Wavelet Correlation Analysis Applied to Wavelet Correlation Analysis Applied to Study Marine SedimentsStudy Marine Sediments

D.A. Yuen Department of Geology and Geophysics and Supercomputing Institute,University of Minnesota, U.S.A.

A.P.Vincent Département de Physique Université de Montreal, Canada

Alexander Kritski Centre for Marine Science and Technology Curtin University. Australia

OutlineOutline

•• AbstractAbstract

•• Wavelet Transform and Wavelet Correlation FunctionWavelet Transform and Wavelet Correlation Function

•• Applications of Wavelet CrossApplications of Wavelet Cross--Correlation Analysis. Correlation Analysis. Synthetic and Experimental dataSynthetic and Experimental data

•• Images of the first and second shear modes and theImages of the first and second shear modes and the ScholteScholte wavewave

•• ConclusionsConclusions

AbstractAbstract

The major motivation and objective of this work is to apply the The major motivation and objective of this work is to apply the wavelet correlation wavelet correlation analysis to study near surface sediments and develop a new diagnanalysis to study near surface sediments and develop a new diagnostic technique ostic technique for extracting physical properties from seismic data. Physical pfor extracting physical properties from seismic data. Physical properties of near roperties of near surface sediments play a prominent role in the geosciences and usurface sediments play a prominent role in the geosciences and underwater nderwater acoustics. Near surface sediments affect acoustic wave fields inacoustics. Near surface sediments affect acoustic wave fields in shallow watershallow waterwaveguideswaveguides and govern conditions for operation of active sonar. Some progrand govern conditions for operation of active sonar. Some progress ess has been made in understanding the propagation and attenuation chas been made in understanding the propagation and attenuation characteristics haracteristics of interface waves in different geological environments. Howeverof interface waves in different geological environments. However, the generating , the generating mechanisms are poorly understood. In particular, what is the acomechanisms are poorly understood. In particular, what is the acousticustic--seismic seismic energy conversion process? As seismic waves involve both time anenergy conversion process? As seismic waves involve both time and space d space parameters , we can relate directly the propagation characterisparameters , we can relate directly the propagation characteristics of the ocean tics of the ocean bottom interface waves to the physical properties of the sedimenbottom interface waves to the physical properties of the sediments over the ts over the propagation area. To address these problems we have applied the propagation area. To address these problems we have applied the wavelet wavelet correlation method to examine the variations of bottom charactercorrelation method to examine the variations of bottom characteristics and their istics and their role in coupling waterborne sound into the sea bottom. To confirrole in coupling waterborne sound into the sea bottom. To confirm the validity of m the validity of the developed modeling technique, we have produced synthetic sethe developed modeling technique, we have produced synthetic seismograms ismograms and applied wavelet correlation analysis for synthetic seismograand applied wavelet correlation analysis for synthetic seismograms ad marine field ms ad marine field data. We display images of the first and second shear modes and data. We display images of the first and second shear modes and thethe ScholteScholtewave (interface wave) component as a function of both arrival twave (interface wave) component as a function of both arrival time and frequency ime and frequency for both synthetic and field data.for both synthetic and field data.

ApplicationsApplications

••Marine AcousticsMarine Acoustics

••Exploration Geophysics. Potential fieldsExploration Geophysics. Potential fields

•• We study the propagation characteristics of the ocean bottom We study the propagation characteristics of the ocean bottom interface and shear waves to extract physical properties of mainterface and shear waves to extract physical properties of marinerinesediments over the propagation areasediments over the propagation area

••The wavelet analysis presents an unique capability for extractinThe wavelet analysis presents an unique capability for extracting the g the most essential scales of structures that change both in time most essential scales of structures that change both in time and space.and space.

••Seismic waves involve both time and space parameters it should bSeismic waves involve both time and space parameters it should be e able to decompose properties of seismic waves over a two dimenable to decompose properties of seismic waves over a two dimensionalsionaltime and period (frequencies).time and period (frequencies).

••We develop a new technique based on the wavelet correlation We develop a new technique based on the wavelet correlation method to examine the variations of bottom characteristics andmethod to examine the variations of bottom characteristics and their their role in coupling waterborne sound into the sea bottom. role in coupling waterborne sound into the sea bottom.

IntroductionIntroduction

dta

bttf

aabWf )()(

1),( ∫

∞

∞−

−= ψ

Wavelet TransformWavelet Transform

Continuous wavelet transform of function f(t) at

location b, relative to wavelet ϕ(t) at scale a:

[Holschneider, 1995]

)2/exp(*)exp(*)( 20

4/1 ttit −−= − ωπψ

Morlet WaveletMorlet Wavelet

Gaussian Gaussian Wavelet Wavelet

)(2

)( 04

0

220

tCosetHt

n ωωαπ α

ω−

=

4

dbabWfabWfT

WC y

T

Txxy ),(),(

1lim

2/

2/

τ+= ∫−

Wavelet Cross-Correlation Function for two different signals fx(t) and fy(t):

τ - time delayWfx(b,a), Wfy(b,a) - wavelet coefficients

[Li, 1998; Yuen et al., 2000]

Wavelet CrossWavelet Cross--Correlation Analysis Correlation Analysis applications to:applications to:

•• Synthetic DataSynthetic Datafrom an environment that lead to dispersion characteristics ifrom an environment that lead to dispersion characteristics in n agreement with the experimental results obtained from the shaagreement with the experimental results obtained from the shallow water llow water trial in May, 2001 (trial in May, 2001 (RottnestRottnest Island).Island).

•• Experimental dataExperimental datashallow water trial, May / 2001 (shallow water trial, May / 2001 (RottnestRottnest Island).Island).

direct arrivals

sourcewater column

marine sedimentsinterface waves

Acquisition scheme for generating synthetic seismograms

Synthetic Data

Physical Parameters of the Model Environment used for generatingsynthetic seismograms for inreface and shear waves. [Schmidt et al., 1984]

Dep

th, m

20

0

40

Cs [m/s]

50 150 250

Cw=1.7 km/sdensity=1870 kg/m3

αp = 0.1 dB/λp

0.3 0.50.1αs[dB/λs]

Synthetic Data

0 0.2 0.4 0.6 0 . 8 1 1.2 1.4 1.6 1.8 2-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7source signal

time, sec

amplitude

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5source signal, spectrum

frequency

amplitude

Synthetic Source

Time signal

Spectrum

Synthetic Data

0 5 10 15 20 25 30 35 40 45-0.5

0

0.5

1

1.5

2

2.5

3Synthetic seismograms

time, sec

range, km

75 m/s

130 m/s

250 m/s

Synthetic Data

Time, sec

Rang

e, k

m

5

4

3

2

1

Trace number

Synthetic Data

Surface wave

1st shear mode

2nd shear mode

Wavelet transform. Signal 5

Wavelet modulus for traces 5 (2.5 km) using Gaussian waveletSynthetic Data

Wavelet transform. Signal 5

Wavelet modulus for traces 3 (1.5 km) using Gaussian waveletSynthetic Data

Wavelet modulus for trace 2 (1.0 km) using Gaussian wavelet

Synthetic Data

0.5 1 1.5 2 2.5 3 3.5 4 4.5

100

150

200

250

300

Gaussian Wavelet analysis. Synthetic data. Receiver at 2.5 km from the Source

Frequency, hertz

Group Velocity, m/sec

Scholte Wave

2nd Share mode

1st Shear Mode

Group Velocity DispersionSynthetic Data

Wavelet Cross Correlation CoefficientsSynthetic data

Synthetic Data

Traces 5 and 3

Wavelet Cross-Correlation for traces 5 (2.5 km) and 4 (2.0km) usingMorlet Wavelet

Tim

e, s

0.0

40..0

10.0

Frequency, Hz0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.5 1.51 2 2.5 3 3.5 4 4.52 2.5 3 3.51.5

Tim

e de

lay,

s

0

5

10

15

20

Trace 4Trace 5Cross-Correlation

Synthetic Data

Experimental Data

Map of the southern Rottnest Basin03 and 04/052001

Map of the southern Rottnest Basin

Trial Area

03 and 04/052001

Experimental Setup

127 m/s

0.5

1.0

0 5 10 15 20 25 30 350

5

10

15bottom hydrophone

time, sec

Rang

e, km

0.5

1.0

1.5

2.0

2.5

1534 m/s

257 m/s 137 m/s77 m/s

Experimental Data

5

4

3

2

1

Trace number

Wavelet Transform of Field Signals

Field Data

Coefficients and Phases of Wavelet Cross CorrelationTraces 5 and 3

Traces 5 and 2Traditional Cross-Correlation

Traditional Cross-CorrelationPhase of WCR

Phase of WCR

WCR Moduli

WCR Moduli

Field Data

Frequency, Hz

Gro

up V

eloc

ity, m

/s

100

250

200

150

300

1.5 2.0 2.5 3.0 3.5

1st Shear mode

Scholte wave

Experimental Data

2nd shear mode

Group Velocity Dispersion

The phase velocity dispersion can be studied directly from The phase velocity dispersion can be studied directly from the phase field of WCRthe phase field of WCR

)]1()2

111

(([),,( 1 xxVV

txWCjj

n

jjxy ∆+−+=Θ ∑ τωω

jω - frequencies

Phase of the wavelet crossPhase of the wavelet cross--correlation function correlation function -- the field the field of phase velocities differences (of phase velocities differences (VV11, V, V22) between two ) between two surface waves measurement at two points (surface waves measurement at two points (XX11, X, X22) on ) on the oceanthe ocean--sediments interface for given frequencies.sediments interface for given frequencies.

Field Data

ConclusionsConclusions

1). The peaks of wavelet correlation coefficients perform the 1). The peaks of wavelet correlation coefficients perform the relative energy distribution in the first two shear modes and relative energy distribution in the first two shear modes and Scholte wave in the experimental seismic data showing Scholte wave in the experimental seismic data showing dispersion.dispersion.

2). Contributions of different periods (frequencies) to the2). Contributions of different periods (frequencies) to thecorrelation are kept reasonably separated.correlation are kept reasonably separated.

3). The phase velocity dispersion can be studied directly 3). The phase velocity dispersion can be studied directly from the phase field of the Wavelet Crossfrom the phase field of the Wavelet Cross--Correlation Correlation Function.Function.

4). The wavelet cross4). The wavelet cross--correlation coefficients can be used for the correlation coefficients can be used for the detecting of groupdetecting of group--velocity dispersion curve over as wide a velocity dispersion curve over as wide a frequency band.frequency band.

5). From the phase field of the wavelet cross5). From the phase field of the wavelet cross--correlation function correlation function the inversion of phasethe inversion of phase-- velocity dispersion can be evaluated over a velocity dispersion can be evaluated over a narrow frequency range where the signal is strong and the phase narrow frequency range where the signal is strong and the phase differences are easily determined.differences are easily determined.

Conclusions (II)Conclusions (II)

Future work:Future work:

••To perform a better resolution in the phase velocity inversion To perform a better resolution in the phase velocity inversion a crossa cross--correlation in spatial domain can be introduced as well. correlation in spatial domain can be introduced as well. In case of a crossIn case of a cross--correlation in correlation in xx direction (distance along direction (distance along the interface) in addition to the time domain crossthe interface) in addition to the time domain cross--correlation correlation the spatial component of phase of the WCR will be:the spatial component of phase of the WCR will be:

•• Inversion forInversion for geacousticalgeacoustical models; density, porosity, attenuationmodels; density, porosity, attenuationprofiles, shear profiles, shear modulimoduli..

∑∑ +=Θn

j

n

k j

jjxy V

WC )(ω

τω

References:References:

1.1. HolschneiderHolschneider, M. Wavelets and Analysis, ‘Oxford Science Publication’, 1995, M. Wavelets and Analysis, ‘Oxford Science Publication’, 19952.2. Li, H. Li, H. Identification of coherent structures in turbulent shear flow wiIdentification of coherent structures in turbulent shear flow with wavelet th wavelet

correlation analysiscorrelation analysis, Transactions of the ASME, Vol. 120, 1998, Transactions of the ASME, Vol. 120, 19983.3. YuenYuen, D. A. , A. P. Vincent, S.Y. Bergeron, F., D. A. , A. P. Vincent, S.Y. Bergeron, F. DubuffetDubuffet, A.A. Ten, V.C., A.A. Ten, V.C. SteinbachSteinbach, ,

L. Strain, L. Strain, Crossing of scales and nonCrossing of scales and non--linearitieslinearities in geophysical processesin geophysical processes, , Problems in Geophysics for the NewProblems in Geophysics for the New MilleniumMillenium, Eds., Eds. Enzo BoschiEnzo Boschi,, Göran EkströmGöran Ekströmand Andreaand Andrea MorelliMorelli, pp. 406, pp. 406--432, 2000.432, 2000.

4.4. GodinGodin, O.A., D.M.F. Chapman, , O.A., D.M.F. Chapman, Dispersion of interface waves in sediments with Dispersion of interface waves in sediments with powerpower--law shear speed profiles. I. Exact and approximate analytical relaw shear speed profiles. I. Exact and approximate analytical resultssults. . J.J.AcoustAcoust. Soc. Am., 110(4), 2001.. Soc. Am., 110(4), 2001.

5.5. Schmidt, H., G. Tango, Schmidt, H., G. Tango, Efficient global Matrix Approach to The Computation of Efficient global Matrix Approach to The Computation of Synthetic Seismograms,Synthetic Seismograms, GeophysGeophys. J.R.Astron. Soc. pp. 331. J.R.Astron. Soc. pp. 331--359, 1984.359, 1984.