Fundamentals of Seismic Tomography

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GEOPHYSICAL

MONOGRAPH

SERIES

David V. Fitterman, Series Editor

Larry R. Lines,Volume Editor

NUMBER

6

FUNDAMENTALS

SEISMIC

OF

TOMOGRAPHY

By Tien-when Lo and Philip L. Inderwiesen

SOCIETY

OF EXPLORATION

GEOPHYSICISTS

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Lo, Tien-when, 1957-

Fundamentals of seismictomography/by Tien-whenLo and Philip L. Inderwiesen.

p. cm. (Geophysicalmonographseries;no. 6) Includesbibliographicalreferencesand index. ISBN 978-1-56080-028-6:$22.00

1. Seismictomography.I. Inderwiesen,PhilipL., 1953II.

Title.

III.

QE538.5.L6

Series.

1994

551.2' 2' 0287--dc20

94-23818 CIP

ISBN 978-0-931830-56-3 (Series) ISBN 978-1-56080-028-6 (Volume)

Societyof ExplorationGeophysicists P.O. Box 702740

Tulsa, OK 74170-2740

¸ 1994by the Societyof ExplorationGeophysicists All rightsreserved.This bookor portionshereof may not be reproduced in anyform withoutpermission in writing from the publisher. Published

1994

Reprinted2000 Reprinted2004 Reprinted 2006 Reprinted2008 Printed in the United States of America

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Contents

Preface I

2

vii

Introduction

1.1 The Concept of SeismicTomography .............

1

1.2

3

Applications ...........................

1.3 Ray vs. Diffraction Tomography ............... 1.4 Suggestionsfor Further Reading ...............

5 6

Seismic Ray Tomography

9

2.1

Introduction

9

2.2

Transform

2.2.1

2.3

2.4

3

...........................

Methods

.......................

10

Projection Slice Theorem ...............

10

2.2.2 Direct-TransformRay Tomography .......... 2.2.3 BackprojectionRay Tomography ........... SeriesExpansion Methods ................... 2.3.1 The Forward Modeling Problem ...........

16 20 22 23

2.3.2

Kaczmarz'

26

2.3.3

ART

...................

.....................

33

............................

39

2.5 Suggestionsfor Further Reading ...............

42

Seismic Diffraction

45

3.1

Summary

Method

and SIRT

Introduction

Tomography

...........................

45

3.2

Acoustic Wave Scattering ................... 3.2.1 The Lippmann-SchwingerEquation ......... 3.2.2 The Born Approximation ............... 3.2.3 The Rytov Approximation ............... 3.2.4 Born vs. Rytov Approximation ............ 3.3 Generalized Projection Slice Theorem ............ 3.3.1 CrosswellConfiguration ................ iii

46 47 51 52 56 58 59

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4

3.3.2 Vertical SeismicProfile Configuration ........ 3.3.3 Surface Reflection Configuration ........... 3.4 Acoustic Diffraction Tomography ............... 3.4.1 Direct-Transform Diffraction Tomography ...... 3.4.2 BackpropagationDiffraction Tomography ...... 3.5 Summary ............................ 3.6 Suggestionsfor Further Reading ...............

72 78 82 84 87 90 •3

Case

95

Studies

4.1

Introduction

4.2

Steam-Flood EOR Operation ................. 4.2.1 CrosswellSeismicData Acquisition .......... 4.2.2

4.3

4.4

4.5

...........................

Traveltime

95

Parameter

Measurements

.........

B

C

99

4.2.3 Image Reconstruction ................. 4.2.4 Tomogram Interpretation ............... Imaging a Fault System .................... 4.3.1 CrosswellSeismicData Acquisition ..........

108 111 125 125

4.3.2

.........

130

4.3.3 Image Reconstruction ................. 4.3.4 Tomogram Interpretation ............... Imaging Salt Sills ........................ 4.4.1 Assumptions and Preprocessing............ 4.4.2 Data Acquisition .................... 4.4.3 Diffraction Tomography Processing.......... 4.4.4 Tomogram Interpretation ............... Suggestionsfor Further Reading ...............

133 135 137 137 139 141 146 150

Traveltime

Parameter Measurements

A Frequency and Wavenumber A.1 Frequency ............................ A.2

95 96

Wavenumber

153 153

..........................

154

The Fourier Transform B.1 Fourier Series ..........................

157 158

B.2 Exponential Fourier Series ...................

159

B.3 FourierTransform- Continuousf(x) ............. B.4 FourierTransform-Sampledf(x) ...............

160 161

B.5

162

Uses of Fourier Transforms

Green's

..................

Function

167

C.1 Filter Theory .......................... C.2 PDE's as Linear Operators .................. C.3 Green's Function Example ................... iv

168 170 173

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C.4 Suggestionsfor Further Reading ...............

INDEX

174

175

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Preface

Motivated by the successfulimplementation of medical tomography in the early 1980s, geophysicistsand production engineershave attempted analogousmethodsusingseismicenergyfor hydrocarbonexploration,reservoir characterization, and production engineering.The theoretical methods and field techniquesemployed are broadly classifiedas "seismictomography" and were fundamentally developedin the late 1980s. Today, seismic tomography is conductedon a commercial basis with its theory anchored on a solid base, and its strengths and limitations known. Field applications have demonstrated that seismictomography can provide valuable services in upstream operations,suchas mapping subsurfacestructures,delineating reservoirs,and monitoring enhancedoil recoveryprocesses. Our book developsthe fundamentalsof seismictomography at the level of a tutorial or practical guide. Considerableeffort has gone into making the book self-containedso that any reader who has had calculuscan easily follow the material. Referencesfor further reading on specifictopics are given at the end of each chapter. We give a short statementfollowingeach referencedetailingits significance as a supplementto this book. In doingso we hope the reader will not feel the referencesmust be read to fully understand a given concept. We use appendicesto review physicalterminology and mathematics required to understand the theoretical presentations. We present various tomographicmethods in a logical and straightforward manner. Unlike many other books on tomography, we use standard notation for variableswhich span the variousmethods,enablingthe reader to easily contrast differences. Also, mathematical steps glossed-overby most research

articles

are filled-in

for our readers.

Sometimes

we deviate

from well-known derivations to provide a deeper physical understanding. However,for completeness,our derivationsare followed-upwith references to the "well-known" derivations at the end of the chapter. In addition, we discussthe limitations of seismictomographyand illustrate successes and pitfalls with casestudies. Our ultimate intent is that after reading this presentation, the reader will exhibit both a greater understandingand appreciation for seismictomography articles presentedin the literature. Chapter I is introductory and summarizes the developmentof seismic tomography and describeshow this new technologycan benefit the oil industry at both the exploration and producing stages. Chapters 2 and 3 are tutorials on the theoretical fundamentals of seismic ray tomography and seismicdiffraction tomography,respectively. Chapter 4 presentsthe vii

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data acquisition,processing,and tomogram interpretation for three seismic tomographycasestudies. Each casestudy has its own unique data acquisition, data processing,and interpretation challenges.They provide useful insight into designingand conductingfuture tomographystudies. The authors thank Texaco for releasing data and results for the case studies published in Chapter 4. We also acknowledgethe releaseof the McKittrick data by Texaco'sjoint partner, Chevron, for that project. Both Eike Rietsch and Bob Tatham of Texaco have encouragedthe authors to pursue this project and have provided support during its progress. We also acknowledgeour fellow boreholeseismologyteam membersat Texaco: Danny Melton, Don Howlett, Ron Jackson, and Stan Zimmer for their contributionsto this field throughout the past few years. In addition, David Fitterman, the SEG monographsserieseditor, and Larry Lines, the volume editor for this book, have been patient with our progressand have provided valuable guidance. Finally, the authors thank Texaco for permission to publish this book.

Philip L. Inderwiesen Tien-when

Lo

E•tP TechnologyDepartment Texaco

Inc.

Houston, Texas

viii

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Chapter

I

Introduction

1.1

The Concept of Seismic Tomography

We definetomography as an imagingtechniquewhichgeneratesa crosssectionalpicture (a tomogram)of an object by utilizing the object'sresponseto the nondestructive,probingenergyof an external source.Seismic tomographymakesuseof sourcesthat generateseismicwaveswhich probe a geologicaltarget of interest.

Figurel(a) is an exampleconfiguration for crosswell seismic tomography. A seismicsourceis placedin one well and a seismicreceiversystem

in a nearbywell. Seismicwavesgeneratedat a sourceposition(soliddot) probe a target containing a heavy oil reservoirsituated between the two wells. The reservoir's responseto the seismicenergyis recordedby detec-

tors (open circles)deployedat differentdepthsin the receiverwell. The reservoiris probedin many directionsby recordingseismicenergywith the samereceiverconfigurationfor different sourcelocations.Thus, we obtain a networkof seismicraypathswhich travel throughthe reservoir. The measured responseof the reservoir to the seismic wave is called

the projectiondata. Tomographyimagereconstruction methodsoperateon

the projectiondata to createa tomogram suchasthe onein Figurel(b). In this casewe usedprojectiondata consistingof direct-arrivaltraveltimes and seismicray tomographyto obtain a P-wavevelocitytomogram.Generally,differentcolorsor shadesof gray in a tomogramrepresentlithology

with differentproperties.The high P-wavevelocities(dark gray/black)in the tomogramin Figurel(b) are associated with reservoirrockof high oil saturation.

Seismictomographyhas a solid theoreticalfoundation. Many seismic

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2

CHAPTER 1. INTRODUCTION

Crosswell Seismic Configuration

o

source

o

receiver

/'-waveVelocity Tomogram

heavy oil (a)

(b)

F•G. 1. (a) Geometry forcrosswell seismic tomography example. P-wave energy traveling alongraypaths probethe geological target. (b) P-wave

velocitytomogramreconstructed from observedtraveltimedata. Different shadesof gray correspond to differentP-wavevelocities.

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1.2.

APPLICATIONS

3

tomography techniqueshave closeties to more familiar seismicimaging methods such as traveltime inversion, Kirchhoff migration, and Born inversion. For example,seismicray tomographyused to determinelithologic velocity is essentially a form of traveltime inversion and seismic diffraction tomographyis closelyrelated to Born inversionand seismicmigration. Thus, seismictomography may actually be more familiar to you at this point than you might think sinceit is just another aspectof the subsurface imaging techniquesgeophysicistshave been using for years. 1.2

Applications

Seismictomography is applicable to a wide range of problems in the oil industry, ranging from exploration to developmentto production. The casestudies presentedin Chapter 4 demonstrate that seismictomography can complement conventional seismicmethods and provide unique, previously unavailable subsurface information. Tomography applied to surface seismicdata can generatesubsurfacevelocity modelsfor explorationproblems. These velocity models can in turn be used as soft information in the geostatisticalinterpolation of well-log data between wells. Seismictomography applied to developmentand production problems is generally implemented by a crosswellconfiguration, as shown in Fig-

ure l(a). Figure 2 illustratesthe benefitof crosswellseismictomography for reservoir

characterization

over conventional

reservoir

characterization

tools. Figure 2(a) representsthe true geologybetweentwo wellsin a producing field where the producing formation is a tar sand layer overlaid by

a thinner, lesspermeablebed (shadedinterval). The heavy oil in suchtar sandsis somewhatimmobile unlessheated using the enhancedoil recovery techniqueof steam flooding. A production engineerplanning to steam flood a tar sand interval needsto know whether the lesspermeablebed is capable of confiningthe steam to the tar sand. In our cartoon the lesspermeable bed is breached by a small fault. Well logging is a conventionalreservoir characterization tool that provides information about the reservoironly a small distancefrom the bore-

holeasdepictedin Figure2(b). Thus,no hardgeological informationabout the unprobed reservoir between the wells can be extracted from conven-

tional well-log data. The well-log data will only show that the low permeability layer exists between 500 and 600 feet in well A and between 400 and 500 feet in well B. Based upon the relative formation dips in each well, the engineermay decidethe low permeability layer is continuousand interpret

the well-logdata usinglinear interpolationas shownin Figure2(c). The small fault is thereforenot detectedand unexpectedsteam flood results will

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CHAPTER 1. INTRODUCTION

(a) true geology

(b) well logging

(c) well logging

interpre[ation

200 ft 4OO ft 600 ft

steam-' 8OO ff

A

B

A

B

(d) tomography

A

B

(e) _tomography Interpretati__on

I logging tool I

source

D receiver

A

B

A

B

FIG.2. (a)Truegeology wewishtoknow.(b)Welllogs sample onlyashort distance intothereservoir, requiring sometypeof interpolation between wells asdepicted in(c).(d)Crosswell seismic records theearth's response toseismic energy between wells thereby permitting animage reconstruction

of thegeology asshownin (e).

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1.3.

RAY

V$.

DIFFRACTION

TOMOGRAPHY

5

occur.

On the other hand, crosswellseismictomography can directly probe the reservoirbetweenthe two wellsas shownin Figure 2(d). A downhole seismicsourcein well A generatesseismicwavespowerful enoughto travel through the reservoirand to be recordedby sensitivedetectorsin well B. After applyingtomographyprocessingto the projectiondata, a tomographic

interpretationliketheonein Figure2(e) mightbe obtained.Thus,the engineer will be aware of the small fault and can make the necessaryalterations to the steam flood operation. Although just a cartoon, Figure 2 illustrates how crosswellseismictomography is a more reliable tool for delineating the reservoir between wells than any interpolation method between well logs. However, crosswellseismictomography becomesan even more significant tool for reservoir characterizationwhen used in conjunctionwith well-log information and core data as is demonstrated in Chapter 4.

1.3

Ray rs. Diffraction

Tomography

To do seismictomographywe must model the seismicwavetraveling through the subsurface. Both ray and diffraction theoretical models are available to us for describingseismicwave phenomena. Which model we use dependsupon the relative sizesof the seismic wavelength and the target we wish to image. A judicial choiceof theoretical model for a given seismicwave and target becomesimportant to the successof the seismic tomography application. If the target's size is much larger than the seismicwavelength,then we may model the propagationof seismicwavesas rays usingray theory. This is similar to using geometrical optics to describe light wave propagation throughlenses.Seismictomographybasedupon the ray theoreticalmodel is discussedin Chapter 2 under the title Seismic Ray Tomography. We subdivide the topic into "transform methods" and "seriesexpansion methods." The transform methods are commonlyused in medical tomography experimentswhile the seriesexpansionmethodsseemuchusein seismictomographyapplications.Currently seismicray tomographyis very popular becauseit is simple to implement under a variety of situations, is computationally fast, and givesgood results. When the size of the target is comparableto the seismicwavelength, then we model the propagationof seismicwavesas scatteredenergy using diffraction theory. Such a target scatters the seismic wave in many directions and only diffraction theory can properly model this response. Seismic tomography based upon the diffraction theoretical model is discussedin Chapter 3 under the title Seismic Diffraction Tomography.As

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6

CHAPTER

1.

INTRODUCTION

you will see,seismicdiffraction tomographypresentedin its simplestform requiresrestrictionson the source-receiver geometry,ignoresmultiplescattering of energy, places limits on the sizes and velocity contrastsof targets, and is computationally intensive. Becauseof these restrictions the method is currently applied only to a few select situations. However, re-

centdevelopments, whichwe list under "suggestions for further reading"in Chapter 3, are overcomingsomeof these restrictions. Our presentationof seismicdiffractiontornographyin its simplestform shouldgive you a solid foundationfor understandingand appreciatingthesedevelopments. Severalcasestudiesare presentedin Chapter 4 to illustrate the application of theory to varioussituations. We emphasizethe need to assimilate as much data as possiblefrom other sources,such as from well logs and core samplesin the crosswelltomographyexamples. Only by integrating all information availablewith the tomogram can one make an optimum assessment

about

the reservoir.

' As a final note, we have crisply divided the application of seismictomographyinto ray and diffractiontomography,dependingupon the relative sizesof the seismicwavelengthand target. However,in reality the probing seismicwave is usually a broad-bandsignal consistingof a large range of wavelengths,and the subsurfacecontainspotential targetswith relative sizesranging from small to large. Thus, this suggestsa blend of seismicray tornographyand seismicdiffractiontornographybe used to optirnallyimage all possibletargets. Although interesting,we pursuethis possibilityno further as it is more of a researchmatter at this point in time. In this book we will concentrateon presentingthe fundamentalsof seismictomography.

1.4

Suggestions for Further Reading Aki, K., and Richards, P., 1980, Quantitative seismology:Theory and methods, Vol. II: W. H. Freeman & Co. Section 13.3.5 o.f Chapter 13 presents a classificationscheme based upon seismic wavelengthand target size which will give you a goodidea when to use ray theory or diffraction theory.

Anderson,D. L., and Dziewonski,A.M., 1984, Seismictomography: ScientificAmerican, October, 60-68. Popular arlicle on seismic ray tomographyapplied to imaging the earth's mantle.

Lines, L., 1991, Applications of tomography to borehole and reflectionseismology:Geophysics:The Leading Edge, 10, 11-17. Overview of seismic ray tomographyapplications.

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1.4.

SUGGESTIONS

FOR

FURTHER

READING

Menke, W., 1984, Geophysical data analysis: Discrete inverse theory: AcademicPress,Inc. Our bookaddressesonly those topics in inverse theory requiredto understand the basicsin seismic tomography. Menke's bookprovides a goodintroduction to inverse theory.

Tarantola, A., 1987, Inverseproblem theory: Methods for data fitting and model parameter estimation: Elsevier. A comprehensivebookon inversetheorywhichincludesmanyprob-

7

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Chapter

2

Seismic Ray Tomography 2.1

Introduction

We begin the study of seismictomographywith image reconstruction methods based on ray theory. We assumethat the sourceproducesseismic wave energy with wavelengthsmuch smaller than the size of the inhomogeneitiesencounteredin the medium. Only when this assumptionis obeyed can the propagation of the seismicwave energy be properly modeled by rays. Otherwise, the seismicdiffraction tomography in Chapter 3 must be applied to solve the problem. Two groups of image reconstruction methods exist for doing seismic ray tomography. The transform methodsin Section 2.2 comprisethe first group. Applicationsof transform methodshave their roots in astronomical and medical imaging problems. They are very limiting as far as seismic imaging problemsare concernedsince straight raypath propagation and full-scan aperture are generally assumed.However, the transform methods make an excellentintroduction to the principlesof tomographybecauseof their simplicity and serve as a bridge between applications of tomography in other fields with applicationsin seismology.Also, the developmentof seismicdiffraction tomography has a closerelationship with the transform methods. The series expansionmethodsin Section 2.3 comprisethe second groupof imagereconstructionmethods. Out of all the methodspresentedin this book the seriesexpansionmethodspresently seethe most usein seismic tomography. Therefore, a large part of Chapter 2 is spent addressingthe seriesexpansionmethods. Beforeproceedingfurther one shouldhave a good graspof the Fourier transform conceptsto understandthe material in Section 2.2. Appendix B

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10

CHAPTER

2. SEISMIC

RAY TOMOGRAPttY

.---i'""• Ox-ray transmitte -

r

FIG. 3. Setup for a medical tomography experiment. X-ray scansare taken

in differentdirectionsabout a person'shead by rotating the transmitterdetector assembly.

presentsa review of the Fourier transform.

2.2

Transform

Methods

The projection slice theorem is presentedfirst in this sectionsinceit providesthe theoreticalfoundationfor the transformmethods. Then, two transform methods are derived from the projection slice theorem' directtransformray tomographyand backprojectionray tomography.

2.2.1

Projection Slice Theorem

The derivationof the projectionslicetheoremis illustratedby a typical medicaltomographyexperiment. Figure 3 showsthe setup for medicaltomography.A donut-shapedx-ray transmitter-detectorassemblysurrounds the target, a person'shead in this example. X-ray intensityis measuredfor a fixed orientationof the assembly.Then the assemblyis rotated about the personso that x-rays pass through the head in a different direction. The experimentis completedwhen the person'shead is scannedin all directions. The objective of the transform methods is to use attenuation information

from the measuredx-ray intensitiesto reconstructa cross-sectional image

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2.2.

'fRANSFORM

METHODS

11

of the person'shead in the z - z plane which containsthe transmitters and detectors. Thus, a brain tumor which attenuates x-rays differently than normal tissuesmay be readily "seen"by the radiologist.

Figure 4 depictsa cross-section of a person'shead. The varying contrasts within the target representnonuniformx-ray attenuation associated with a tumor, normal tissues,and the skull. For the purposeof deriving

the projectionslicetheorem,we definethe modelfunctionM(z, z) • as the spatial distribution of the attenuation. In general, the model function representsthe unknown distribution in spaceof some physical property of the

target medium which affectsthe propagatingenergy in some observable manner. A typical model function usedin seismictomographyis the reciprocal compressional-wavevelocity, or slowness,which has a direct influence on the observedtraveltime of the propagatingenergy. The projectionslicetheoremrequiresthat observationsof the propagating energy be taken along a given projection which is perpendicular to the raypaths. Figure 4 illustrates a singleprojection in the medical tomography experiment. X-rays emitted by the transmitters travel along the parallel rays and are recordedby detectorspositionedalongthe u-axis. The rotated

spatialcoordinatesystem(u,v) is introducedto describeall of the possible orientationsfor the transmitter-detectorassemblyabout the target. The v-axis is defined parallel to the direction of x-ray propagationand the uaxis, defined perpendicular to the v-axis, is the direction along which the x-ray intensity is measured. If the u- v coordinatesystemsharesthe same origin as the z- z coordinate system, then the relationship between the two coordinatesystemswhen one is rotated through an angle 0 relative to the other

is

z

sin 0

cos 0

u]

ß

(1)

For a givenray in Figure 4 we defineP(u, O) as the decimalpercent drop in x-ray intensity,

P(u, o) =

[.'o-

0)l/o,

where I(u,O) is the intensity measuredby the detector at (u,O) and Io is the x-ray intensity at the transmitter. We refer to P(u, O) as the data 1Other literature on tomographymight refer to the model function as an image function or as an object function. We chose "model function" to be consistent with inverse problem terminology.

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12

CHAPTER

2.

SEISMIC

RAY

TOMOGRAPHY

x X-RAY

SOURCES

v

P(., O)

Z

U

FIG. 4. Cross section of a person's head where varying contrasts repre-

sentnonuniformx-ray attenuation.The projectionP(u, t•) is the decimalpercent drop in x-ray intensity measuredalong the rotated coordinateaxis, u. The u-axis is perpendicularto the v-axiswhich alwaysparallelsthe x-ray

propagationdirection. The model function M(x, z) providesa numerical value

for the attenuation

and is an unknown

from the observedprojectionsP(u, •).

which

must

be determined

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2.2.

TRANSFORM

METHODS

13

function2. If the attenuationis small overthe raypath, then data function P(u, O) is linearlyrelatedto the attenuationM(z, y) as the line integral,

e(,,, o)-

(2)

ay

taken overthe raypath.s Note that eachobservationof the data functiona P(u, O)providesan empiricalsolutionto equation(2) alongthe givenraypath without actually knowingthe model function M(x, z). If there is negligiblex-ray attenuation outside the target, then equation (2) givesthe same measuredprojectionfor any transmitter-detector separationas long as each transmitter and detector remainsoutsideof the

target. We usethis assumptionto rewriteequation(9•)with infinitelimits; a mathematical step which will be taken advantageof later in this section. Thus,

0)- f::

z)a.

(a)

To obtain a simpler and more meaningful relationship between the

modelfunctionM(x, z) and the data functionP(u, 0), we transformeach into the Fourier

domain.

The 2-D Fourier

transform

of the model function

M(z, z) is

117I(k•, k,.)- f;: ];: M(x, z)e-J(k•x +k,.z)dxdz ' (4) where k• and kz are spatial frequenciesalong the x- and z-axes, respectively. Spatial frequency,or, wavenumber,is definedas k = 2•r/A where is wavelength. Figure 5 representsthe 2-D Fourier transform's amplitude

spectrum • ofa hypothetical model function M(x, z). Notethatif •(k•, 2P(u, O)is a projection in the ray tomo•aphyproblem,but is c•ed a data]unction in inv•e

theory te•nology.

The,

we c•

the v•iable P a "data f•ction"

sine re.on we c•ed M(x,y) the "modelf•ctioff' data f•ction

with the v•iable

P to re.rid

for the

e•Ser. However,we representthe

you that the me•ed

data •e projections.

awe o•y considerthe 5ne• inv•se problem in t•s book. Thus, the data f•ction will •waya be •ne•ly related to the model f•ction, even if • approximation is req•red to force the •ne• relatio•p. The •amption of am• x-ray attenuation is req•red for the x-ray tomo•aphy problem.

•Although the actuMobservation is the x-ray intensityl(u,O) in t•a c•e, we wi• frequently refer to "the observation of the data f•ction" from

the observed

since it is in•rectly

obt•ned

data.

•Althoughthe2-D Fo•ier tryfore of themodelf•ction •(x,z) h• bothmp•tude •d ph•e spectra,we representthe 2-D Fo•ier tryafore M(kx,k•) with o•y the

mpftude spect•

component, desi•ated • ]•(k,,

k,) I.

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14

CHAPTER

2.

SEISMIC

RAY

TOMOGRAPHY

K FIG. 5. The essenceof the projection slice theorem is represented. The

2-D Fouriertransformof a hypotheticalmodelfunctionM(x, z) produces the amplitudespectrumI M(ks, k,) [. The contoursin the ks - k, plane connectequal valuesof amplitude. The amplitudespectrum[ P(fi, d) [ from the 1-D Fouriertransformof the data functionP(u, t•) representsa

sliceof [ J17/(ks, k,) [ alongthefl-axisin theks- k, plane.

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2.2.

TRANSFORM

METHODS

15

is known,then the unknownmodelfunctionM(z, z) can be foundby the 2-D inverse Fourier transform

M(x,z) = 1fj: fj: 11•'I(k• k,)eJ(k•x +k,Z)dk•dk, (5) 4•r2

,

ß

Nowlet•5(f•,0) represent the1-DFourier transform ofthedatafunction P(u, O) alongthe u-axis,shownin Figure4. The 1-D Fouriertransformis written

0)- f/: where f/ is spatial frequencyalongthe u-axis. Substitutingequation(3) into equation(6) gives

We nowwishto put equation(7) entirelyin termsof z and z. The variable • is replacedwith z and z usingthe inverseof equation(1) givenby v

- sin 0

cos 0

z

'

Usingequation(8) andreplacing dvduwith dzdzin equation(7) weget,

•5(f•, O)-- f:: f:: M(x, z)e-Jf•( xcos 0+zsin O)dxdz = f:: f:: M(x,z)e-j[(f•cosO)x +(ftsinO)z]dx ' (9) Comparingthe integrands of equation(9) andequation(4) weseethat equation(9) is simplythe 2-D Fouriertransform of M(x, z) wherekx and k, are restricted to the Q-axis by setting k•

=

•cos0, and

k,

=

f/sin0.

(10)

This relationship is evident in Figure 5.

Substitutingequation(10) into equation(9) we write

•5(fi, O)- /:: /:: M(x,z)e-J(k:•x +k,Z)dxdz, (11)

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16

C•APTER

2.

SEISMIC

RAY

TOMOGRAPHY

whereks and kz are definedby equation(10). Comparingthe integrands of equations(11) and (4) showsthat we haveachieveda simplerelationship betweenthe data functionP(u, O) and the modelfunctionM(x, z) in the spatial frequencydomain,

P(•, O) -

A7/(k,,, kz).

(12)

In words,equation(12) statesthat the 1-D Fouriertransformof the pro-

jectionrepresented bythedatafunction P(f],_0)isequalto onesliceofthe 2-D Fouriertransformof the modelfunctionM(kz, k•) definedon the loci: kz = f] cos0 and k, = f]sin 0. Equation(12) is calledthe projectionslice theorem.

The projection slice theorem givesonly one slice of the model function per projection as shown in Figure 5. We will now show how many projections at different angles of 0 are used to reconstruct the entire model function via the projection slice theorem. The two techniquespresented are the direct-transform ray tomography method and the backprojection

ray tomographymethod. In Chapter 3 we will define an analogoustheorem for the reconstructionmethods in diffraction tomography called the generalizedprojectionslice theorem.

2.2.2

Direct-Transform Ray Tomography

Direct-transform ray tomography utilizes the projection slice theorem in a straightforward manner. We showedin the previous section that the application of the projection slice theorem to the 1-D Fourier transform

of a singleprojection represented bythedatafunction /5(12, 0) determines onlyonesliceofthemodel function A•(k•- 12cos0,k• - f] sin0). Figure 5 illustrates such a slice through the model function. To recoverthe entire model function, the target must be probed from many different directions. Figure 6 showsthree directions along which x-rays probe the head of our make-believepatient. The observeddata functionsfor thesethree pro-

jectionsare P(u, 0•), P(u, 02), and P(u, 0a). After applyingthe projection slice theorem to the 1-D Fourier transformsof these data functions,we obtain the three slicesthrough the model function's amplitude spectrum shownin Figure 7. Now the 2-D Fourier transform of the model function M(k•, k,) is better definedthan by the singlesliceshownin Figure5, but is still inadequatefor imagereconstruction.We must probe the target with x-rays from all directionsletting 0 range from 0 degreesto 180 degrees. Only then will the k•- kz plane in Figure 7 be completelycoveredby slices

M(f] cos0,f]sin0), where0 rangesfrom 0 degreesto 180 degrees.After such an experiment, the complete 2-D Fourier transform of the unknown

modelfunctionM(x, z) is determinedalongradial linesin the k•-k, plane.

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2.2.

TRANSFORM

METHODS

17

•X

Z

02 FXG. 6. The cranium of our make-believepatient is probed with x-rays in three different directions: 0•, 09.,and 03. Application of the projection slice

theorem to the data functions resulting from thesc projectionsyields the slicesdepicted in Figure 7.

ToobtainM(z, z) from.g/(f•cosO,f• sin0), analgorithm employing the direct-transformray tomography method must first interpolate the data

from a polar grid (f•cosO,f•sinO) onto a Cartesiangrid (k•,,kz) in the k• - kz plane, or

AT/(f•cosO, f•sinO)in,•r__•ot.,• A•(k•,kz).

(13)

One must exercisecaution in performingthe interpolation sincelarge errors introduced by the operation could obscurethe true solution.

Lastly, a 2-D inverseFouriertransformof M(k•,,k;•) is performedto obtain M(x, z),

M(z, z) = 4•rI 2 '•,•

,•

.•l(k• k,)eJ(k•z 4-k•Z)dk•,dk ' ß (14) '

Thus, the image reconstructionis completedand the technicianmay give the tomogram of the patient's head to the radiologistfor interpretation. The direct-transformray tomographymethod is easilysummarizedin five steps:

Step 1: Acquirethe data functionP(u, O) of the target with the projectiondirection,0, rangingfrom 0 degreesto 180 de-

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18

CHAPTER

2. SEISMIC RAY TOMOGRAPHY

M(•cosO 3, • sin63 )1

• sin61 )

K

Kz F•(•. 7. Plot showing slicesthroughthe unknownmodelfunction'samplitudespectrumI M(k=, k•) I foundby applyingthe projectionslicetheorem to the data functionsfoundfor the x-ray projectiondirections 01, 0•, and 03 shownin Figure 6.

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2.2.

TRANSFORM

METHODS

19

grees.Rememberthat the unknownmodelfunctionM(z, z) correspondingto the target representsa physicalproperty which affectspropagatingenergyin somemanner(e.g., attenuation which affectspropagatingx-ray intensity in medical tomographyor slownesswhich affectsseismicwavetrav-

eltimesin seismictomography). Thus, a data functionis just the line integral of the model function along each ray, or

Step 2: Perform a 1-D Fourier transform along the u-axis for each data function given by

P(n, o)- f:: o)-J Step 3: Use the projection slice theorem to obtain slicesof the 2-D

Fourier

transform

of the model

function.

Each slice is

defined by

cos0,iqsin0) -- J5(i2,0). Step 4: Convert the 2-D Fourier transform of the model func-

tion in the k• -kz planefrom a polar grid (f• cos0, f•sin 0) to a Cartesiangrid (k•, kz),

Step 5: Perform a 2-D inverseFouriertransformon M(k•,, k•) to obtain M(x,z), the reconstructedimage of the target. The inversetransform is given by

47i-2

The direct-transformray tomographymethod would be quick to implement if it were not for the fourth step above requiring interpolation of the model function in the frequency domain. In the next section we present backprojectionray tomographywhich obviates the need for interpolation resultingin a faster and more accuratealgorithm.

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CHAPTER

2.2.3

Backprojection

2.

SEISMIC

RAY

TOMOGRAPtIY

Ray Tomography

Backprojection • ray tomography usesthe samedata acquisition scheme asthe direct-transform ray tomography:recordthe data functionP(u, O)by experimentallymeasuringthe line integral of the unknown mode] function

M(:r,z) along differentraypathsor projections. The differencebetween backprojectionray tomographyand direct-transformray tomography is how we compute the model function from the data function. To derive the backprojectionray tomography method, we first write

downthe2-D inverse Fouriertransform of themodelfunction•(k•, k,),

•(• ' z) = 4•2•

•

•(• ' •)•j(•+•z)••

(•5)

ß

Next we makea changeof variablesin equation(15) by replacingk• with • cos0 and k• with •sin 0, and by changingthe integration from dk•dk,

to ] • l d•dO.• This gives,

M(x,z) = 4•2 M(•cos0,•sin0) • ej•(xcosO + zsin 0)I•ld•dO.

(1•)

Integrationwith respectto 0 in equation(16) can be rewritten • two integrals,

1 • M(r,z) = 4=•

• (• cos 0,• sin 0)

• ej•(xcos 0+ zsin0) I•ld•dO +•

• '

•[• •o•(0 +•), • •in(0 +•)]

• •j•[• •o•(0+ •) + z•in(0 + •)] I• [•0.

(17)

Using the fundamentaltrigonometricangl•sum relations,cos(0+ •) = - cos0 andsin(0+ •) = - sin0, we may rewritethe secondmodelfunction •The te•

"b•mjection"

imp•es the inve•e problemwherewe st•t with the pr•

jection •d •lve for the model f•ction. Here the projections •e t•en Mong raypat•. • Section 3.4.2 the me•ed projectio• of scattered energy •e described by the wave equation •d we •e • •Mogo• te•, "backpropagation."

•The ch•ge • inte•ation is •Mogo• coor•n•es is the r•M

to compute the •ea of a •sk. •st•ce

from

c•e to prese•e the si• negative vMu•.

the •sk's

to goingfrom C•tesi•

coorSnatesto pol•

For a •sk we replace dxdz by rdOdr, where r

center.

The

absolute

vMue

of •

is t•en

of the •fferentiM •ea when we will shortly •ow

in o•

• to t•e

on

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2.2.

TRANSFORM

21

METHODS

on the right-handsidein equation(17) as,

.g/[f•cos(0 + •r),f•sin(0 + •r)] = .tfl(-f•cosO,-f• sinO).

(18)

The secondsetof integralson the right-handsidein equation(17) is rewritten by replacingthe modelfunctionwith equation(18), applyingthe anglesum relationsusedin obtainingequation(18) to the exponential,setting f• - -f• and dfl - -dfl, and reversingthe direction of integration with respectto fl. With theseoperationsequation(17) is written,

M(x z) = 4•21 • ,

•(fl cos 0,flsin 0)

x ej•(xcos 0+ zsin0)

+•

•(•

cos•, • sin•)

• •j•(• •o•• • • •i. •) I • I ••. Combiningthe integralswith respectto the variable• we get, '

4•

'

• d•(• •o•• • • •i. •) I • I ••.

(•)

Usingthe projection slicetheorem,wereplaceM(• cos•,•sin •) in equastruction

formula

ß

,

4•2

'

We cansummarizethe backprojection ray tomographyreconstruction method in just three steps:

Step 1: Data acquisition.Let the modelhnction M(x, z) representthe unknownparameter(such• seismicwaveslow-

ness)at position(x, z). Experimentally determinethe line integralof the modelfunctionalongeachray whichyieldsa setof datahnctions(such• seismic wavetraveltime),

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22

CHAPTER

2.

SEISMIC

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TOMOGRAPttY

Step 2: Take the 1-D Fourier transform of each data function

along the u axis.

P(fl, O)- /:: P(u, O)e-Jfl udu. Step 3: Use the backprojectionformulaequation(20) to compute the unknownmodelfunctionM(x, z), or

=

4•.2

x

'

+

[ I

Unlike direct-transformray tomography,backprojectionray tomography does not require a 2-D interpolation in the wavenumberdomain, and therefore,is in generalfaster and more accuratethan direct-transformray tomography. It shouldbe mentionedthat mostcommercialCAT (ComputerizedAxial Tomography)scannersuse the backprojectionray tomographyor its modificationas their image reconstructionalgorithm. 2.3

Series Expansion Methods Seriesexpansionmethods comprisea group of computation algorithms

which,like the transformmethods,determinethe modelfunctionM(x, z) of the target area. However,unlike the transformmethods,thesealgorithms easily allow curved raypath trajectories through the target area and are therefore well suited for applicationsin seismictomography. As before, we

restrictthe discussion to a 2-D problemsothat the modelfunctionM(z, z) is determined in a plane which cuts through the target and containsall of the sources and receivers.

Our discussionof the seriesexpansionmethods is divided up into three subsections. Section 2.3.1 presentsthe forward modeling problem which permits us to predict the tomographydata in terms of a system of linear equationswhich explicitly contain an estimate of the true model function. Section2.3.2 showshow the true model function is determinedusing Kaczmarz' method. The method devised by Kaczmarz in 1937 is iterative and determinesan approximate solution to the true model function. Drawing an analogy,the Kaczmarz method is to the seriesexpansionmethod as the

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2.3.

SERIES

EXPANSION

METHODS

23

projection slice theorem is to the transform methods. We exploit Kaczmarz' method in Section 2.3.3 to derive two seriesexpansionalgorithms:

the algebraicreconstruction technique(ART) and the simultaneous iterative reconstruction technique(SIRT).

2.3.1

The Forward Modeling Problem

As will be shown shortly, the seriesexpansionmethod iteratively up-

datesan estimatedmodelfunctionM e't so that it converges towarda true model function M true. The updates are found by comparingthe observed data functionpolo with a predicteddata functionppre. Forwardmodeling is required to determine the predicted data function and is the subject of this section.

Equation (2) in Section2.2 definesthe experimentalprocessfor the transformmethodsas the line integral of the function M(a:,z) along a straight raypath in the v-axis direction. Accordingto the abovenotation equation(2) couldbe written,

Pøbø(u, O)-- i Mt"u•(z' z)dv' ay

We did not requirea forwardmodelingprocedurein Section2.2 becausethe raypathswere straight and the projectionslicetheoremcouldbe employed

directlyto determinethe true modelfunctionMtrue(x,z). For the seriesexpansionmethodswe wish to include curvedraypaths.

Equation(2) is easilytransformedto accommodate curvedraypathsby rewriting the model function in terms of a position vector r. Thus, for a

givensource-receiver pair the line integralof the modelfunctionM(r) over the raypath is

pobo = f• MtrU•(r)dr, ay

wherethe observed projectiongivenby the datafunctionpolo represents the measured lineintegral(observed tomography data) and MtrUe(r) is the true model function which remains to be determined. The last equation is used to formulate the forward modeling by setting

P- i M(r)dr, ay

(21)

where P is now the predicteddata function and M(r) is the estimated

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CHAPTER

24

2.

SEISMIC

RAY

MI

M2

M3

M4

M5

M7

M8

M9

M10 Mll

TOMOGRAPtIY

M6 M12

M13 M14 M15 M16 M17 M18 M19 M20 M21 M22 M23 M24 FIG. $. The seriesexpansionmethodsusea discretemodelfunctionMj, j - 1,..., J, whereMj is the averagevalueof the continuous modelfunction M(r) within the jth cell. Here J - 24.

modelfunctions. Thus, forwardmodelingis definedas determiningthe predicted data function from the line integral along the raypath through a known, but estimated, model function. Just as was done with the transform methods, the model function in the seriesexpansionray tomographyis discretizedto allow computationby digital computer. Figure 8 showsan image area of a target divided into many small cells. Each cell is assignedthe averagevalue of the physical parame-

ter (e.g., x-ray attenuation,slowness, etc.) represented by the continuous modelfunctionM(r) within that cell. The modelfunctionin Figure 8 is divided into 24 cells and is written discretely as M•, where j - 1,..., 24.

Thus, Mj represents the averagevalueof M(r) within the jth cell. Figure 9 depicts a single ray traveling through the discretizedmodel function. Equation(21) is rewritten in discreteform, to describeray travel through the discretemodel function, as J

Pj=l

whereMj is theestimatedmodelfunctionfor thejth cell,,5' i is the raypath SHere we will symbolize the predicted data function as P and the estimated model function a• M for brevity. Thesesymbolswill be changedto ppre and M est, respectively, in the following section.

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2.3.

SERIES

EXPANSION

•

Z

METHODS

25

r

7

6

..•_1.. ••

source

receiver • •._•..1•.. ••15 M16M17IV116 M19 M20 M21 M22 M23 M24 FIG. 9. Ray travel through a discretemodel function. The resulting data function, determinedby the line integral through the discretemodel function, is definedby equation(22).

length of the ray within the jth cell, and J is the total number of cells in the gridded target. The example in Figure 9 has J - 24 cells, but the ray

penetratesonly sevencells(j - 12, 11, 10, 16, 15, 14, and 13). To keep equation(22) consistentwith equation(21) we set Sj - 0 for all cellsnot penetratedby the ray. After all, the ray'spath lengthSj for the jth cell is obviously zero if the ray did not traverse that cell. Figure 9 shows17 cellsfor which we don't have information becausethe singleraypath did not traversethem. By addingmore sourcesand receivers around the unknowntarget region, differentrays samplethe 17 unsampled cellsin addition to someof the cellsalready sampled. The addition of extra rays is depicted in Figure 10. Now all of the cells are interrogatedby this network of rays.

We must modify the index notationof equation(22) to includea projection value for every ray. If Pi representsthe projection, or line integral,

predictedfor the ith ray, then equation(22) is rewritten, J

Pi = E Mj$ij,for/- 1,...,1,

(23)

whereI is the total numberof rays,$ij is the path lengthof the ith ray throughthe jth cell, and, asbefore,Mj is the discreteestimateof the model functionfor the jth cell and J is the total numberof cells.Equation(23) is

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26

CHAPTER

2.

SEISMIC

RAY

TOMOGRAPHY

source

ray I

receiver ray 2

!

ray I

FIG. 10. Generally, a single ray does not provide information on all of the model function's cells. However, by using more source and receiver locations around the target all of the cells can eventually be sufficiently interrogated. I rays were found sufficienthere.

the formulation of the "forward modeling problem" usedin seriesexpansion ray tomography.

Equation (23) can effectivelymodel the data acquisitionprocessif we let the projectionsPi, i - 1,..., I, be the observeddata (i.e., traveltime or decimalpercentdecreasein x-ray intensity)and the modelfunction j: 1,..., J, be the true, but unknown model function, or J

Piø•' = • M]"•'*Sii, fori- 1,...,I.

(24)

j=l

Kaczmarz' method providesthe theoreticalframeworkfor indirectly solving equation(24) for the true modelfunction. 2.3.2

Kaczmarz'

Method

In this section we introduce Kaczmarz' method to indirectly solve

equation (24)forthetruemodelfunction M]ru*,j = 1,..., J, whichisthe tomogram. As stated in the previoussection,forward modelingis required to determine the true model function. Thus, before proceedingwe will reformulateequation(23) into a matrix form to simplifythe mathematical discussion.Sinceequation(23) is discreteits elementsare easily put into matrices.In matrix form equation(23) becomes,

P

-

SM,

(25)

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2.3.

SERIES

EXPANSION

METHODS

27

where the predicted projections in data vector P are, P1

P

-

.

,

Pl

the discrete estimated model function values in model vector M are, ml

m

-

.

,

Ma

and the raypath lengthsfor I rays and J cells in S are, S• S•.•

s

S•. S•

-

... ...

S• S•

. ß

S•i

S•2

-..

Note that S in equation (25) can be thought of as a linear operator that operates on the estimated model vector M producing the predicted data vector

P.

We couldalsoformulateequation(24) in matrix form as

po•, = SM'•.

(29)

Althoughwe will not directlysolveequation(29), we wouldwant to determinethe true modelvectorM truegivenpob•and S. The problembecomes oneof findinga generalized inverseoperatorS-•.9 Then wecouldapplythe generalizedinverseoperatorS-• to both sidesof equation(29) to determine the true model vector, or S-ap oh, = m

S-aSM '•e M true .

Theoretically the last equation is true, but in practice it is very often difficult to determine S-a for two reasons. First, S is usually quite large and 9We write

S-g

rather

squaxe and because S-9S

tha•n S -1

as in usual matrix

notation

is not always the identity matrix.

since S-g

need not be

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28

CHAPTER

2.

SEISMIC

RAY

TOMOGRAPHY

sparse,which makes computation of S-• costly. Second, S is usually "ill conditioned"whichmakescomputationof S-• very unstable.iø Kaczmarz' method circumvents the problems associated with the inversion of a large and sparsematrix and provides an efficient means for determiningan approximatesolutionto equation (29) using an iterative procedure. Figure 11 presentsa flowchart outlining the method in which there are three basicstepsto the iterative part of the algorithm. An initial

estimateof the modelvectorM init is input to the iterativeloopof the algorithmandservesasthe first "currentestimate"M •st of the true solution M true. For now we assume that the initial model vector M init is known.

However,morewill be saidon the selectionof the initial modelvectorM in the casehistoriespresentedin Chapter 4.

With the current estimate of the model vector M est known, the first step is to usethe forwardmodelingproblemdefinedby equation(25) to determinea predicteddata vectorppre. This stepis carriedout by applying the linearoperatorS (determinedby someray tracingtechniqueof personal choice)definedby equation(28) to the estimatedmodelvectorM •s•,

ppr• = SM•t.

(30)

In the second step the predicted data vector PPr• is compared with

the observed data vectorpob, by takingthe difference betweenthe two. A small differenceor good agreementbetweenthe predicted and observeddata vectorsimpliesgood agreementbetweenthe estimated model vector M • and the true model vector M t•"•. Thus, if the differenceis smaller than a specifiedtolerance, then the current estimate of the model vector M •t is

output as the solutionto equation(29) in the final step of the algorithm. The selection of a suitable tolerance for the difference is discussed with the

case histories in Chapter 4. The third step of the iterative portion of Kaczmarz' method comesinto play when the difference between the predicted and observed data vec-

tors is larger than the specifiedtolerance. This important step essentially

makesuseof the differenceinformation,pob•_ pp,.e,to updatethe current estimated model vector M •

with a new estimate of the model vector

M("ew)• whichhopefullyis closerto the true modelvectorM •r"•. This third step is written in equation form as

M ("•w)•t = M e'•+ A/M, fori

-

(31)

1,...,I,

Ill conditioned meanssmallchanges in S producelargechangesin the modelfunction true or in S-g howeveryou wishto look at it

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2.3.

SERIES

EXPANSION

METHODS

29

Initial estimate

Minit

currenf estimafe est

Step

1

predicted

observed

data vector

data vector

Ppre Step

2

Step

3

pObS

FIG. 11. Flow chart for Kaczmarz' method. M i"i' is the initial estimate

of the model vector; M e'• is the current updated estimate of the model vector; ppre is the predicted data vector from the forward modelinggiven

by equation(30); andpobois the observed data vector.M eø•is iteratively updateduntil ppre matchespoboto within a specifiedtolerance.

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CttAPTER

3O

2.

SEISMIC

RAY

TOMOGRAPttY

whereAiM represents the incrementalupdate to the currentestimateof the modelvectorand superscripti meansapplyingequation(31) whenthe ith row of the pob• and ppre vectorsare compared. The new estimate M (new)e•t is then taken as the current estimate for the next iteration. As

you will soonsee,equation(31) bringsthe current modelestimatetoward the true solution, at least in theory.

The methodof computing AiM in equation(31) is obviously a crucial factor

in the success of Kaczmarz

'method.

Kaczmarz' method computes

AiM with the equation' AiM1 A i M2

-

.

,

(32)

ß

AiMj where

pi.øb s __piPre

(33) Note that the summation in the numerator is just the predicted data vector

ppre found in the forward modelingin equation(30). Now we will geometricallyderiveequation(33) and showthe convergenceof Kaczmarz'methodthrougha simpleexample.Let two rays(I = 2) travel througha two-cellmodel(J = 2) so that the vectorequation(29) for the problem can be written as

p•,b, = Sll M1 + S12M2,for ray 1, and p•b, __ S21 M1+ S22M2,forray2,

(34) (35)

whereP•'b•and P•*• are observed data and M1 and M2 are unknown TM. Rememberthat $i1 is just the ith ray's path lengththroughthe jth cell and is generallyknownfrom the forwardmodeling.We plot equations(34) and (35)in Figure12 as lines(hyperplanes) on a 2-D spacewith axesM1 and M2 .12 The solutionto the equationsoccursat point X where the two • • Here M• and M2 are unknown and therefore defined as independent variables. Only

whenthe solutionis foundare theyreferredto as M[ •ue andM• •ue as in equation(29). •2If I = 3 and J = 3, then we would be looking for the intersection point of three planes in a 3-D model space. For situations where J > 3, equation (29) representsa J-dimensionM space and we would look for the solution at the intersection of I = J

hyperplanes where a hyperplane has J - 1 dimensions. Note that we must have at least I = J hyperplanesto solve equation (29) and generally I > J.

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2.3.

SERIES

M2

EXPANSION

METHODS

31

obs

M2= P1 o\ ,,....__/ S12

•

obs

H

12M2

FIG. 12. For two rays (I = 2) in a two-cellmodel (d = 2), possiblevalues

for M• andM2 aredefined by [hetwolines(hyperplanes). p•,b,and are [he observed data from [he two rays. Point X represents[he desired

modelvalueswhichlie at the intersection of the twohyperplanes.A• M• and A•M2 for ray 1 are geometrically derivedsothat point B is the projection of point A ontothe hyperplanedefinedby equation(34). The resultof this geometricalderivationis equation(33).

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32

CHAPTER

2. SEISMIC

RAY TOMOGRAPItY

lines(hyperplanes) intersect. Sincethe solutionat point X with coordinates(M•"•'e,M•"•'e) is unknown, we must start with an initial estimate at point A given by the coordinates (M1A, M•A). Thisinitialestimatebecomes the currentestimate as shownin the flow chart for Kaczmarz'methodin Figure 11. The geometricalstepin Figure 12 is to find the perpendicularprojectionof point A

ontothehyperplane •:•defined byequation (34) at pointB. Mathematically this step is given by,

s = M+AM,and

(36)

whereA•M• and A•M2 are the corrections soughtgeometrically and defined by equation(33) for i- 1 and j- 1,2. The first geometricalrelationshipto note in Figure 12 is the similarity of trianglesAABC, AFED, and AGEH. Using these similaritieswe can immediately write

AIM• = AB cosc•= DFcos c• =

EF cos2 c•

=

__GH • EF '•E:Z'

=

EF

A • M2 = AB sinc•= DF sinc•

--GH

and

(38)

cos c• sin c•

EH

= EFG---•• .

(39)

Our task now is to determineEF, GH/•--•, and EH/GE in equations (38) and (39).

On the line segmentEF, point E is located at

(M• = P•'b'/sI•,M2 = 0). PointF is the intersection with the M•-axisof a linewhichis parallelto equation(34) andcontains the point(MIA,M•). The equation for this line is

S11M• A + S•2M•A - S••M1 + S1:•

(40)

Fromequation(40) we determinethe coordinateof point F alongthe M•-

axisas (M• - (SI•M1A+ S•2M•A)/S•I,M2 - 0). Thus,the lengthof the line segment EF is given by 1

(41) laAlthoughequations(34) and (35) both representlines,we will continueto refer to them as hyperplanes since that is what they are called when J > 3.

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2.3.

SERIES

EXPANSION

METHODS

33

The ratio GH/GE is simply, GH GE

(42)

ß

Similarly, EH/GE

is givenby,

EH

P•ø/Szz +

=

S12 .

(43)

+

results in the model corrections due to the first ray,

p•b•

I:•p re

i and AiM1-- •'!1•_--' S121 .•.S122 p•bs I•Prß

- -• AiM2- $1•$•---•+$• ,

(44)

(45)

where,rre _ SzzMi• + $z•M• t ß Equations(44) and (45) are the sameas a I

equation (33) wheni = 1 andj = 1,2. Thus,weseethat equation (33) simplydeterminesthe projectionof a modelestimateonto oneof the hyperplanesdefinedby equation(29). Carryingthisexampleonestepfurther,wecandeterminepoint I in Figure 12, the projectionof point B ontohyperplane2 definedby equation(35) for the secondray usingthe indicesi - 2, j - 1, 2 in equation(33). If we alternateprojectionsof the modelestimatesbetweenthe two hyperplanes, then the updatedmodelestimates(step3 in Figure 11) must convergeon point X as depictedin Figure 13. Thus, Kaczmarz'methodwill converge to the solutionof equation(29). 2.3.3

ART

and

SIRT

The algebraicreconstruction technique(ART) and the simultaneous iterativereconstruction technique(SIRT) are the two commonimplementations of Kaczmarz' method in seismicray tomography. This sectiondescribesthe basic features of both algorithms.

ART is a computational algorithmfor solvingequation(29) that directly uses Kaczmarz' method. Thus, the ART algorithm is comprisedof the

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34

CHAPTER

2.

SEISMIC

RAY

TOMOGRAPHY

M2

I

Hyperplane2

:-M I Hyperplane

I

FIG. 13. By applyingequation(31) to alternatinghyperplanes,the model estimateof (M•, M2), starting at point A, convergestowardsthe solution for equations(34) and (35) at point X. The iterative updating of the model estimate correspondsto the loop in Kaczmarz' method shown in the flow chart in Figure 11.

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2.3.

SERIES

EXPANSION

35

METHODS

stepsshownin Figure11. Wefirstsettheestimated modelfunction M• 't,

j-

1,..., or, to theinitialmodelestimate Mj"i' j - 1,

or Thenthe

followingthree stepsare iterated cyclicallyfrom one hyperplaneto the

next until the observed data p/ob,matches the predicteddata pfre, for i=

1,...,I.

Step 1: Conductforwardmodeling(ray tracing)for the ith ray usingequation(23) or equation(25), restatedfor reference here as, J

Only one ray is traced out of a total of I rays sincewe are determining the projection of the current model estimate onto only one hyperplane. Note that if our model consists

of slownesses, thenthe predicteddata P•'reare calculated traveltimes fromtheforwardmodeling andp/oh,areobserved traveltimes.

Step 2: Subtractthe predictedith ray data p•0•, from the observed ith data p/o•,,anduseequation(33) to findcorrectionsfor all of the J cellscomprisingthe model function estimateTM,

=

'

Step 3: Apply the correctionsto the model estimaterecommended by the ith ray to all orcells,

SIRT differsfrom ART in that all I rays are traced throughthe model

sothat all AiMj corrections determined forthe I hyperplanes areknown. 14Note that the model adjustment dependsupon the discrepancybetween the predicted aaadobserveddata values and the raypath length through the cells for the ith ray.

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36

CHAPTER 2. SEISMIC RAY TOMOGRAPtIY

Thenan average of AiMj withrespect to indexi is takenfor eachmodel

cellj - I , ..., J,togetnew model estimates

,j-1,...,J.

As

withART,themodel estimates M[•t areupdated untilthepredicted data Pf•ecompares favorably withtheobserved datap/oh,, i- 1,..., I. After settingthe currentmodelfunctionestimateequalto the initial

model function, orMf øt- M'. i"itforj = I

are iterated to update the model estimates:

J thefollowing threesteps

Step1: Conductforwardmodeling (ray tracing)usingequation (23)or equation(25),

for all raysi

-

Step 2: Findthecorrection foreachcellbyexamining therays cut throughthat celland averaging the corrections recommendedby eachray. Thisoperation is definedfor the jth cell by, I

= W,..= 1•AiM 1 _

forj

-

1

I

BlObS 1 s,s • - EsS=

(46)

i=1

1,...,J.

The weightWj is the numberof raysintersecting the jth cellor someothersuitableraydensityweightusedto obtain an averagecorrectionAMj.

Step 3: Determine thenewmodelestimate fromthe average modelcorrections AMj, or

M«"ew)e" = M;"+AM./,j - 1,...,J. Figure14illustrates howequation (46)makes SIRTdifferent fromART.

Asin Figure 12weuseonlytworays(orI - 2 hyperplanes) andtwomodel cells(or J - 2 modelspace) in orderto visualize theproblem.TheART algorithm isshown inFigure 14(a)andtheSIRTalgorithm inFigure 14(b).

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2.3.

SERIES

EXPANSION

M2

37

M2

E

.

D A

METHODS

•

C B Hyperplane I

(a)

(b)

FIG. 14. Comparisonof ART andSIRT algorithmsfor the two-ray(or I 2 hyperplanes) and two-cellmodel(or J - 2 modelspace)examplegiven in Figure 12' a) Convergence of model estimatesfor the ART algorithm startingwith an initial modelestimateat pointA. b) Convergence of model estimatesfor the SIRT algorithm starting with an initial model estimate at point A. The iteratively updated model estimates, determined from the

averagecorrections AMj in equation(46), are alongthe solidline defined by points A, B, C, D, and E. Each estimated point is the average of the same letter's primed and double primed projection points located on hyperplanesI and 2, respectively.

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CHAPTER

38

2.

SEISMIC

RAY

T¸M¸GRAPHY

The ART algorithmin Figure 14(a) finds the solutionby alternatelyprojecting the current model estimate onto each hyperplane. The model estimate moves along the solid line from the initial model estimate at A to B, to C, to D, and so on. On the other hand, the SIRT algorithm finds point A's projections on both hyperplanes,points B t and B •t, then moves the initial estimate from point A to point B, the midpoint between B t and Btt. For the next iteration the SIRT algorithm finds point B's projections on both hyperplanes, points C t and C", then moves the current estimate

from point B to the midpoint betweenC t and C t•, or point C. Starting with the initial model estimate at point A, SIRT convergestowardsthe solution along the solid line from point A to B, to C, to D, to E, etc. If a true model solution exists and is unique, then both ART and SIRT will convergeto that solution. However,one shouldnote that the convergence of ART depends upon the ordering of the hyperplane projections while the convergence of SIRT doesnot. You may seethis by projecting point A onto hyperplane2 first in Figure 14(a). The ART and SIRT methods, as we have stated, are intended to solve

linear systemsof equationslike thoserepresentedby equation(29) which explicitly relate the model function to the data function. But just because

equation(29) explicitly relatesthe modelfunctionto the data functionin a linear form does not imply a linear relationship for all types of model and data functions.

Take

for instance

a model

function

of slowness and

a data function of observeddirect-arrivaltraveltime. Equation (29) does not provide a linear relationshipin this casebecausethe raypat.h lengths in S are also dependent upon the slownessesdefined in the model function. Thus, we do not know the true raypath lengthsin S until the true slowness field is also known.

To solvethe nonlinearproblemin practice we computeestimatedraypath lengthsusing the estimatedslownesses in the model function and use the estimatedraypath lengthsin the ART or SIRT algorithm. This is called an iterative linear approach to solving a nonlinear problem. We can use Figure 12 to visualizewhat happensto the hyperplaneswhen solvinga nonlinear problemby an iterative linear approach. The two hyperplanesshown in Figure 12 are the true hyperplaneswhen we know the raypath lengthsin S. When we useestimated raypath lengthsin S the estimated hyperplanes will not be coincident with the true hyperplanes. The resulting projection will be different from the projection shown in Figure 12 as point A is projected onto an estimated hyperplane. Each time we update the model function slownesses,using either the ART or SIRT method, the new estimated hyperplanes will be located differently in the model space since we also have new estimated raypath lengths in S. What we hope happens is that the estimated hyperplanes will not be wildly repositioned to a new

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2.4.

SUMMARY

39

location each time the model slownessesare updated. Then as we iterate further towards the true model function slownessesthe estimated hyperplanes should become more coincident with the true hyperplanessince the estimated raypath lengths will be approachingthe true raypath lengths. Thus, we can get convergenceto a solution even though the problem is nonlinear.

2.4

Summary

1. Seismicray tomography attempts to solvethe inverseproblem formulated by the line integral equation, p

__

•.yM(r)dr,

taken over the raypath. P is called the data function and represents

the observeddata. M(r) is called the model function and representsthe spatial distribution of somephysicalproperty of the medium

whichaffectsthe propagatingenergyin someobservablemanner. The model function is unknown and the goal of seismicray tomography is to determine an estimated model function M e•t of the true model function M tr•e.

2. Transform methodsin seismicray tomography are of limited usesince straight raypaths and full scan aperturesare generallyassumed.However, they serve to introduce the tomography concept and terminology,and provideinsightinto seismicdiffractiontomographypresented in Chapter 3.

3. The projectionslicetheoremis the basisfor the transformmethods. The theorem states that the 1-D Fourier transform of the data func-

tion/5(f•,0) provides a sliceof information in thek• - k, wavenumberplaneof themodelfunctionA74(k•,kz) defined ontheloci'k• = f• cos0 and k, - f•sin 0 as shownin Figures4 and 5. Equation(12) definesthe projection slice theorem as,

4. Direct-transformray tomographyappliesthe projectionslicetheorem

tomany projections ofthedatafunction/5(f•, 0•for0 degrees _(0 _( 180 degrees.The resultis the modelfunctionM(f•cosO, f•sinO) definedon a polar grid. Interpolationof the modelon the polar grid onto a rectangularks - k, grid is requiredto take the 2-D inverseFourier

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CHAPTER

40

2.

SEISMIC

RAY

TOMOGRAPHY

transformof the modelfunctionwhichyieldsM(x, z). Interpolation error may distort the resultingestimated model function. 5. Backprojection ray tomography is another transform method which utilizes the projectionslice theorem. However,by making a change of variableswe were able to do the image reconstructionwithout the interpolationstep requiredby direct-transformray tomography.The

reconstruction formulagivenby equation(20) is

M(x,z) = 4•r• , x ej•(xcosO + zsin0)]•]dC2da. Backprojectionray tomographyor its modificationis usedas the image reconstructionalgorithm in computerizedaxial tomographybecause it is both accurate and fast.

6. Seriesexpansionmethods are the most frequently used seismictomographymethods. The model function is divided up into small cells where each cell is assignedan averagevalue of the continuousmodel function within that cell. Thus, the ith observation of the data function is related to the discretemodel function by the equation, J

Piø•' -- Z$ijM]•"•,i- l,...,I, j=l

where I is the total number of rays or observations, J is the total

numberof cellsin the discretemodel function, and $ij is the path length of the ith ray in the jth cell. The inverseproblem is to deter-

mineanestimated modelfunction M• '• of thetruemodelfunction

M]ruegiven theobserved datafunction Piø•s. 7. Kaczmarz' method iteratively solvesthe system of equations defined for the seriesexpansionmethods for the estimated model function

Mf s•. Theiterativepartof thealgorithm consists of threestepsas shownin Figure11andaninitialestimateof themodelfunctionY init must be input.

Step i requiresthat the current estimate of the model function be

usedin forwardmodeling(i.e., ray tracing) to get a predicteddata functionP/P• for the ith ray. The forwardmodelingis definedby the equation, J

j=l

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2.4.

SUMMARY

41

Step2 compares the predicteddata functionpfre with the observed datafunctionPiøb•.If theobserved andpredicted datafunctions agree

to withina specified error,thentheestimated mode] function Mf • istakena.sa goodestimate ofthetruemodelfunction M]rue. Step3 updates thecurrent estimate ofthemodel function M•• if the observedand predicteddata functionsdo not comparefavorably.The updatecorrectionis determinedby projectingthe currentestimateof the model function onto the hyperplane defined by the ith ray. The correctionis given by,

pfbz__pfre 1

The corrections to the model estimate recommended by the ith ray are applied to all J cells,

j

-

1,...,J.

Then, theupdated estimated model function M«"ew)e• becomes the current estimated model function back in step 1.

8. The arithmeticreconstruction technique(AP•T) is a seriesexpansion method which directly usesKaczmarz' algorithm.

9. The simultaneousiterative reconstructiontechnique(SIP•T) usesa modified Kaczmarz' method. Instead of updating the model after

tracing each ray in the forward modelingstep a.sis done in AP•T, all raysare tracedthroughthe currentestimatedmodeland a model correctionfound for eachray. Then the correctionsto eachmodel cell are averagedaccordingto the equation, I

i=1

forj

-

1,...,J.

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CHAPTER

42

2.

SEISMIC

RAY

TOMOGRAPHY

The weightWj is the numberof raysintersecting the jth cellor some other suitableray densityweightusedto obtain an averagecorrection

AM 1. The averagecorrectionis appliedto the currentestimated model function to get an updated or new estimated model function.

2.5

Suggestions for Further Reading Anderson, A. H., and Kak, A. C., 1982, Digital ray tracing in two-dimensional refractive fields: J. Acoust. Soc. Am., 72, 1593-1606. Addressesray tracing througha griddedrefractiveindex model commonly used in ultrasound computerizedtomography.

Berryman,J. G., 1990,Stable iterative reconstructionalgorithm for nonlinear traveltime tomography: Inverse Problems, 6, 21-42. Discussion of the nonlinear problem associatedwith traveltime tomography.

Bishop,T. N., Bube, K. P., Cutler, R. T., Langan, R. T., Love, P. L., Resnick, J. R., Shuey, R. T., Spintiler, D. A., and Wyld, H. W., 1985, Tomographicdetermination of velocity and depth in laterally varying media: Geophysics,50,903923. Traveltime tomographyapplied to reflection seisinology. Chapman, C. H., and Pratt, R. G., 1992, Traveltime tomography in anisotropicmedia- I. Theory: Geophys. J. Int., 109, 1-19. Expandstraveltime tornographyapplicationsfrom isotropic media to anisotropic media, a subject which has much current interest. A companionpaper immediately follows this paper on the applicationsof the theory. Dines, K. A., and Lytle, R. J., 1979, Computerizedgeophysical tomography: Proc. IEEE, 67, 1065-1073. First application of ray tornographyto subsurfaceimaging. Discusses both ART

and SIRT.

Langan, R. T., Lerche, I., and Cutler, R. T., 1985, Tracing of rays through heterogeneousmedia: An accurate and efficient procedure: Geophysics,50, 1456-1465. We do not elaborateon how to determine the raypath lengths through a griddedvelocitymodelfor the matrix S. This referenceis more than adequatein addressingthe problemas appliedto direct arrivals in seismic ray tomography.

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2.5.

SUGGESTIONS

FOR

FURTHER

READING

Moser, T. J., 1991, Shortest path calculation of seismicrays: Geophysics, 56, 59-67. Utilizes network theory to get raypaths and traveltimes of first arrivals. Very robustfor defining the matrix S, whether the first arrival is a direct arrival or a head wave.

Peterson, J., Paulsson, B., and McEvilly, T., 1985, Application of algebraic reconstruction techniquesto crossholeseismic data: Geophysics,50, 1566-1580. ART applied to crosswell seismic

data.

Phillips, W. S., and Fehler, M. C., 1991, Traveltime tomography: A comparisonof popular methods: Geophysics,56, 1639-1649. Comparison of various linear inversion methods.

Vidale, J. E., 1988, Finite-difference calculation of traveltimes: Bull. Sets. Soc. Am., 78, 2062-2076. Approximates the eikonal equation using finite differences to compute traveltimes for first arrivals.

43

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Chapter

3

Seismic

Diffraction

Tomography 3.1

Introduction

Seismicdiffractiontomographyis usefulfor reconstructingimagesof subsurface inhomogeneities whichfall into two categories.The first categoryincludesinhomogeneities that aresmallerin sizethanthe seismicwavelength and have a large velocitycontrastwith respectto the surrounding medium.Imagingtheseinhomogeneities with the seismicray tomography methodspresentedin Chapter 2 is generallyout of the question.The second categoryincludesinhomogeneities that are muchlargerin sizethan the seismicwavelengthand have a very small velocity contrastwith the surroundingmedium. Although seismicray tomographyis valid for imaging theseinhomogeneities, it worksbest when the velocitycontrastsare large. Note that both categories of inhomogeneity are capableof producingmeasurablescatteredwavefieldsof similar power. The large velocitycontrast of the first categoryinhomogeneity offsetsits smallsizewhilethe largesizeof the secondcategoryinhomogeneity makesup for its smallvelocitycontrast. The outlinefor this chaptercloselyparallelsthat of Chapter 2. First, in Section3.2 we review acousticwavescatteringtheory and derivetwo independentlinear relationshipsbetweendata functionsrepresenting scattered energyand the modelfunctionM(r). The modelfunctionM(r) usedin this chapteris a measureof the velocityperturbationcausedby an inhomogeneityat vectorpositionr from a constantbackground velocity.Second, usingeither of the linear relationshipsbetweena data function and the modelfunctionM(r), the generalized projectionslicetheoremis derived 45

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46

CHAPTER

3.

SEISMIC

DIFFRACTION

TOMOGRAPHY

in Section 3.3 which servesas the foundation for the image reconstruction algorithms used in seismic diffraction tomography. Finally, two seismic diffraction tomography image reconstruction algorithms are presented in Section3.4' direct-transformdiffraction tomographyand backpropagation diffraction tomography. Before proceeding further one should have a good understanding of

the followingmathematicalconcepts: frequencyand wavenumber(AppendixA); Fouriertransform(AppendixB); Dirac deltafunctionandGreen's function (Appendix C). Also, sincewe presentseismicdiffractiontomography for inhomogeneitiesembedded in a constant velocity medium, the implementation of diffraction tomography is actually very similar to the transform methods for ray tomography. Thus, a review of Section 2.2 on ray tomography'stransformmethodsmight be of somebenefit.

3.2

Acoustic Wave Scattering

The propagationof an acousticwavefieldP(r,t) through a medium consistingof a variablevelocityC(r) and constantdensityis modeledby the acoustic wave equation,

1 O2P(r,t) V•P(r,t)C•(r) at• _ - 0,

(47)

where r is a vector position within the model and t is time. The Laplacian

operator X7• is definedin terms of the vector operator •7 which in the Cartesian coordinatesystem is given by

v -

+ 0

where•, j, and• aremutuallyorthogonal unitvectors. We use the Helmholtz form of the acoustic wave equation to describe acousticwave scattering. The Helmholtz acousticwave equation, found by

taking the temporalFouriertransformof equation(47), is

VaP(r,w)+ k•(r,w)P(r,w) = 0.

(48)

The variablek(r, w) is the magnitudeof the wavenumberat positionr and is defined by

k(r,w)= C(r)'

(49)

Notethat equation(48) dependsuponthe valuesetfor angularfrequencyw. Henceforth,for the sakeof brevity,we will write both P(r,w) and k(r,w)

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3.2.

ACOUSTIC

WAVE

SCATTERING

47

as P(r) and k(r) with the understandingthat there is a dependenceon angular frequency w. Two nonlinear integral-equation solutionsfor the scalar Helmholtz wave

equationdefinedby equation(48) arederivedin thissection.The LippmannSchwinger integral equation is one such solution which, because of its prominencein quantum mechanicalscattering, is presentedby itself in Section 3.2.1. The Lippmann-Schwinger equation nonlinearly relates the

data function P0(r), calledthe scatteredwavefield,to the modelfunction M(r). The Born approximationin Section3.2.2 linearizesthe LippmannSchwingerequation. Another nonlinear integral-equation solution is formulated in Section 3.2.3 using exponentials. Although the definition of the model function M(r) remainsthe same, the data function is different from the scatteredwavefieldP0(r). The Rytov approximationis used to linearize this nonlinear integral-equation solution. Except for the data functions, both linearized integral-equationsolutionsare identical in form. Section 3.2.4 provides a comparison between the Born and Rytov linear integral solutions.Either solutionenablesus to derive the generalizedprojection slice theorem in Section 3.3 which serves as the foundation for the diffraction tomographyimage reconstructionalgorithmsin Section3.4.

3.2.1

The Lippmann-Schwinger Equation

We begin the formulation of the acousticwave scatteringproblem with Figure 15. The acousticwave velocity is representedby C(r), where r is the vector position of a point within the model. The shaded region in Figure 15 depictsan inhomogeneityimbeddedin an otherwisehomogeneous medium. The acoustic velocity of the inhomogeneity varies spatially and can be thought of as a velocity perturbation from the constantbackground velocity Co of the homogeneousmedium.

An incidentwavefieldPi(r) is initiated by an acousticsourceand propagatesoutward in the homogeneous medium. No scatteringof the incident wavefield takes place until the inhomogeneityis reached. At that point any velocity contrast as a result of the inhomogeneitycausesthe creation of a secondwavefieldcalled the scatteredwavefieldP•(r). Each point in the inhomogeneitymay be consideredas a secondarysourceof seismic acousticenergy. Note that once acousticenergyis scatteredfrom one inhomogeneity,then that scatteredenergymay be scatteredagain from another inhomogeneitywhich leads to higher order sourcesof seismicacousticenergy. As you will see, we ignore multiple scatterings in Section 3.2.2 to linearize the Lippmann-Schwingerequation and assumethat the scattered

wavefieldP•(r) arisesonlyfromscatteringthe incidentwavefieldP•(r) from the source as depicted in Figure 15. Therefore, for layered media we as-

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48

CHAPTER

3.

SEISMIC

DIFFRACTION

Receiver

TOMOGRAPIIY

Source

C(r) = Co

FIG. 15. Acousticwavescatteringproblem.The incidentwavePi(r) propagatesfrom the sourceat the constantbackgroundvelocityCo. The velocity inhomogeneity, depictedby the shadedarea, actsas a secondarysourceand

scattersthe incidentwavefield.The scatteredwavefieldPj(r) travelsaway from the inhomogeneous regionat the backgroundvelocityCo whereit is recordedby the receiver.

sume that multiple reflectionsare negligiblewhen comparedto primary reflections.

The wavefield recorded by a receiver in the model consistsof both the

incidentwavefieldPi(r) and the scatteredwavefieldP,(r) whichwe call the total wavefieldPt(r) , or

P•(r) -

Pi(r) -[-P,(r).

(50)

For a constantdensitymodel,equation(48) describesthe propagationof the total wavefieldPt(r) throughan inhomogeneous medium,or

[V'• + k•(r)]Pt(r) = O.

(51)

At thispointwereformulate k2(r) in equation(51) asa perturbation to a constantko • for the homogeneous background mediumwherethe magnitude of ko is given by

=

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3.2.

ACOUSTIC

WAVE

SCATTERING

49

We begin by writing

-

+

[k•(r)1].

(53)

Substitutingequations(49) and (52) into the bracketedterm on the righthand sideof equation(53) for k(r) and ko, respectively, gives

ke(r)

C• 1] - ko •+k•[C•.(r) = ko 2-ko 2[1 C2(r ½;] ).

(54)

If we definethe modelfunctionM(r) asthe bracketedterm in equation(54),

thenwe get the desiredreformulation of k2(r),

k2(r) -

k• - ko•M(r),

(55)

whereM(r) is definedby

M(r)- 1 C•(r C•).

(56)

The model function M(r) in equation(55) definesa perturbationto an

otherwiseconstantko • of the background medium. When C(r) - Go in equation(56), M(r) - 0 and thereis no perturbationof ko • (i.e., by equation (55), k•(r)We now want to establish a relationship between the scattered wave-

field P,(r) and the modelfunctionM(r) usingequation(51). First, equations(50) and(55) aresubstituted intoequation(51) for Pt(r) andk2(r), respectively, giving

[•7• + k•- k•M(r)][Pi(r)+ Po(r)] -

O.

(57)

We rearrangeequation(57) so that terms involvingsourcesof scattered energyt are on the right-handside. Thus,

[V2+ ko2]Pi(r)+[V2+ ko•]Po(r)-

ko2M(r)[Pi(r)+Po(r)]. (58)

Note that the term ko•M(r)[Pi(r) + P,(r)] is the sourceof the scattered wavefieldP0(r) and, as previouslystated, dependsupon both the incident 1Termscontainingthe modelfunctionM(r) are sourcesof scatteredenergy.

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5O

CHAPTER

3.

SEISMIC

DIFFRACTION

TOMOGRAPItY

wavefieldand scatteredwavefieldat r. The left-handsideof equation(58) describes the propagationof the incidentwavefieldPi(r) and the scattered wavefieldP,(r) in the backgroundmedium, both of which travel at a constant velocity Co.

The incidentwavefieldPi(r) is generatedby a sourcein the homogeneousbackgroundand containsno scatteredenergy. Therefore, the incident wavefieldtravelsthrough the model at the backgroundacousticvelocityCo and contributesto the scattered wavefieldthrough the scatter sourceterm

in equation(58) when M(r) •- 0. Thus, the Helmholtzacousticwave equation for a constant velocity medium describesthe propagationof the incidentwavefieldPi(r) and is givenby

[V'2+ko•]Pi(r) -

O.

(59)

Equation(59) permitsus to reduceequation(58) to

[v +

-

(60)

Equation (60) describesthe propagationof the scatteredwavefieldat the backgroundvelocity when inhomogeneitiesoccur which scatter both the

incidentwavefieldPi(r) and existingscatteredenergyPs(r). Note that if no inhomogeneities occur, then M(r) = 0 and the right-hand side of equation(60) is zero,implyingthereis no sourcefor the scatteredwavefield and P•(r) = 0. Solvingequation(60) directlyfor P,(r) is difficult. A simpleapproachis to formulatean integralsolutionusingthe propertiesof the Green'sfunction developedin Appendix C. For the Helmholtz equation the Green'sfunction is the responseof the differential equation to a negative impulse source

function. 2 Thus,theGreen'sfunctionbecomes thesolutionto equation(60) if we replacethe sourceterm with a negativeimpulsesourcefunction-5(rr t) • or

[V2+ ko•]a(r[r')

-

-5(r - r').

(61)

The Green'sfunctionG(r Jr') givesthe solutionat positionr for a negative impulse at r' which correspondsto the location of a point scatterer.

The solutionto equation(61) for a 2-D spacecontainingan infinite-line scattererat r' and infinite-line receiverat r, where both lines are perpendicular to the plane containingr • and r, is

a(rIr') -- j4 Ho(X)(ko Ir- r' l)'

(6:2)

• Traditionally the Green'sfunction solutionfor the Helmholtz equation is determined for a negativesourcedensity on the right-haxtdside, or X72P + ksP = -p. In order for the Dirac delta function to represent a source density impulse, the sign in front of the Dirac delta function must also be negative.

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3.2.

ACOUSTIC

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SCATTERING

51

where Ho (•) isthezero-order Hankel function ofthefirstkind.In a 3-Dspace containingpoint scatterersand field points, the solution to equation (61) is

G(rlr')= 4•rlr-r' I '

(63)

With the Green'sfunction for equation(61) known, the solution to equation(60) is foundby multiplyingthe Green'sfunctionby the negative of the sourceterm in equation(60) and integratingover all spacewhere

M(r •) •- 0, or 3

P,(r)- -ko •/ G(r [r')M(r')[P•(r') +P•(r')]dr'. (64) Equation (64) is called the Lippmann-Schwinger equation,which is the desiredintegral solutionfor the acousticwave-scatteringproblem. We point out in Appendix C that suchan integral is analogousto obtaining an output from a filter systemwhenthe impulseresponse of the filter (i.e., the Green's

function)and the input (i.e., sourceterm fromequation(60)) are known. Equation(64) is just a convolutionintegral,whichis easilyseenif the Green'sfunctionfrom either equation(62) or (63) are substitutedin for I,?). The Lippmann-Schwingerequationnonlinearlyrelatesthe model func-

tion M(r) to the data function(scatteredwavefield)P,(r). The nonlinearity is a result of the scatteredwavefieldP,(r) insidethe integrandof

equation(64) whosevaluedepends on the modelfunctionM(r). Because of this nonlinearity,it is difficult to use equation(64) to perform either forwardmodeling(computeP,(r) from M(r)) withoutresortingto computationally extensiveapproachessuchas finite differencemethodsor to derive diffractiontomographyimage reconstructionalgorithms(compute M(r) from P,(r)). One way to get aroundthis problemis to linearize equation(64) by makinga simplifyingapproximation calledthe Born approximation.

3.2.2

The Born Approximation

The Born approximationlinearizesequation(64) by assumingthat the

scattered wavefield P,(r) is muchweakerthanthe incidentwavefield Pi(r), 3In this book we use special integration notation which should not be confused with

a line integral. Equation(64) couldbe either an integrationovera plane (2-D space) or a volme (3-D space)dependingupon which Green'sfunctionis used. Thus, to

remain general, if weareintegrating overa plane, thenf dr'• f dx'dz'; andif weare integrating overa voltune, thenf dr'::•f dx'dy'dz'.

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52

CHAPTER

3.

SEISMIC

DIFFRACTION

TOMOGRAPHY

or

P,(r) • P•(r).

(65)

If the conditiongivenby equation(65) is true, then the Born approximation states that

Pi(r) + P,(r)

•

Pi(r).

(66)

The Lippmann-Schwinger equationis linearizedby substitutingequation(66) into equation(64),

P,(r) • -ko 2/ G(r [r•)M(r•)Pi(r•)dr •. (67) Note that the integrandno longercontainsthe scatteredwavefieldP•(r) and the data function P•(r) and mode]function M(r) are now linearly related. If the primary acoustic source is a negative impulse located at vector position r,, then, by the definition of a Green's function, we can

directlyrepresentthe incidentwavefieldP•(r') in equation(67) by a Green's function,

Pi(r')

:

G(r' [ rs).

(68)

If a pressure-sensitive receiver(e.g.,hydrophone) is locatedat positionr = rp, thenby substituting equation(68) intoequation(67) for P•(r') wefind,

P,(r,,rp) • --k• 2/ M(r')G(r' lr,)G(r pIr')dr', (69) whereP,(r,, rp) is the scattered wavefield observed at positionrp whenthe negativeimpulsesourceis located at positionr•. Both Green'sfunctions are definedby either equation(62) or (63).

Equation(69) is the Lippmann-Schwinger equationlinearizedby the Born approximation. This equation establishesthe linear relationship be-

tweenthe data function(scatteredwavefield)h(r,,rp) and the unknown modelfunctionM(r) requiredby the diffractiontomographyproblem.Note that sincewe exploitedthe Born approximationin derivingequation(69) that the modelfunctionM(r) must be a weak scatterer. Only then will the scattered

wavefield

be much weaker than the incident

wavefield

and the

conditionspecifiedby equation(65) be satisfied.

3.2.3

The Rytov Approximation

In this sectionwe derivea nonlinearintegral-equationsolutionto equa-

tion (48) usingexponentials. Althoughthe modelfunctionM(r) is defined

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3.2.

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53

the same as for the Lippmann-Schwingersolution, the data function representingthe observedscatteredwavefieldis different. The Rytov approximation establishesa linear relationship between the data function and the model function. The resulting linearized integral solution strongly resemblesequation(69) for the Born approximation. We begin the derivationby returning to equation(51) which describes the propagationof the total wavefieldP•(r) through a constantdensity, variable-velocity medium. Usingequation(55) andequation(56) asthe def-

initionsfor k2(r) andthemodelfunctionM(r), respectively, equation(51) is rewritten

as

Iv • + • - •4(•)]P,(•)

-

0.

(70)

Exceptfor settingPt(r) = Pi(r)+ P,(r), equation(70) is the sameas equation(57). We deviatefrom the earlierderivationof the LippmannSchwingerequationby representingthe total wavefieldwith the exponential equation,

P•(r)-

e•,(r),

(71)

where•b•(r) is calledthe "complextotal phasefunction." Note that, as in previous sections, variables which are a function of the position vector r

carryan impliedfrequencydependence. We wish to substituteequation(71) into equation(70) to obtain a differentialequationin termsof •,(r). The Laplacianof P,(r) mustbe taken to achieve the this result.

We start with

v•P,(•)

= v. [vP,(•)].

Substitutingequation(71) in for Pt(r) and performingthe differentialoperationsgives,

v•P,(•) = = =

v. b•,(•)v•,(•)], e•'(•)v-v•,(•)+ Ve•'(•).v•,(•), •,(•)v•,(•) + •,(•)v•,(•). v•,(•), e•'(r)[v•,(r)+V•,(r).V•,(r)].

(72)

Equations (71) and(72) aresubstituted intoequation(70)for V2P,(r) and P,(r) giving,

•,(•)[v•,(•) + v•,(•). v•,(•)]+ [• - •(•)1•,(•)

- 0. (73)

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54

CHAPTER

3. SEISMIC

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TOMOGRAPHY

Dividing through byeqb'(r), equation (73)becomes, V•6 (r) + [V6t(r) V6t(r)]+ ko • ko•M(r)-

0,

(74)

whichis the desireddifferential equationin termsof •bt(r). At this pointweintroducethe incidentwavefield Pi(r) expressed

Pi(r)-

e0i(r),

(75)

where$•(r) is calledthe "complex incidentph•e function."The complex incidentph•e function•i(r) is relatedto the complextotal ph•e function •t(r) throughthe "complex ph•e difference function,"definedby

0a(r) =

0t(r)- 0,(r).

(76)

Sincethetotalwavefield P•(r) variesfromthe incidentwavefield P•(r) only whenscatteringoccurs,we can surmisethat the complexph•e difference function$a(r) is a meansof accounting for scatteredenergy.

Wecontinue thederivation byreplacing 0,(r)in equation(74) by •a(r) • definedby equation(76). Carryingthis operationout yieldsa differentialequationin termsof 0,(r) and

V•0i(r)+ V•0•(r) + [V0i(r). V0i(r)]+ 2[V0i(r). +[V$•(r). V&a(r)]+ k• - k•M(r) - 0.

(77)

The termsin equation(77) are rearrangedin a form whichwill proveconvenient later on,

=

+

(78)

The termsinsidethe squarebracketson the left-handsideof equation (78) are all relatedto the incidentwavefield and havea sumequal to zero. This is e•ily showntrue by usingequation(59) whichdescribes the propagation of the incidentwavefieldPi(r) throughthe background medium.Equation(75) defining Pi(r)is substituted intoequation (59)to get the differentialequationin termsof •i(r). Followingthe sameprocedurewhichgavethe Laplacianof Pt(r) in equation(72), the Laplacianof

V•P•(r)- e0•(r)[v•0•(r) + V0•(r).V0,(r)].

(79)

Substituting equations (79) and (75)into equation(59) for V•P•(r) and P•(r), respectively, gives

e•i(r)[v2•i(r) + V•i(r)-V•i(r)] + k•e•i(r)- O.

(80)

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3.2.

ACOUSTIC

WAVE

SCATTERING

55

Dividing equation (80)bye•i(r)results inanequation which demonstrates that the bracketedterms in equation(78) havea zerosum,or

V2•i(r) + V•i(r).V•i(r)

+ ko • -

O.

(81)

+ ••(•).

(8•)

By equation(81), equation(78)is rewritten •

•v•,(•). v•(•)

+ v•(•)

- -v•(•).

v•(•)

Equation(82) providesa crucialrelationshipwhichwe will uselater on in this derivation. Now we turn our attention to the important product

P•(r)•(r)

which will becomethe data functionfor this derivationor a

me•ure of the scattered wavefield. For purposesof the derivation the Laplacian is taken of this product. Performingthe differentiationwe have,

V•[P,(•)•,(•)]

- V. [•,(•)VP,(•) + P,(•)V•,(•)] = •,(•)V•(•) + •V•(•). V•,(•) +P,(r)V2•d(r).

(83)

Rearrangingthe termsin equation(83) and makinguseof the fact that

V•P•(r)-

-k•P•(r) by equation(59), weget the following,

2VP,(r). V•(r)+

P,(r)V•(r)

- V•[Pi(r)•(r)]- •(r)V•P,(r) = V2[pi(r)•d(r)]+ •d(r)koP•(•) 2 = [V• + k•]P,(r)•d(r). (84)

Switchingthe left- and right-handsidesof equation(84) and using the definitionof P•(r) givenby equation(75), we can write the following,

Iv • + •]P,(•)o,(•)

-

•v•,(•). v•,(•) + P,(•)v•o,(•)

= •(•)vo•(•). v•,(•) + p•(•)v•o,(•) = •P,(•)v•,(•). v•,(•) + P•(•)v•,(•) = p,(•)[•v•,(•). vo,(•) + V•d(•)].

(SS)

The quantity inside the square brackets on the right-hand side of equation (85) is definedby the earlier "crucial"relationshipgiven by equa-

tion (82). Substitutingequation(82) into equation(85) gives,

IV• + •]P,(•)o,(•)

-

-P,(•)[v•,(•).

vo,(•)-

•(•)].

(86)

Just • w• donefor equation(60), equation(86) can be solvedfor P•(r)•d(r) in termsof an integralequationby exploitingthe propertiesof the Green'sfunction. The resultingsolutionis

P,(•)o,(•) - f P,W)[vo,(•'). v•,(•')•(•')]a(• I•')d•',(87)

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CHAPTER

56

3.

SEISMIC

DIFFRACTION

TOMOGRAPHY

where G(r Ir') is a Green'sfunctiongivenby either equation(62) or equation(63). The data functionP•(r)da(r) is nonlinearlyrelatedto the modelfunction m(r) sinceda(r) appearsinsidethe integrandof equation(87). Both forward modeling(computingP•(r)•a(r) from M(r)) and image reconstruction(computingm(r) from P•(r)&a(r)) are difficultbecauseof this nonlinearity. At this point in the derivation the Rytov approximation is formulated to remedy this situation. The Rytov approximation linearizes

equation(87) by assuming the conditionV•a(r) (k) -

Equations(B-22) and (B-23) enableusto computeoneof the Fouriertransform pairsgiventhe other. Equation(B-22) is the inverseFouriertransform operationrepresented by equation(B-4) and equation(B-23) is the Fourier transformoperationrepresented by equation(B-3). B.4

Fourier Transform-

Sampled f(x)

The use of digital computersto carry out computationsrequiresthat we samplethe continuousfunctionf(x). We may take a samplefrom the continuousfunction f(x) at intervalsof Ax giving the discretesamples: ß.., f-2, f- •, f0, f•, f2,. ß., where the subscriptsindicate the sample number. If the sample number is n, then the positionof the sampleis x - nAx.

However,a sampledfunctioncan no longercontainwavenumbers (or frequencies)out to 4-00 as demonstratedby the integrallimits for the continuousfunction f(x) in equation(B-22). The shortestwavelengthrepresented by a sampled function is 2Ax which gives the highest possible wavenumber(or frequency),

knyq= Az'

(B-24)

referredto as the Nyquistfrequency. a Note that this alsoappliesto temporally sampled functions. aThe Nyquist frequencyhas a wavelengthor period which is sampled by two points; or we may state that the shortest wavelength or period in a sampled signal must contain at least two sample points.

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162

APPENDIX

B.

THE FOURIER

TRANSFORM

In sectionsof this book where we are referring to a sampledfunction, you will see the limits on the inverse Fourier transform equationswritten

as :t:•r. Equation (B-24) showsthat thesefrequencylimits correspondto a unit sample interval, Az = 1. A unit sample interval is commonly usedin

digital computationsto avoidmultiplyingby Az, whichbecomesnothing more than a scalefactor. Thus, you will seethe Fourier transformequations written as 4

f(x)- •1 , p(k)eJkXdk ,

(B-25)

P(k) -

(B-26)

and

f(x)e-Jkx dz,

whenf(x) and•(k) aresampled functions. Again,theclueto f(x) being sampledis the noninfinite frequencylimits. Of course,in actual computer

computations we usuallydo not useequations(B-25) and (B-26) directly. Instead, the Fast FourierTransform(FFT) method is utilized. B.5

Uses

of Fourier

Transforms

FromequaLion (B-26) weseethat F(k) is generallya complexvalued function, called the complex spectrum, which may be written as,

P(k) -

•Re{P(k)} + jlm{P(k)},

(B-27)

whereRe and Im designatesthe operationof taking the real and imaginary

partsof P(k), respectively. FigureB.1shows thelocation of onepointof the complexspectrumin the complexplane;so calledbecausethe real part of the complexfunctionis plottedon oneaxisand the associatedimaginary part on the other axis. The figure demonstratesthat the complexspectrum can be also describedin terms of polar coordinates.In polar form we can write the complexspectrum as,

P(•) - Ip(•)leJ•(•),

(s-28)

whereI P(•) I iscalledtheamplitude spectrum givenby,

]#(•)] - V/ee{#(•)} •+t,•{#(•)}•, 4Mar•y times geophysicists cha•ngethe sign in the exponentialof the forward and inverse Fourier transforms

when the transform

involves time.

This is done so that we

stay consistentwith the physics, that is, a wave traveling in the +x direction is described

byA(w,kx)eJ(kxxwt)andnotA(w,kx)eJ( kxx+ wt).

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B.5.

USES OF FOURIER

TRANSFORMS

163

Complex Plane

Im {F(k)}l (Re {.•(k)} ,lm {•(k)} ) > Re {F(k)}

FIG. B.1.

Complex plane with an arrow directed to the point

(Re{17'(k)},Im{17'(k)}). Thepointin polarcoordinates is represented by the magnitudeI F(k) I definedby equation(B-29) and the phase(I)(k) definedby equation(B-30). The magnitudeis calledthe amplitudespectrum and the phaseis called the phasespectrumwhen eachis plotted separately as a function

of k.

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164

APPENDIX

B.

THE

FOURIER

TRANSFORM

and (I)(k) is calledthe phasespectrumgivenby,

(I)(k)- tan -1 Im{F(k)} •

ß

(B-30)

The amplitude spectrumgivesthe magnitudeof a particular sinusoidat frequencyk while the phasespectrumgivesits shift in spaceor time. Most often we are interested in the frequency content of a signal and a plot of the amplitude spectrum servesthis purpose. Throughout this book we work in tl•e Fourier domain when it simplifies derivations. In addition to the above concepts of amplitude and phase

spectra, we encounterthree other applicationsof the Fourier transform' the 2-D Fourier transform, the Fourier transform of the time derivative,

andthe Fouriertransform of eJkox. Supposewe wish to take the Fourier transform of a function f(x,z) with respect to the spatial variables x and z. This operation is called a 2-D Fouriertransform. We beginby taking the Fouriertransformof jr(x, z) with respect to x, or

P(k•,z) -

f(z,z)e-Jk•zdz,

(B-31)

wherethe wavenumberalongthe x-axis is denotedby ks. Next we apply a Fouriertransformto equation(B-31) with respectto z, or

•(k,,,kz) -

•(k,:,z)e-JkzZdz,

(B-32)

where the wavenumberalong the z-axis is denoted by kz. Substituting

equation(B-31) into equation(B-32) givesthe definitionof a 2-D Fourier transform,

•(k•,k,) -

f(x,z)e-J(k•x + k•Z)dxdz ' (B-33)

Similarly, the inverse2-D Fourier transform may be definedas,

f(x,z)

/_•o /_••(k•, k•)eJ(k•x +k•Z)dk•dk• ' (B-34)

471-2

Time derivativesare simpler in the Fourier domain which is one reason many differentialequationsare solvedin the frequencydomain. Look at the inverse Fourier transform,

= 1

f'(w)e-Jwt dw,

(B-35)

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B.5.

USES OF FOURIER

TRANSFORMS

165

wheret is time andw is frequency.The first time derivativeof equation(B35) is, ef(t)

= -• • /_•o j.•(•)• _

dt

a•, -J•t

= FT-•[-j•P(•)].

(B-36)

A second-orderderivative is found by taking a time derivative of equation (B-36) or,

dt 2

(B-37) Thus, an nth order derivative is defined,

rr-•[(-i•)-p(•)].

dtn

(B-38)

WesimplymultiplyP(w) bytheappropriate factor s (-jw)" andtakethe inverse Fourier

transform.

ThelastitemtonoteistheFourier transform oftheexponential e-Jkox, which comesup many timesin diffractiontomography.We determinethis in a roundabout way. We use the Dirac delta function defined in Appendix C

in thisderivation. Taketheinverse Fouriertransform of .b(k)- 5(k- ko), wherethe Diracdeltafunctionis locatedat wavenumber koin the spatial frequencydomain. By equation(B-22) we find that, f(x)

= -2•r

5(k- ko)eJkxdk,

= l,j•o•. 2•r

Thus, theFourier transform ofeJkoxmust be2•rS(k-ko).

SUse(+jkx) n and (+jk,) n for spatialderivatives.

(B-39)

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Appendix

C

Greens

Function

Equation(60) in Chapter3 on seismicdiffractiontomographydescribes the propagationof the scatteredwavefieldP0(r) at a constantbackground velocitywhen inhomogeneities scatter both the incidentwavefieldPi(r) and existingscatteredenergyP0(r). We rewrite equation(60) here for your reference as

Iv • + •o•]•,(•) -

•o•U(•)[•,(•)+ •,(•)1.

(C-l)

Through the use of Green'sfunctionswe end up with an integral solution to thisequationgivenby equation(64), the Lippmann-Schwinger equation, which we rewrite here for your referenceas,

?o(,,) - -• f •(,,I•")•(,?)[?,(,?) +•0(•')]a•'. (c-•) G(rl rt) is the Green'sfunctionand the integralis takenovera planein 2-D space or a volume in 3-D space.

The intent of this appendixis to provideyou with an intuitive feelingfor Green's functions. With this understandingyou will know how to immediately write down an integral solution to any inhomogeneousdifferential equation with constant coefficients,such as the solution to the partial dif-

ferentialequation(C-l) givenby the integralequation(C-2). We do NOT go overtechniques for determiningthe Green'sfunctionG(r I r•) whichare readily found in many texts with chaptersdevoted to the subject, such as referencedat the end of this appendix. However, an example problem is

workedlater on to giveyousomeideaof howonemay solvefor G(r I r•) . The approachwe take for conveyingthe ideas of Green's functions is throughthe conceptsinvolvedwith filter theory. We do this for two reasons: 167

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168

APPENDIX

C.

GREEN'S

FUNCTION

Gt LINEAR OPERATOR

FIG. C.1. Gt is the impulseresponseof the linear operator giventhe input discrete impulse 50 at time sample t - 0.

(1) its easyto see, and (2) many of you are alreadyfamiliar with filter theory,especiallythe conceptof convolution.We will look at the discretely sampledcasefirst and then extrapolate to the analogcase.

C.1

Filter Theory

In filter theory we can input a discrete impulse 5• of unit height at samplet into a linear operator or systemwhich givesan output, calledthe impulse responseGr. Figure C.1 showsa plausibleimpulse responseof a linear operatorwhen a unit impulse5o is input at sample! - 0.• A linear systemis time invariant when the impulse responseG• is found at the same time lag relative to the time sample of the input impulse. For example, Figure C.1 showsan input impulse50 with an impulseresponseGt starting at time sample0. If we had usedan impulse5•, then the impulseresponse G• would have begun at time sample5 which may be written as Gt-•. The impulseresponseG• in Figure C.1 can also be scaled. We may multiply the unit impulseinput 5oby a scalarf0 whichresultsin the impulse response G•-0 beingscaledby f0 asshownin FigureC.2(a). The t-0 in the subscript of G•-0 indicates the delay of the impulse responseGt because of a delay in the input. Here no delay occurs and the output foG• is the responseto the input fo5o, where f0 ---1. We can scale the impulse input 51 by a scalar fl - 1 which results in the linear operatorresponseflG•-i - G•-i in Figure C.2(b). The subscript t-

I indicates that G• is delayed by one time unit in accordancewith the

time invarianceprinciplestatedearlier.Similarly,FigureC.2(c) showsthe impulseinput 52 scaledby f2 - -1/2 resultsin a linear operatorresponse 1We assumea unit interval, At = 1, betweensamples.

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C. 1.

FILTER

THEORY

'

169

fog t.o

-1 •o o OPERATOI•.

t

(a)

+

1Gt.1

1•1 LINEAR

0

!

OPERATOI•

o

t

•) +

-1/2õ2

f2 Gt.2 LINEAR OPF•ATO•

(c)

01--1 ft llU-

LINEA• OPF•ATOI•

!øi

Pt

(d)

FIG. C.2. Illustration of the principle of superpositionfor a linear operator. The input consistsof a seriesof scaled, time-shifted impulsesgiven by

ft -- j•050-}'f151 -}'j•252,wherefo = -1 in (a), fl ----1 in (b), and/2 = -1/2 in (c). The resultingoutput to f• is the sum of the responses in (a), (b), and (c): P• - foG,-o 4-fxG,_x + f2G,-:• shownin (d). This is just an example of convolutionof an input signal ft with an impulseresponseof a linear filter

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APPENDIX

170

C.

GREEN'S

FUNCTION

FigureC.2(d)is thesumof the inputsandoutputsfromFiguresC.2(ac). The discretizedinput is thoughtof as a sum of scaledimpulsesgiven by,

ft -- fo•o q-fl•l q-f2•2-- (-1)50 + (1)51 q-(-•1)5:•.

(C-3)

The linear operator'sresponseto the input function ft is given by, ---- foGt-o q- flGt-1 q- f2Gt-2

= (-1)G,_0 q-(1)G,_l q-(-•

(c-4)

Equations(C-3) and (CL4)combinedillustratethe principleofsuperposition for a linear operator. That is, the same output results whether a linear

operatoractson the entire input as depictedin Figure C.2(d), or on each individual componentof the input as shownin Figures C.2(a-c) and the results summed.

Generalizingequation(C-4) we have,

This equationis the well-knownconvolutionequationfor two discretized signals. Here the convolutionis between the input function .It and the impulse responseof the linear operator Gt which givesthe responseof the linear operator Pt. If the above functions are continuous,then the sum in equation(C-5) becomesan integralor

P(t)- • f(t')G(t - t')dt',

(C-6)

wheredr' is explicitlywritten in placeof At' in equation(C-5) whichwas assumedequal to one.

C.2

PDE's as Linear Operators The partial differentialequation(C-l) can be cast in the samelight as

thelinearoperatorin filtertheory.Thesourceterm,ko•M(r)[Pi(r)+P,(r)], is analogousto the input to the linear operator. The linear operator is the

bracketedpart of the functionon the left-handside,IX72+ ko•], and the solutionto the differentialequation,P,(r), canbe thoughtof asthe output from the linear operator.

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C.2.

PDE'S

AS LINEAR

-6 (r-r')

OPERATORS

•

171

•2+k2 ] • G(rlr' oI )

- -5(r-r') [•2+k2o] G(rlr') FIG. C.3. The Green's function is the impulse responseof a linear differential equationoperatorto a sourceterm (input) givenby a negativeDirac delta function. Comparethis with the impulseresponsein filter theory depictedin Figure C.1.

As in the previous section, the first step is to determine the impulse responseof the linear operator to an impulsive input. For the partial dif-

ferentialequationwe usea negativeDirac deltafunction 2 -5(r-

r •) for

the impulsiveinput and the Green'sfunctionG(r- r •) for the impulseresponseto the negative Dirac delta function. Thus, using the analogiesin the previousparagraph,equation(C-1) is rewritten as,

IV2+ ko•lG(r I r') -

-5(r-r'),

to get the impulseresponse,or Green'sfunction,G(r I r') = •(r-

(C-7) r').

Note that the space vector r is the independent variable here instead oftime used in the previoussectionon filter theory. Figure C.3 summarizes the analogy made between the differential equation operator and the filter theory operator in Figure C.1.

The Green'sfunctionsolutionG(r I r •) to equation(C-7) is the impulse responseto the negative Dirac delta function. Just as with filter theory,

we may weight the negativeDirac delta function,say -f(r')5(rr'), on the right-handsideof equation(C-7) whichresultsin the solution(output) f(r')G(r Jr'). Using the principleof superpositionshownin Figure C.2, the resultingresponse isjust the integration(sum) overall outputresponse 2The Dirac delta function6(x) has ax•infiaitesi•

width, infiniteheight,and an

area equal toone atx = 0. It isdefined asf_+•S(x)dx = 1. When combined with another function, f_+•!(z)a(z-Xo)= l(Zo),where theDirac delta function islocated at 3• '-' 3•o.

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172

APPENDIX

C.

GREEN'S

FUNCTION

componentsfrom the linear operator, or

: - fr,

(c-s)

wherethe negativesignin front of the integralis presentbecausethe Green's function is the impulse responseto a negative Dirac delta function. Equa-

tion (C-8) is similarto equation(C-6) exceptfor the negativesignandthe independent variable.

Settingf(r') in equation(C-8) to the right-handsideof equation(C-l),

becomesthe sourceterm (input to the linear operator) in equation(C8) and we get the integralsolutiongivenby equation(C-2). Thus, the integralsolutionto equation(C-l) isjust the negativeof the convolution of the sourceterm (input) f(r) with the Green'sfunction(impulseresponse), analogousto filter theory. The r- r • part of the Green'sfunction(see equations(C-9) and (C-10) below)givesthe "lag"in the impulseresponse as a result of the weightednegativeDirac delta function being located at a position other than r • = 0. This is a space-invariantproperty equivalentto the time-invariant property of the linear operator in the previous section on filter theory. Chapter 3 on seismic diffraction tomography makes extensiveuse of Green's functions. However, closeinspection reveals that only two Green's

functionsare actuallyevermentioned,both aresolutionsto equation(C-7). The 2-D solutionto equation(C-7) is exclusivelyusedin Chapter 3 and is given by,

[r - r' I), a(rl') - JHo(1)(ko whereHo (1)is thezero-order Hankelfunction of thefirstkind. Herethe negative Dirac delta function representsan infinite-line seismicsourceor an infinite-line scatterer at r' which causesa cylindrically shaped seismic disturbance

that is determined

at an infinite-line

field location

r. Both the

infinite-line source and infinite-line field location are perpendicular to the

plane representingthe 2-D space. A 3-D spacesolutionto equation(C-7) is givenby,

ejkolr-r'l

G(rlr')= 4•rlr-r']'

(C-10)

This equation is not usedin Chapter 3 sincemost data acquisitionschemes, suchas crosswellseismic,are gearedto 2-D spaceor cross-sectional studies

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C. 3.

GREEN'S

FUNCTION

EXAMPLE

173

rather than volumetric investigations.The 3-D spacesolution represents point seismicsourcesor point scatterersat r t which causea sphericalseismic disturbancethat is determined at a point field location r.

C.3

Green's Function Example

In this section we find the integral solution of a simple 1-D differential equation to reinforcethe ideas presentedin the last section. The method of determining the Green's function here is just one example of many and we refer you to the referencesat the end of this appendix for other methods. The differential equation is,

[d-••a2]p(x) - f(x), for -cx>