An Analysis of Least-squares Velocity Inversion

8/15/2019 An Analysis of Least-squares Velocity Inversion

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R e d i s t r i b u t i o n s u b j e c t t o S

E G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p : / / l i b r a r y . s e g . o r g /


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GEOPHYSICAL MONOGRAPH SERIES

Don W. Steeples, Editor

NUMBER 4

AN ANALYSIS OF LEAST-SQUARES

VELOCITY INVERSION

By Fadil Santosa and William W. Symes

Edited by Raymon L. Brown

SOCIETY OF EXPLORATION GEOPHYSICISTS


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Library of Congress Cataloging-in-Publication Data

Santosa, Fadil.

An analysis of least-squaresvelocity inversion / by Fadail Santosa

and William W. Symes: edited by Raymon L. Brown.

p. cm.- (Geophysical monograph series:no. 4)

Bibliography: p.

ISBN 0-931830-89-8: $20.00

1. Seismic waves -- Measurement. 2. Inverse problems (Differential

equations) I. Symes, William W., 1949- . II. Brown, Raymon L.,

1944- .III. Society of Exploration Geophysicists. V. Title.

V. Title: Least-squares velocity inversion. VI. Series.

QE538.5. $25 1989

551.2'2--dc20

ISBN 0-931830-56-7 Series

ISBN 0-931830-78-8 Volume

Society of Exploration Geophysicists

P.O. Box 702740

Tulsa, OK 74170-2740

¸ 1989 by the Society of Exploration Geophysicists

All rights reserved

Published 1989

Printed in the United States of America


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To Amelia, Jan, and Lene

iii


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This page has been intentionally left blank


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Contents

P reface

vii

1 Introduction

2 The Model

3 The SH-wave Inverse Problem

15

4 The Output Least-squares Principle

17

5 Generalities on Nonlinear Least-squares Problems

21

Perturbations About A Slowly Varying

Reference Velocity

25

Perturbations About A Rapidly Changing

Reference Velocity

41

Implications for the Solution of the Least-squares

Inverse Problem

65

9 Computing the Least-squares Solution

83

10 Numerical Experiments

91

11 Conclusion

105

References

109

A Least-squares and the Velocity Spectrum

117

B Relation Between the L2-norms of (x,t)

and (p, r) Sections

123


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C Perturbational Seismogram for Two-layer References

127

D Hough-background WKBJ Perturbational

Seismograms for Layered Media

135

E The Optimum Coherency Principle

141

F An Example: Impedance Trends are not Determined 149

vi


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Preface

This book grew out of our attempt to understand he mechanismshrough

which band-limited reflectionseismograms etermine velocity distributions

in elastic models of the earth's crust. We were especially interested in the

feasibilityof recovering eryslowlyvarying out-of-passband)elocitycom-

ponents rom band-limited high-frequency)eflectiondata. Our interest

was spurred by reports of successful nversions or layered media, which

came to our attention in late 1984 and 1985, ust as we began this project.

By the fall of 1986, we felt that we had assembled cogent, convincing,

and so far as we knew, uniquely complete analysis of the least-squares

approach to velocity estimation, which explained both its feasibility and

its computational pitfalls. This difficult aspect of least-squares nversion

usuallymanifeststselfasslow or no) convergencef iterativeminimization

algorithms, and still stands in the way of extensiveand reliable application

of the technique.

Between he first circulationof this manuscript November1986) and

the presentwriting (August1988),the volumeof published aperson least-

squares nversionhas perhapsdoubled, and the techniquehas gained much

wider visibility and interest in both the exploration and academic geo-

physicscommunities. Nonetheless,no published analysis has appeared of

the essential ssue--determination of velocity trends from band-limited and

aperture-limited data---of sufficient depth and detail to provide a quanti-

tative understandingof the strengths and limitations of least-squares n-

version, and so to suggestwhat, if anything, could be done to remedy its

deficiencies. We hope that the present manuscript, with its emphasison

simple examples and relevant concepts from computational mathematics,

will go some distance toward filling this gap.

We date the beginning of the work reported here to a conversationwith

Jeff Resnick n the fall of 1984, who pointedout to us that field geophysicists

daily estimate velocity trends from band-limited data. Many of the ideas

developedhere have heir roots n our subsequent tudy of velocityanalysis,

seismic omography,and other topics, both conventionaland experimental,

vii


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fromthe literatureof reflection eismology.

Many of our colleagues ere generous ith their insights nto these

matters:We thank especially en Bube,Guy Canadas, amesCarazzone,

Guy Chavent, rancisCollino, ohnDennis,PierreKolb, PatrickLailly,

Alan Levander, uanMeza,and Paul Sacksor manyhoursof illuminat-

ing conversation n variousaspectsof our work. We owe thanks to Al-

bert Tarantolawhose isionand energyhave nspiredmuchwork on the

seismicnverse roblem.We are gratefulo Enders obinsonor bring-

ing this manuscripto the attentionof the SEG Publications ommittee,

and o theCommittee embersor heirbroad-mindednessn considering

manuscriptriginating o ar outsidehe geophysicalainstream. aymon

Brownwent throughour manuscriptwith great careand madenumerous

usefulsuggestions hich have made this material more readable. We thank

him for his good work.

We beganour workunder he auspicesf the SRO-III project Inverse

problemsf acoustic•ndelastic aves t Cornell niversity,rincipal

investigators.H. Pao and L.E. Payne, undedby the Officeof NavalRe-

searchcontract umber -000-14-83-K0051)uring he period 982-1985.

We are grateful o CharlesHolland, hen programmanager f Applied

Mathematics t ONR, for his encouragement,nd to Professorsao and

Payne or theirguidancendadvice.Our workwas urthersupportedy

the Officeof NavalResearchnder ontract -000-14-85-K0725,ndby

the NationalScience oundation ndergrantsDMS-8403148 nd DMS-

8603614.

The SVD computationsn Chapters through werecarriedout at

theCray-XMP acility t theNavalResearchaboratoryourtesyf ONR.

Many of the othercalculationsereperformed n the Pyramid-90Xmade

available y the Computer cience epartmentt RiceUniversity. ivian

Choiexpertlyeducedeams f illegiblecrawlo thebeautifullyype-set

manuscript sing •TEX.

F. S.

Newark, Delaware

W. S.

Houston, Texas

August 1988

..o

VIII


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Chapter

Introduction

Th e output least-squares approach t o inverse problems in seismology has at-

tracted a great deal of attention in recent years. Also known as least-squares

inversion, nonlinear iterative inversion, generalized linear inversion, and a

variety of other names, it involves a systematic search for an earth model

in some class which best fits some type of seismic data in a least-squares

root-mean-squares, rms) sense. Recent contributors include Bamberger et

al. 1979 and 1982), Tarantola and Valette 1982a and 1982b), Lesselier

1982), Keys 1983), Tarantola 1984 and 1986), Lailly 1984), McAu lay

1985 and 1986), Gauthier et al. 1986), Kolb et al. 198 6), Canadas and

Kolb 1986 ), Mora 1987a and 1987b), Chapman and Orcu tt 1985 ), Shaw

and Orcutt 198 5), and Pan et al. 1988). For a lucid discussion and many

older references consult Lines and Treitel 1984). Further discussion may

be found in a new monograph by Tarantola 1987).

There seems to exist considerable confusion regarding the sort of in-

formation about the subsurface which one might expect to extract using

the least-squares approach to the inverse problem of reflection seismology,

and also regarding the quality of least-squares inversion results relative

to the output of conventional processing methods. Th e relation between

least-squares inversion and m igration of both stacked and unstacked data is

now well understood, at least in principle [see Lailly 198 4), also Tarantola

1984 ), and Beylkin 1985)]. On the other hand, the possibility of reliable

subsurface parameter estimation from realistically band-limited data seems

to arouse various opinions. One reads several compelling argum ents in

the literature that extraction o f velocity trends the main out^of-passband

com ponents of interest in reflection seismology) from band-limited low -

cut) data by least-squares inversion is impossible, or so difficult as to be

infeasible Tarantola, 1986). One also encounters convincing simulations in

1


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2 LEAST-SQ UARES INVERSION

which suchband-extrapolation s accomplishedKolb et al., 1986;Canadas

and Kolb, 1986; McAulay, 1985 and 1986).

The purpose of the present work is to describe the circumstances under

which least-squares nversion might be expected to succeed n a reasonably

accurate recovery particularly of out-of-band components)of a layered

velocity profile from an idealized band-limited common-shot gather, and

also to explain those factors which make this approach computationally

difficult. These aspects of the least-squaresapproach may be understood

from consideration of very simple examples, and we shall devote most of

our attention to these, leaving the mathematically involved general case to

appendices.

We concentrateon the layered acousticor SH-wave) velocity model

because the ideas are most clearly expressed and illustrated numerically

in that context, and because,at present, rigorous mathematical backup is

available for that case only. We do so in full consciousnesshat the model

studied here is so simple as to rule out immediate application to field data

processing.

Our purpose is didactic and limited: to explore the mathematical ca-

pabilities and limitations of the output least-squaresapproach in a simple

and revealing context in which some of the most important features of the

reflection seismic experiment are modeled. Consequently our arguments

will be simple and our examples almost toy-like. However, we emphasize

that the conclusions eached regarding this simple model have direct impli-

cationsabout more seriousmodelsof seismic xploration e.g., nonlayered

elasticmedia). Natural conjectures ill emergeconcerninghesemoregen-

eral and realistic models, some of which have already been explored in the

references cited above. The reader may also consult the reference list for

applications of the least-squaresapproach to field data.

We make no claim, explicit or implied, that least-squaresor any other

sort of) inversion s useful. Such a claim would necessarily est on two

propositions: that the mechanical model underlying the inverse problem

adequately represents he propagation of seismic waves, and that the re-

lation between mechanical parameters and lithology is sufficiently unam-

biguous to yield geologicallymeaningful conclusions.We offer no opinions

concerning either proposition, noting merely that the former is an active

subject of discussion n the literature, and that the latter is essential to

the practice of seismologyper se. Note also that the subject of our work -

the feasibility/reliability of data-derived model parameters- is basic o the

resolution of both issues.

The backbone of our analysis of least-squares nversion is fbrmed of

familiar ideas, which underlie the everyday practice of exploration seisinol-

ogy. We devote the remainder of this introduction to an overview of these


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1. INTRODUCTION 3

ideas, and their less widely familiar computational consequences, nd to a

description of our principal conclusions.We end the chapter with a sketch

of the organization of our book.

The most important and influential insight into the relation between

earth models and seismic reflection data comes from the Wentzel-Kramers-

Brioullin-JeffreysWKBJ) (or geometric ptics/acoustics,r high-frequency

asymptotics) nalysisof the perturbationalseismogramor Born approx-

imation, or linearized orward map). See Clayton and Stolt (1981), for

example. This analysis also lies at the heart of the confusionover the effec-

tivenessof the least-squares pproach n extracting out-of-band information

about earth models, most especiallyvelocity trends. In fact, according o

this analysis, it should not be possible to infer out-of-band components

of the model perturbation at all: such components are filtered out in the

seismogram.

A convenientand precisedescription of the filtering action of the seis-

mogram is provided by the languageof numerical linear algebra which we

shall use throughout. Small changes n the seismogram re linearly related

to small changes n the velocity, to good approximation. After suitable

parameterization of the perturbations, any such inear relation is expressed

by a matrix A. The normalrnatrizATA is symmetric, enceadmitsa

principal-axesnalysis. he square-rootsf theeigenvaluesf ATA are he

singular values of A, and are called collectively the singular spectrum. A

smallsingular alue r, then,correspondso an eigenvectorof ATA, which

yieldsa (relatively) small result when multipliedby A. The lengthof Az

is just •r times the length of z.

Thus we can restate the apparent result of the WKBJ analysis of the

perturbational seismogram: out-of-band componentscorrespond o small

singularvaluesof the perturbational velocity-to-seismogramelation, hence

have little influenceon the seismogram.Conversely, uchout-of-band com-

ponents of the velocity perturbation cannot be estimated reliably from the

seismogramperturbation.

By extension, an iterative solution method for the nonlinear least-squares

problem which relies on repeated solution of the linearized problem should

not be able to update the out-of-band components.

One must remain uneasy, despite the compelling nature of this argu-

ment, because t is common practice in exploration geophysicso estimate

velocity trends from band-limited reflection data. Of course many veloc-

ity analysismethodsand morerecentlyseismic eflection omography e.g.

Bube et al., 1985) rely on traveltimepicks;nonetheless,heseare inherent

in the data, so should somehowplay a role in least-squares nversion, which

purports to make use of the entire seismic record.

The key to the paradox is the assumption, absolutely essential in the


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WKBJ perturbational analysis, that the reference medium, about which

the seismogram s expanded to first order, is slowly varying. A reasonable

model earth velocity profile, or sonic log, is generally not slowly varying:

it contains many reflectors, .e. thin zones of rapid variation in velocities

and other mechanical arameters hopefully) ocatedat geologicallyignif-

icant interfaces. A perusal of the cited referenceson least-squares nversion

reveals that in every caseof successful and-extrapolation into the low fre-

quency regime, the targel profile contained a fairly dense set of reflectors.

Thus for these problems, the WKBJ analysis does not apply at the solution.

We will show that the singular spectrum of the perturbational seismo-

gram about a sufficiently rough referencevelocity profile has a completely

different character than that about a smooth profile. Provided that reflec-

tors are sufficiently dense, a modest part of the precritical perturbational

seismogramsuffices o determine velocity trend perturbations which there-

fore do not correspond o very small singular values. This result, which is

already evident from analysis of a simple, single-layer example, stands in

complete contrast to the slowly varying background situation.

In fact, velocity trend perturbations correspond o rather large singular

values, or a sufficiently ough background. This is understandable,as trend

perturbations give traveltime perturbations, i.e., time shifts, which have a

drastic effect on high-frequencycomponents.This coupling between bands

is at the heart of velocityspectrumanalysis Taner and Koehler, 1969),

and is responsible for both the successand the computational difficulty

encountered in least-squares nversion.

Recall that the eigenvectors orresponding o large singular values are

directionsin modelspace) n which he seismogramhangesapidly,so hat

the graph of the mean-squareseismogramerror is very steep in these direc-

tions. On the other hand, many smallersingularvaluesexist, corresponding

to directions in which the mean square seismogramerror changesslowly.

(A linear map, suchas the linearizedseismogram, avinga very largerange

of singularvalues, s called ill-conditioned).Thus the graph of the mean-

square error has the shape of a long, narrow valley near the solution. The

bottom of this valley is curved, moreover, reflecting the nonlinear nature

of the model-seismogram.Finally, the range of values encounteredby the

mean-squareerror is far smaller over a reasonable ange of models, than is

predicted by the quadratic with principal curvaturesgiven by the singular

values. Thus the mean-squareerror must diverge from its quadratic ap-

proximation, quite rapidly in the steep directionscorrespondingo large

singularvalues/velocity rend perturbations,and fiatten out.

This Grand Canyon shape narrow valley surrounded by undulat-

ing plateau of the mean-squareseismogramerror greatly reduces the

efficiencyof gradient-based terative methods: the iterates tend to zig-


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1. INTRODUCTION 5

zag rather than proceeddirectly to the solution.

This zig-zag effect is enormously greater if postcritical energy is in-

cluded in the residual seismogram. Such refracted arrivals are very much

larger than typical reflectionsas are their responseso trend perturbations

(when hey arrive within the same emporaland spatialwindow).Thus the

least-squaresnverseproblemposed n terms of the entire (z - t) seismo-

gram, including refracted arrivals, is very stiff, i.e. has very ill-conditioned

linearization. We believe that this fact explains some of the numerical

difficulties reported in the literature.

We partly overcome his obstacleby redefining he least-squares rob-

lem: we attempt to match only the precritical part of the data. This projec-

tion onto the precritical components s most naturally accomplished n the

p-tau (Radon transform,plane-wavedecomposition, lant-stack)domain,

so we set most of our development n this domain.

We have implemented a quasi-Newton code which solves his precritical

least-squaresproblem. Its behavior conforms to the predictions of the the-

ory. In particular, it convergesor exampleswhich have causeddifficulties

for codesbasedon matching he full seismogram, nd is (sometimes) ble

to extract rather precisevelocity trend information from precritical p-tau

sections.t alsoexhibits he same ype of inefficiencyfailureto converge t

a reasonableate) reported n the above-cited eferencesor other versions

of the output least-squaresapproach.

It is worth emphasizing he importance of velocity trends in determin-

ing the quality of output-least-squares nversion results, quite apart from

their role in algorithmic efficiency.Besides he obvious mportanceof slowly

varying components n determining phase nformation, i.e., time-to-depth

conversion, hey also have a more subtle but profound nfluenceon ampli-

tude information of rapidly varying components, .e., reflectivity.

This effect manifests tself in two generallydifferent ways, depending

on aperture. First, least-squares nversion amplitudes for wide aperture

data are critically dependenton correct velocity trends, for essentially he

same reason that the amplitudes of a conventional CMP stack are criti-

cally dependent n the velocityused n the NMO correction. The intimate

relation between least-squaresnversionand NMO/stack is explained in

AppendixA.) This effectwill be evident n someof the examples iscussed

in Chapter 10, and is also displayed for example in the work of Ikelle et

al. (1988), which ncludes n exampleshowinghow surprisingly ensitive

are least-squares mplitudes o velocity trends. This sensitivitydisappears,

on the other hand, or inversionrom smallaperture small-or zero-offset)

data, which may account or the almost total lack of attention paid to the

trend-to-reflectivity connection in earlier work on linearized inversion

e.g., Cohenand Bleistein 1979), which was mostly concernedwith CMP


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data. Note that linearized inversion is essentially the solution of the

linearized least-squaresproblem about a reference model usually assumed

to include approximately correct velocity trends (Lailly, 1983 and 1984;

Tarantola, 1984; kelle et al., 1988). Recently, ncreasing ttentionhasbeen

paid to before-stackinearized nversion Clayton and Stolt, 1981;Beylkin,

1985; Parsons,1986; Weglein and Foster, 1986; Bleistein, 1987; Beylkin and

Burridge, 1987 and 1988), largely in an attempt to extract multiparame-

ter estimates, i.e., acoustic or elastic reflectivity models, which in turn is

an attempt to provide a rational basis for amplitude-versus-offsetanalysis

for direct hydrocarbon detection. As has been thoroughly established n

Santosaand Symes 1988b), Clayton and Stolt (1981), and Beylkin and

Burridge 1988), suchmultiparameter stimates re grossly nreliable rom

small-aperturedata. Spratt (1987) has recentlynoted that trend inaccura-

ciesspecifically ausegrossamplitude anomalies n small-apertureRs/Rp

estimation, n the contextof conventional VO (which is an approximation

to linearizedelastic nversion).

To summarize: the interest in amplitude recovery lies mostly in estima-

tion of multiparameter models, especially elasticity. Reliable multiparam-

eter amplitude estimation demands wide data aperture, and therefore also

quite accurate velocity trends. Even though we do not specificallyaddress

multiparameter models in this monograph, this conclusion underlies our

emphasis on the possibilities and difficulties of velocity trend recovery in

least-squares nversion.

A number of limitations on the information content of reflection data

emerge from our analysis. For example, in order for the band-limited data

to contain trend information, it must correspond o a velocity profile con-

taining a sufficiently dense set of reflectors. Intuitively, this condition is

necessary n order that sufficienttraveltime information be present to de-

termine velocitycomponents t spatial wavelengths elow he passband. t

also ollows rom the singularvalue analysis or the perturbational problem,

for which we give both theoretical description and numerical illustration.

Finally, we have given a rigorous mathematical definition of sufficiently

dense set of reflectors, which we review in an appendix.

A second imitation stems from the role of moveout in uniquely speci-

fying the traveltime-depth elation. We show hat a sufficiently arge data

aperture in p-tau (at least 60 percentof the total precriticalrangeat each

depth) is necessaryo ensure he effectivedeterminationof velocity rends.

In particular, the structure of large very low velocity zones may be en-

tirely undetermined. We illustrate this effect numerically. Since no rays

are near turning in suchzones,our restriction o the precritical regime s

not responsible or this indeterminacy.

We emphasize that these limitations (density of reflectors, aperture


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1. INTRODUCTION 7

needed o determinevelocity trends) are proper to the reflectionseismol-

ogy problem over a layered earth--not to our choiceof problem formula-

tion (leastsquares), pecificmodel layeredacoustics, lane-waveesponse),

analysis,or algorithm. They represent undamental limits of the reflection

method, at least insofar as the elastic model is appropriate.

Our book beginswith a definition of the SH-wave model, the plane-wave

decomposition,and the precritical p-tau section in Chapter 2. Chapter 3

introduces the inverseproblem, and posesnatural requirementsof stability

and feasibility to be satisfied by any practical formulation. Chapter 4 in-

troduces the output least-squares formulation, and Chapter 5 reviews the

general features of nonlinear least-squares roblems,and reveals he singu-

lar spectrum of the linearized problem as the key determinant of stability.

Chapters 6 and 7 form the heart of our book. In Chapter 6 we describe

the spectrum of the linearized problem about a homogeneous ackground,

and state a general theorem concerning he singular spectrum of the lin-

earizedproblemabout a slowlyvarying smooth)background.n Chapter

7 we perform a similar analysis for perturbations about a homogeneous

layer over a homogeneous alf-space. The quantitative differencebetween

the perturbation spectra in the smooth and nonsmooth cases s evident in

this very simple example. We also state a general theorem concerning his

point and exhibit singular value decompositionsor a number of examples,

determined by numerical simulation. Chapter 8 outlines the implications of

the spectral analysis or the performanceof least-squaresoptimization, and

includesa discussion f the role of refracted energy, llustrated by singular

value decompositionSVD) of 2-D perturbational seismograms.Chapter

9 describes he implementation of our least-squarescode, and Chapter 10

presents some examples. We restate our conclusions n Chapter 11.

We cover several ancillary points in Appendices. We discuss he close

relation between the output least-squaresormulation and the velocity spec-

trum analysisof Tuner and Koehler (1969) (Appendix A), the relation

between ms error measuresL2 norms) or (x, t) and (p-tan) sectionsap-

pendix B), and give the detailed calculationsof the single-layer xample

(AppendixC). In AppendixD we sketchour generalization f the rough-

reference pectral analysis o a large classof arbitrary media. This is the

only theoretical novelty in our work: it is a technicalextensionof geomet-

ric acoustics. In Appendix E we briefly describe an alternate formulation

of the inverse problem, based on an optimum coherencyprinciple which

is closely related to the scan technique of velocity analysis. This alter-

nate approach voids he stiffness ill-conditioning)-inducednefficiency

inherent n least-squaresnversionwhile producing he samesort of model

estimate. A more extensivediscussion,with examples, s given in Symes

(1988). Finally, in AppendixF, we show hat moveout s entirelyresponsi-


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ble for the estimation of velocity trends. We give an acoustic2-D example in

which two models with identical velocities but completely different density

trends yield virtually identical band-limited common-sourceseismograms.

Of coursedensity rends for fixed velocity)haveno effecton moveout.


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Chapter 2

The Model

We consider a linearly elastic isotropic medium confined to the half-space

{z • 0], subject to prescribed raction on {z - 0]. We introducethe

notation

P

-- positionvector - (Xl, x•2, 3) or -- (•, y, z) interchangeably;

- displacement vector;

- material density;

- Lam• parameters;

- •V. u• + •(V•u• + V•u•) - •u•,•,• + •(u•,• + u•,•)

= (i, j)-component of stress

- sourcewavelet time function).

Then the responseo a horizontallypolarizedshear ine load with wavelet

f, extended n the y- (x2-) direction, s the solutionof the initial boundary

value problem consistingof the equationsof motion

piii- •rij,j z • O, i-1,2,3

j=l

together with boundary conditions

i- 1,2,3

and initial conditions

u-0, t((0.

The field u_ is necessarily ndependentof y, and obeys the equation of


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motion and the boundary and initial conditions

p/•2 - V. ItVu2

•-•(•, O, ) -- •(O)V•,•(•, O, ) -- f(t)•(• -- •o)

u2--O, t


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2. THE MODEL 11

Clearly we can, if we wish, view the seismogram s a functionalof c, rather

than of p:

s[c, , x0l(x, t) := u(x, 0, t; x0).

Second,we introduce he slant-stackp-tau, Radon ransformed)ield.

Formally,

•(p,,) f &00,•(0,,- p. 0;0).

The t-derivative under the integral sign is a technical conveniencewhich

partly offsets he smoothing nfluenceof the z integration. That is, we

transform he SH velocity ield. SeeChapman 1978), Treitel et al. (1982),

and Appendix B for more information.

For reasonsto be explained below, we wish to restrict our attention to

the precritical egionof (p, r)-spacedefined n terms of the vertical ravel-

time

. T(z,) - dC/1

c(½) '

Note that this vertical traveltimediffers rom the traveltimealonga ray at

slownessp, which would be

z 1 .

j•0 Cc(C)N/1ca(C)p

It is in fact the time required for a point on a plane wavefrontat slowness

p with fixed horizontalcoordinate o reach depth z from the surfacez - 0.

For any safetymargin A with 0 < A < 1 we define he (A) precritical

depth function

Za(p)-maxz' for


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12 LEAST-SQUARES INVERSION

finitecableength. n SantosandSymes1988b),weshow hat for a suit-

ably arranged utoff unction /(x,p), the truncated adon ransform f the

surface trace

t•(p,,) f •0•(•0,)0,•(0,,- p. 0,0)

is, up to an error whichdecaysaster han any (negative) owerof fre-

quency,he boundary alueof the solutionof the plane-wave quations

(1

•(•) p• o•t•(•,t;p) o•t•(•,t;p)0, • >0

O•U(O,t;p) = F(t) := f'(t) (2.5)

U _= O, t•O.

We define he plane-wave SH) seismogram ,r p-tau section,as

Sic,y](•, v) = u(0, •; v) =

f a•oO,•[c,, 0](0,v. 0),(•0,), v,) r.

$ is the Radon ransform r slant-stackf the SH-linesource eismogram.

We shalloftensuppress in our notation or $, regardingt as fixed.

One of our principal ools, or both theoretical nd computationalur-

poses,will be the perturbational nalysis f the seismogram. he formal

linearizationdifferentialeismogram,nhomogeneousackgroundornap-

proximation) is given by

where5U solves he perturbationalproblem

(1_p:•)t•62

- -O•SU =

o•u(o, t; p) •

$u •

2•c tu

c 3

0

0, t


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2. THE MODEL 13

Remark. Because f the translational dentity (•.3a),

v(o,;v)f d•O,(•, ,+v-•,o).

This equation allows us to interpret U either as a plane-wave componentof

the responseo the line source ocated t xo -O, or as the response rw of

the medium to the plane-wave sourceboundarycondition

v•,,,,,o•, o, ; r) - •'(t - r-

at x-O:

•(z, t, r) - •(0, z, t; r).

In fact one can even interpret U as the 3-D slant-stack of the point-

source esponse _at vectorslownessp, 0)'

U(z,; ) f f dzoyoOtu•(O,,, pzo,o,o)

where the line load boundary condition above is replaced by a shear point

load at (xo, yo) with the same ime dependence.We shall occasionallyake

advantage of this equivalence,as certain calculations with the Radon trans-

form are easier n 3-D than in •-D (AppendixB).


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Chapter 3

The SH-wave

Problem

Inverse

The most primitive version of the seismic nverse problem, specialized to

the layered,/• = const. SH line-sourcemodel, is the functional equation

Sic]- G. (3.1)

The right hand sideof (3.1) is meant to represent he Radon transform

of the measured line source SH-velocity surface trace, and we demand that

the equationhold at least n the precriticalpart P of the (p, r) domain.

Generally, quation 3.1) is overdetermined,nd will haveno solutions

at all if G is producedby slant-stackinga noisy data set. Data G for which

a solutionc of equation 3.1) exists are called consistent. Generic data

are inconsistent. Therefore we must modify the statement of the inverse

problem.

Before stating alternative formulations, we shouldmake explicit the goal

which any reformulation is to achieve. A modest objective might be the

stability of the solution for near-consistent data. We could reasonably hope

that any reasonable data set is near a consistent data set: otherwise, our

model of the basic physics s erroneous. Thus suppose hat

G=G*+N

where

S[c*] =

for some easonable elocity profile c*, and N is a small noise erm (we

shallbe moreexplicitabout the meaningof small ater). Then we should

15


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consider a restatement of the inverse problem successful f it enables us to

derive from G a velocity estimate c which is close to c*.

A second useful quality of a formulation of the inverse problem would

be that computation of its solution should be easy or relatively inex-

pensive and, at least, feasible).

The above criteria are stated in vague terms. They are nonethelessuse-

ful as guides in developinga theory of the inverseproblem and algorithms

for its solution, and will be made more precise as we see how to meet them.


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Chapter 4

The Output

Least-squares Principle

As mentioned in the introduction, a popular approach to the problem of

inconsistent ata is to seeka model i.e., velocityprofilec) for which he

mean-square error

IlS[c]-11-f fr"'"rls[c](r,) •(r,) (4.1)

is as small as possible. This is the output least-squares ormulation of the

inverseproblem, speciahzed o the precritical p-tau section.

Actually an even more popular approach s to attempt to minimize the

mean-square rror in the full (x, t) seismogram:

ils[c]_[[2Jdx t[[c](x,t)-(x,)I•. (4.2)

The error defined n equation 4.1) is essentiallydentical to the error in

the pointsource eismogramseeAppendixB; the extra "p" in the integral

in (4.1) is not a misprint, ut is the correctweight o bringequation4.1)

and the point source eismogramrror as closeas possible).The relation

of the point-source eismogramrror with the line source eismogramrror

(4.2) is morecomplicated. lso t is possibleo introduceweights"data

covariancematrix") to reflect the presumed tructureof data errors. See

Tarantolaand Valette (1982a and 1982b) or details.

As we will explain, he minimizationof equation 4.1) is easier han the

minimizationof equation 4.2), so we shallconcentratemostlyon equation

(4.1).

17


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Unfortunately, the solution of even this "precritical" output-least-squares

problem is excessivelysensitive to data noise, for several reasons. This is

so even in the "perfect" caseof impulsivedata, i.e., f(t) - 5(t). In that

case, a stable estimate of c requires that highly oscillatory componentsbe

constrained a priori, i.e., independently of the data. Thus resolution must

be sacrificed to obtain stability. We emphasize that this loss of resolution

is intrinsic to the problem: it stems from the nonlinear nature of the rela-

tion between the coefficient and the solution of the wave equation, which

strongly couples he high- and low-frequency egimes,and is present or any

inverseproblem n wavepropagation, n which wavevelocitiesare (among)

the unknowns. For an extensivediscussion ee Symes (1986a). We note

also that this bandcoupling s the essence f velocity analysis, and likewise

will explain most of the features, positive and negative, of least-squares

inversion.

The subject of our present discussion s the inverse problem for band-

limited data, i.e., f(t) y• 5(t). In fact, a commonmodel wavelet , in a

scale reasonable or exploration seismology, s depicted in Figure 1. It is a

so-called ero phaseRicker wavelet scaledsecond erivativeof a Gaussian

pulse)with peak frequency ear 20 Hz. As inspectionof its powerspectrum

(Figure 2) reveals, t has little energycontentabove35 Hz or below4 Hz.

The lack of energy content above 35 Hz causes ittle difficulty in con-

structing approximate solutions o the inverse problems. It simply results

in loss of resolution, which may be compensatedby a priori constraints on

2 i I i i

- 1

o

-1

-2

--3

0.00 0.05

1

0.10 0.15

FIG. 1. Ricker wavelet; peak frequency s approximately 20 Hz.


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4. THE 0 UTP UT LEAST-SQ UARES PRINCIPLE 19

15

10

25

20

_

_

-

-

-

_

_

_

_

_

_

_

o •

o

I i i i i i i i ] i i i

20 40 60

FIG. 2. Power spectrum of Ricker wavelet; peak frequency s approximately 20

Hz.

highly oscillatory components n c, as must already be done to eliminate

the intrinsic instability mentioned above. In this book we will restrict c to

lie in a spaceof functions splines)with suitably imited frequency ontent.

The absence of low-frequency energy in f has much more interesting

(and very well known)consequenceshichare harder to understand han

the effect of the lack of high-frequencyenergy. We begin to explore there

consequencesn Chapter 6. Before doing so, we shall consider n a general

way the factors which influence the stability of solutions of nonlinear least-

squares problems.


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Chapter 5

Generalities on Nonlinear

Least-squares Problems

In this chapter we examine the sensitivity of the least-squares nversion to

data noise or perturbation. We include this well-known material for the

sake of completeness;t has been treated with great care, for instance, in

Linesand Treitel (1984). In the notationof the previous hapters,we want

to determine he effecton a (the) minimumof

1 i:

caused by a perturbation •G in the data G.

We shall use here the standard notation from advanced calculus D

for derivative: as applied to the plane-waveseismogram$ for a reference

velocity c and a perturbation •c,

DS[c]acnm {s[c eat]$[cl}.

•---0 I[

ThusDS[c]•Sc the directional erivative f S at c in the direction 5c )

is exactly the first order perturbation in S due to the perturbation •c in

velocity.DS[c] itself is a linear operator,mappingvelocityperturbations

•c to corresponding erturbations n S.

In Chapter 2, we defined he formal first orderseismogram erturbation

•S as the solution f the perturbational oundary alueproblem 2.6). If

the limit definingD$[c]•5c xists, hen t ought o be true that D$[c]•5c

•S. Circumstances nder which this relation holdsare discussedn Symes

(1986a)and Sanrosa nd Symes 1988b).

21


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22 LEAST-SQ UARES LYVERSION

Now J is a real valued unction of two arguments c and G), hencehas

a gradientwith respect o each. Let ( , )e denotea suitable nner product

for velocity perturbations, possibly ncorporating some weighting. A simple

(but not the only) choice s

(•5Cl,5C2)- dzCl(Z)•C2(Z)

Then the c-gradient is that function gradcJ satisfying he identity: for any

5c,

DeJ[c,]Sc lime_•0[J[c eSc,] J[c, ]]

= (gradcJ[c,G],

(For functionsdependingon more than one argument,we use subscripts

to denote the partial derivatives and gradients with respect to the various

arguments,as above.)

From calculus, f c is a (local) minimum of J, then the c-gradientvan-

ishes:

gradeJ[c,G]- DS[c]*(S[c] G) - 0 (5.1)

and this remains true at the minimum c q-5c corresponding o G q- 5G.

Here DS[c]* is the adjoint operator o DS[c]--see e.g., Tarantola 1984)

and Lailly (1984).

If all small data perturbations 5G correspond to small model perturba-

tions5c, then we are ustified n usinga perturbationexpansionfirst order

Taylor series)of the gradientabout c:

gradcJ[cq- 5c,G + 5G] -

grad•S[c,G] + Degrad•S[c,G]Sc+ Dagrad•S[c,a]aa +...

where the omitted terms are of secondand higher order in the perturbations

5c, 5G. Differentiating again with respect to c defines he Hessian operator

(DcgradcJ[ca])Sc -. HesscJ[c, ]Sc

which is given explicitly as

Uessa[c, ]ec OS[c]*OS[c]ec Sic]

On the other hand, the G-derivative of the gradient is

DagradcJ[c,oleo - -OS[c]*ea.


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5. NONLINEAR LEAST-SQUARES PROBLEMS 23

Thus we obtain the perturbational relation between small tic and tic:

to first order,

HesscJ[c, ltic = DS[cl*tiC. (5.2)

If we assume also that G is consistent data, i.e., that

sty] = c,

then this relation simplifies, and we obtain

=

(5.3a)

which we recognize s the normal equationof the linear least-squares rob-

lem

min IDS[c]acacll (5.3b)

•c

[seeGoluband van Loan (1983, p. 138)].

To summarize: if small tic yields small tic as the solution of the linear

problem 5.2) [or problem 5.3), in the caseof consistentata], then the

linearization is valid and in fact sufficiently small tic will yield small tic

for the nonlinearproblem 5.1) as well. A partial converses alsotrue: if

we can producea solutionof equation 5.2) with tic large relative to tiC,

then we cando the samewith equation 5.1) (thoughboth perturbationsn

generalwill be small). Thus for small perturbations, he relation between

tic and tic is determined y the linear equation 5.2).

Recall that our aim is to assess he stability of the least-squaressolu-

tion for near-consistentata (Chapter 3). A very detaileddescription f the

sensitivityof the seismogramo model perturbations s contained n the sin-

gular valuedecompositionSVD) of the perturbational eismogramS[el.

For an extensive general discussionof this important concept, see Golub

and van Loan (1983, p. 16-20 and p. 174-175). For examples f its use

in the investigation f geophysical roblems,seeBube et al. (1985), Breg-

man et al. (1986), Fawcett 1985), Linesand Treitel (1984), (many older

referencesited there), Nolet (1985) and Van Riel and Berkhout 1985).

Briefly,the singularvaluesof DS[c] are the square-roots f the eigen-

valuesof the symmetric normal ) operator

os[d'os[]

[comparewith equation 5.3a)]. The correspondingigenvectorsre called

(ri9ht) sintlular ectors f DS[c]. The collection f singular alues f DS[c]

is called its sintlular spectrum.


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We have ust seen hat the effect of small data perturbationson the

nonlineareast-squaresolutions governed y the linearnormalequations

(5.3). In general, ecansay hat, for solutionsf equation5.3),

where as usual)II II denoteshe L2-normand O'mi is the leastsingular

valueof DS[c]. Thus 1/•rmi is the worst rrormagnificationactorpossible

between6G and 6c, and we can say with certainty that the criterion of

stability for near-consistentdata is satisfied f amin s not too small. This

leadsus to study in the next two chapters he dependence f t7min n c.

A further consequencef the preceding easonings that for consistent

data (S[c] = G) the shape f the graph of

IIS[c

is determined by the shapeof the quadratic model

IIDS[c]acll

The singularvaluedecompositionSVD) of DS[c] givesexactly he princi-

pal axes of this quadratic. Large singularvaluescorrespondo directions

(principal xes) n which he quadraticmodel 5.5) [hencehe rms error

(5.4), locally] s changing ery rapidly. Similar nterpretation pplies or

smallsingular alues,which orrespondo directions f smallchangesn the

rms error. Variousversions f Newton'smethod the only obvious venue

of solution f the east-squaresnversionroblem, ecausef its size) esult

fromrepeated olution f equation5.2) or equation5.3a). The accuracy

and efficiencyand thus the effectiveness f Newton-like iteration with which

these quationsanbesolved ependsnthedistributionf singular alues

of DS[c]. Moreover,as pointedout in the introductionand discussedmore

thoroughlyn Chapter , the distribution f singular alues f DS[c]at a

minimum f J has mplicationsor the shapeof the graphof J evenoutside

the regionn which t is closeo the quadratic5.5). Thuswewill need o

consider arefully he dependencef the extreme ingular alues max nd

tYmi n on c.


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Chapter 6

Perturbations About A

Slowly Varying Reference

Velocity

The singularspectrumof the normal operator D$[c]*D$[c] is easiest o

analyze in case the referencevelocity c is slowly varying, i.e., smooth. In

fact, this analysis underlies much of the heuristic reasoning n reflection

seismology.

We temporarily restrict our attention to a single plane-wavecomponent,

whichmay as well be the normal ncidence omponentp-- 0), and write

s0[•](t) := s[•](0, t).

For constant reference velocity c ---- const., the derivative of $0 may

be computed in closed form: This is the famous "Born approximation,"

or convolutional model, which forms the basis of much seismic data pro-

cessing.Seefor exampleWaters (1981), Cohen and Bleistein 1979), and

Gray (1980). The following s also a specialcaseof the formuladerived n

Appendix C:

D$o[c]Sc(t)2 dz t 2_•)c(z) (6.1)

-- cF . Sc(t)

where5C(T) - 5C(•-) is the reparameterizationf 5c by two-way ime.

Suppose for example that we take for F the 20 Hz Ricker wavelet of

Figure 1, which has very little energybelow w• - 4 Hz. If 5c is very smooth,

25


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so that it has almostall of its energy nside he band [0, f•] cycles/m, hen

5c has very little energyoutside he [0,cf•/2] Hz band, and the convolution

with F will be very small; provided hat c•/2 _<

IIDo[C]Scll


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6. SLOWLY VARYING REFERENCE VELOCITY 27

3

-2

0.0

I I I I I I I I I I I I

0.5 1.0 1.5

t

FIG. 4. Perturbational seismogramwith background velocity of c = 1500 m/s

and perturbation shown in Figure 3. The vertical scale used is the same as

that in Figure 5.

approximation xpression6.1) shouldbe modified o

D$[c]Sc-dz t- Sr(z)

where

dSc

tir ----

dz

is the reflectivity profile, for comparisonwith the figures.

Nonetheless, we shall stick with the displacement seismogram rather

than the velocity seismogram, or our theoretical discussions.

For comparison, the reference seismogram s shown in Figure 5, and

the impulsive perturbational seismogram n Figure 6. The band-limited

perturbational seismogram s negligible, and a substantial multiple of this

ticcouldbe added o a minimumof equation 4.1) without muchdisturbing

the value of the mean-square residual.

As explained in Chapter 5, the sensitivity question is well-addressedby

the SVD of DSo[c].

We have computed he SVD of DSo restricted o a spaceB/v of (rel-

atively) smooth perturbations ic. B/v is spannedby the cubic B-splines


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• i I i i

o

-1

0.0 0.5 1.0 1.5

t

FIG. 5. Seismogramor c-- 1500 m/s.

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

0.0

i i i i i i i i i

0.5 1.0

t (s)

FIG. 6. Perturbational seismogramwith an impulsivesource. The background

and perturbation are the same as for Figure 4. The "hair • is discretization


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0.8

,.½ o.2 -

0 120 240 360

FIG. 7. The trial space of cubic B-splines, used to õenerate sinõular value de-

compositions.

(DeBoor,1978)depictedn Figure7, eachof which s a translateof a single

representative B-spline of ½:

n=l

zo + (n -- 1)Az.

Of course, in performing numerical computations we must necessarily

restrict D$o to some finite dimensional space. The B-splines provide a

convenient way to generate such a finite-dimensional trial space, as they

are quite localized, with no long oscillatory "tails," but also maximally

smooth among piecewisepolynomial trial functions.

We applied a Ricker wavelet plane-wave traction peaked at 35 Hz, sim-

ilar to that in Figure 1, to the surface of a homogeneous alf-spacewith

constant reference) elocityc = 2500 m/s. Using he finite difference ode

mentionedabove, we computed he perturbationsD$o[c]5cas 5c ranged

over the B-spline basis:

5c,•(z) -- ½(z- z,•), n -- 1,... ,N.

We stored he perturbationDSo[c]Sc,.,s the nth columnof the matrix A.

The calculationof D$o[c] was performedusingthe finite difference ode

described above.


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Remark. In this and other singular value calculations described here, we

defined (and DS) to be he plane wavedisplacement eismogram,ather

than the velocityseismogramin contrast o the other illustrations,and in

commonwith the theoreticaldiscussion).

Thus A was an NT x N matrix, where NT was the number of time

steps. We choseNT (and the spatial step size) to ensureaccuracy n the

finite difference calculation; with a seismogramduration of 0.5 s we chose

N2 • - 1000.

Havingcomputed , we computedhe N x N normalmatrix ATA, as

well as the mass matrix

- f -

of •he splinebasis. We •hen calculated he eigenvalues,•) and eigenvec-

•ors • of the generalized igenvalue roblem

A TAv -- AMv

using he IMSL (InternationalMathematicaland Statistical Library) rou-

tine EIGZS.

Finally, the singularvaluesof DS'o[c], estricted o our splinespace,are

given by

o.•-- X/•, n-- 1,...,N

and the (v,•) are the correspondingright) singularvectors.

In the simulations reported here we have used parameters z0 - 60 m,

Az- 16 m. Unless otherwise noted, we chose N- 15.

The results for the constant reference velocity case are summarized in

Figures8 and 9. Figure 8 displays he singularvaluesof D$o[c] arranged

in increasingorder and plotted against index. These range from o.• =

1.18 • 10 4 to o.1• - 6.53 x 10 3.

Note that the scale of the source wavelet F corresponds o an overall

scaledegreeof freedom n D$o[C], hence n the o.'s. Consequently more

meaningful measure of spectral spread is the condition number

o.n

O'1

For the constant-background xample above, tc- 55.3.

Note that the singular value distribution appears to be an asymmet-

ric rearrangementof the sourcepower spectrum. This is not surprising:

as equation 6.1) shows,D$o[c] amounts o convolutionwith F, which is

diagonalized by the Fourier transform.


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0.015

O.OLO

0.005

0.000

0

5 10 15

index m

FIG. 8. Singularvalue distribution or c ---- 500 m/s.

In Figure 9, we depict the right singularvectorscorrespondingo the

smallest a) and largest b) singularvalues.As expected.rom the Fourier

analysis, heseare (roughly) he least and mostoscillatoryunit vectors n

the trial space of B-splines.

Of course,we have consideredsofar only a singleplane-wavecomponent.

One might hope that the redundancyof the line-source ata set, consisting

of an infinity of plane-wavecomponents,might ameliorate the instability

outlined above. Recall that the plane-wave component U solves

1 ) a2U

_p• u

t= az •

-0

ou

U-0, f


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0.15 ............

- I I

_

_

0.10 --

_

-

-

-

0.05 --

_

_

_

-

0.00

-O.O5

-0.10

-0.15 • • I , ,

0 120

_ _

240 360

(a)

0.15 ............

- I

0.10

0.05

o.oo

-0.05

-O.lO

-0.15

I I I I I I I i

0 120 240 360

z (m)

(b)

FIG. 9. (a) Right singularvector correspondingo smallestsingularvalue; (b)

Right singular vector corresponding o largest singular value. The back-

ground velocity s c = 2500 m/s.


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the plane-wave field V solves

1 •92U

c2

OU

•9z

•9•U

=0

1

If [a•t,•a]s he assbandfF, henx/1- :•p•t,/1- :•p•a]s he

passband f Fp. That is, as p approachesritical slowness,he lowerband-

limit of F is effectively extrapolated toward 0 Hz.

1.0

o.6

o.4

0.•

0.0

0.0 0.2 0.4 0.6 0.8 1.0

normalized slowness

FIG. 10. Extrapolation factor to scaleeffectivesourceband, as functionof cp.

To illustrate the extent of this effect, we display in Figure 10 a plot

of V/1- ca/9versusp. Clearly pmustbe ratherargen order hat

V/1- ca/9 be significantlymallerhan 1. For exampleor cp - .87,

V/1- ca/9 .5 whereasorcp- .98,V/1- c2p - .2. That s, n or-

der to movethe passband oward 0 Hz by a factor of .2, we must probe the

medium with a plane wave at essentially ritical angle.

We exhibit in Figure 11 a plot of the plane wave racesat cp - 0,..., .84

for the example of Figure 4. Plotted on the same scaleas the wavelet, the

perturbation barely showsup in the cp = .84 trace. Even with the vertical

scaleexpanded y a factorof 20 (Figure 12) the responses negligible elow

cp = .84. In fact, a significant ortionof this perturbation Figure3) occurs


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1.0

0.5

.0

0.0 0.6

normalized slowness

FIG. 11. Perturbational wave traces, cp = 0, .12, .24,..., .84. The source, refer-

ence velocity, and velocity perturbation are the same as for Figure 4.

1.0

0.5

0.0

0.0 0.6

normalized slowness

FIG. 12. Same as Figure 11, verticalscaleexpandedby factor of 20.


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8OOO

?ooo

6000

5OOO

4000

3000

2000

lOOO

i i i i i i i i i i i i i i i i

0 2 4 6 8 10

k (l/m)

FIG. 13. Powerspectrum f/ic (Figure3).

at wavelengths f 1500 m or more, correspondingo a temporal frequency

of .5 Hz. Sincehe ower and-limits 5 Hz, V/1- c2p - .1 is required

to produce significant nteraction, which corresponds o cp - .99. See

the powerspectrumof •c displayedn Figure 13. Clearly the information

concerning he low-frequencyvelocity perturbation is minimal, and could

easily be overwhelmedby noise.

So far we have considered nly the perturbational problem about con-

stantbackground.Actually,smoothly aryingbackgroundsieldessentially

the samespectralstructure,as is evident rom the expression

/ [ /0 ]

So[c]•c(t)- dzF(t- 2T(z)) •c(z) + dz'K(z,z')•c(z') (6.3)

T( fodz'

which s a directgeneralizationf the convolutionalormula 6.1). For a

derivation f formula 6.3) seeSanrosa nd Symes 1988b). The correc-

tion term involving he integralkernelK affectsonly the lower requency

components,so i

An Analysis of Least-squares Velocity Inversion

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