lectures wavefield i2 oristaglio blok

1Waveeld Imaging and Inversion in

Electromagnetics and Acoustics

Michael Oristaglio and Hans Blok

Abstract

These notes study the inverse problems that arise when electromagnetic and acoustic wavesare used to probe material objects. The methods studied have many dierent practicalusesin medicine, engineering, and the earth sciences. We concentrate on the mathemati-cal and computational techniques that are needed to understand how these methods work,in theory and practice. Topics covered include: the formulation of forward and inverse scat-tering problems using partial dierential and integral equations, the relationship betweeninverse source and inverse scattering problems, the classication of dierent inverse prob-lems and inversion methods, and the analysis of inversion algorithms based on linearization(the Born approximation) and non-linear optimization.

Contents

Preface 5

1 Introduction 71.1 Waves as tools for probing inside objects . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Seismic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.2 Electrical and electromagnetic methods . . . . . . . . . . . . . . . . 141.1.3 Inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.2 Basic equations of electromagnetics and acoustics . . . . . . . . . . . . . . . 281.2.1 Scalar wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.2.2 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.2.3 Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.2.4 Electromagnetics: Two-dimensional formulation . . . . . . . . . . . 341.2.5 Electromagnetics: Green functions . . . . . . . . . . . . . . . . . . . 35

1.3 Reciprocity theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.3.1 Reciprocity and Green functions . . . . . . . . . . . . . . . . . . . . 371.3.2 Reciprocity theorems in electromagnetics . . . . . . . . . . . . . . . 38

1.4 Formulation of direct scattering problems . . . . . . . . . . . . . . . . . . . 401.4.1 Integral equations and scattering . . . . . . . . . . . . . . . . . . . . 42

2 Inverse Problems 472.1 Denition of inverse source and scattering problems . . . . . . . . . . . . . 47

2.1.1 Relationship between inverse source and scattering problems . . . . 502.1.2 Inverse source and scattering problems as integral equations . . . . . 54

2.2 Some elements of inverse theory . . . . . . . . . . . . . . . . . . . . . . . . . 582.2.1 Linear rst-kind integral equations . . . . . . . . . . . . . . . . . . . 592.2.2 Minimum-norm solutions: Annihilating the annihilator . . . . . . . . 642.2.3 Regularized least-squares . . . . . . . . . . . . . . . . . . . . . . . . 67

2.3 Non-uniqueness in the inverse problems . . . . . . . . . . . . . . . . . . . . 70

3 Inversion Methods 733.1 Classication of inverse problems . . . . . . . . . . . . . . . . . . . . . . . . 733.2 Methods based on linearization . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.2.1 Inverting the Born approximation . . . . . . . . . . . . . . . . . . . 813.2.2 General models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.2.3 Imaging the Born approximation . . . . . . . . . . . . . . . . . . . . 993.2.4 Beyond the Born approximation . . . . . . . . . . . . . . . . . . . . 102

3

4 CONTENTS

3.2.5 Modied extended Born approximation . . . . . . . . . . . . . . . . 1063.3 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.3.1 Partial dierential equations . . . . . . . . . . . . . . . . . . . . . . 1083.3.2 Inverse scattering as abstract nonlinear inversion . . . . . . . . . . . 1113.3.3 Integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.4 Nonlinear (iterative) methods . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.4.1 Direct iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . 1253.4.2 Optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.5 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

MOM matrix algebra and the extended Born approximation . . . . . . . . . 141Modications of the extended Born approximation . . . . . . . . . . . . . . 143

4 Reciprocity and Inversion 1494.1 Time-domain reciprocity theorems for scalar waveelds . . . . . . . . . . . . 1494.2 Inverse scattering problem in the time domain . . . . . . . . . . . . . . . . . 1514.3 Reciprocity and sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1514.4 Sum-of-squares representing the error functional . . . . . . . . . . . . . . . 1534.5 Time-domain reciprocity theorems for EM elds . . . . . . . . . . . . . . . 1554.6 Time-domain inverse EM scattering problem . . . . . . . . . . . . . . . . . 1574.7 Reciprocity and sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1584.8 Reciprocity and optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 1604.9 Inversion versus migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Preface

These notes were the basis of a course of lectures of the same title1 given by the authorsat Delft University during the spring term (April to June) of 1995. The goal of the coursewas to present a survey of practical methods for solving the inverse problems that arisewhen electromagnetic and acoustic waves are used to probe material objects. Aside fromtheir intrinsic mathematical interest, these inverse problems have fascinating and importantapplications in all areas of applied science. In particular, during the past twenty years,the elds of medical imaging, geophysical exploration, and non-destructive testing andevaluation have all seen spectacular progress based not only on better ways of solving real-world inverse problems, but also on better understanding of what the solution of aninverse problem means.In putting together this survey, we tried to emphasize a few themes that we thought would

help students understand the many dierent inversion and imaging techniques that alreadyhave been developed and the new ones that (undoubtedly) will be developed in the yearsahead. These themes are: the interaction between partial dierential and integral equationformulations of inverse problems for acoustic and electromagnetic waves; the relationshipbetween inverse source and inverse scattering problems (especially, the implications of non-uniqueness in inverse source problems); the use of reciprocity in inverse scattering problems;and the unity and natural progression of dierent inversion algorithms when set in theframework of modern optimization theory. The second section of the notes develops therst two themes; the third, and longest, section tries to capture the last two. Of course,like any such attempt at present, it does not capture them fully! The rst section containsbackground material: a survey of electromagnetic and acoustic methods in the earth scien-ces, with examples drawn mainly from exploration for natural resources; the equations of(linear) acoustics and electromagnetism; reciprocity theorems; and the formulation of directscattering problems.The notes themselves capture only part of the coursenamely, (most of) the material that

was presented in the 16 lectures (of roughly 45 minutes each) that comprised the formalcourse, following the standard Delft University graduate curriculum. The class, whichconsisted of 15 graduate studentsmainly doctoral candidates in electrical engineering,mathematics, physics, and the geosciencesalso met for 6 additional sessions (of about 90minutes each) to discuss a selection of recent papers. (Two other sessions were devoted toproblems and review.) These informal sessions were prepared and led by students and werein many ways the most interesting and lively parts of the course. The papers discussedwere:

1Wavefield Imaging and Inversion in Electromagnetics and Acoustics, Centre for Technical Geoscience(CTG) course tg 101; ASEE course et 01-82; Electrical Engineering course et 01-38.

5

6 PREFACE

Devaney, A. J., 1978, Nonuniqueness in the inverse scattering problem, J. Math. Physics,19, 1526-1531.

Devaney, A. J., 1982, A ltered backpropagation algorithm for diraction tomography,Ultrasonic Imaging, 4, 336-350.

Tarantola, A., 1984, Inversion of seismic reection data in the acoustic approximation,Geophysics, 49, 1259-1266.

Kleinman, R.E., and van den Berg, P.M., 1992, An extended range modied gradient tech-nique for prole inversion, Radio Science, 28, 877-884.

Miller, D., Oristaglio, M., and Belykin, G., 1987, A new slant on seismic imaging: Migrationand integral geometry, Geophysics, 52, 943-964.

Habashy, T., M., Oristaglio, M. L., and de Hoop, A. T., 1994, Simultaneous nonlinearreconstruction of two-dimensional permittivity and conductivity, Radio Science, 29,1101-1118.

This selection is of course very personal, but is nonetheless representative for the range oftechniques that have been developed (and are being used) for solving practical acoustic andelectromagnetic inverse problems.The course was given while one of us (MLO) was a visiting lecturer in the Laboratory of

Electromagnetic Research (Faculty of Electrical Engineering) and the Centre for TechnnicalGeoscience at Delft University. We thank Delft University for providing both the oppor-tunity and nancial support for the visit and Schlumberger for allowing one of its researchmanagers to try such a thing. We also thank Rob Remis, who served as a teaching assistantfor the course, and Kasper Haak for their careful proofreading of the rst draft of thesenotes. Finally, we thank Toke Hoek for helping to organize the course materials.

Michael OristaglioHans BlokDelft, 30 June 1995

Chapter 1

Introduction

This course is about using electromagnetic and acoustic waves to make images of objects.Most of the images that we encounter in everyday life, and in science and technology, areformed from electromagnetic or acoustic (sound) waves. (And, of course, everything thatwe see is an image formed from the high-frequency electromagnetic waves called light.)The course will concentrate on methods that have been developed recently for seeing in-side objects using electromagnetic and acoustic waves. These methods have many dierentapplicationsin medicine, material science and manufacturing, civil engineering, and theearth sciences. The course will emphasize the mathematical and computational techniquesthat are needed to understand how these methods work, and will study some of their appli-cations in exploring below the earths surface for natural resources, such as hydrocarbons(oil and gas), minerals, and groundwater. The next section briey describes some of thesemethods.

1.1 Electromagnetic and acoustic waves as tools for probinginside objects

Electromagnetic and acoustic waves are often used to probe the insides of objects, includingthe earth. The goal of this is simple: to nd out whats inside the object without having tocut it open (or dig down) and sample the insides directly. The process of seeing inside anobject without opening it up goes by dierent names in dierent disciplines: non-invasivetechniques in medicine, non-destructive testing & evaluation in manufacturing and ma-terials science, and geophysical exploration or remote sensing in the earth sciences.Two words that are now commonly used in all these applications are tomography and

imaging (sometimes combined into the phrase tomographic imaging). Tomography, whichcomes from the Greek word tomos meaning section or slice, was the name originally given tothe method in which a series of X-ray scans are combined to form an image of a plane section,or slice, through an object. Because a computer was used to process the dierent X-ray scansand produce the nal image, the technique became known as computer-aided tomography,or computerized axial tomography, or just computerized tomography (Kak and Slaney, 1988).The images themselves are called CAT-scans or CT-scans (see Fig. 1.1). When introducedwidely into medicine in the 1970s, CAT revolutionized medical diagnosis. The revolutioncontinues with the techniques of magnetic resonance imaging (MRI), ultrasound imaging

7

8 CHAPTER 1. INTRODUCTION

(Fig. 1.2) (Macovski, 1983), and electrical impedance tomography (EIT) (Fig. 1.3) (Dijkstraet al., 1993; Kohn and McKenney, 1990).A similar tomographic revolution has been occurring in the earth sciences. X-rays can

only penetrate a few centimeters into rock or soil, but electromagnetic eldsfrom directelectrical currents (DC) to waves at MHz frequenciesand acoustic wavesat frequenciesfrom about 5 Hz to 1 kHzcan penetrate deep into the earth and be used to form imagesof its interior. (Very low-frequency acoustic and electromagnetic waves with periods ofminutes or days can penetrate all the way through the earth and are our main sources ofinformation about its deep interior.)The ability to make accurate images of the earths shallow interior with acoustic and

electromagnetic waves has led to tremendous improvements in our ability to locate andproduce hydrocarbons (oil and gas), minerals, and groundwater from inside the earth. Twobasic congurations are used (Fig. 1.4):

In surface methods, the electromagnetic or acoustic sources and receivers are movedalong the earths surface and probe downward into the earthto depths of severalmeters, in environmental studies, or to several kms, in oil and gas exploration. Surfacemethods allow rapid surveying of large areas, but their ability to resolve structuresdeep in the earth is limited.

In borehole or logging methods, devices called sondes are moved along a slim hole,usually about 10-20 cm in diameter, that is drilled into the earth to a depth thatdepends on the application. A typical well in mineral exploration is about 100-500 mdeep; in oil exploration, about 1-5 km deep. The sondes contain electromagnetic andacoustic sources and receivers that probe outward to sense the region within about 2m of the borehole. (Nuclear sources and receiversemitting and detecting -rays andneutronsare also used in logging but they probe only a few centimeters away fromthe borehole.) Borehole methods can measure with high resolution (on the order ofcentimeters) how the earths properties vary with depth at a given location, but their

X-ray source

detectorarray

Figure 1.1: (Left) Fan-beam geometry of X-ray tomography. The source and detector arrayrotate around the object to generate dierent scans. (Right) CAT-scan of a human head(from Kak and Slaney, 1998).

1.1. WAVES AS TOOLS FOR PROBING INSIDE OBJECTS 9

ultrasound source

receiver array

reflected waves

surface of the body

Figure 1.2: Medical ultrasound reection imaging. A source of high-frequency (10 MHz) soundwaves is placed on the surface of the body and reections are recorded by a receiver array.

lateral coverage is limited.

There are also hybrid methods that try to combine the lateral coverage of surface methodswith depth resolution of logging methods by placing sources or receivers both on the surfaceand in boreholes or in two dierent boreholes (cross-well methods). The following sectionsgive brief descriptions of some of the above methods and the types of images they generate.More detailed descriptions of these methods can be found in Telford et al. (1976).

1.1.1 Seismic methodsSurface seismic methods

In the surface seismic method (Fig. 1.5), a source of acoustic energy near the earths surfacegenerates sound waves which propagate both along the surface and down into the earth.A useful approximate model for analyzing the seismic method is that the earth consists ofhomogeneous regions (sedimentary layers or lithologic units) where the acoustic properties

1 2

3

4

5

i1 2

V3 4

electrode

Figure 1.3: Electrical impedance tomography. Current is injected sequentially between adjacentpairs of electrodes on an array placed around the body, and the potential drops across the remainingpairs are measured.


source receivers

logging sonde

wireline

Figure 1.4: Surface methods and borehole (or logging) methods for exploring the earthsshallow interior.

are constant, separated by interfaces where the properties change sharply. The downgoingacoustic waves generated by the source are partially reected and transmitted at theseinterfaces. The reected waves or echoes travel back to the surface and are recorded by anarray of acoustic receivers placed near the source (that is, near compared to the depthof the structures investigated). The source is then moved to a new location along a surveyline and the process is repeated.In land seismic surveys, the source is either an explosive charge buried in a shallow hole

or a large truck that vibrates a plate against earths surface. The receivers are geophoneswhich measure the vibrations caused by the acoustic waves. (Geophones have directionalcapabilities, and can measure vibrationslocal particle velocityalong three orthogonalaxes.) In marine seismic surveys, the sources and receivers are towed in streamers behinda ship that moves continuously along the survey line. The acoustic sources are arrays ofair-guns or water-guns which release a pulse of compressed air or water into the sea;the receivers are pressure sensors called hydrophones. The receiver streamer can containseveral thousand hydrophones and stretch a km or more behind the ship. Marine surveysallow rapid surveying in areal patterns called 3D seismic surveys, where the survey linesare closely spaced and the data are processed to provide 3D subsurface images (3D seismicsurveys are also carried out on land, but the process is much more tedious).Although sound waves in the solid earth are elastic waves, with both compressional and

shear components, seismic methods are often analyzed in the acoustic approximation whichneglects shear wave energy. (Shear energy is important even in marine surveys, becausethe pressure waves in the water are converted at the sea bottom into both compressionaland shear waves in the solid earth.) Reection (scattering) of acoustic waves is causedby changes in the acoustic wave speed c or density of the medium . For example, thereection coecient for an acoustic plane wave normally incident at the interface betweentwo materials is

R =c22 c11c22 + c11

=Z2 Z1Z2 + Z1

, (1.1)

where Z, the product of the acoustic velocity and density, is called the acoustic impedance.


The goal of seismic processing and imaging is to convert the recorded reections into a mapof structures within the earth (see Fig. 1.6). The useful frequency bandwidth in surfaceseismic surveys is about 550 Hz, and the range of (compressional) sound speed in rocksis about 25005000 m/s. So, the wavelengths of the acoustic waves in seismic surveys areabout 50 m or more, which determines the spatial resolution of seismic images. Traditionally,seismic data are processed so that the nal image emphasizes the location of interfacesroughly, changes in acoustic impedancebetween dierent rock types. A more recent andambitious goal, called seismic inversion, is to map quantitatively the (3-D) variation of theearths elastic properties.

acoustic source

acoustic receivers (geophones)

survey lines

hydrophone arrays

Figure 1.5: (Top) Schematic of surface seismic method. (Bottom) 3-D marine seismicsurvey.


Figure 1.6: Surface seismic images. (Top) Section through two salt domes. (Bottom) Sectionthrough growth fault. Horizontal scale in each image is approximately 20 km. Vertical scaleis two-way travel time.


surface source

hyrdophonearray

geophone

acousticsource

interface wave

Figure 1.7: (Left) Sonic logging method. (Right) Vertical seismic prole or VSP.

Sonic Logging

In many ways, sonic logging resembles a miniature marine seismic survey turned on itsside (with sources and receivers conned to a tube of uid in a solid; see Fig. 1.7). Anacoustic source (usually, a piezo-electric element) in the sonde generates a pressure wave inthe borehole uid that propagates out to the borehole wall, where it is converted to elasticwaves at the uid-solid interface. Although some of the acoustic energy propagates out intothe formation to interact with structures away from the borehole, the main eect is thegeneration of strong interface wavescompressional and shear head waves and guided-wavemodes, such as Stonely and pseudo-Rayleigh waveswhich propagate along the boreholeand re-enerate pressure waves in the borehole uid. These pressure waves are detected byan array of pressure sensors also housed in the sonde. Traditionally, sonic logging data areprocessed to obtain the variation of the compressional and shear wave speeds along thetrack of the borehole, generating a 1D prole called a sonic log. A more recent goal,called sonic imaging, is to process the echoes that are returned from structures away fromthe borehole into high-resolution images of the region near the well. Since sonic loggingtools use frequencies around 10-20 kHz, their resolution can be on the order of cms.

Borehole seismic methods or vertical seismic proles

Borehole seismic surveys (also called vertical seismic proles or VSPs) are a hybrid of surfaceseismic methods and sonic logging. An acoustic source is placed on the surface of the earthnear a borehole while a sonde containing geophones is lowered into the well and anchoredto the borehole wall at a particular depth to record the waves generated by the source. Ina true VSP, the source is placed very close to the wellhead, and recordings are made withthe sonde at dierent depths covering the entire length of the well, with a 5-15 m spacingbetween levels. Although the recordings are made by lowering the sonde to dierent depthssequentially and ring the source repeatedly, the full set of data can be treated as if thesource were red once and recordings were made simultaneously by an array of geophonesin the well (provided of course that the source is truly repetitive). A VSP records boththe waves travelling down into the earth and the upcoming echoes generated by interfaces.It provides a way of connecting the high-resolution local measurements of sonic logging


to the low-resolution large-scale measurements of surface seismics. The useful frequencybandwidth of VSPs is about 5-150 Hz, so it is in fact more like a local, high-resolutionseismic survey than a sonic log. Variations of the standard VSP involve moving the sourceto dierent positions at the surface in order to illuminate (and ultimately, image) a largerregion around the well.

1.1.2 Electrical and electromagnetic methods

Surveys with electromagnetic sources and receivers are used to map the earths electricalproperties:

, the electrical conductivity, or its inverse, electrical resistivity , , the electrical permittivity (or dielectric constant), and , the magnetic permeability.

In most regions, the dominant electrical property of the earth is the electrical conductivity, which can vary over many orders of magnitude in rocks. In fact, most (dry) rockshave virtually zero conductivity, but this is changed dramatically by the addition of smallquantities of metallic minerals to the rock matrix (e.g., in mineral deposits) or by saturatingthe pore space of the rocks with conductive uid (salt water). Even fresh ground water isconductive enough to raise the electrical conductivity of dry rocks signicantly. Some rockssuch as shales are naturally conductive.Two broad classes of electrical methods are used: direct-current (DC) or resistivity meth-

ods and induction methods. Sometimes, the term electrical method is reserved for DCsurveys and electromagnetic method for induction surveys, but we will use these termsinterchangeably to denote any method using electromagnetic elds.

Resistivity methods

In the surface DC or resistivity method, direct electrical current is injected into the groundthrough an electrode and the variation of electrical potential along the earths surface ismeasured with an array of electrodes or a roving electrode. There are many variations of theresistivity method, depending on the areal pattern of the survey and on whether currentis injected and potentials are measured with isolated electrodes (poles)that is, withthe current return electrode and potential reference electrode at innity (far away)orwith a closely-spaced electrode pairs (dipoles). Surface resisitivity methods are generallyused to nd conductive regions, associated with mineralization or groundwater, in otherwiseresistive host rock. But, in general, the goal of resistivity methods is to map the spatialvariation of electrical conductivity in the earth.Figure 1.9 shows an example of a combined surface and borehole resistivity survey that

was used to monitor an in-situ vitrication experiment (Ellis and Spies, 1995). In thisprocess, a region of the ground believed to contain hazardous waste is rst melted andthen alowed to solidify into a glassy state (vitried) with the goal of immobilizing thehazardous material.


i V

Figure 1.8: Surface resistivity method. Direct electrical current is injected into the groundwith a current electrode, and variation of electrical potential is measured along the earthssurface. In practice, low-frequency (10 Hz or less) current is used in order to average outnoise. The survey geometry in this gure is called the pole-dipole method.


.

.

.

.

.

.

.

.

Pre-melt ModelOffset y = 6.8m Misfit = 20%

.

.

.

.

.

.

.

.

.

.

Melt ModelOffset y = 6.8m Misfit = 1.3%

.

.

.

.

.

.

.

.

.

.

Post-melt ModelOffset y = 6.8m Misfit = 2.2%

ohm-m

10000

1000 -

100 -

10 -

1 -

x

z

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10 11

Depth

electrode

Figure 1.9: Images of electrical conductivity from DC resistivity surveys monitoring an in-situvitrication experiment for immobilizing hazardous waste. (Top) Image of resistivity in a cross-section through the region of the experiment. Image was obtained by inverting measurements froma survey in which an electrode array was laid out on the surface and in two boreholes anking theregion. Current was injected and potentials were measured sequentially in each of the electrodes ofthe array. (Middle) Resistivity image during the melting process. (Bottom) Resistivity image afterthe melt had cooled for several days. All of these images are slices from a full 3D inversion of themeasurements (from Ellis and Spies, 1995). Scale is indicated on the middle image in meters.


Resistivity logging

In resistivity logging, the current injection and potential measuring electrodes are packagedinto a sonde and moved along the well, making nearly continuous measurements. In astandard resistivity logging tool, the electrodes do not make contact with the formation.Current is injected into the borehole uid, and the current then passes into the surroundingformation (provided the borehole uid itself is reasonably conductive). Modern logging tools(see Fig. 1.10) actually use an array of extended current injection electrodes in combinationwith potential monitoring electrodes to dynamically focus the injected currents. One setof focussing conditions produces a pattern of currents that senses the region close to the

current injectionelectrode

A

A

0

1

A2

M1M2

A'1

A'2

i

Vpotential measuringelectrodes

well casing

M'1M'2

azimuthalelectrodes

Figure 1.10: (Right) Resistivity logging. An electrode in the borehole injects current that penetratesinto the formation and returns to a second electrode on the surface (or on another part on the loggingcable). Another pair of electrodes records the variation of electrical potential along the borehole.Current and potential electrodes are moved continuously along the borehole. (Left) Shematic of amodern resistivity logging tool. The electrode section of the tool is approximately 30 ft long. Thedark regions labelled A0, etc., are extended current-injection electrodes, the narrow regions, labelledM1, etc., are potential monitoring electrodes. The light regions are insulating sections. The dierentelectrodes are used to focus currents at two dierent depths into the formation, producing logs ofshallow and deep apparent resistivity.


borehole (within the rst 25 cms); a second set produces a current pattern that senses theregion between 25 cm - 1 m away from the borehole. This is needed because the electricalproperties of the region close to the borehole can be changed signicantly when the well isdrilled. The main eect is caused by the drilling uid leaking out of the borehole into thesurrounding rocks, a process called invasion. The two sets of readings are plotted as alog of deep and shallow apparent resistivity of the formation (see Fig. 1.11).Resistivity logging is one of the chief ways of detecting hydrocarbons in wells drilled for

oil and gas reservoirs. In the sedimentary basins where most oil and gas deposits are found,the pores of the rocks are saturated with saline uids lowering their electrical resistivity tothe range of about .1 - 1 m. If the pore space is (partially) lled with oil or gas, however,the resistivity rises dramatically to much higher values (10 m or more). Thus, boreholeresistivity methods are generally used to detect resistive regions in otherwise conductiverocks. An active research topic in electrical logging is the conversion of the raw (focussed)measurements of electrical logging into an accurate map of electrical resistivity around theborehole, including the invaded zone.


Figure 1.11: (Next Page) Modern resistivity log from the Azimuthal Resistivity Imager (Mark ofSchlumberger). The track at the far right contains the logs: the dotted curve (LLS) is the shallowresistivity, and the solid curve (LLD) is the deep resistivity. These curves are produced by twofocussed current patterns on the sonde. The gray curve is a high-resolution deep measurementobtained by averaging the readings on the short azimuthally-segmented electrode (see Fig. 1.10).The track on the far left contains curves from each of the azimuthal electrodes. The images inthe center, which show the variation of resistivity around the borehole, are obtained by processingreadings from these azimuthal electrodes. On the left is the raw image; on the right, the image hasbeen dynamically normalized to emphasize weak features.


ARI Image


Surface induction methods

Electromagnetic induction methods use a time-varying magnetic eld to induce electricalcurrents (by Faradays law) to ow in conductive regions of the earth (Fig. 1.12). Thesource is usually a loop of wire (or solenoid) through which is passed a controlled time-varying electrical current. Often the current loop is small enough to be modelled as anideal magnetic dipole. The receivers are magnetometers (or induction coils) which measurethe total magnetic eld (or its rate of change). The total magnetic eld consists of themagnetic eld of the source in free-space and the magnetic eld of any secondary currentsthat are induced in the ground. Again, there are many variations of the electromagneticmethod, which depend on the areal survey patterns and on the size and relative orientationof the magnetic eld sources and receivers (see Nabighian, 1991). In addition, inductionmeasurements can be made with a harmonic (sinusoidal) source operating at one or morediscrete frequencies, which is called the frequency-domain or harmonic electromagneticmethod (HEM), or with a transient source, which is called the time-domain or transientelectromagnetic method (TEM). Since direct electrical contact does not have to be madewith the ground, surveys can be made with induction sources and receivers carried byhelicopters or small airplanes (airborne electromagnetic methods). Induction surveys, likeresistivity surveys, are usually done for specic reasonssuch as mineral, groundwater, or

3-axis induction coils

helicopter system

large fixed-loop

i(t)

H = (Hx, Hy, Hz)

i(t) H(t)

Figure 1.12: Surface transient electromagnetic (TEM) surveys. (Top) Large xed-loop TEM sur-veys: a steady current owing through a large loop is switched o rapidly to induce currents in theearth. The decaying magnetic eld generated by the induced currents is measured along survey lineswith induction coils or magnetometers. The entire conguration is then moved to a new locationand the process is repeated. Small transmitter and receiver coils can also be mounted into a packagethat is towed beneath a helicopter or small plane.


depth(m)

airborne TEM image

surface TEM image

0

500

0

500

1000

1000 3000 5000 7000 m

Figure 1.13: Images of electrical conductivity in a 2D cross-section through the earth obtained byinverting airborne (top) and surface (bottom) TEM measurements along a 7 km survey line (fromMacnae et al., 1991). Light areas are regions of high conductivity.

hydrocarbon explorationbut their general goal is the same: to map the spatial variationof electrical conductivity in the earth (see Figure 1.13).

Remark. The range of useful frequencies of induction methods in geophysical exploration is about100 Hz to 100 kHz. This range is determined by the fact that time-varying electromagnetic elds areattenuated very rapidly in a conductive earth. The length scale of this attenuation is measured by aquantity called the skin-depth , which is the distance in which the amplitude of a plane harmonicelectromagnetic wave decays to e1 .37 of its intial value:

=

2

503.3m

f, (1.2)

where = 2f is the angular frequency of the plane-wave, and in the approximate numerical formula,we have used the value of in free-space (4 107H/m). Thus the range of useful frequenciesdepends on the average conductivity (or resistivity) of the ground and the scale of the investigation.For example, in a region of 1 m average resistivity, the skin depth at 1 kHz is about 16 m. The100 Hz100 kHz bandwidth quoted above encompasses most induction surveys.


Induction logging

Induction logging complements resistivity logging in oil and gas wells. These wells areoften drilled with drilling uids having an oil-base (oil-based muds) to give better lu-brication of the drill bit and better control over the reaction of the drilling uid with theformation (shales in particular tend to absorb water-based drilling muds and break downcausing the hole to collapse). Oil-based muds are insulators and prevent the ow of directelectrical current into the formation from an electrode suspended in the well. In inductionlogging (Doll, 1949), currents are induced in the formation with a time-varying magnetic

coil transmitter

array of coil receivers

formation layers

borehole with drilling mud

2 m

v

multi-turninduction coils

20 cm

magnetic field lines

Figure 1.14: Induction logging. A multi-turn coil carrying a time-varying current generates amagnetic eld that induces electrical currents in the formation (if the formation is symmetric aroundthe borehole axis, the currents ow in circular loops around the borehole). An array of receiver coilsmeasures the magnetic eld of the source and the secondary currents. Measurements at dierentspacings and frequencies are inverted to produce an image of electrical conductivity in the regionaround the borehole.


0.0

0.00.0

60.0

0.50

1.00

4.00

7.00

10.00

13.00

16.00

19.00

22.00

25.00

50.00

resistivityOHM M

borehole axis

in radial depth into formation

0.2 2000

10 m

Figure 1.15: Induction log from the Array-induction Imager Tool (Mark of Schlumberger) (left)Image of electrical conductivity varying with depth and distance from the borehole axis, obtained byinverting induction logging measurements. There is an extreme exaggeration of the horizontal scale;the image extends to only 1.5 m (60 in) from the borehole, but covers almost 100 m in depth. Theinvasion of borehole uid into the formation is prominent in the middle section of the image. (right)Induction logs showing the average (estimated) resistivity at dierent distances from the borehole.

eld, and the strength of the magnetic eld generated by the induced currents is used toestimate the resistivity around the borehole (see Figure 1.14). The constraints imposed bythe borehole limit the conguration of sources and receivers in induction logging. A moderninduction tool consists of a (multi-turn) transmitter coil with its axis along the long axisof the sonde and an array of induction coil receivers at distances ranging from about 20cm to 2 m from the transmitter. Measurements of the total magnetic eld at each of thereceiver coils are made at several frequencies between 10 kHz and 100 kHz. A completeset of readings is recorded about every 15 cm as the tool moves along the borehole axis.Measurements at the dierent separations and frequencies allow the formation to be probedto dierent depths, and can be converted (assuming the formation is symmetric around theborehole axis) into a 2D image of the electrical conductivity near the borehole (Fig. 1.15).

High-frequency electromagnetic methods

Electromagnetic elds in the earth are mainly diusive in nature until frequencies of about1 MHz (106 Hz). At these and higher frequencies, the dielectric properties of rocks becomeimportant and the electromagnetic eld propagates as a strongly attenuated wave (the skindepth in a 1 m earth at 1 Mhz is about 0.5 m). High-frequency electromagnetic methodssuch as ground-penetrating radar (GPR) or borehole radar are, however, becoming popularfor shallow, high-resolution environmental studies. In addition, dielectric logging (which


uses electromagnetic waves at frequencies up to a GHz) is a standard high-resolution loggingtechnique.

1.1.3 Inverse problems

A feature common to all the above methods is that a source of electromagnetic or acousticenergy generates a eld inside an object (in this case, the earth), and the eld is recordedat dierent positions on or inside a small region of the object by one or more receivers ofelectromagnetic or acoustic energy. The goal is to make an image of the objects interior.An image means a picture or map of the spatial variation of a physical property of theobject, such as its density, acoustic velocity, electrical conductivity, or index of refraction(dielectric function). It also implies that there is enough resolution to distinguish, withsome detail, structures inside the object and not just a few average properties.This course will study the mathematical and computational techniques that are necessary

to understand these methods of imaging the earths interior. All of these methods involveinverse problems for electromagnetic and acoustic wave elds. Moreover, they can all beconsidered as types of inverse scattering problems, if we use the term in a broad sense toinclude any inverse problem involving a physical eld generated by an applied source (evena static source, as in the inverse problems of static current ow).1

Inverse problems are the inverse of the usual mathematical problem, called a forwardproblem, in which the physical properties of a model are specied rst, and the goal isto calculate the models responseusually in order to develop an understanding of how aphysical system behaves or to compare the computed response with measurements of anactual objects response. This process of going from the models properties to its response iscalled solving a direct scattering problem, or solving a forward problem, or forward modeling,or just modeling.In the inverse scattering problem, the goal is to calculate the properties of the object

from its response to a source (or sources). Some questions that arise naturally in dealingwith such inverse problems are:

Does a solution exist? In the forward problems of electromagnetics and acoustics,this question has been answered completely in terms of general boundary and initialconditions under which the partial dierential or integral equations governing the eldshave solutions. The situation is less clear in inverse scattering problems. Compatibilityrelations must be imposed on the data (the models response) to guarantee that therecan be a realizable physical model that generates the dataan example of such arestriction is causality: the measurements should have been zero before the sourcehad actedbut the complete set of compatibility relations is not yet known. Theexistence problem is compounded in practical inverse scattering problems involvingmeasurements with random and systematic errors.

Is the solution unique? In the forward problems of acoustics and electromagnetics, ifa solution exists, it is unique. Perhaps surprisingly, this may also be true in inversescattering problems with complete data; that is, when the objects response isknown for all possible positions of the sources and receivers (outside the object, of

1In a narrow sense, inverse scattering applies only to the inversion of classical scattering data in whichthe wave field scattered by a localized object is measured in the far-field; see, eg, Colton and Kress (1992).


course) and all frequencies. In fact, proving uniqueness for certain inverse scatteringproblems has been one of the triumphs of mathematics in the last few decades, andwe will review some of this work in a later section. In practice, though, the objectscomplete response can never be measured, and a more relevant question is the nextone.

Is the solution stable? This is probably the biggest dierence between forward andinverse scattering problems. The forward problems of acoustics and electromagneticsare stable (or well-posed in the sense of Hadamard): a small change in the datathe initial or boundary conditionsgenerates a small change in the solution. Nearlyall the inverse scattering problems of acoustics and electromagnetics are ill-posed: asmall change in the data can generate (arbitrarily) large changes in the model (thisis true even for complete data). Another way of saying this is that there are manydierent models than can match the data, and the choice of a particular model requiresadditional constraints. This process is called regularization.

Is there an algorithm for computing the solution? Well-understood analytical and nu-merical techniques (separation of variables, eigenfunction expansions, nite-dierence,nite-element, moment methods, etc.) are available for solving forward problems. Sev-eral good algorithms for inverse scattering problems have been developed in recentyears, but there is still much room for improvement. Inverse scattering problems arenonlinear inverse problems that can have thousands or millions of unknowns: exam-ples are the inverse scattering problems whose solutions generated the images shownin the previous section. In fact, most images such as these are obtained by solvinga linearized version of the inverse scattering problem, which is accurate only underlimited conditions. Iterative methods are now being applied to the full nonlinear in-verse problem, but the convergence properties of these algorithmsdo the iterationsconverge and to what?are still only vaguely understood.

We will touch upon all of these questions in this course, but will concentrate mainly onthe last question: the development and analysis of algorithms for solving practical inversescattering problems in electromagnetics and acoustics.

Inverse source problems

In studying the inverse scattering problem, we will also consider another type of inverseproblem in which the goal is to obtain information about an unknown electromagnetic oracoustic source embedded in an object. The information sought might be the sourceslocation (for a point source) or spatial variation (for a distributed source) and its frequencycontent. This information is to be recovered from measurements of the eld outside theregion where the source is located. In this case, the assumption is made that the mediumselectrical or acoustic properties are known; only the source is unknown.This type of problem is called an inverse source problem. Two classical inverse source

problems in the earth sciences are determining the location and nature of earthquake sourcesfrom the seismic waves they generate and determining the distribution of mass in the earthsinterior from measurements of the gravity eld at the earths surface or in boreholes. An-other practical inverse source problem is the problem of antenna synthesis in electricalengineering; that is, the design of an antenna with a specic radiation pattern.


Inverse source and inverse scattering problems are related, but they have very dierentproperties. In one way, inverse source problems are easier than inverse scattering problemsbecause they are linear inverse problems (the equations of electromagnetics and acousticsare linear in the source terms). Inverse source problems are, however, inherently non-unique:there are always innitely many dierent sources that can generate the same eld outside theregion occupied by the source. In contrast, inverse scattering problems, although nonlinear,can be unique with complete data: there is one and only one model that matches the objectscomplete response. But complete measurements are never available, and practical inversescattering problems are both non-unique and unstable. As we will see, the non-uniquenessof the inverse source problem can help illuminate the behavior of practical algorithms forsolving inverse scattering problems.

Mathematical structure

The mathematical setting for the problems that we will study is that of recovering the coef-cients or right-hand-side of a partial dierential equation from knowledge of its solutions.We will work mainly with two equations:

The wave equation

2U 1c2(x)

ttU = S(x, t), (1.3)

for a scalar waveeld U = U(x, t), where c(x) is the local wave speed in the medium.The scalar wave equation is the basic equation of linear acoustics.

The diusion equation

2U 1D(x)

tU = S(x, t), (1.4)

for a eld U(x, t), which we will also (somewhat loosely) call a waveeld, whereD(x) is the local diusion constant. The diusion equation, with 1/D = ,applies to electromagnetic elds in a conductive medium when displacement currentsare neglected, and is the starting point for the analysis of electromagnetic inductionmethods.

In fact, for the most part, we will work with these two equations in their time-harmonicform, when U(x, t) = Re{u(x, )eit}, and both equations reduce to the Helmholtz equa-tion

2u+ k2(x)u = s(x, ), (1.5)

where k = kw = /c for wave phenomena and k = kd =i/D for diusion phenomena.

In dealing with problems of static current ow, we will also need to consider Poissonsequation

(x)u = s(x), (1.6)

where u(x) will be identied with the electric potential V .


The inverse problems for all of these equations are special cases of the inverse problemfor the general partial dierential equation,

a(x)u + b(x)u+ c(x)u = f(x), (1.7)

where the goal is to recover the coecients a(x) and b(x), and c(x) or the right-hand-side f(x) from (partial) knowledge of the solution u in some domain (Isakov, 1993). Themathematical structure of these problems has been studied intensively in this century, inthree waves of activity:

In the 1950s, in connection with the inverse scattering problems of quantum mechan-ics, which led to the classical analytical methods, associated with names Gelfand-Levitan and Marchenko, for reconstructing potentials in quantum mechanical scatter-ing (Newton, 1982);

In the 1970s, when it was discovered that these classical inverse scattering methodscould be used to study and construct the nonlinear waves called solitons (Lamb, 1980);and

In the 1980s and 1990s, in connection with the problem of electrical impedance to-mography, whose mathematical study was spurred by the famous paper of Calderon(1980), which lead to the remarkable uniqueness proofs of Kohn and Vogelius (1984,1985) and Sylvester and Uhlmann (1987).

Although these studies have rarely yielded practical algorithms, the proofs of uniquenessor non-uniqueness help to clarify what one can expect from practical algorithms. We willbriey review some of this work in a later section.In the rest of this section, we review the basic equations of electromagnetics and acoustics,

show how they lead to the scalar Helmholtz equation (or its generalization) in special cases,and discuss general features of the Helmholtz equation.

1.2 Basic equations of electromagnetics and acoustics

In both electromagnetics and acoustics, the eld quantities satisfy:

[1] Field equations, which in dierential form can be written as

[F ] [F ] = [S] , (1.8)

in which F(x, t) is a eld matrix, S(x, t) is a source matrix, and

F = {x, t} (1.9)

is a matrix partial dierential operator.

[2] Constitutive equations, which express the inuence of matter on the eld;

[3] Causality conditions; that is, initial conditions in the time domain.

1.2. BASIC EQUATIONS OF ELECTROMAGNETICS AND ACOUSTICS 29

Many of the problems in waveeld imaging and inversion will make use of time-harmonicwaves, so we will also introduce the eld quantities in the frequency-domain and write,

[F ] = Re {[F(x;)] exp(it)} , (1.10)

where = 2f denotes the angular frequency of the eld. In the frequency-domain, theeld quantities satisfy

[F ] [F ] = [S] , with F = x.

The constitutive equations, which for general (linear) media, are convolutions in the time-domain, reduce to simple products in the frequency-domain, but causality does requireappropriate regularity conditions on the frequency response of materials.

Electromagnetics

The eld equations of electromagnetics are Maxwells equations (see, e.g., Chew, 1990):

H+ iD Jc = Je, (1.11)E iB = Ke, (1.12)

D = , (1.13) B = 0, (1.14)

with the constitutive equations

D = (x;) E, (1.15)Jc = (x;) E, (1.16)B = (x;) H, (1.17)

where

H = magnetic eld strength (A/m),D = electric displacement (As/m2),E = electric eld strength (V/m),B = magnetic ux density (V s/m2), = charge density (As/m3),Jc = volume density of electric conduction currents (A/m2),Je = volume density of external electric currents (A/m2),Ke = volume density of external magnetic currents (V/m2), = permittivity (tensor) (F/m), = conductivity (tensor) (S/m), = permeability (tensor) (H/m).

Generally, we will work with the eld equations in which the constitutive relations withisotropic material propertiesscalar , , and have been incorporated

H+ (i)E = Je, (1.18)E iH = Ke. (1.19)


Thus, for the electromagnetic eld matrix we have

[F ] =[EH

],

while the source matrix is found to be

[SV] =[JK

]and [SS] =

[ nHnE

],

where SS refers to sources specied on a surface S with n as the unit (outward) normalto S. In later sections, we will consider the reduction of these equations to simpler formsin particular cases; for example, when the elds are purely diusive or static in nature, orwhen only a few of the vector components of the elds are active.Sometimes it is convenient to use Maxwells in subscript notation rather than in vector

notation. We then have

kijiHj + ( i)Ek = Jek , (1.20)

kijiEj iHk = Kek, (1.21)

where i = /xi denotes the partial derivative with respect to a (Cartesian) spatial coor-dinate xi (i=1,2,3) is the anti-symmetric Levi-Civita tensor dened by kij

kij =

1 if (k,i,j) is an even permutation of (1,2,3),0 if not all subscripts are dierent,1 if (k,i,j) is an odd permutation of (1,2,3).

(1.22)

(See for an extensive discussion De Hoop, 1995)

Acoustics

The eld equations (Navier-Stokes equations) for acoustic waves in uids in linearized formare

p+ iv = f , v iK p = , (1.23)

in which

p = acoustic pressure (Pa) = (N/m2),v = particle velocity (m/s),f = volume density of body force (N/m3), = volume density of injected volume (s1), = volume density of mass (kg/m3),K = compressibility (Pa1).

The eld matrix is

[F ] =[pv

],

while the source matrix is

[SV] =[f

]and [SS] =

[ n vn p

].


1.2.1 Scalar wave equation

To focus attention on features of the inverse problem, we shall start with a scalar formulationand introduce a (complex-valued) scalar wave function u(x;), which can be one of thecomponents of the eld matrix [F ] and which corresponds to a (real-valued) space-timefunction U = U(x, t) through

U(x, t) = Re {u(x;) exp(it)} . (1.24)

We will concentrate rst on the scalar wave equation, where the space-time function satisesthe wave equation,

2U c2(x)ttU = S(x, t), (1.25)

with S a space-time source function and c(x) the local speed of wave propagation in themedium (the wave speed). With minor modications, the same considerations apply tothe diusion equation. The wave function u(x;) satises the Helmholtz equation (alsocalled the reduced wave equation),

2u + 2c2(x)u = s(x, ). (1.26)

Notations

The Helmholtz equation is often presented in dierent forms, varying in the way the secondterm is written. Before proceeding, it is worthwhile to review some of these forms and theirnotations. The product of (angular) frequency and the inverse of velocity, which appearsin the second term,

c1 =

c

has the dimensions of wavenumber (angular frequency/length) and is usually denoted by k,giving

2u + k2(x)u = s(x, ). (1.27)

The reason for this, of course, is that when k is constant, the source-free (s = 0) version ofthe Helmholtz equation,

2u + k2 u = 0 (1.28)

supports plane wave solutions of (xed) wavelength, = 2/k; in other words, k really isthe wavenumber of these waves.Sometimes it is convenient to measure all velocities with respect to an (arbitrary) ref-

erence velocity, which will usually be noted by co. For example, in optics, co is naturallytaken as the speed of light in free-space, and it is conventional to specify (indirectly) thespeed of light in other materials by giving their plane-wave index of refraction,

n =coc. (1.29)


We can then write,

k2(x) =2

c2(x)=

2

c2o

c2oc2(x)

= k2o n2(x),

where ko = /co is the wavenumber at the reference velocity (of a harmonic wave of angularfrequency ). The Helmholtz equation becomes

2u + k2o n2(x)u = s(x, ). (1.30)In all of these formulas, the term that varies spatially appears as a square, which can

clutter subsequent expressions (especially series expansions). To remove it, k2 is sometimeswritten,

k2(x) = k2o [1 + (x)] , (1.31)

where is called the susceptibility function of the medium.2 The formula for is

(x) = n2(x) 1 = c2o

c2(x) 1, (1.32)

and the Helmholtz equation is then

2u + k2o [1 + (x)] u = s(x, ), (1.33)or

2u + k2o u+ k2o(x)u = s(x, ). (1.34)These last two forms are useful for scattering problems in which an inhomogeneous objectthat is, c, or n, or varies inside the objectis situated or embedded in a homogeneousmedium. If the reference speed co in (1.32) is taken as that of the surrounding medium, then(x) contains all the necessary information about the scattering object (and, moreover, iszero outside the object). For this reason, (x) is sometimes called the object function,and written as O(x):

2u + k2o u+ k2o O(x)u = s(x, ). (1.35)We already have (too) many ways of writing the same equation (remember that this

equation is one of the most studied in all of physics and each discipline has its own twiston notation), but we will introduce one more. There are also times when it is useful toconsider that the actual model is embedded in a spatially-varying background model ce(x)(e for the embedding model). Let

Oe(x) = e(x) =c2o

c2e(x) 1 (1.36)

be the object function of the embedding model, which is also measured with respect to theconstant absolute reference velocity co. Then, we can write the Helmholtz equation for thefull model c(x) as

2u + k2o [1 +Oe(x)] u+ k2o O(x)u = s(x, ), (1.37)2The terminology comes from electromagnetics, where the in this formula is the classical susceptibility.


where O is the dierence in object functions between the actual medium and the embeddingmedium:

O(x) = O(x)Oe(x). (1.38)

This form is somewhat awkward, however, in that two dierent reference velocities areinvolved (the constant reference velocity co and the varying reference velocity ce). We willoften write this case more simply as

2u + 2c2o (x)u + 2 m(x)u = s(x, ), (1.39)

where m(x) is an un-normalized contrast function,

m(x) =1

c2(x) 1c2o(x)

. (1.40)

If all this seems confusing (it is!), just go back to the original equation (1.26) to which allof these forms reduce.

1.2.2 Acoustics

The Helmholtz equation arises directly in acoustics as the equation for the pressure eldwhen the density of the medium is the same everywhere. Then, taking the divergence ofthe rst equation in (1.23) and substituting for v from the second equation, gives

2p + 2K(x) p = f + i, (1.41)

which shows that the local speed of acoustic (sound) waves in a medium is

c(x) =

1

K(x). (1.42)

If the density also varies in the uid, then p satises a slightly more complicated equationof Helmholtz-type,

1(x)p + 2K(x) p = 1(x)f + i. (1.43)

1.2.3 Green functions

Solutions of the reduced wave Equation (1.26) or (1.27) can be formally written using Greenfunctions

u(x;) =DG(x,x;)s(x;)dx. (1.44)

The Green function satises

2G(x,x;) + k2(x)G(x,x;) = (x x). (1.45)

G(x,x;) represents the response of a point source located at x = x.


1.2.4 Electromagnetics: Two-dimensional formulation

Vector Helmholtz equations follow directly from Maxwells Equations (1.18) and (1.19).Assuming that the permeability is a constant and that

= i = (x) (1.46)varies in space, we nd for E the vector Helmholtz equation

2E+(

E) + k2E = iJe ( 1 Je) +Ke. (1.47)

The equation for the magnetic eld strength H is even more complicated. In homogeneousmedia these equations are simplied to

2E+ k2E = iJe 1( Je) +Ke, (1.48)

2H+ k2H = Ke 1i

( Ke) Je. (1.49)

In Equations (1.47)(1.49), k is dened as

k2 = i = i( i) = 2+ i. (1.50)In two-dimensional formulations, where we assume that conguration and sources and sub-sequently the eld quantities are independent of one spatial coordinate, say x2, the electro-magnetic eld separates into two independent elds: a transverse electric (TE) eld with{E2,H1,H3}(x1, x3) = 0 and a transverse magnetic (TM) eld with {H2, E1, E3}(x1, x3) =0. The TE-eld can be excited by an electric current density Je2 (x1, x3). The correspondingequations are

11E2 + 33E2 + k2E2 = iJe2 , (1.51)H1 = (i)13E2, (1.52)H3 = (i)11E2. (1.53)

The fundamental eld quantity E2 = E2(x1, x3) satises a scalar Helmholtz equation. No-tice that = constant and = (x1, x3).The TM-eld can be excited by a magnetic current source Ke2 = K

e2(x1, x3). The corre-

sponding equations are found as

11H2 + 33H2 (ln ) H2 + k2H2 = Ke2 , (1.54)E1 = 13H2, (1.55)E3 = 11H2. (1.56)

In equation(1.54) the operator = (i11+ i33). For homogeneous media, the fundamentaleld quantity H2 = H2(x1, x3) satises the standard scalar Helmholtz equation

11H2 + 33H2 + k2H2 = Ke2 . (1.57)The presence of the logarithmic derivative of in Equation(1.54) makes TM-elds to behavedierent and more dicult to compute than TE-elds.

1.3. RECIPROCITY THEOREMS 35

1.2.5 Electromagnetics: Green functions

Solutions of Maxwells Equations (1.18) and (1.19) can be formally written using dyadic(tensor) Green functions

E(x;) =DGEJ(x,x;) Je(x, )dx +

DGEK(x,x;) Ke(x, )dx,

H(x;) =DGHJ (x,x;) Je(x, )dx +

DGHK(x,x;) Ke(x;)dx. (1.58)

In subscript notation these relations read

Em(x;) =DGEJmn(x,x

;)Jen(x;)dx +

DGEKmn (x,x

;)Ken(x;)dx,

Hm(x;) =DGHJmn(x,x

;)Jen(x;)dx +

DGHKmn (x,x

;)Ken(x;)dx (1.59)

The meaning of each Green tensor is clear from the relations: for example, GEJmn(x,x;)is the m-th component of the electric eld generated by an impulsive point electric-currentdipole located at position x, pointing in the n-th direction. The Green tensors satisfy

kijiGHJj,n + (x)GEJk,n = k,n(x x),

kijiG

EJj,n i(x)GHJk,n = 0, (1.60)

and

kijiGHKj,n + (x)GEKk,n = 0,

kijiG

EKj,n i(x)GHKk,n = k,n(x x). (1.61)

1.3 Reciprocity theorems

In this subsection, we present a number of reciprocity theorems pertaining to the scalar waveequation. A reciprocity theorem interrelates in a specic way the eld quantities associatedwith two dierent physical states that (can) occur in the same domain of space. Thesereciprocity theorems apply to a time-invariant conguration; the media in the physical statesare also assumed to be time-invariant and linear in their physical behavior (consititutiveproperties). In the time-domain, we distinguish between reciprocity theorems of convolutiontype and reciprocity theorems of the correlation type. Their frequency domain counterpartsare the classical reciprocity theorem (which in electromagnetics is named after H. A. Lorentzand in acoustics, after Lord Rayleigh) and the power reciprocity theorem, respectively.Further details and a full derivation of the equations of this section can be found in Blokand Zeylmans (1987).We consider a time-invariant bounded domain Do in which two dierent scalar states can

occur. The two states will be distinguished by the subscripts A and B. To avoid writingout separate expressions for the two dierent states, we will use the subscript {A,B} to


mean that either of A or B can be picked consistently in an expression to generate the twostates. The Helmholtz equations pertaining to both states are given by

2u{A,B} + k2{A,B}u{A,B} = s{A,B}, (1.62)

in which

k2{A,B} = 2c2{A,B}(x)

= k2o(1 + {A,B}). (1.63)

Here ko = /co; co is an (arbitrary) reference velocity; and

{A,B} = {A,B}(x;) =c2o

c2A,B(x) 1 (1.64)

is the susceptibility function (see Fig. 1.16).

Remark. We allow the susceptibility function to depend on , which implies of course that c(x)or co (or both) also is frequency dependent, even though we have omitted this from their arguments.Frequency dependence of any of these quantities also implies that they are complex valued andsatisfy causality constraints in the frequency-domain. In electromagnetics, frequency-dependence ofc arises naturally when the electrical conductivity is non-zero, since then

c2() = (

i

);

and c is frequency-dependent even if the material properties , , and are not.

With this conguration, the scalar-wave reciprocity theorem of the convolution type is found

DoA(x, )B(x, )

{uA, sA} {uB, sB}

Do

Do

nn

Do

State A State B

Figure 1.16: Domain Do and the two states A and B in the reciprocity theorem

1.3. RECIPROCITY THEOREMS 37

to beDo

[uA(x;)nuB(x;) uB(x;)nuA(x;)] dS(x) =

k2o

Do[A(x;) B(x;)] uA(x;)uB(x;) dV (x)

+Do[uA(x;)sB(x;) + uB(x;)sA(x;)] dV (x), (1.65)

where Do is the boundary surface of Do and

n = n

is the normal derivative, with n the unit normal directed away from Do. This classicalreciprocity theorem can be regarded as the most fundamental theorem of scalar wave theoryin the frequency-domain.An alternative form for the reciprocity theorem (1.65) is found when the complex conju-

gate of the wave eld uB(x;) is used in state B. The resulting theorem isDo

[uA(x;)nuB(x;) uB(x;)nuA(x;)] dS(x) =

k2o

Do[A(x;) B(x;)] uA(x;)uB(x;) dV (x)

+Do[uA(x;)sB(x;) + uB(x;)sA(x;)] dV (x). (1.66)

This relation corresponds to the time-domain reciprocity theorem of correlation type andis sometimes called a power reciprocity theorem.

1.3.1 Reciprocity and Green functions

As an application of the reciprocity theorem, we prove the symmetry property (betweensource and receiver points) of the Green function for the Helmholtz equation. The Greenfunction satises

2G(x,x;) + k2(x)G(x,x;) = (x x), (1.67)

where the partial derivatives operate on the x dependence of G(x,x;). As a function of x,G(x,x;) represents the response of a point source located at x. The symmetry propertyof G holds that the same value is obtained if the source is placed at x and the response ismeasured at x.The equation for G needs to be supplemented by boundary conditions to specify a unique

Green function. For simplicity, let G satisfy homogeneous Dirichlet boundary conditions onDo,

G(x,x;) = 0, x Do.


Then apply (1.65) to the two states:

State A: uA = G(x,x1;), sA = (x x1), A = (x), and (1.68)

State B: uB = G(x,x2;), sB = (x x2), B = (x). (1.69)

State A is a state with a point source at x1 and State B is a state (for the same medium)with a point source at x2 (Figure 1.17). The boundary terms in (1.65) vanish because of theboundary conditions and the term involving vanishes because A = B. There remains,

0 =Do[G(x,x1;)(x x2) +G(x,x2;)(x x1)] dV (x), (1.70)

or

G(x2,x1;) = G(x1,x2;). (1.71)

This symmetry has nothing to do with the symmetry of the medium; it holds for arbitarymodels, k(x) = /c(x). It also holds for other boundary conditions on G; in particular, itholds when Do is a wholespace and G satises radiation conditions.

1.3.2 Reciprocity theorems in electromagnetics

Following the discussion of reciprocity theorems for scalar wavefunctions in Subsection 1.3,we will present here the two reciprocity theorems of the convolution and correlation type inelectromagnetics. For derivation and illumination of these theorems, see De Hoop (1995).The global reciprocity theorem of the convolution type for Maxwells Equations (1.18) and

Do(x, )

n

Do

Do(x, )

n

Dox1

x2

State A

uA = G(x, x1;)

sA = (x x1)State B

uB = G(x, x2;)

sB = (x x2)

Figure 1.17: Two states in the application of the reciprocity theorem to prove symmetry ofthe Green function.

1.4. FORMULATION OF DIRECT SCATTERING PROBLEMS 39

S = D

B , B , B

{JB ,KB}

{EB ,HB}D

State B S = D

A, A, A

{EA,HA}

{JA,KA}D

State A

Figure 1.18: Two dierent electromagnetic states (elds, media and sources) occupy thesame space domain and interact in a reciprocity theorem.

(1.19) in domain D surrounded by surface S = D isSn (EA HB EB HA) dS(x) =

+D[i(B A)HB HA (B A)EB EA] dV (x)

+D[HA KB HB KA EA JB +EB JA] dV (x), (1.72)

where {EA,HA,JA,KA} and {EB ,HB ,JB ,KB} are two electromagnetic states with corre-sponding material properties {A, A, A} and {B , B , B}, respectively; n is the outwardnormal to S = D (see Figure 1.18). We will often arrange states in our use of (1.72) sothat the surface integral and the terms involving material properties vanish, leaving onlythe interaction of elds and sources in the two states:

D(HA KB HB KA) dV (x) =

D(EA JB EB JA) dV (x). (1.73)

The global reciprocity theorem of the correlation type isSn(EA HB EB HA) dS(x) =

+D[i(B A)HB HA (B A)EB EA] dV (x)

D[HA KB +HB KA +EA JB +EB JA] dV (x) (1.74)

Again, when the terms not involving sources vanish, the reciprocity theorem (1.74) becomesD(HA KB +HB KA) dV (x) =

D(EA JB +EB JA) dV (x). (1.75)


1.4 Formulation of direct scattering problems using reciprocity

This section shows how reciprocity can be used to formulate the direct scattering problem asan integral equation for the total eld. The direct scattering problem is shown in Fig. 1.19.There is a large domain Do with constant properties, c = co (later we will relax thisassumption), in which is embedded an object with varying properties c = c(x). The objectoccupies the region D Do. The term embedded here means the following: Removingthe object from D does not leave a hole in Do; instead the region D assumes the sameproperties as the rest of Do. The medium that occupies Do when the object is removedis often called the background medium. The source function si(x;) is located in a regionDi Do.We rst need the Green function for the Helmholtz equation in Do when the object in

not present,

2Go(x,x;) + k2o Go(x,x;) = (x x). (1.76)Again, we need appropriate boundary conditions for Go on the boundary Do of Do, or aradiation condition if Do extends to innity in certain directions. To be concrete, let usassume that Go (as a function of x) vanishes on the boundary Do. The Green function Go iscalled the background Green function, and for simple domains can be expressed analytically.Next we consider the eld when the object is present. This is the total eld u, which

satises

2u + k2o [1 + (x)] u = si(x;), (1.77)where

(x) =c2o

c2(x) 1, (1.78)

Do

(x, )

Do

D

si

Diuin

ko

embedding domain

scattering domain

source domain

Figure 1.19: The direct scattering problem


is the susceptibility function of the object and is non-zero only in the region D occupied bythe object. We take as boundary conditions for (1.77) that u also vanishes on Do.We can now apply the basic reciprocity theorem (1.65) to the two states:

State A: uA = u(x, ;), sA = si(x;), A = (x), and

State B: uB = Go(x,x;), sB = (x x), B = 0 .The boundary terms in the reciprocity theorem vanish because of the boundary conditionson u and G; there remains

0 = k2o

D(x;)u(x;)Go(x,x;) dV (x)

+Do

[u(x;)(x x) +Go(x;x;)si(x;)] dV (x). (1.79)The rst integral collapses to the domain D because is zero elsewhere. Rearranging gives

u(x;) =DiGo(x;x;)si(x;) dV (x) + k2o

DGo(x,x;)(x;)u(x;) dV (x), (1.80)

where the integral involving the source term si collapses to the domain Di where the sourceis non-zero. Now use the symmetry of the Green function, proved in the previous section,

Go(x,x;) = Go(x,x;),

to rewrite the above equation,

u(x;) =DiGo(x;x;)si(x;) dV (x) + k2o

DGo(x,x;)(x;)u(x;) dV (x). (1.81)

Finally, relabeling variables so that the integration is over the primed variablesthis stepis just adjusting notation; it has nothing to do with symmetrygives

u(x;) =DiGo(x;x;)si(x;) dx + k2o

DGo(x,x;)(x;)u(x;) dx, (1.82)

where we have also reverted to a simple notation dx for the volume element.This equation expresses u as the sum of two integrals, each of which has the form of a

source distribution radiating in the background medium with the Green function Go (as ifthe object were not present). The rst term on the right-hand-side is just the eld of theoriginal source si radiating in the background medium; we call this the incident eld uin.It is exactly the eld generated by the source when the object is absent (so it is the eldincident on the object before any scattering takes place),

uin(x;) =DiGo(x,x;)si(x;) dx. (1.83)

The second term in (1.82) is the eld of a set of sources which occupy the domain of theobject and also radiate in the background medium,

usc(x;) = k2o

DGo(x,x;)(x;)u(x;) dx. (1.84)


This term is called the scattered eld usc, and we discuss it in the next section.It is conventional to replace the rst term in (1.82) with uin giving the nal equation,

u(x;) = uin(x;) + k2o

DGo(x,x;)(x;)u(x;) dx. (1.85)

Equation (1.85) is the basic integral equation of scattering theory.

Remark. In most forward and inverse scattering problems, the domain Do is all of space, awholespace (or, as is common in geophysics, a halfspace). The appropriate boundary conditionfor the Helmholtz equation in a wholespace is an outgoing or causal radiation condition of the form,

limR

R

(u

R iku

)= 0

where R is the distance from the origin. This boundary condition, which guarantees uniqueness forthe solutions of the Helmholtz equation in innite domains, basically ensures that all waves froma localized source are outgoing at innity. It also ensures (again for a localized source) that theterm in parentheses above, when evaluated on a sphere encircling the origin, vanishes faster thanthe radius of the sphere. This property makes the surface integrals that appear in the reciprocityformula vanish at innity. When the domain is a halfspace, u satises the above radiation conditionon the semi-circle bounding the halfspace plus an ordinary boundary condition at the surface of thehalfspace, where u or its normal derivative (or a combination of the two) is specied.

1.4.1 Integral equations and scattering

Equation (1.85) (repeated here)

u(x;) = uin(x;) + k2o

DGo(x,x;)(x;)u(x;) dx,

holds everywhere in the domain Do; i.e., x in this equation can be any point in Do. Thisequation is equivalent to the original partial dierential equation for the total eld (1.77) andincludes the boundary conditions, which are implicit in the boundary conditions satised bythe Green function Go (and in the boundary conditions on uin, if uin is not dened directlyby 1.83). Although we derived this equation using reciprocity, it can also be derived usingthe principle of superposition and the denitions of the total, incident, and scattered elds.We will do this derivation in the next section, when we begin our discussion of inversescattering problems.Equation (1.85) serves two functions. It is both an equation to be solved for u and a

formula for computing u:

[1] When x D, the equation is a true integral equation, because x and x then occupythe same domain D and the function u(x;) outside the integral is the same as thefunction u(x;) inside the integral. Equations of this form, in which the unknownappears both under the integral and outside the integral, are called Fredholm integralequations of the second kind. There are many techniques (moment methods, nite-element methods, etc.) that can be used to solve such equations.


[2] After the equation is solved and u is known in D, the same equation can then beused to compute the eld anywhere in Do. In this case, equation (1.85) is an integralrepresentation of the eld.

In particular, the integral term in (1.85) represents the scattered eld, which is thedierence between the total and incident elds,

usc(x;) = u(x;) uin(x;) = k2oDGo(x,x;)(x;)u(x;) dx. (1.86)

The scattered eld contains all the eects of scattering by the object. According to thisrepresentation, the scattering object can be replaced by a set of distributed sources, whichare the product of the susceptibility function and the total eld inside the object,

(x;)u(x;). (1.87)

These sources are called constrast sources, because they involve the dierence in materialproperties between the background medium and the object through . In electromagnetics,the constrast sources are also called scattering currents because they have the form of aninduced current in the object. Equation (1.86) says that the scattering by the object canbe fully represented by the contrast sources radiating in the background medium. This is aremarkable feature of the scattering process, and contrast courses are a very useful conceptin scattering. But they do not help much in solving the forward scattering problem, becausethey are unknown until the eld inside the scatterer is known (in other words, until theproblem is already solved!).

Iterative methods and the Born approximation

A simple method of solving (1.85) is to iterate the equation. When x D, this equationcan be substitued into itself, giving,

u(x;) = uin(x;) + k2o

DdxGo(x,x;)(x;)uin(x;)

+ k4o

DdxGo(x,x;)(x;)

DdxGo(x,x;)(x;)u(x;). (1.88)

Repeating the process gives an innite series for u,

u(x;) =n=0

u(n)(x;), (1.89)

where

u(0)(x;) = uin(x;) (1.90)

u(n+1)(x;) = k2o

DGo(x,x;)(x;)u(n)(x;) dx. (1.91)

The series (1.89) is called the Neumann series expansion; it satises the integral equation(that is, represents u) when it is uniformly convergent (see Newtown, 1982). Successiveterms involve repeated integrals of over the domain D and higher powers of ko. The series


has a nice physical interpretation as a multiple scattering series (Fig. 1.20). If the scatterer isweakthat is, if |(x)|


where e(x;) is the susceptibility of the background medium. Equation (1.92) is thenproperly called the Distorted-Wave Born approximation or just the Distorted-Born approx-imation. Usage, however, is relaxing on this point, and in many modern papers the termBorn approximation is used to denote a linearization of the scattering equation aboutany background model.


References

Blok, H., and Zeylmans, M.C.S., 1987, Reciprocity and the formulation of inverse prolingproblems, Radio Science, 22, 1137-1147.

Born, M., 1926, Z. Phys., 38, 803.Calderon, A., 1980, On an inverse boundary value problem, Seminar on Numerical Analysis

and Its Applications to Continuum Physics, Rio de Janeiro, 65-73.Chew, W., 1990, Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold.Dijkstra, A.M., Brown, B.H., Leathard, A.D., Harris, N.D., Barber, D.C., and Edbrooke,

D.L., 1993, Clinical applications of electrical impedance tomography, J. Med. Eng. Tech.,17, 89-98.

Kohn, R. V., and McKenney, A., 1990, Numerical implementation of a variational methodfor electrical impedance tomography, Inverse Problems, 6, 389-414.

Doll, H. G., 1949, Introduction to induction logging and application to logging of wellsdrilled with oil base mud, J. Pet. Tech, 1, 148-162.

Ellis, R., and Spies, B., 1995, Geophysics, to appear.Fokkema, J.T., and van den Berg, P.M., 1993, Seismic Applications of Acoustic Reciprocity,

Elsevier.Isakov, V., 1993, Uniqueness and stability in multi-dimensional inverse problems, Inverse

Problems, 9, 579-621.Kak, A., and Slaney, M., 1988, Principles of Computerized Tomographic Imaging, IEEE

Press.Kohn, R., and Vogelius, M., 1984, Determining condutivity by boundary measurements,

Commun. Pure Appl. Math., 37, 281-298.Kohn, R., and Vogelius, M., 1985, Determining condutivity by boundary measurements,

interior results II, Commun. Pure Appl. Math., 38, 643-667.Lamb, M., 1980, Soliton Theory, Wiley.Macnae, J.C.., Smith, R., Polzer, B.D., Lamontagne, Y., and Klinkert, P.S., 1987, Imaging

quasi-layered conductive structures by simple processing of transient electromagneticdata, Geophysics, 52, 545-554.

Macovski, A., 1983, Medical Imaging Systems, Prentice-Hall.Nabighian, M.N. (ed.), 1991, Electromagnetic methods in Applied Geophysics, Part A, vol.

2, Society of Exploration Geophysicists.Newtown, R.G., 1982, Scattering Theory of Waves and Particles, 2nd ed., Springer-Verlag.Sylvester, J., and Uhlmann, G., 1987, Global uniqueness theorem for an inverse boundary

value problem, Ann. Math., 125, 153-169.Telford, W.M., Geldart, L.P., Sheri, R.E., and Keys, D.A., 1976, Applied Geophysics,

Cambridge U. Press.

Chapter 2

Inverse problems inelectromagnetics and acoustics

We begin the study of inverse problems for electromagnetics and acoustics with an overviewof inverse source and scattering problems for the scalar Helmholtz equation, including def-initions using both dierential and integral equations. We also study general properties ofinverse problemsnon-uniqueness, ill-posedness, and regularizationand show how theseproperties apply to inverse source and scattering problems. We present the theorems onnon-radiating sources (Devaney and Wolf, 1973; Friedlander, 1973; Cohen and Bleistein,1977), which lead to non-uniquessness in the inverse source problem. We then discuss theimplications of non-radiating sources for inverse scattering problems (Devaney, 1978; De-vaney and Sherman, 1982). We conclude with a review of recent mathematical results aboutuniqueness for certain inverse scattering problems.

2.1 Denition of inverse source and inverse scattering prob-

lems

We start by considering the inverse problems for a scalar waveeld u(x;) that satises theHelmholtz equation with a variable wave speed c(x) and a source term s(x;)

2u + 2c2(x)u = s(x;), x Do (+ ABC). (2.1)

Here x Do indicates that the dierential equations holds in some domain Do and thenotation (+ ABC) stands for plus appropriate boundary conditions. This is to indicatethat boundary conditions are always needed to give a well-dened problem for these partialdierential equations, but also that the specic boundary conditions are unimportant fordeveloping the ideas at this stage.Although it is usual to talk (and think) about the waveeld u as the quantity that satises

this equation, it is really the triplet of eld, wave speed, and source function,

{u, c, s},

that satises the equation (plus boundary conditions). Given any two of these quantities,the equation enables the calculation of the third. The forward problem is to determine

47

48 CHAPTER 2. INVERSE PROBLEMS

the eld u(x;) within a known model that is specied by the spatial variation of thewave speed c(x), when the eld is excited by a source function s(x;) (+ ABC). In otherwords, the forward problem is to solve equation (2.1) for u by any of the standard analyticalor numerical techniques for solving partial dierential equations: separation of variables,transform methods, nite-dierence or nite-element methods, etc.Loosely dened, the inverse source problem for the Helmholtz equation is to determine the

source s(x;), when the eld u(x;) and the wave speed c(x) are known (or measured).And the inverse scattering problem for the Helmholtz equation is to determine the wavespeed c(x), when s and u are known.

Remark. We can also consider a combined inverse-source/inverse-scattering problem, which wouldbe to determine both c and s, given u. Without further restrictions on c and s, it is not clear that aninteresting theory can be developed for such problems because one is left in eect with one equationin two unknowns. Nevertheless, in practical inverse scattering problems, imperfect knowledge of thesource or incident eld is common. For example, a chronic problem in seismic exploration is controlof the eective source waveform or spectrum. In this case, we have s(x;) = S(x)W (), wherethe spatial dependence of source S(x)its location, extent, etc.is well-known (that is, subject tonormal measurement errors), but its frequency spectrum W () is poorly known. Estimating W ()prior to, or as part of, inverting or imaging u plays a important role in most seismic processingschemes.

An important, but usually unstated, assumption in all of these problems is that the rightpartial dierential equation is being used to relate the dierent quantities u, c, s. (Imagineusing (2.1) to model u when u actually satises Laplaces equation!) If, in this example, theHelmholtz equation is indeed the right equation, then a simple, but important, observation isthat, when the u and c or u and s are known everywhere (that is, for all x), the inverse sourceand scattering problems are trivial. For example, when u and c are known everywhere, s isobtained by simply reading (2.1) backwards. Similarly, when u and s are known everywhere,rearrangement of (2.1) gives

c(x) =( 2u(x;)2u(x;) + s(x;)

)1/2. (2.2)

Thus, when the known quantities are available everywhere in space, the only non-trivialproblem is the forward problem!1 This simple observation suggests that one way to solvethe inverse problem when the quantities are only known in a limited region would be toextrapolate the eld from the region where it is known into the region where it is unknownand then apply a formula similar to (2.2). Waveeld extrapolation is, in fact, the mainapproach to solving the inverse problem of seismic imaging (but the formula for computingc is dierent).Non-trivial inverse source and scattering problems arise when the region where one or

more of the three quantities {u, c, s} are known, or can be measured, is restricted (see Figure2.1). Typically, we will work with two regions or domains: a large domain Do in whicheverything takes place (often this will be all of space) and a smaller domain D Do, whereone of the quantities of interest is unknown. Generally, the domain D is either inaccessible

1What happens if the known quantities are given everywhere but imperfectlythat is, with randomerrorsso that, for example, that direct application of (2.2) would give the square root of a negativenumber, but it is known that c(x) is real? See the interesting paper by Kohn (1991?).

2.1. DEFINITION OF INVERSE SOURCE AND SCATTERING PROBLEMS 49

or inconvenient for direct measurement (for example, the interior of the earth or the insideof the human body). Outside D, all quantities are known, or can be measured. We willuse the standard set notation x D and x D to indicate points that lie inside or outsidethe region of interest. Sometimes we will write Do Do D for the entire region (in Do)which is outside D.The goal of inverse source or scattering problems is to discover what is happening inside

the region D from measurements outside the region. In the inverse source problem, thegoal is to characterize sources in D from measurements of the waveeld outside the sourceregion (Figure 2.2).

Inverse source problem. Let {u, c, s} satisfy

2u + 2c2(x)u = s(x;), x Do (+ ABC). (2.3)

Assume that s(x, ) is contained (is non-zero) in D Do, but is unknown, and thatc(x) is known everywhere in Do. The inverse source problem is to recover s(x;)from measurements of u(x;) outside D.

In the inverse scattering problem (Figure 2.3), the roles of s and c as knowns and un-knowns are reversed. We will need several iterations to build up a full-denition of theinverse scattering problem. We start with the simplest.

source receivers

D

D

D = Do D

D

Do

Do

?

Figure 2.1: Domains for inverse source and scattering problems. The large domain Do iswhere everything takes place. A smaller region D in Do will be investigated. Receiversoutside D record a eld that is either generated by an unknown source ins

lectures wavefield i2 oristaglio blok

Documents

inverse scattering problems

inversion methods

inverse scattering problems

denition of inverse

elements of inverse

electromagnetic methods

seismic methods

inversion inelectromagnetics