SL)

Bolinichthyslongipes


5.0(SL)

Bolinichthysphotothorax


7.3(SL)

Bolinichthyssupralateralis


11.7(SL)

Boreogadussaida

Polarcod


Le

40.0(TL)

Bramabrama

Atlanticpomfret

Bramidae(breams)

D

Brevoortia

tyrannus

Atlanticmenhaden


50.0(TL)

menhadens)

Brosm

ebrosm

eTusk

Lotidae(hakesandburbots)

Buenia

jeffreysii

Jeffrey’sgoby

Gobiidae(gobies)

C

Buglossidium

luteum

Solenette

B

Caelorhinchuscaelorhinchus

Hollowsnoutgrenadier

Macrouridae(grenadiersorrattails)

Callionymuslyra

Dragonet

Callionymidae(dragonets)

B

Callionymusmaculatus

Spotteddragonet

Callionymidae(dragonets)

Centrolabrusexoletus

Rockcook

Labridae(wrasses)

Centrolophusniger

Blackfish

Centrolophidae

Appendix C 681Chelonlabrosus

Thicklipgreymullet

Mugilidae(mullets)

Chim

aeramonstrosa

Rabbitfish

Chimaeridae(shortnosechimaeras

A

orratfishes)

Chirolophisascanii

Yarrel’sblenny


Ciliata

mustela

Fivebeardrockling


Ciliata

septentrionalis

Northernrockling


Clupea

harengusharengus

Atlanticherring


Sfm

D45.0(SL);24(L50)

menhadens)

Clupea

harengusmem

bras

Balticherring


24.2(TL)

menhadens)

Clupea

pallasiipallasii

Pacificherring


46.0(TL)

menhadens)

Conger

conger

Europeanconger

Congridae(congerandgardeneels)

A300(TL)

Coregonusartedi

Cisco

Salmonidae(salmonids)

Sm

57.0(TL)

Coryphaenoides

arm

atus

Abyssalgrenadier


102(TL)

Coryphaenoides

rupestris

Roundnosegrenadier


110(TL)

Cottunculusmicrops

Polarsculpin

Psychrolutidae(fatheads)

30.0(SL)

Cottunculusthomsonii

Pallidsculpin

Psychrolutidae(fatheads)

35.0(SL)

Crystallogobiuslinearis

Crystalgoby

Gobiidae(gobies)

C

Ctenolabrusrupestris

Goldsinny-wrasse

Labridae(wrasses)

Cyclopteruslumpus

Lumpsucker

Cyclopteridae(lumpfishes)

A

(continued)

682 Appendix C

TableC.7(cont.)


Commonname

Family

Bladder

Yang

Max.length/cm

code

group

(TL,SL,orFL);

L50/cm

Cyclothonebraueri

Garrick

Gonostomatidae(bristlemouths)

3.8(SL)

Diaphustheta

Californianheadlightfish


11.4(TL)

Dicentrarchuslabrax

Europeanseabass

Moronidae(temperatebasses)

Le

103(TL)

Echiichthysvipera

Lesserweever

Trachinidae(weeverfishes)

B

Echiodondrummondii

Pearlfish

Carapidae(pearlfishes)

A

Enchelyopuscimbrius

Fourbeardrockling


A

Engraulisanchoita

Argentineanchoita

Engraulidae(anchovies)

17.0(SL)

Engraulisaustralis

Australiananchovy


15.0(SL)

Engraulisencrasicolus

Europeananchovy


20.0(SL)

Engrauliseurystole

Silveranchovy


15.5(TL)

Engraulisjaponicus

Japaneseanchovy


18.0(TL)

Engraulismordax

Californiananchovy


24.8(SL)

Engraulisringens

Anchoveta


Sm

20.0(SL)

Entelurusaequoreus

Snakepipefish

Sygnathidae(pipefishesandseahorses)

Etm

opterusspinax

Velvetbellylanternshark

Dalatiidae(sleepersharks)

A

Euthynnusaffinis

Kawaka

Scombridae(mackerels,tunas,bonitos)

Nbi

100.0(FL)

Euthynnusalleteratus

Littletunny


Nb

122(TL)

Appendix C 683Euthynnuslyneatus

Blackskipjack


Nb

84.0(FL)

Eutrigla

gurnardus

Greygurnard

Triglidae(sea-robins)

B60.0(TL);19L50

Gadiculusargenteus

Silverycod


15.0(TL)

argenteus

Gadiculusargenteusthori

Silverypout


D15.0(TL)

Gadusmorhua

cod


Lfm

A200(TL);70L50

Gaidropsarusvulgaris

Three-beardedrockling


A

Galeorhinusgaleus

Topeshark

Triakidae(houndsharks)

A

Gasterosteusaculeatus

Three-spinedstickleback

Gasterosteidae(sticklebacksand

Le

aculeatus

tubesnouts)

Glyptocephaluscynoglossus

Witch

Pleuronectidae(righteyeflounders)

B

Gobiusniger

Blackgoby

Gobiidae(gobies)

C

Gobiusculusflavescens

Two-spottedgoby

Gobiidae(gobies)

C

Gymnammodytes

Smoothsand-eel


C

semisquamatus

Gymnelusretrodorsalis

Auroraunernak

Zoarcidae(eelpouts)

14.0(TL)

Halargyreusjohnsonii

Slendercodling

Moridae(moridcods)

56.0(TL)

Helicolenusdactylopterus

Blackbellyrosefish

Sebastidae(rockfishes,rockcods,and

47.0(TL)

dactylopterus

thornyheads)

Hippocampusguttulatus

Long-snoutedseahorse

Syngnathidae(pipefishesandseahorses)

Le

16.0(TL)

Hippoglossoides

platessoides

Americanplaice


B82.0(TL);17L50

(continued)

684 Appendix C

TableC.7(cont.)


Commonname

Family

Bladder

Yang

Max.length/cm

code

group

(TL,SL,orFL);

L50/cm

Hippoglossushippoglossus

Atlantichalibut


B

Hoplostethusatlanticus

Orangeroughy

Trachichthyidae(slimeheads)

75.0

Hygophum

benoiti

Benoit’slanternfish


5.5(SL)

Hyperoplusim

maculatus

Greatersandeel


C

Hyperopluslanceolatus

Greatsandeel


C

Katsuwonuspelamis

Skipjacktuna


Nbi

108(FL)

Labrusbergylta

Ballanwrasse

Labridae(wrasses)

Lampadenaanomala


18.0(SL)

Lampadenaspeculigera

Mirrorlanternfish


15.3(SL)

Lampanyctuscrocodilus

Jewellanternfish


30.0(SL)

Lampanyctusintricarius


20.0(SL)

Lampanyctusmacdonaldi

Rakerybeaconlamp


16.0(SL)

Latesniloticus

Nileperch

Latidae(lates,perches)

Sm

193(TL)

Lebetusguilleti

Guillet’sgoby

Gobiidae(gobies)

C

Lebetusscorpioides

Diminutivegoby

Gobiidae(gobies)

C

Lepidioneques

NorthAtlanticcodling

Moridae(moridcods)

Lepidopuscaudatus

Silverscabbardfish


Appendix C 685Lepidorhombusboscii

Fourspottedmegrim

Scopthalmidae(turbots)

40.0(SL)

Lepidorhombuswhiffi

agonis

Megrim

Scopthalmidae(turbots)

B

Lesuerigobiusfriesii

Fries’sgoby

Gobiidae(gobies)

C

Lim

andalimanda

Dab


40.0(SL);12L50

Lophiuspiscatorius

Angler

Lophiidae(goosefishes)

A

Lumpenuslampretaeform

isSnakeblenny


A50.0(TL)

Lycenchelysalba

Zoarcidae(eelpouts)

26.7(SL)

Lycenchelysmuraena

Zoarcidae(eelpouts)

22.6(SL)

Lycenchelyssarsi

Sars’swolfeel

Zoarcidae(eelpouts)

Lycodes

esmarkii

Greatereelpout

Zoarcidae(eelpouts)

Lycodes

eudipleurostictus

Doublelineeelpout

Zoarcidae(eelpouts)

Lycodes

frigidus

Zoarcidae(eelpouts)

69.0(TL)

Lycodes

pallidus

Paleeelpout

Zoarcidae(eelpouts)

Lycodes

reticulatus

Arcticeelpout

Zoarcidae(eelpouts)

36.0(TL)

Lycodes

seminudus

Longeareelpout

Zoarcidae(eelpouts)

51.7(TL)

Lycodes

squamiventer

Scalebellyeelpout

Zoarcidae(eelpouts)

26.0(TL)

Lycodes

vahlii

Vahl’seelpout

Zoarcidae(eelpouts)

A

Lycodonusflagellicauda

Zoarcidae(eelpouts)

19.9(SL)

Macquarianovemaculeata

Australianbass

Percichthyidae(temperateperches)

Le

60.0(TL)

(continued)

686 Appendix C

TableC.7(cont.)


Commonname

Family

Bladder

Yang

Max.length/cm

code

group

(TL,SL,orFL);

L50/cm

Macrourusberglax

Onion-eyedgrenadier


Macruronusnovaezelandiae

Bluegrenadier

Merlucciidae(merluccidhakes)

120(TL)

Mallotusvillosus

Capelin

Osmeridae(smelts)

Lm

Maurolicusmuelleri

Pearlsides

Sternoptychidae

Melanogrammusaeglefinus

Haddock


Lem

A100.0(TL);30L50

Melanonusgracilis

Pelagiccod


18.7(SL)

Melanonuszugmayeri

Arrowtail


28(TL)

Merlangiusmerlangus

Whiting


A70.0(TL);20L50

Merlucciusalbidus

Offshorehake


40.0(TL)

Merlucciusaustralis

Southernhake


Lm

126(TL)

Merlucciusgayigayi

SouthPacifichake


Lm

87.0(TL)

Merlucciusgayiperuanus

Peruvianhake


68.0(TL)

Merlucciusmerluccius

Europeanhake


A

Merlucciusproductus

NorthPacifichake


Lm

91.0(TL)

Microchirusvariegatus

Thickbacksole

B

Microgadustomcod

Atlantictomcod


Le

38.0(TL)

Micromesistiusaustralis

Southernbluewhiting


Lm

90.0(TL)

Appendix C 687Micromesistiuspoutassou

Bluewhiting


Lm

Microstomuskitt

Lemonsole


B65.0(TL);20L50

Molvadipterygia

Blueling


A

Molvamolva

Ling


A

Moronesaxatilis

Stripedbass

Moronidae(temperatebasses)

Le

200(TL)

Mullussurm

uletus

Redmullet

A

Myctophum

punctatum


Myoxocephalusscorpius

Bull-rout

B

Myxineglutinosa

Hagfish

B

Neoscopelusmacrolepidotus

Large-scaledlanternfish

Neoscopelidae

25.0(SL)

Neoscopelusmicrochir

Neoscopelidae

30.5(SL)

Nerophisophidion

Straight-nosedpipefish


Nesiarchusnasutus

Blackgemfish

Gempylidae(snakemackerels)

130(SL)

Nezumia

aequalis

CommonAtlanticgrenadierMacrouridae(grenadiersorrattails)

36.0(TL)

Notacanthuschem

nitzii

Deep-seaspinyeels

Notacanthidae(spinyeels)

120(TL)

Notoscopelusjaponicus

Japaneselanternfish


Notoscopeluskroyeri

Lancetfish


14.3(SL)

Oncorhynchusgorbuscha

Pinksalmon


Oncorhynchusnerka

Sockeyesalmon


Sm

84.0(TL)

(continued)

688 Appendix C

TableC.7(cont.)


Commonname

Family

Bladder

Yang

Max.length/cm

code

group

(TL,SL,orFL);

L50/cm

Onogadusargentatus

Arcticrockling


Onogadusensis


Orcynopsisunicolor

Plainbonito


Nb

130(FL)

Oryziaslatipes

Japanesericefish

Adrianichthyidae(ricefishes)

Le

4.0(TL)

Osm

erusmordaxdentus

Arcticrainbowsmelt

Osmeridae(smelts)

32.4(TL)

Osm

erusmordaxmordax

Atlanticrainbowsmelt

Osmeridae(smelts)

Sm

35.6(TL)

Pagrusmajor

Redseabream

Sparidae(porgies)

Pk

100.0(SL)

Perca

fluviatilis

Europeanperch

Percidae(perches)

Le

51.0(TL)

Pholisgunnellus

Rockgunnel

Pholidae

A25.0(SL)

Pholislaeta

Crescentgunnel

Pholidae

25.0(TL)

Phrynorhombusnorvegicus

Norwegian(topknot)

B

Phycisblennoides

Forkbeard


Pleuronectesplatessa

Europeanplaice


B100.0(SL);33L50

Pollachiuspollachius

Pollack


A

Pollachiusvirens

Pollock


Lm

A130(TL)

Pomatoschistusmicrops

Commongoby

Gobiidae(gobies)

C

Pomatoschistusminutus

Sandgoby

Gobiidae(gobies)

C

Appendix C 689Pomatoschistusnorvegicus

Norwaygoby

Gobiidae(gobies)

C

Pomatoschistuspictus

Paintedgoby

Gobiidae(gobies)

C

Protomyctophum

arcticum


Pterycombusbrama

Silverpomfret

Bramidae(breams)

40.0

Raja

batis

Skate

B

Raja

circularis

Sandyray

B

Raja

clavata

Roker

B

Raja

fullonica

Shagreenray

B

Raja

montagui

Spottedray

Rajidae(skates)

B80.0(TL)

Raja

naevus

Cuckooray

B

Raja

radiata

Starryray

B47L50

Ranicepsraninus

Tadpolefish


Pc

Rastrineobola

argentea

Silvercyprinid

Cyprinidae(minnowsorcarps)

Sm

9.0(SL)

Rhinonem

uscimbrius


Salm

osalar

Atlanticsalmon

Salmonidae(salmons,trouts)

150(TL)

Salm

otruttatrutta

Seatrout


140(SL)

Salvelinusalpinus

Charr


Sardaaustralis

Australianbonito


Nb

180(FL)

Sardachiliensischiliensis

EasternPacificbonito


Nb

102(TL)

(continued)

690 Appendix C

TableC.7(cont.)


Commonname

Family

Bladder

Yang

Max.length/cm

code

group

(TL,SL,orFL);

L50/cm

Sardachiliensislineolata

Pacificbonito


102(FL)

Sardaorientalis

Stripedbonito


Nb

102(FL)

Sardasarda

Atlanticbonito


Nb

91.4(FL)

Sardinapilchardus

Europeanpilchard


25.0(SL)

menhadens)

Sardinopssagax

SouthAmericanpilchard


Sm

39.5(SL)

menhadens)

Sciaenopsocellatus

Reddrum


155(TL)

Scomber

japonicus

Chubmackerel


64.0(TL)

Scomber

scombrus

Atlanticmackerel


Nf

E60.0(FL);30L50

Scomberesoxsaurus

Skipper

Scomberesicidae(sauries)

Scopelengystristis

Pacificblackchin

Neoscopelidae

Nb

20.0(SL)

Scopthalm

usmaxim

us

Turbot

B

Scopthalm

usrhombus

Brill

B

Sebastes

fasciatus

Acadianredfish


30.0(TL)

thornyheads)

Sebastes

marinus

Oceanperch


Lm

A100.0(TL)

thornyheads)

Appendix C 691

Sebastes

mentella

Deepwaterredfish


55.0(TL)

thornyheads)

Sebastes

schlegelii


Sm

65.0(TL)

thornyheads)

Sebastes

viviparus

Norwayhaddock


A

thornyheads)

Sillagociliata

Sandsillago

Sillaginidae(smelt-whitings)

Le

51.0(TL)

Soleasolea

Commonsole

Soleidae(soles)

B70.0;27L50

Sparusaurata

Giltheadseabream

Sparidae(porgies)

Le

70.0(TL)

Spinachia

spinachia

Fifteen-spinedstickleback

Gasterosteidae(sticklebacksand

tubesnouts)

Sprattussprattusbalticus

Balticsprat


16.0(TL)

menhadens)

Sprattussprattussprattus

Europeansprat


Sm

D16.0(SL);10L50

menhadens)

Squalusacanthias

Spurdog

A

SymbolophoruscaliforniensisBigfinlanternfish


11.0(SL)

Symphodusmelops

Labridae(wrasses)

Synaphobranchuskaupii

Kaup’sarrowtootheel

Synaphobranchidae(cut-throateels)

100.0(TL)

Syngnathusacus

Greaterpipefish


Taractes

asper

Roughpomfret

Bramidae(breams)

Theragra

chalcogramma

Walleyepollock

Lfm

(continued)

692 Appendix C

TableC.7(cont.)


Commonname

Family

Bladder

Yang

Max.length/cm

code

group

(TL,SL,orFL);

L50/cm

Thunnusalalunga

Albacore


140(FL)

Thunnusalbacares

Yellowfintuna


Lr

Thunnusatlanticus

Blackfintuna


108(FL)

Thunnusgermo

Pacificalbacore

Pi

Thunnusobesus

Bigeyetuna


Lr

250(TL)

Thunnusthynnus

Northernbluefintuna


Pb

458(TL)

Trachinusdraco

Greaterweever

B

Trachuruscapensis

Capehorsemackerel

Carangidae(jacksandpompanos)

Lm

60.0(FL)

Trachuruspicturatus

Bluejackmackerel


Lm

60.0(TL)

Trachurussymmetricus

Pacificjackmackerel


Lm

Trachurustrachurus

Atlantichorsemackerel


E70.0(TL);24L50

Trachyrinchusmurrayi

Roughnosegrenadier


37.0(TL)

Trigla

lucerna

Tubgurnard

B

Triglopsmurrayi

Moustachesculpin

B

Trisopterusesmarkii

Norwaypout


Lm

A35.0(TL);13L50

Trisopterusluscus

Bib

A

Appendix C 693

Trisopterusminutus

Poorcod


A

Urophycistenuis

Whitehake


Valenciennellus

Constellationfish

Sternoptychidae

3.1(SL)

tripunctulatus

Xiphiasgladius

Swordfish

Xyphiidae

Zeusfaber

Dory

A

C.2 REFERENCES

Bertrand, A., Josse, E., and Masse, J. (1999) In situ acoustic target-strength measurement of

bigeye (Thunnus obesus) and yellowfin tuna (Thunnus albacares) by coupling split-beam

echosounder observations and sonic tracking, ICES J. Marine Science, 56, 51–60.


Egloff, M. (2006) Failure of swim bladder inflation of perch, Perca fluviatilis L. found in

natural populations, Aquatic Sciences, 58(1), 15–23.

Foote, K. G. (1997) Target strength of fish, in M. J. Crocker (Ed.), Encyclopedia of Acoustics

(pp. 493–500), Wiley, New York.

Froese, R. and Pauly, D. (Eds.), FishBase, version (01/2007), available at http://www.

fishbase.org/search.php (last accessed March 23, 2009).

Iversen, R. T. B. (1967) Response of yellowfin tuna (Thunnus albacares) to underwater sound,

in W. N. Tavolga (Ed.), Marine Bio-acoustics (Vol. 2, pp. 105–121), Proceedings Second

Symposium on Marine Bio-Acoustics, American Museum of Natural History, New York,

Pergamon Press, Oxford, U.K.

Kitajima, C., Tsukashima, Y., and Tanaka, M. (1985) The voluminal changes of swim bladder

of larval red sea bream Pagrus major, Bull. Japanese Soc. Scientific Fisheries, 51, 759–764.

Knijn, R. J., Boon, T. W., Heessen, H. J. L., and Hislop, J. R. G. (1993) Atlas of North Sea

Fishes: Based on Bottom-trawl Survey Data for the Years 1985–1987 (ICES Cooperative

Research Report No. 194), International Council for the Exploration of the Sea,

Copenhagen, 1993.

Simmonds, E. J. andMacLennan D. N. (2005) Fisheries Acoustics (Second Edition), Blackwell,

Oxford, U.K.

Tavolga, W. N. (Ed.) (1967)Marine Bio-acoustics (Vol. 2), Proceedings Second Symposium on

Marine Bio-Acoustics, American Museum of Natural History, New York, Pergamon Press,

Oxford, U.K.

Whitehead, P. J. P. and Baxter, J. H. S. (1989) Swimbladder form in clupeoid fishes, Zoological

Journal of the Linnean Society, 97, 299–372.

Yang, J. (1982) An estimate of the fish biomass in the North Sea, J. Cons. int. Explor. Mer, 40,

161–172.

694 Appendix C

Index(bold indicates main entry)

absorption cross-section 245

see also extinction cross-section; scattering

cross-section

acidity see pH

acoustic deterrents 523, 524, 525, 632

acoustic intensity 32 ff, 56 ff, 95 ff, 209, 245,

417, 550, 663

acoustic power 31 ff, 37 ff, 56 ff, 80 ff, 97,

209, 245

acoustic pressure 31ff, 41, 58, 96 ff, 192 ff,

233ff, 430, 492, 525ff, 548 ff, 563ff,

661

see also pressure (RMS)

acoustic sensors

communications equipment 88 ff, 523,

526, 575, 599 ff, 632

echo sounder see echo sounder

fisheries sonar 5, 22, 519, 520, 575

minesweeping sonar 520, 522, 575

navigation sonar 22, 528, 575

oceanographic sensors 5, 22, 528, 575

search sonar 519, 522, 575

seismic survey sensors 534 ff, 575

sidescan sonar 515 ff, 575

acoustic waveguide 462

bottom duct 478, 483, 508

channel axis 21, 138, 462 ff, 496ff

convergence zone 474

cut-off frequency 449, 458, 472, 490

deep sound channel 148, 462

multipath propagation 308, 452, 525

SOFAR channel 20�21, 23surface duct 459ff, 462 ff, 471 ff, 478ff, 502

adiabatic pulsations (of air bubble) 230 ff,

240, 367

see also isothermal pulsations; polytropic

index

Airy functions 204, 448, 648

Albersheim’s approximation 315�316,330ff, 597 ff

ambiguity

ellipse 301, 304

function 300�301, 304 ffsurface 301ff

volume 300

amplitude threshold 51, 63, 312, 313, 326 ff,

346, 531ff

see also detection threshold (DT); energy

threshold

analogue to digital converter (ADC) 251

analytic signal 281�282see also envelope function; Hilbert

transform

APL-UW High-Frequency Ocean

Environmental Acoustic Models

Handbook 175, 364ff, 372 ff, 392 ff,

411, 424ff, 622

array gain (AG) 62 ff, 69 ff, 76, 85 ff, 90, 98 ff,

102ff, 107, 114 ff, 122, 252, 271 ff,

308, 580ff, 594 ff, 611ff, 622 ff, 629

asdics 12, 16, 17

asdivite 12

ATOC 528

attenuation coefficient of compressional

wave 197, 199

see also attenuation coefficient of shear

wave; volume attenuation coefficient

in pure seawater see attenuation of sound

in seawater

in rocks 183

in sediments 172ff, 377 ff, 604

in whale tissue 156

attenuation coefficient of shear wave 197,

199

see also attenuation coefficient of

compressional wave

in rocks 183

in sediments 180

attenuation of sound in seawater 18, 28 ff,

146�148, 471audibility of sound in seawater 29, 615ff

see also attenuation of sound in seawater;

visibility of light in seawater

audiogram 550

see also hearing threshold

of cetaceans 551ff, 619 ff

of fish 555ff

of human divers 554 ff

of pinnipeds 551 ff

of sirenians 554

autocorrelation function 296 ff, 651

narrowband approximation 299

autospectral density 287, 296

background energy level 97, 98, 113

background level (BL) 610, 625ff

backscattering cross-section (BSX) 41, 106,

209, 400, 491, 493, 607

see also scattering cross-section; target

strength (TS)

of fish 219, 223, 246

of fluid objects 214�215of gas bubble 216, 246

of metal spheres 211

of rigid objects 210 ff

backscattering strength

bottom 391 ff

Chapman�Harris model 372

Ellis�Crowe model 224, 396, 398

Ogden�Erskine model 371�372surface 371ff, 502

Bacon, Francis 3

Ballard, Robert see historical vessels

(Titanic)

Balls, R. (Captain) see historical sonar

equipment (fish finder)

bandwidth 42, 61 ff, 68, 73, 76, 80 ff, 104,

112 ff, 279, 283ff, 306, 345, 346, 577,

579, 587, 604, 612 ff

see also critical bandwidth; critical ratio;

effective bandwidth

Batchelder, L. 17 see also historical

institutions (Submarine Signal

Company); sound speed profile

(thermocline)

bathymetry 126ff, 142

bathythermograph see also

conductivity�temperature�depth(CTD) probe

expendable (XBT) 129 ff

Spilhaus 17

beamformer 44 ff, 252 ff

array response 45, 61 ff, 84 ff, 98, 114

array shading see shading function

beam pattern 45 ff, 61, 252 ff, 272 ff, 576ff,

602, 607ff, 625

beamwidth 46, 69 ff, 261, 264, 265, 496,

513, 626

broadside beam 46, 71 ff, 87 ff, 102ff,

115 ff, 253ff, 267 ff

endfire beam 47, 114, 253ff, 267 ff, 580

sidelobe 257 ff, 264, 265, 627

steering angle 46, 252ff

Beaufort wind force 159ff

da Silva et al. 160, 162, 165

Lindau 160

WMO CMM�IV 165

WMO code 1100 159, 164, 165

Beauvais, G. A. see historical sonar

equipment (Brillouin�Beauvaisamplifier)

Behm, Alexander 16

Bessel function 261, 316, 645�646, 648see also modified Bessel function

696 Index

Beudant, Francois see historical events

(speed of sound in water, first

measurement of )

binary integration see M out of N

detection

bistatic sonar 96, 493 ff, 508, 587

Blake, L. I. 14

Boltzmann constant 126, 549

bottom reflected path 443 ff, 462

bottom refracted path 444 ff

Boyle, Robert William 10 ff

see also historical institutions (Applied

Research Laboratory); historical

institutions (Board of Invention and

Research)

Bragg scattering vector 206, 224

Bragg, W. H. (Professor) see historical


Research)

Brillouin, Leon see historical sonar

equipment (Brillouin�Beauvaisamplifier)

bubble pulse 537 ff

bulk modulus 193, 194 ff

adiabatic 367

see also polytropic index

of air 367

of dilute suspension 225

of gas bubble 229, 230 ff

isothermal 367

see also polytropic index

of saturated sediment 227

of water 8, 32, 192, 225, 228, 649

carrier wave 280

caustic 445 ff, 468ff, 504 ff

characteristic impedance 58, 429, 552, 576,

663

see also impedance

of air 37, 417

of seabed 172

of water 417, 550

chemical relaxation 18, 146

boric acid 18, 147

magnesium carbonate 18, 147

magnesium sulfate 18

Chilowski, Constantin 10 ff

see also Langevin, Paul

chi-squared distribution 51, 328

coherent addition 35ff

coherent processing 51, 64 ff, 99 ff, 279,

312ff, 346, 574 ff, 606ff

Colladon, Daniel see historical events

(speed of sound in water, first

measurement of)

column strength (CS) 410, 412�413complementary error function (erfc) 49 ff,

85 ff, 339 ff, 482, 597, 637�638compressibility see bulk modulus

compressional wave 179, 193ff, 379


compressional wave

speed of compressional wave

Conan Doyle, Arthur 311

conductivity�temperature�depth (CTD)

probe 129, 134

see also bathythermograph

convergence zone (CZ) 474

see also acoustic waveguide

convolution 281, 344, 651

theorem 651

correlation

function 206

length 205, 206, 362, 370

radius 207, 224

cosine integral function (Ci) 640

Cox�Munk surface roughness slope see

roughness slope (surface)

critical angle see reflection coefficient

critical bandwidth 557 ff

critical ratio 557 ff

cross-correlation function 297, 298

cross-correlation theorem 650

CTD see conductivity�temperature�depth(CTD) probe

cubic equation, roots of 240, 476, 655

Curie, Jacques and Pierre see historical

events (piezoelectricity, discovery of )

cusp 468

da Vinci, Leonardo 18, 53

damping coefficient 216, 229, 237, 243 ff,

373

see also damping factor

damping factor 229ff

see also damping coefficient

Index 697

decibel (dB) 29, 58, 175, 525�526, 661�663see also logarithmic units

deep scattering layer 402, 412

density 192 ff, 492 ff

of air 30, 151, 237

of fish flesh 153, 155, 222

of metals 210, 212

of rocks 180ff

of seawater 8, 28, 127 ff, 233

of sediments 172 ff, 176, 178, 203, 227,

377ff, 393, 441, 449, 500, 604

of whale tissue 156

of zooplankton 156, 157

detection area 590

detection probability 47 ff, 71 ff, 85 ff, 92 ff,

103ff, 107 ff, 115 ff, 313ff, 329 ff

cumulative 354

detection range 77 ff, 90 ff, 107 ff, 117ff,

585ff, 605, 614 ff

detection theory 21, 47 ff, 311

detection threshold (DT) 63ff, 74 ff, 85 ff,

89 ff, 103 ff, 107 ff, 115ff, 279, 315 ff,

326ff, 347 ff, 355 ff, 581ff, 597 ff,

612ff

detection volume 587 ff

DFT see discrete Fourier transform (DFT)

dilatation 193 ff

dilatational viscosity see viscosity (bulk)

dipole source 38, 69, 419ff, 424 ff, 485,

535ff, 621

see also monopole source

Dirac delta function 62, 222, 314ff, 412,

591, 636, 650

Dirac distribution 314 ff

directivity factor 115, 266 ff, 580ff, 611 ff,

622

see also directivity index (DI)

directivity index (DI) 62, 69, 266, 580 ff,

594ff, 611

see also directivity factor

Dirichlet window see shading function

(rectangular window)

discrete Fourier transform (DFT) 43ff,

651ff

Doppler autocorrelation function (DACF)

299ff

Doppler effect 99, 298

Doppler resolution 294, 295, 301ff

see also frequency resolution; range

resolution

dose�response relationship 563

duct axis see acoustic waveguide (channel

axis)

echo energy level 606

echo level (EL) 400, 493, 508, 607 ff

echo sounder 5, 16, 22, 516, 575

see also acoustic sensors

multi-beam 516, 518, 519, 575

single beam 515, 516

effective angle 457

effective bandwidth 283ff

effective pulse duration 282 ff

effective water depth 457 ff

electromagnetic wave

radar 17, 21, 311 ff, 476

visibility of light 10, 29, 163

ellipsoid 212

surface area 155

volume 155

elliptic integrals 155, 467 ff, 644

energy density 32, 663

kinetic energy density 32

potential energy density 32

energy threshold 51, 63, 328

see also amplitude threshold; detection

threshold (DT)

envelope function 282 ff, 298

see also analytic signal; Hilbert transform

equivalent plane wave intensity (EPWI) 58,

493, 552ff, 661ff

equivalent target strength 607 ff

see also target strength (TS)

error function (erf ) 453, 481, 494, 508,

533 ff, 637

Ewing, Maurice see historical events

(SOFAR channel, discovery of )

explosives 431, 538ff

scaled charge distance 539ff

shock front 539 ff

similarity theory of Kirkwood and Bethe

539

exponential integrals 39, 66, 100, 297, 639

extinction cross-section

see also absorption cross-section;

scattering cross-section

698 Index

of fish 246

of gas bubble 246

facet strength 399

false alarm 7, 48, 54

false alarm probability 50 ff, 72, 87, 103,

115, 312, 328, 345, 350 ff, 582, 597ff,

613

false alarm rate 104, 115, 582, 613

far field 209, 400, 418, 431, 514ff, 576, 608

Fay, H. J. W. 13

Fessenden, Reginald 9 ff, 516

see also historical sonar equipment

(fathometer); historical sonar

equipment (Fessenden oscillator)

figure of merit (FOM) 69, 75 ff, 85, 91 ff,

101, 113, 121, 585, 619 ff

filter

anti-alias 42, 251, 594

band-pass 80

Doppler 99, 297

see also discrete Fourier transform

(DFT); Fourier transform

flat response 62, 84, 594 ff, 610

high-pass 42

low-pass 42, 251, 289

matched 280, 296ff, 345, 508, 606, 612

passband 42, 43, 62, 80, 264, 474, 558

pre-whitening 595

spatial see beamformer

temporal 42, 594

filter gain (FG) 593 ff

FishBase 152, 673

Fisheries Hydroacoustic Working Group

(FHWG) 560, 563

form function 210

see also scattering cross-section

Fourier transform 206, 281, 286, 289, 296,

649, 651, 652

Franklin, Benjamin 13, 18, 573

frequency modulation (FM) 22

hyperbolic (HFM) 283 ff, 305

linear (LFM) 283ff, 304 ff

frequency resolution 44, 90

see also Doppler resolution

frequency spread 285 ff

Fresnel integrals 288, 293, 636, 653

full width at half-maximum (f.w.h.m.) 44 ff,

256ff, 287

fusion gain 350 ff

gamma function 498, 640

incomplete 291, 328, 342, 497, 628, 642

Stirling’s formula 641

Gaussian distribution 47ff, 71, 208, 312,

316, 322

Gerrard, Harold 10, 14

see also historical institutions (Board of

Invention and Research)

grain size 172 ff, 180, 377, 392 ff, 454, 583,

599, 665

Gray, Elisha 14

see also historical institutions (Submarine

Signal Company)

grazing angle 38, 114, 116, 198ff, 205 ff,

224, 362ff, 376 ff, 428, 448ff, 464 ff,

495ff, 607, 626

Hall�Novarini bubble population density

model 169, 231 ff, 367

Hamming, Richard 259

see also shading function (Hamming

window)

Hayes, Harvey 13 ff

see also historical institutions (Naval

Experimental Station); historical

institutions (Naval Research

Laboratory)

Heard Island Feasibility Test (HIFT) 22,

528

hearing threshold 418, 550 ff, 619

see also audiogram; permanent threshold

shift; temporary threshold shift

Heaviside step function 281, 452, 635, 650

HFM see frequency modulation (FM)

HIFT see Heard Island Feasibility Test

(HIFT)

Hilbert transform 281

historical events

1918 Armistice 12

Cold War 4, 21

echolocation, first demonstration of 12

echo ranging, conception of 8, 13

First World War (WW1) 4, 7, 10 ff

piezoelectricity, discovery of 10, 13

Roswell incident 21

Index 699

Second World War (WW2) 4, 12 ff, 408 ff,

418

SOFAR channel, discovery of 20

‘‘sonar’’, coining of 17

speed of sound in water, first

measurement of 8

historical institutions

Anti-Submarine Division 12

see also asdics

Applied Research Laboratory (ARL) 16

Board of Invention and Research (BIR)

10, 14

British Admiralty 12, 13

California, University of 17

Columbia, University of 17

Lighthouse Board, U.S. 14

Manchester, University of 10

Marine Studios, Florida 23

National Defense Research Committee

(NDRC) 17

Naval Experimental Station, New

London 14

Naval Research Laboratory (NRL) 16, 17

Oxford University Press 12

Public Instruction and Inventions,

Ministry of 13

Submarine Signal Company 14, 16, 17

Woods Hole Oceanographic Institution

(WHOI) 17

historical sonar equipment

Brillouin�Beauvais amplifier 12, 13

‘‘eel’’ 15

fathometer 15

Fessenden oscillator 10, 11, 516

fish finder 15

gruppenhorchgerat (GHG) 17

JK projector 16

M�B tube 14, 15

M�V tube 15

QB 16

recording echo sounder 16

rho-c rubber 16

Rochelle salt 10, 16

sound fixing and ranging (SOFAR) 21

see also RAFOS

sound surveillance system (SOSUS) 21

towed fish 14

U-3 tube 15

underwater bell 8, 10, 14

historical vessels

Glen Kidston 16

Nautilus, USS 22

Prinz Eugen 17

Titanic, RMS 4, 10, 22

Hooke’s law 193

Hunt, F. V. 5, 7, 17, 515

see also historical events (‘‘sonar’’, coining

of )

Huxley, Thomas Henry 251

hydrophone

sensitivity 54, 514, 545, 594

hydrophone array

horizontal line array 55, 69 ff, 87 ff, 102ff,

116, 253, 267ff

line array 44, 114, 252ff, 267 ff, 580

planar array 261, 266ff

vertical line array 602

hypergeometric functions 232, 320, 648

in-beam noise level 74 ff, 92, 107, 584ff,

601 ff

in-beam noise spectrum level 92, 105

in-beam signal level 584 ff, 601 ff

incoherent addition 51, 80, 335, 341, 343,

578

incoherent processing 51, 80, 112, 327, 591

instantaneous frequency 283, 285 ff, 291ff

integration time 72, 89, 316, 345, 346, 597,

599, 602

Iselin, Columbus see historical institutions

(Woods Hole Oceanographic

Institution)

isothermal pulsations (of air bubble) 229 ff,

367

see also adiabatic pulsations; polytropic

index

K distribution 348

Kirchhoff approximation 208, 212

Lame parameters 195

Langevin, Paul 10 ff

see also historical events (echo location,

first demonstration of)

LFM see frequency modulation (FM)

700 Index

historical events (cont.)

Lichte, H. 19, 20, 439

see also historical events (SOFAR

channel, discovery of)

Liebermann, L. 18, 139

Lippmann, Gabriel see historical events

(piezoelectricity, discovery of)

Lloyd mirror 36, 64, 443, 474, 592

logarithmic units

see also pH

bel 660

see also decibel (dB)

decade 664

neper (Np) 29, 30, 660, 665

octave 264, 420, 558, 562, 595, 596, 665

phi unit (�) 173, 665

see also grain size

third octave 420 ff

longitudinal wave see compressional wave

M out of N detection 356

Marcum function 21

generalized Marcum function 330, 644

Marcum Q function 314 ff, 644

Marcum, J. 21

Marley, Bob 513

Marti, P. see historical sonar equipment

(recording echo sounder)

matched filter gain (MG) 306 ff, 612

mean square pressure (MSP) see pressure

(mean square)

see also pressure (RMS)

Mersenne see historical events (echo

ranging, conception of)

Michel, Jean Louis see historical vessels

(Titanic)

Minnaert, Marcel 191

modified Bessel function 314, 320, 326, 329,

644, 646�647see also Bessel function

monopole source 31ff, 418ff, 428, 491ff,

576

see also dipole source

Mundy, A. J. 14

see also historical institutions (Submarine

Signal Company)

M-weighting 559

Nash, G. H. see historical sonar equipment

(towed fish)

natural frequency 215, 216

see also resonance frequency

near field see far field

Neptunian waters 146

noise

ambient 55, 66, 73, 309, 415 ff

background 37 ff, 55, 61, 67, 427, 485, 557,

578, 614, 629

colored 596

dredger 490

flow 545, 549

foreground 578, 579

gain (NG) see array gain (AG)

isotropic 61 ff, 269 ff

level (NL) 61, 483, 585, 593, 605, 621, 624

non-acoustic 549, 578, 579

platform 579

precipitation 414, 415, 426, 489, 578, 596

self 55, 545, 550, 579

shipping 425, 427, 484, 485, 599

spectrum level 75

thermal 415, 484, 485, 488, 489, 545, 549,

578, 579

wind 115, 424�426, 484, 560, 614, 621,624

non-SI units 659, 662

see also logarithmic units; SI units

Nyquist frequency 42

Nyquist interval 87, 345, 346

Nyquist rate 306, 345

Ockham, William of 27

one-dominant-plus-Rayleigh distribution

313, 318, 322, 342

Painleve, Paul 13

see also historical institutions (Public

Instruction and Inventions, Ministry

of)

particle velocity 32, 192, 209, 550, 557, 630

permanent threshold shift (PTS) 558 ff

pH 664

free proton scale 664

National Bureau of Standards (NBS) scale

138, 147, 664

of seawater 28, 138, 147

Index 701

seawater scale 138, 664

total proton scale 138, 664

Physics of Sound in the Sea 17

physoclist 152, 158, 220, 401, 402, 619, 673,

674

physostome 152, 157, 158, 220, 401, 402,

673ff

Pichon, Paul 12

Pierce, G. W. 13

Plancherel’s theorem 652

Planck, Max 361

plane propagating wave 58, 552

Poisson’s ratio 195, 196

polytropic index 230, 234 ff

pressure

acoustic 31 ff, 58, 96, 97, 192, 198, 233,

243, 430, 492, 529, 531, 540, 550, 661

atmospheric 31, 60, 126, 127, 139, 151,

177, 216ff

complex 32, 35, 96

gauge 31

hydrostatic 30, 127, 151, 220, 230, 231

peak 431, 539, 540, 560, 565

peak to peak 525

peak-equivalent RMS (peRMS) 431, 531,

533, 534, 548, 617

RMS 32, 59, 415, 417, 431, 515, 529, 531,

533, 534, 549, 550, 576, 663

static 30, 31, 127, 231, 239, 367

zero to peak 525, 537

Principles of Underwater Sound 5, 18, 514,

576

prior knowledge 357, 583, 589, 591

probability of detection see detection

probability

probability of false alarm see false alarm

probability

processing gain (PG) 308, 593, 602, 610,

628

propagation factor 33, 59, 80, 608

see also propagation loss (PL)

coherent 64, 577

cylindrical spreading 452, 454, 481, 483,

494, 498

differential 452, 478, 496, 500, 576, 607,

608

incoherent 36, 82, 593

Lloyd mirror 443, 476

mode stripping 453, 494, 498

multipath propagation 443ff, 452ff, 478ff,

one-way 101, 106, 107, 116, 491, 494, 616,

619, 625, 627

single mode 457

spherical spreading 452

two-way 96, 100, 104, 113

Weston’s flux method 464 ff, 478ff

propagation loss (PL) 58, 60, 66 ff, 83 ff,

96, 101ff, 113 ff, 307, 365, 418ff,

440, 483, 493, 504, 506, 544, 576,

583 ff, 592ff, 607 ff, 663

see also propagation factor

PTS see permanent threshold shift (PTS)

pulse duration 96 ff, 115, 285, 287, 291,

294, 295, 302, 303, 305, 306, 345,

346, 495, 497, 531, 539, 565

see also effective pulse duration

p-wave see compressional wave

Q-factor 216ff, 244ff

quadratic equation, roots of 655

quartic equation, roots of 656

radiant intensity 33, 34, 60, 428, 429, 576,

592, 608

scattered 40, 99, 209, 396, 397, 400, 608

radiation damping 242, 243, 244, 373

see also damping coefficient; damping

factor

radius of curvature 466, 472, 476, 480, 504,

505

RAFOS 528

see also SOFAR

raised cosine spectrum see shading function

(Tukey window)

range resolution 115, 301 ff, 613

see also doppler resolution

Rayleigh distribution 51, 71, 317, 323, 340,

345, 347, 352, 355, 612

Rayleigh fading 317, 319, 582

Rayleigh parameter 205, 208, 373

Rayleigh�Plesset equation 232, 242

receiver operating characteristic (ROC)

curve 71, 85, 103, 115, 315 ff, 344ff,

581

reciprocity principle 492

702 Index

pH (cont.)

rectangle function 70, 253, 280, 285ff, 635,

636

reduced target strength 402, 406

see also target strength (TS)

reference distance 59, 60, 420, 431, 514, 544

reference pressure 59, 415, 554, 556, 663

reflection coefficient 408

see also reflection loss

amplitude 198, 201, 221, 222

angle of intromission 378

bottom 172, 177, 202ff, 375 ff, 447 ff,

452ff, 480

coherent 205, 207, 209

critical angle 378

cumulative 480

energy 200

Rayleigh 199, 375, 455

surface 30, 35, 37, 362 ff, 466

total internal reflection 377

reflection loss

see also reflection coefficient

bottom 375 ff, 445, 454ff, 508

surface 364 ff, 467ff

relaxation frequency 29, 147

resonance frequency 152, 216, 219 ff, 232 ff,

238, 239, 246, 409, 412, 413

see also resonant bubble radius

adiabatic 234

isothermal 235

Minnaert frequency 216, 232, 234, 236,

237, 238, 239

resonant bubble radius 232ff, 241

see also resonance frequency

reverberation level (RL) 495, 508

Reverberation Modeling Workshop 498,

503

Rice, Stephen 21

see also Rician distribution

Richardson, Lewis 10

see also historical vessels (Titanic)

Rician distribution 21, 317, 318, 319, 322

Rician fading 318, 319, 321

right-tail probability function 329, 637

rigidity modulus see shear modulus

RMS pressure see pressure (RMS)

see also acoustic pressure

ROC curve see receiver operating

characteristic (ROC) curve

rock 180

igneous 179, 180, 182, 183

metamorphic 180, 182

sedimentary 179, 180, 181, 182, 183, 184

roughness slope

bottom 225

surface 374

roughness spectrum 206, 224, 369, 392, 398

see also wave height spectrum

Gaussian 207, 224

Rutherford, Ernest (Lord) 10, 11, 14, 125

see also historical institutions

(Manchester, University of )

Ryan, C. P. (Captain) 14

see also historical institutions (Board of

Invention and Research)

salinity 20, 128, 129, 133, 139, 146,

absolute 129

practical 129

profile 134, 136, 439, 461

surface 135

scattering coefficient 41, 223, 224

see also scattering strength

backscattering coefficient 224, 225, 371

bottom 391ff, 496, 497

surface 42, 116, 369 ff, 625ff

scattering cross-section

see also absorption cross-section;

backscattering cross-section (BSX);

extinction cross-section

differential 40, 41, 209, 210, 214, 494, 607

of gas bubble 216, 243

total 209, 245

scattering strength

see also scattering coefficient

backscattering strength 371, 391

bottom 391ff

Ellis�Crowe model 398

Lambert’s rule 396

McKinney�Anderson model 399

surface 371ff

sea state 165ff

search sonar


coastguard sonar 522

helicopter dipping sonar 521, 575

hull-mounted sonar 15, 521, 575

sonobuoy 522, 575

towed array sonar 15, 522, 575, 579

Index 703

sediment

biogenic 172

chemical 172

clastic 172

consolidated 180, 385

unconsolidated 172 ff, 375ff, 583

seismic survey sources


air gun 535 ff, 560, 562, 575, 662

boomer 537, 538, 575

sleeve exploder 537, 538

sparker 537, 538, 575

sub-bottom profiler 514, 516, 520, 575

water gun 537, 538, 575

shading degradation 264, 269, 270

shading function 252, 259

cosine window 257, 264

Hamming window 259, 202, 264

Hann window 254, 258, 270, 583, 585

raised cosine window 258, 260

rectangular window 253, 254, 264

Taylor window 261, 264

triangular window 264

Tukey window 259, 264

shadow zone 459, 462

shear modulus 153, 192 ff, 219, 227

see also bulk modulus

shear speed see speed of shear wave

shear viscosity see viscosity (shear)

shear wave 172, 179ff, 194ff, 227, 379ff,

457

see also attenuation coefficient of shear

wave; speed of shear wave

SI units 39, 128, 141, 164, 659, 661

see also logarithmic units; non-SI units

sign function 635

signal energy level 97, 105

signal excess (SE) 63, 67, 84, 100, 112, 121,

322, 346, 357, 583 ff, 602ff, 621 ff

see also detection threshold (DT); figure of

merit (FOM)

signal gain (SG) 63, 69, 273, 584, 602

see also array gain (AG)

signal level 74 ff, 92 ff, 95, 109ff, 117, 414,

491, 585, 629

signal to background ratio (SBR) 55, 98,

100, 112, 116

signal to noise ratio (SNR) 5, 41, 51, 62,

67, 84, 104, 271, 272, 314, 324 ff,

332, 340ff, 345ff, 400, 545, 595, 598,

599

signal to reverberation ratio (SRR) 508

sine cardinal function (sinc) 43 ff, 253 ff,

296 ff, 636, 650

sine integral function (Si) 267, 640

sinh cardinal function (sinhc) 82, 91, 203,

383, 636

Smith, B. S. see historical institutions

(Applied Research Laboratory)

snapping shrimp 429

Snell’s law 19, 171, 199, 366, 377, 449, 459,

471, 480, 481, 505

SOFAR see acoustic waveguide (SOFAR

channel); historical sonar equipment

(SOFAR); see also historical sonar

equipment (SOSUS)

sonar equation 5, 6, 53, 573, 666

active (Doppler filter) 100

active (energy detector) 112

active (matched filter) 606

broadband passive (incoherent) 84, 279,

591

narrowband passive (coherent) 67, 279,

574

use of (worked examples) 74, 88, 105, 117,

583, 599, 613

sonar oceanography 27, 125

sound exposure level 559 ff

sound pressure level (SPL) 58, 417, 418,

663

sound speed see speed of compressional

wave; speed of sound in seawater

sound speed profile 145, 383, 459 ff, 474ff,

490

afternoon effect 16, 17

downward refracting 459, 462, 474 ff,

478 ff, 500, 502

isothermal layer 599

solar heating 459, 474

sound speed gradient 20, 28, 177ff, 389,

440, 445ff, 459, 471 ff, 478, 494, 506

summer 20, 459ff, 474

thermocline 129, 474, 479

see also Batchelder, L.;

bathythermograph

upward refracting 20, 462 ff, 478 ff, 500

wind mixing 365, 459, 462

winter 459 ff, 583

704 Index

source factor 60, 65, 74, 80, 81, 89, 100,

106, 419 ff, 424, 426, 429, 485, 491,

492, 496, 528, 548, 575, 576, 592,

607, 615

source level (SL) 60, 68, 85, 96, 97, 101,

113, 417, 493, 514, 525, 529, 531,

575, 592, 608

of acoustic cameras 523, 527

of acoustic communications systems 523,

526

of acoustic deterrent devices 523, 524, 525

of acoustic transponders 523, 527

dipole 419ff, 424 ff, 535ff

of echo sounders 515, 516, 518, 519

energy 96, 430, 525, 540, 544

of explosives 541

of fisheries sonar 519, 520

of marine mammals 542 ff, 616ff

of military search sonar 519, 521, 522

of minesweeping sonar 520

monopole 419ff, 428

of oceanographic research sonar 523, 528

peak to peak 430, 431, 540ff, 616

peak-equivalent RMS (peRMS) 431, 533,

548, 617

of seismic survey sources 534 ff

of sidescan sonars 515, 517, 519

of sub-bottom profilers 516, 520

zero to peak 431, 540

source spectrum level 88 ff, 417, 424 ff, 483,

599, 604

spatial filter see beamformer

specific heat ratio of air 150, 217, 230, 234,

235

spectral density

level 57, 61, 65, 67, 68, 81, 84, 488, 602,

663

power 66, 75, 89, 287, 424, 428, 595

speed of compressional wave 193

see also speed of shear wave; Wood’s

equation

in air 30, 148

in bubbly water 228, 365

see also Wood’s equation

in dilute suspension 226

see also Wood’s equation

in fish flesh 153, 155, 221

in metals 212

in rocks 181�183

in seawater see speed of sound in seawater

in sediments 172ff, 176, 177, 178, 196,

203, 227, 377 ff, 389, 393, 445 ff, 455,

500

in whale tissue 156

in zooplankton 156, 157

speed of shear wave see also speed of

compressional wave

in metals 212

in rocks 181�183in sediments 179, 379

speed of sound in seawater 8, 13, 19, 28,

126, 139, 145, 379

Leroy et al. formula 144

Mackenzie’s formula 140, 459

spherical wave 31�34spheroid

oblate 155

prolate 153, 154, 155, 215, 405, 407

Spilhaus, Athelstan 17

see also bathythermograph; historical

institutions (Woods Hole

Oceanographic Institution)

Spitzer Jr., Lyman see Physics of Sound in

the Sea

SPL see sound pressure level (SPL)

standard atmosphere 126, 220, 662

see also standard temperature and

pressure (STP)

standard gravity 126

standard temperature and pressure (STP)

126, 128, 151, 216, 219, 220, 662,

670

stationary phase approximation 284, 289,

290, 291�295, 296, 447, 448, 652statistical detection theory 21, 47, 311

Stirling’s formula see gamma function

Stokes, G. 18

STP see standard temperature and pressure

(STP)

Sturm, Charles-Francois see historical

events (speed of sound in water, first

measurement of)

surface area

of ellipsoid 155, 405

of fish 153, 405

of fish bladder 153, 401, 409

surface decoupling 459, 471, 474

surface tension 151, 230 ff

Index 705

surface wave spectrum 362

Neumann�Pierson 166, 167, 363

Pierson�Moskowitz 166, 168, 362, 364,

369, 370

SURTASS 519, 522

s-wave see shear wave

Swerling distributions 21

Swerling I 328

Swerling II 327, 340, 341, 344, 357

Swerling III 328

Swerling IV 327, 342, 343, 344, 347, 357

swim bladder 675, 676, 678

taper function see shading function

target strength (TS) 99, 101, 105, 113, 400,

493, 607, 610

of cetaceans 402, 403

of euphausiids 404

of fish 222, 401, 404, 619, 620, 624

of fish shoal 400

of gastropods 406

of human diver 402

of jellyfish 407

of marine mammals 402, 403

of mine 408

of siphonophore 407

of squid 406

of submarine 408

of surface ship 408

of torpedo 408

temperature

profile 127, 129, 131, 134, 135

see also sound speed profile

(thermocline)

potential 133

surface 20, 128, 129, 130, 459

temporary threshold shift (TTS) 558ff

Texas at Austin, University of see

Reverberation Modeling Workshop

thermal conductivity of air 151, 237, 244

thermal damping 243


factor

thermal diffusion frequency 236, 239

thermal diffusion length 235

thermal diffusivity 151

of air 151, 217, 235, 237

thermohaline circulation 128

third octave see logarithmic units

total internal reflection see reflection

coefficient

total path loss (TPL) 96, 97, 100, 101, 113

transmission coefficient

amplitude 199, 202

energy 200

transmission loss 60

transverse wave see shear wave

triangulation 15, 21

TTS see temporary threshold shift (TTS)

tunneling 472, 474

Udden�Wentworth sediment classification

scheme 173, 174

see also grain size

underwater acoustics 30, 191

Urick, R. J. 13, 19

see also Principles of Underwater Sound

viscosity 18, 146

see also attenuation of sound in seawater;

viscous damping

bulk 139, 155, 217, 244

shear 139, 155, 217, 232, 242, 244

viscous damping 242, 246


factor

visibility of light in seawater 29

see also audibility of sound in seawater

volume

see also surface area

of arthropods 152

of euphausiids 152

of fish 153

of fish bladder 153

volume attenuation coefficient


compressional wave; attenuation of

sound in seawater

of bubbly water 411

of dispersed fish 411

volume backscattering strength 399, 409 ff

volume viscosity see viscosity (bulk)

von Hann, Julius 257

see also shading function (Hann window)

wake strength 413

Washington, University of see APL�UWHigh-Frequency Ocean

706 Index

Environmental Acoustic Models

Handbook

wave equation 192, 193, 194, 200, 491

wave height 37

spectrum 166, 367

see also roughness spectrum

mean peak-to-trough 167

RMS 167, 168, 169

significant 166, 167, 168

waveguide see acoustic waveguide

Weibull distribution 348

Wells, A. F. 16

Weston, David E. 245, 439, 464, 468

see also propagation factor (Weston’s flux

method)

whispering gallery 468

Wiener�Kinchin theorem 651

wind speed 40, 159, 162�165, 166 ff, 367ff,424ff, 471 ff, 478, 484ff, 585, 621,

623, 624

window function see shading function

WMO see World Meteorological

Organization (WMO)

WOA see World Ocean Atlas (WOA)

Wood, Albert Beaumont 10, 11, 13, 14, 16

see also historical institutions (Applied

Research Laboratory); historical


Research); historical sonar

equipment (recording echo sounder);

Wood’s equation

Wood’s equation 226, 228, 366

World Meteorological Organization

(WMO) 159, 160, 162, 163, 164,

165, 166, 168

World Ocean Atlas (WOA) 129, 130, 131,

133, 135, 136, 137, 145

XBT see bathythermograph (expendable)

Young’s modulus 196

Zacharias, J. (Professor) see historical sonar

equipment (SOSUS)

Index 707