Seismic True-Amplitude Imaging

Jorg Schleicher, Martin Tygel, and Peter Hubral

Campinas, January 2006

II

Preface

Any method concerned with the determination of an image of subsurface reflectors from seismicreflections or diffractions is, within the world of seismic exploration, called Seismic Depth Migration.Required for this method is an a priori given reference velocity model, which plays the role of aninitial guess of the actual depth velocity model to be constructed. Depending on the geologicalcomplexity of the Earth, the reference velocity model, generally called the macrovelocity model,needs to be vertically and/or laterally inhomogeneous, elastic isotropic or anisotropic.

If the migration procedure consists only of transforming interpreted and picked traveltimesof selected reflections (like the primary reflections from some sought-for key horizons), the methodis termed Map Migration.

Manipulating the seismic reflections to be migrated with algorithms based upon or derivedfrom the wave equation (assuming any given propagation medium), leads to what has becomeknown as Wave-Equation Migration.

The fact that seismic traces as a whole, and not only interpreted (picked) reflection events,can be used by wave-equation migration methods has substantially contributed to simplify andimprove the seismic imaging and inversion processes, as well as the subsequent interpretation ofthe migrated results. It has also given the amplitudes of the migrated events a certain physicalsignificance, which map migration cannot provide.

In close correspondence to seismic migration, there are a variety of other seismic imagingmethods (e.g., the dip movement (DMO), migration-to-zero-offset (MZO) or redatuming processes,etc.) that also transform one image or section, which may be in the time or depth domain, intoanother. In this sense, any collection of traces (e.g., a constant-offset or time-migrated section) wegenerally call an image. Therefore, it is necessary to refer to the process that has created an image(e.g., a common-offset depth migration, zero-offset time migration, etc.). In this way, we will speakof the migrated image, MZO image, etc. The whole set of imaging methods can be referred to asSeismic Imaging.

In many of the wave-equation migration methods the geometrical simplicity of map migration(which in general involves constructing rays, wavefronts, isochrons or maximum-convexity surfaces)is largely lost. Many geophysicists and seismic interpreters unfortunately have become accustomedto this situation. Some are inclined to believe that good migrated images can only be achieved atthe expense of losing the geometrical insight. This is, however, not true. A kinematic conception,as, for example, proposed by Hagedoorn for migration, can and should be maintained in connectionwith all imaging processes.

III

IV

In fact, the general belief among seismic explorationists until not too long ago was that whilethe ray method is quite valuable in connection with forward seismic modeling (i.e., the constructionof synthetic seismograms for a given earth model), traveltime inversion (i.e., the construction of aninitial macrovelocity model from picked traveltimes), or reflection tomography (i.e., the refinementof an initial macrovelocity model with the help of picked traveltimes of some key primary reflec-tions), it has little to offer in the realm of Seismic Reflection Imaging or wave-equation migration.This situation has now dramatically changed.

The authors of this book have during more than 10 years used the seismic ray method todevelop imaging algorithms. To some extent, what we present here is a didactic reorganization ofour previous works including our up-to-date view of the subject. One of our principal aims is toconfirm that ray theory should no longer be considered a stepchild in the fields of wave-equationmigration and seismic imaging. It is, in fact, a very useful part of these fields, being able to handlethe kinematic (related to reflection traveltimes) and dynamic (related to reflection amplitudes)aspects of wave-equation migration in a geometrically and physically appealing exact way.

Subsurface images in either the time or depth domain can be constructed, as will be shown, ona ray-theoretical basis from specific (e.g., compressional-primary or shear-wave) reflections recordedwith various measurement configurations (e.g., zero-offset, common-offset, common-shot, commonreceiver, or vertical seismic profiling (VSP) measurements). All reflections imaged by seismic mi-gration methods provide, in addition to the subsurface reflector positions, “migrated amplitudes.”In this book, some emphasis will be put on correctly handling the amplitudes of specific elementaryreflections in a 3-D prestack migration. For definiteness, we principally consider primary reflectionsof the P-P type. However, we will also show that the same approach easily handles also waves ofshear or converted type. From the depth-migrated elementary reflections, a quantitative measure ofangle-dependent plane-wave interface-reflection coefficients can be obtained. This is most desirableas it provides the input to the so-called AVO (amplitude-versus-offset) techniques. We know of noother approach than the one based on ray theory, where wavefield amplitudes can be handled inreflection imaging in a similar geometrically easy way. The basically simple principles of ray theoryare equally valid in the presence of an inhomogeneous, 3-D layered earth and for arbitrary measure-ment configurations. The earth and the distribution of the petrophysical parameters is geometrical.Ray theory, which is also a geometrical wave-equation theory, provides an ideal complementation.

Clearly, much of the seismic world restricts the meaning of “wave-equation migration” to“differential wave-equation migration.” We find this use misleading since it implicitly suggests thatray-based migration methods are something entirely different that have nothing to do with thewave equation. However, as we have tried to point out above, ray-theoretical approaches are alsobased on the wave equation and it is proper to refer to them as such.

In principle, the theory presented in this work can be looked upon as a generalization ofHagedoorn’s original ideas, where time-to-depth migration was performed with the help of ei-ther maximum-convexity curves or isochrons. Hagedoorn’s purely kinematic migration conceptsfound, as is well known, already a full wave-equation-based equivalent formulation in what hasbecome known as Kirchhoff Depth Migration. This is a wave-equation migration technique basedupon the weighted summation (or stacking) of seismic trace amplitudes on seismic records alongmeasurement-configuration-specific “diffraction-time curves (surfaces).” These are auxiliary sur-faces constructed using the a priori given macrovelocity model. As shown below, also much moregeneral imaging tasks (e.g., MZO, redatuming, etc.) can be achieved in a similar way. All thesemethods can be generally addressed as Kirchhoff-type imaging procedures. All of them are based on

V

weighted stacking (summing) the seismic data along auxiliary surfaces that are constructed withinthe given macrovelocity model and are specific to the imaging purpose under consideration.

Strictly speaking, the attribute “Kirchhoff” for this depth migration technique is slightlymisleading. In fact, the Kirchhoff (modeling) operator consists of a wavefield forward extrapolationoperator. It is based upon the idea of extrapolating true physical wavefields recorded upon a surfacein direction away from the sources. This is realized as a superposition of Huygens elementary waves.

Depth migration has nothing to do with this forward extrapolation. In the early days of wave-equation migration (restricted to a homogeneous macrovelocity model), the migration operation wasconceived as wavefield extrapolation designed to “propagate” the recorded wavefield backwards intime towards fictitious exploding sources at the reflectors. As this extrapolation was derived upon amodification of the Kirchhoff integral, the new operation was called Kirchhoff migration. R. Bortfeldvehemently rejected the term Kirchhoff migration. His reason was, and we very much share his view,that with the (original) Kirchhoff integral one can only perform a wavefield forward extrapolation.Seismic post-stack migration (of a common-midpoint (CMP) stack or NMO/DMO/stack section,which is assumed to approximate the response at the earth’s surface of a hypothetical exploding-reflector-model wavefield) is evidently a backward extrapolation of an hypothetical wavefield. It canonly be achieved with an operator which resulted from the Kirchhoff integral after applying a trick.This trick involves the change of the propagation direction of the elementary Huygens waves, whichformally appear in the Kirchhoff integral upon the “measurement surface” surrounding the real orsecondary sources. The Kirchhoff-type wavefield backward extrapolation operator, which resultedfrom applying the trick, has in fact nowadays in the non-geophysical community become known asthe Porter-Bojarski Integral.

Moreover, all seismic records, apart from the common shot-record, do not represent theresponse of one wavefield, as they cannot be described by one single physical experiment. As a con-sequence, we cannot claim that Kirchhoff migration is based on the Kirchhoff (modeling) integral.What we can, however, accept is that Kirchhoff migration can be regarded as a “physical inverse”to Kirchhoff modeling. This is because Kirchhoff migration recovers the Huygens elementary wavesthat are the input to Kirchhoff modeling.

We have to acknowledge, however, that the term “Kirchhoff migration” has now becomecommon use in the seismic community. It is for this reason, and because of the above physicalconsiderations, that we accept the term.

It is to be recalled, however, that before “Kirchhoff migration” was introduced, there existedthe term Diffraction-Stack Migration. This was also very much based upon the original ideas ofHagedoorn (1954), but involved no more than summing the amplitudes of a CMP stack sectionalong maximum convexity curves, i.e., diffraction-time curves or surfaces. No weights were used inthe diffraction-stack migration. No wavefield extrapolation concepts were even necessary, and noexploding reflector model was required to describe the CMP stack section in order to justify thediffraction-stack-migration operation mostly performed in the time domain.

It is, therefore, absolutely legitimate to call a Kirchhoff migration as proposed in this book aWeighted Diffraction-Stack Migration. As we will describe here a Kirchhoff migration that is alsoconcerned with amplitudes, we require weights and descriptions of the reflections in the seismicrecords in terms of solutions of the wave equation. Moreover, we require shots and receivers to bereproducible, i.e., to have identical characteristics, when moved along the measurement surface.

VI

In fact, it was P. Newman who first realized back in 1975 the need to modify the ordinarydiffraction stack in order to handle migration amplitudes correctly and give time-migrated primaryreflections a quantitative, physically well-defined value, which at that time was referred to as a trueamplitude. We understand the attribute “true” in the sense of “faithful” rather than of “in accor-dance with verity.” A “true amplitude” is nothing more than the amplitude of a recorded primaryreflection (e.g., a zero-offset reflection) compensated (i.e., multiplied) by its geometrical-spreadingfactor. Since all depth-migrated seismic reflections aimed at in this book are true-amplitude re-flections, we feel justified to call the proposed wave-equation Kirchhoff migration method also aTrue-Amplitude Migration.

We have to admit that many geophysicists think that a “true amplitude” is the designationfor an “unmanipulated amplitude” of, e.g., a primary reflection as recorded in the field. Some alsoassume that if an amplitude is kept unchanged in a certain seismic process, then the process isa true-amplitude process. In our terminology and that of P. Newman, this is not so. In fact, asthe reader will learn in this book, the construction of a true-amplitude reflection from a reflectionrecorded in the field implies, not only a scaling of the considered reflection amplitude with thegeometrical-spreading factor but also the reconstruction of the analytic source pulse multiplied bythe reflection coefficient. Elementary-wave reflections (like primary reflections) may have sufferedmodifications due to caustics along the ray path between source and receiver. The reason for theresulting source-pulse distortion (and the need for the source-pulse reconstruction) is, as the theoryshows, due to the fact that the geometrical-spreading factor may not necessarily be a real positivebut also a negative or imaginary quantity. In short, migrated true-amplitude reflections providea good and physically well-defined measure for estimating angle-dependent plane-wave reflectioncoefficients that may laterally vary along a curved target reflector.

In this book, we also describe other true-amplitude imaging processes, such as, e.g., a true-amplitude MZO. In these more general cases, the term true-amplitude, loosely meaning that geomet-rical spreadings are accounted for in the best-possible way, require a more precise, problem-specificdefinition. This will be provided in Chapter 1. For example, true-amplitude MZO means that thegeometrical-spreading factor of an input common-offset primary reflection is transformed into thegeometrical-spreading factor that pertains to the corresponding zero-offset reflection obtained afterthe MZO transformation. Note, however, that just like in true-amplitude migration, reflection andtransmission coefficients of primary reflections remain unaltered by any true-amplitude process.

Up-to-date, comprehensive collections of research publications devoted to the subject of thisbook are contained in the volumes 2 and 6 of the Seismic Applications Series, entitled “Am-plitude Preserving Seismic Reflection Imaging”, edited by P. Hubral (1998), and “Seismic TrueAmplitudes”, edited by M. Tygel (2002), both published by Geophysical Press. Another importantcontribution to the subject is also the book “Mathematics of Multidimensional Seismic Migration,Imaging, and Inversion” by N. Bleistein, J.K. Cohen and J.W. Stockwell Jr. (2001).

The proposed theory of true-amplitude migration very much makes part of what scientistsin other disciplines, e.g., non-destructive testing, radar, etc., also may call Reflection Tomography.Each of these important subjects has developed its own specific terminology to describe very similarmethods and results.

In the terminology of S. Goldin (1987a,b, 1990), the theory described below could be alsoformulated with the method of discontinuities. He also uses the concept of a true-amplitude mi-gration. In the terminology of G. Beylkin (1985a,b) and N. Bleistein (1987) the methods proposedhere may also be called Seismic Migration/Inversion.

VII

As we consider ourselves, however, exploration geophysicists at heart we opted for the chosensimple title of this book, giving up any desire to be the most general and universal. We are fullyaware that both terms “true-amplitude” and “migration” are unfortunately terms very specific toour profession. As indicated above, the techniques described in this book resemble in parts thosedeveloped by G. Beylkin, N. Bleistein, R. Bortfeld, S. Goldin, and P. Newman and many others.They summarize our research performed over the last decade. Nevertheless we hope that this bookwill offer sufficient new and compact results of interest to readers that look for conceptionally andgeometrically appealing seismic full-wave equation migration methods in 3-D media. The proposedtrue-amplitude migration and imaging methods provide not only a good understanding of thegeometry involved in the imaging process. They also give the imaged amplitudes a lithologicallysignificant value.

Ray theory is confined to the description of seismic waves in smooth elastic or acoustic mediaseparated by interfaces along which the medium parameters change in a discontinuous manner.Consequently, the migration procedures also expect the real earth to be representable by a mediumof this type. As ray theory nowadays is well developed for more complex media (say anisotropic,absorbing, slightly scattering, etc.) we are convinced that, in principle, the approaches offered herefor an isotropic elastic medium can be extended to all such media in which ray theory offers a gooddescription of the seismic wave-propagation phenomena. The reader, who likes to learn more aboutthe ray method and its use in forward seismic modeling should consult the excellent book “SeismicRay Theory” by Cerveny (Cambridge University Press, 2001).

As already indicated, true-amplitude migration is not the only topic dealt with in this book.There exists a variety of additional seismic imaging procedures, which can all more or less keep theamplitudes well controlled. The first of these procedures to be cited is true-amplitude demigration.Under a given measurement configuration, demigration transforms a true-amplitude depth-migratedimage into its corresponding true-amplitude seismic record. True-amplitude migration and demi-gration provide the building bricks for the unified theory of reflection imaging developed in thisbook. All true-amplitude imaging processes (see Chapter 9) result from chaining or cascading atrue-amplitude migration and demigration. Popular in seismic reflection imaging is the process ofMigration to Zero Offset (MZO) that is closely related to the dip-moveout (DMO) process. TheMZO and DMO processes can also be handled in a true-amplitude manner as can the remigration(velocity continuation or residual migration) and other imaging processes. Remigration or residualmigration involves improving a true-amplitude depth-migrated image by taking a better macrovel-ocity model into account and using the roughly depth-migrated image as an input. Other imagingprocesses that can be treated analogously are shot continuation and true-amplitude redatuming.Shot continuation simulates a displaced common-shot section from a neighboring one. Redatumingrequires changing the seismic traces from one measurement surface to another. Taking into con-sideration that we treat various true-amplitude imaging processes, we therefore consider it fullyjustified to have given this book the title Seismic True-Amplitude Reflection Imaging.

Finally, we stress, once more, that the theory offered in this work relies on the validity ofthe ray-theoretical description of the seismic wave propagation in the media under consideration.In particular, and in conformity with the ray assumptions, it addresses the imaging of selectedelementary (i.e., essentially primary) reflections.

Given the comprehensive scope and versatility of the ray method in the formulation of thetheory of seismic wave propagation, we feel there lies a good future ahead in seismic exploration andreservoir imaging under the present approach. Its consistent use should improve the understanding,

VIII

not only of seismic ray theory, but also of the seismic-reflection imaging problem in general.

Acknowledgment: During the preparations of this book, we have relied on the help of agreat number of colleagues and friends. We are grateful for innumerable scientific discussions andtechnical contributions, including those of (in alphabetical order): Norm Bleistein, Ricardo Biloti,Robert Essenreiter, Alexander Gortz, Sonja Greve, Valeria Grosfeld, Christian Hanitzsch, ZenoHeilmann, Thomas Hertweck, Christoph Jager, Herman Jaramillo, Makky S. Jaya, Frank Liptow,Jurgen Mann, Volker Mayer, Claudia Payne, Mikhail Popov, Rodrigo Portugal, Matthias Riede,Lucio T. Santos, Robert H. Stolt, Kai-Uwe Vieth, Andrea Weiss, Yonghai Zhang.

List of symbols

The symbols and indices used in this book are listed in this chapter. Conventional symbols werechosen whenever possible. More extensive explanations of all indices and symbols can be found inthe text at those places where they appear for the first time. In this list, we restrict ourselves toa short definition and a reference to the page, section, appendix, figure, or equation where moredetailed information can be found.

Variables and symbols

Latin lowercase letters

aS constant source position vector describing the configuration; Section 2.2

aG constant receiver position vector describing the configuration; Section 2.2

c acoustic velocity; page 54

cijkl components of the elastic tensor; Appendix G

c conversion coefficient vector, describes the recorded components of the particle displace-ment on a free surface, page 194, Appendix B

ek coordinate unit vectors of the ray-centered coordinate system q in the qk direction; page75

f [t] seismic source wavelet, source pulse, or source signal; assumed to be reproducible, if morethan one experiments or shots are involved; equation (3.2.8)

fm scalar model parameter; equations (E-3)

g source strength, directional characteristics, radiation pattern; page 72

gm scalar model parameter; equations (E-3)

h half-offset vector; equation (2.2.5)

h elastic polarization vector; Appendix G

hS

polarization vector of the source ray; Appendix G

hG

polarization vector of the receiver ray; Appendix G

href

polarization vector of the reflected ray, Appendix G

hB Beylkin determinant; equation (5.6.15)

i imaginary unit, i =√−1

IX

X LIST OF SYMBOLS

ik coordinate unit vectors of the global Cartesian coordinate system r in the rk direction;page 154

jk coordinate unit vectors of the local Cartesian coordinate system x in the xk direction;page 154; Section 3.2.5

k bulk modulus; Table 3.1

`0 length scale of the inhomogeneities of the medium

`ψ characteristic length of medium; equation (3.2.16)

m(r) prestretch factor; equation (9.1.3)

mD stretch factor of migration; equation (5.3.14a), Section 8.2

m midpoint vector; equation (2.2.8)

n number of transmitting or reflecting interfaces in a system of seismic layers; Section 3.13.1

nI stretch factor of demigration; equation (5.3.14b), equation (9.1.9)

n unit normal vector to the ray;page 75

nM unit normal vector to a (real or hypothetical) interface at depth point M ; Figure 7.3

nR unit normal vector to the reflector at the specular reflection point MR; Figure 7.3

n(x) surface normal in local Cartesian coordinates; equation (3.11.2)

p ray parameter or horizontal slowness; the ray parameter is only defined in laterally homo-geneous media or with respect to some reference direction; Appendices A and B (only)

p acoustic pressure; equation (3.1.3)

p 3-D ray slowness vector; equation (3.4.2)

p0 3-D ray slowness vector of the central ray; Section 3.11.1

pp 3-D slowness vector of a paraxial ray; Section 3.11.1

p(x)p 3-D slowness vector in local Cartesian coordinates; equation (3.11.28)

p(q) 3-D slowness vector in ray-centered coordinates; page 78

pG slowness vector of the receiver ray at the scattering point; Appendix G

pref slowness vector of the incident ray after specular reflection at the scattering point; Ap-pendix G

pT 3-D projection of pp into the tangent plane at P ; first step of the double projection;Section 3.11.5

p 2-D vector, that represents a measure for the 3-D slowness vector of the paraxial ray, pis obtained through double projection of the latter; an index indicates the point where itis taken; equation (3.11.20)

pp 2-D projection of the 3-D slowness vector pp of a paraxial ray into the tangent plane atP0 by a single projection; Section 3.11.5

p0 2-D projected slowness vector of the central ray; Section 3.11.1

p(q) 2-D slowness vector in ray-centered coordinates; observe that p(q) is a paraxial quantitysince for a central ray, always p(q) = 0; Section 3.10.1

p0(q) initial value of p(q); equations (3.10.19)

VARIABLES AND SYMBOLS XI

q 3-D ray-centered coordinate of a point on the paraxial ray; Section 3.9

q 2-D ray-centered coordinate of a point on the paraxial ray; page 80

q0 initial value of q; equations (3.10.19)

r 3-D coordinate vector in global Cartesian coordinates, with index location vector of therespective point; Section 2.1

rS 3-D source coordinate vector; page 55

rG 3-D receiver coordinate vector; page 56

r upper 2-D sub-vector of r; horizontal coordinate vector; with index horizontal locationvector of the respective point, Figure 2.4 and Section 2.1

rS 2-D source coordinate vector; Section 2.2

rG 2-D receiver coordinate vector; Section 2.2

rM depth point coordinate; page 119

rP horizontal coordinates of a generic point P ; Section 2.1

r0 horizontal coordinates of a central point P0; equation (3.11.14)

rR horizontal coordinates of the reflection point MR; Figure 2.4 on page 28

rCT stationary point of configuration transform; page 253

r∗ stationary point of remigration; page 242

s arclength of a ray; equations (3.4.8b) and (3.4.8)

t time variable

t unit tangent vector of the ray; one of the Frenet vectors; equation (3.4.12)

u real elastic particle displacement vector; equation (3.1.1)

v wave propagation velocity of the elementary wave mode under consideration; is specifiedas α, β, or c, a possible index indicates the location at which it is taken; defined inconnection with equation (3.2.5)

vS wave velocity at the source point S

vG wave velocity at the receiver point G

vM wave velocity at an arbitrary depth point M

vR wave velocity at a reflection point MR

v0 wave velocity at the coincident source-receiver point of a normal ray; Section 7.6

v(r, z) input velocity field for remigration; Section 2.4.3

v(ρ, ζ) ouput velocity field for remigration; Section 2.4.3

w multiple migration weight vector; Section 8.4.2

x 3-D local Cartesian coordinate system; x = (x, x3), where the x3-axis points in the di-rection normal to a given surface that passes through its origin; its index indicates thelocation of its origin; Section 3.11.1

x 2-D local Cartesian coordinate system defined in the tangent plane to a given surfacethat passes through its origin; its index indicates the location of its origin; represents ameasure for the distance of the paraxial ray from the central ray in the plane tangent tothe surface; Section 3.11.1

XII LIST OF SYMBOLS

xS 2-D local Cartesian coordinate system in the plane tangent to the measurement surfaceat S; Section 2.2

xG 2-D local Cartesian coordinate system in the plane tangent to the measurement surfaceat G; Section 2.2

xR 2-D local Cartesian coordinate system in the plane tangent to the target reflector at MR;Section 6.2

xM 2-D local Cartesian coordinate system in the plane Ω at an arbitrary depth point M ;Section 3.5.2

z vertical (depth) coordinate; Section 2.1

zR vertical (depth) coordinate of the reflection point MR; Section 8.2

Latin capital letters

A aperture of seismic migration; equation (7.1.4); generally equal to the aperture of theseismic experiment, i.e., the surface in which all end points of the parameter vector ξ liein the seismogram section, thus areas over which data exist; Section 2.2

A˜

upper left 2 × 2 submatrix of a propagator matrix ˆT˜

of a paraxial ray in the vicinity ofa known central ray; describes the dependence of the coordinates of the endpoint of theparaxial ray on those of its initial point; without an index, A

˜refers to the whole primary

reflected ray, with index 0, 1, or 2, it refers to the corresponding ray segment; equation(3.11.37a)

B migration input amplitude; Section 8.2

BCR configuration transform input amplitude; Appendix H

B˜

upper right 2 × 2 submatrix of a propagator matrix ˆT˜

of a paraxial ray in the vicinityof a known central ray; describes the dependence of the coordinates of the endpoint ofthe paraxial ray on its slowness vector at its initial point; without an index, B

˜refers to

the whole primary reflected ray, with index 0, 1, or 2, it refers to the corresponding raysegment; equation (3.11.37b)

Ca reflection (a = r) or transmission (a = t) coefficient; Appendix F

C˜

lower left 2 × 2 submatrix of a propagator matrix ˆT˜

of a paraxial ray in the vicinity of aknown central ray; describes the dependence of the slowness vector at the endpoint of theparaxial ray on the coordinates of its initial point; without an index, C

˜refers to the whole

primary reflected ray, with index 0, 1, or 2, it refers to the corresponding ray segment;equation (3.11.37c)

CMP symbol for a seismic experiment, in which source and receiver are dislocated such thattheir Common MidPoint remains fixed; all rays of the involved ray family are assumed topertain to the paraxial vicinity of the normal ray that emerges at the common midpoint;Section 2.2

CMPO symbol for the CMP experiment if the rays do not belong to the paraxial vicinity of thenormal ray at CMP; the O reminds us that the central ray is now an arbitrary offset ray;Section 2.2

CO symbol for a seismic experiment, in which source and receiver are dislocated such thattheir Common Offset remains fixed; Section 2.2

VARIABLES AND SYMBOLS XIII

CR symbol for a seismic experiment, in which the sources are dislocated along the seismicline and the Common Receiver remains at a fixed location; Section 2.2

CS symbol for a seismic experiment, in which the receivers are dislocated along the seismicline and the Common Source (or Shot) remains at a fixed location; Section 2.2

DG denominator of the free-surface conversion coefficients; equations (B-4)

DR denominator of the elastic, isotropic reflection coefficients; equation (A-3)

D˜

lower right 2 × 2 submatrix of the propagator matrix ˆT˜

describing a paraxial ray in thevicinity of a known central ray; describes the dependence of the slowness vector at theendpoint of the paraxial ray on that at the start point; without index, D

˜refers to the

whole primary reflected ray, with index 0, 1, or 2, it refers to the corresponding raysegment denoted with this index; equation (3.11.37d)

EY Young’s modulus; Table 3.1

E demigration aperture; equation (9.1.1)

F˜

surface curvature matrix; equation (3.11.1)

F [t] analytic source signal assigned to f [t]; equation (3.2.10)

G geophone (or receiver) position; Section 2.2

G receiver position in the (paraxial) vicinity of G; Section 2.2

G˜

transformation matrix from local Cartesian coordinates x to ray-centered coordinates q;Section 3.11.3

G˜

(r) transformation matrix from local Cartesian coordinates x to global Cartesian coordinatesr; Section 3.11.4

G˜

upper left 2 × 2 submatrix of G˜

; Section 3.11.3

G˜

(r) upper left 2 × 2 submatrix of G˜

(r); Section 3.11.4

H˜

transformation matrix from ray-centered coordinates q to global Cartesian coordinates r;Section 3.9.1

H˜CC Hessian matrix of TCC(ξ, r; N ); equation (9.2.18), Appendix I

H˜D diffraction-traveltime Hessian matrix; equation (5.3.8b)

H˜F Fresnel zone matrix; equation (4.5.3)

H˜I Hessian matrix of TI ; equation (5.3.11)

H˜IS Hessian matrix of δIS ; equation (9.1.8)

H˜P projected Fresnel zone matrix; equation (4.5.8)

H˜R reflection-traveltime Hessian matrix; equation (5.3.8a)

H˜

∆ traveltime difference Hessian matrix; equation (5.6.10a)

H˜

Σ Hessian matrix of TΣ(ξ, r); equation (5.3.9)

H˜

Φ Hessian matrix of the migration output Φ; equation (8.3.1)

ˆI˜

4 × 4 unit matrix; equation (3.10.25)

I˜

3 × 3 unit matrix; equation (3.3.11)

I˜

2 × 2 unit matrix; equation (2.2.2)

J ray Jacobian; equation (3.5.3)

XIV LIST OF SYMBOLS

K˜

surface curvature matrix; Appendix C

K˜I curvature matrix of the isochron; equation (5.6.8c)

K˜R curvature matrix of the target reflector; equation (5.6.8c)

K˜

∆ difference of curvature matrices K˜I −K

˜R; equation (5.6.10c)

KDS true-amplitude weight function (or kernel) of Kirchhoff migration in integral (7.1.4); page181, general form in equation (7.2.25)

KIS true-amplitude kernel for Kirchhoff demigration; equation (9.1.23)

KCR true-amplitude kernel for cascaded remigration; equation (9.2.25b)

KCC true-amplitude kernel for cascaded configuration transform; equation (9.2.7b)

KCT true-amplitude kernel for single-step configuration transform, (9.2.22)

KRM true-amplitude kernel for single-step remigration; equation (9.2.31b)

M arbitrary point in the depth domain; Section 2.3

MCT depth point where the isochrons of the input and output configurations in a configurationtransform are tangent; page 253

MI subsurface point on the isochron; Section 2.3

MID point on the isochron z = ZI(r;ND) of ND; page 256

MR reflection point, where the central ray is reflected in keeping with the rules of Snell’s law

MR paraxial reflection point; Section 4.5.3

MRM the dual point in the depth domain of NRM ; page 259

MΣ reflector point; Section 5.2.1

Mp P-wave modulus; Table 3.1 on page 54

M˜

3 × 3 traveltime Hessian matrix in ray-centered coordinates; equation (3.11.24)

M˜

(x) 3 × 3 traveltime Hessian matrix in local Cartesian coordinates; equation (3.11.22)

M˜

2 × 2 submatrix of M˜

; equation (3.11.27)

N arbitrary point in the time-trace domain (the seismic section); Section 2.3

ND point on the output model Huygens surface TD(ξ; M); page 256

NR dual point to MR on the reflection time surface; Section 2.3

NRM time point where the Huygens surfaces of input and output models in a remigration aretangent; page 259

NΓ point on the reflection time surface; Section 5.2.1

N˜

second-derivative (or Hessian) matrices of two-point traveltimes in local Cartesian coor-dinates; meaning of indices as follows: N

˜AB: point source at point A, second derivatives

taken at point B; N˜AB: mixed second derivatives with respect to first the coordinates of

A, then those of B; equations (4.2.26)

NIP point, where the normal ray meets the reflector (normal (ray) incidence point); Section4.6.2

O˜

2 × 2 zero matrix; equation (2.2.2)

P generic (ray) point; Section 3.6.2

VARIABLES AND SYMBOLS XV

P0 initial point of a ray; Section 3.6.2

P1 point on ray 1; Section 4.2.1

P2 point on ray 2; Section 4.2.1

Pw point on a wavefront on ray 2; Section 4.2.1

P paraxial ray point; Section 3.9

PG factor appearing in the elastic free-surface conversion coefficients; equations (B-4)

P˜

dynamic-ray-tracing matrix; equation (3.10.11)

P˜

1 2×2 submatrix of the propagator matrix in notation of Cerveny; describes the dependencyof the slowness vector in ray-centered coordinates at the end point of the paraxial ray,p(q)′, on the coordinates of the initial point, q; equations (3.10.23)

P˜

2 2×2 submatrix of the propagator matrix in notation of Cerveny; describes the dependencyof the slowness vector in ray-centered coordinates at the end point of the paraxial ray,p(q)′, on that at the initial point, p(q); equations (3.10.23)

PV symbol for the principal value of an integral; equation (3.2.11)

Q˜

3× 3 transformation matrix from ray coordinates γ to ray-centered coordinates q; Section3.9.2

Q˜

(r) 3 × 3 transformation matrix from ray coordinates γ to global Cartesian coordinates r;Section 3.5.2

Q˜

ray Jacobian matrix; dynamic-ray-tracing matrix; upper left 2× 2 submatrix of Q˜

; equa-tion (3.9.12)

Q˜

1 2×2 submatrix of the propagator matrix in notation of Cerveny; describes the dependencyof the ray-centered coordinates at the end point of the paraxial ray, q ′, to those of theinitial point, q; equations (3.10.23)

Q˜

2 2×2 submatrix of the propagator matrix in notation of Cerveny; describes the dependencyof the ray-centered coordinates at the end point of the paraxial ray, q ′, on the slownessvector at the initial point, p(q); equations (3.10.23)

Rc reflection coefficient; Section 3.13.1 and Appendix A

Rc reciprocal reflection coefficient at the target reflector; equation (3.13.8)

R˜

configuration rotation matrix; Section 2.2

S source position; Section 2.2

S paraxial source point; Section 2.2

SG factor appearing in the elastic free-surface conversion coefficients; equations (B-4)

ˆS˜

ray-tracing system matrix; equation (3.10.15)

T period of a mono-frequency wave traveling along the ray; equation (4.5.1)

Tk transmission coefficient at interface k; Section 3.13.1 and Appendix A

Tk reciprocal transmission coefficient at interface k; equation (3.13.8)

ˆT˜

local Cartesian surface-to-surface propagator matrix; 4 × 4 matrix of a paraxial ray innotation of Bortfeld (1989), made up of four 2×2 matrices A

˜, B˜

, C˜

and D˜

; describes theassumed linear connection between the parameters of the paraxial ray in the beginningand the end point of the central ray; without an index, it refers to the whole primary

XVI LIST OF SYMBOLS

reflected ray, with index 0, 1 or 2 to the ray segment associated with this index; equation(3.11.38)

U seismic trace, i.e., the recording of the scalar amplitude of the principal component of theparticle displacement as an (analytic) function of time; Section 2.2, equations (3.13.15)and (7.1.1)

U0(ξ) approximate seismic trace amplitude for negligible A; equation (7.1.2b)

U(η, τ) desired simulated seismic record in the output space; equation (9.2.23)

U analytic elastic particle displacement vector; equation (3.2.13)

V an arbitrary volume under investigation; Section 3.6 and Chapter 6

V3 vertical component of the particle velocity; Appendix B

V˜

velocity derivative matrix; equation (3.10.4)

ˆW˜

4 × 1 or 4 × 2 matrix of paraxial ray quantities; equations (3.10.13) and (3.10.16)

X˜

2 × 2 residual matrix; equation (3.11.33)

XP symbol for a seismic experiment, in which source and receiver are dislocated perpendicularto each other, i.e., a cross profile; Section 2.2

XS symbol for a seismic experiment, in which source and receiver are dislocated on perpen-dicular lines, i.e., a cross spread; Section 2.2

Y˜R 2 × 2 auxiliary matrix; equation (5.5.10)

Z˜

spatial Hessian matrix; Appendix C

Z˜CR Hessian matrix of ZCR(ξ, r; M ); Appendix I

Z˜I isochron Hessian matrix; equation (5.3.12a)

Z˜R reflector Hessian matrix; equation (5.3.12b)

Z˜

∆ Hessian matrix difference Z˜I − Z

˜R, i.e., Hessian matrix of difference function ZI − ZR;

equation (5.6.10b)

Z˜

Γ Hessian matrix of depth function ZΓ; equation (5.3.13)

ZO symbol for a seismic experiment, in which source and receiver are coincident, i.e., haveZero Offset, and are dislocated jointly along the seismic line; Section 2.2

Calligraphic capital letters

A amplitude factor describing the accumulated transmission losses along the ray; equation(3.13.7)

B˜

3 × 3 rotation matrix involved in the transformation from local Cartesian coordinates xto global Cartesian coordinates r; equation (3.11.13)

B˜

upper left 2 × 2 submatrix of B˜

; projection matrix from local Cartesian coordinates x tothe global Cartesian coordinates r; equation (C-10)

D˜

2×2 traveltime difference matrix between the Hessian traveltime matrices of the CMPO-diffraction and CMPO experiments; equation (4.6.19)

F general function of six coordinates (r, z, ξ, t) that defines the Huygens and isochron sur-faces; equation (5.3.3)

VARIABLES AND SYMBOLS XVII

G Green’s function, i.e., solution of the wave equation for a point source with a δ-impulseas time signal; equation (6.1.1)

H Hamiltonian of the ray equations; equation (3.4.4)

HT symbol for the Hilbert transformation; equation (3.2.11)

L point-source geometrical-spreading (GS) factor (normalized), also called spherical diver-gence; equation (3.6.15)

N kernel of the anisotropic, elastic Kirchhoff-Helmholtz integral; equation (G-15)

OD depth obliquity factor; equation (5.6.9)

ODS obliquity factor of the diffraction-stack integral; equation (7.2.24)

OF Fresnel obliquity factor; equation (4.6.4)

OKH obliquity factor of the Kirchhoff-Helmholtz integral; equation (6.1.10)

P acoustic amplitude factor; equation (3.2.15)

T traveltime along a ray, frequently also called eikonal; Section 3.2.3

T1 traveltime of the descending ray segment (source ray); Section 4.3.2

T2 traveltime of the ascending ray segment (receiver ray); Section 4.3.2

T0 traveltime of the entire central ray; Section 4.2.3

T01 traveltime of the descending ray segment of the central ray; Section 4.3.2

T02 traveltime of the ascending ray segment of the central ray; Section 4.3.2

TCC ensemble of stacking lines for the cascaded configuration transform; equation (9.2.7a)

TCT stacking line for the single-step configuration transform; equation (9.2.21)

TD diffraction-traveltime function; Section 2.3, Section 5.3

TI 4-D function representing the ensemble of Huygens surfaces for all points MI on theisochron ΣN ; equation (5.3.6)

TR reflection-traveltime function; Section 2.2

T coR common-offset reflection time; Section 2.4.3

T zoR zero-offset reflection time; Section 2.4.3

T∆ traveltime difference TD − TR; equation (7.1.7)

TΣ 4-D function representing the ensemble of Huygens surfaces for all points MΣ on thetarget reflector ΣR; equation (5.3.4)

Tε length of the source wavelet, pulse length: f [t] = 0 ∀ t /∈ (0, Tε); equation (4.5.4)

U (0) zero-order amplitude coefficient; equation (3.2.17)

U (1) first-order amplitude coefficient; equation (3.2.17)

U Generic amplitude factor of the principal component of the wavefield, U = |U |; equation(3.3.26)

U (P ) P-wave amplitude factor of the principal component of the wavefield; equation (3.3.16)

U (S)1 S-wave component amplitude factor; equation (3.3.20)

U (S)2 S-wave component amplitude factor; equation (3.3.20)

XVIII LIST OF SYMBOLS

U (S) S-wave amplitude factor of the principal component of the wavefield; equation (3.3.24)

U vectorial displacement amplitude factor; defined in connection with equation (3.2.5)

Uc

displacement amplitude at a free surface; Appendix B

X amplitude factor after Kirchhoff migration; equation (8.2.8)

ZCR ensemble of stacking lines for cascaded remigration; equation (9.2.25a)

ZI isochron surface function; Section 2.3, Section 5.3

ZM measurement surface function; Section 2.2

ZR(r) target reflector function; Section 2.2

ZR(ρ) remigrated (more accurate) reflector image

ZRM stacking line for single-step remigration; equation (9.2.31a)

Z∆ depth surface difference ZI −ZR; equation (9.1.13)

ZΓ 4-D function representing the ensemble of isochron surfaces for all points NΓ on thereflection-time surface ΓR; equation (5.3.7)

Greek lowercase letters

α P-wave velocity, possesses an index according to position; Table 3.1 and equation (3.3.14)

β S-wave velocity, possesses an index according to position; Table 3.1 and equation (3.3.15)

βP in-plane surface dip angle at a point P ; Section 3.11.4

βR in-plane reflector dip angle at MR

βM in-plane dip angle at M

γ 3-D ray coordinate vector; Section 3.5.2

γ 2-D ray coordinate vector; Section 3.5.2

δjk Kronecker delta; equals one for j = k, else zero

δ(t) Dirac delta function

δIS Phase function of the demigration integral; equation (9.1.6)

δCC Phase function of the cascaded transformation transform; equation (9.2.16)

δCT Phase function of the single-step transformation transform; equation (H-8)

δCR Phase function of the cascaded remigration; equation (9.2.33)

δRM Phase function of the single-step remigration; equation (H-23)

ζ output depth coordinate after remigration; Section 2.4.3

η output configuration parameter after configuration transform; Section 2.4.3

ϑ−k incidence angle at a reflection/transmission point at interface k; Section 3.13.1

ϑ+k scattering angle at a reflection/transmission point at interface k; Section 3.13.1

ϑS ray emergence angle at point S; equation (3.13.6c)

ϑG ray emergence angle at point G; equation (3.13.6c)

VARIABLES AND SYMBOLS XIX

ϑ±M incidence and reflection angles between the ray segments at M and the interface normalnM ; equation (4.6.2b)

ϑ±R incidence and reflection angles between the ray segments at MR and the interface normalnR; Section 5.6

ϑP emergence angle at P ; Section 3.11.3

ϑ0 the angle the normal ray makes with the surface normal at the coincident source-receiverposition S0 = G0; equation (4.6.9)

κ compressibility; Table 3.1 (only)

κ KMAH-Index; this index counts the number of caustics along a ray path, i.e., the numbersof points where the ray tube shrinks to zero; note that a focus point increases the KMAHindex by two, as the ray tube shrinks to zero in two dimensions; an index may indicate aspecific ray segment

λ Lame parameter; Section 3.1

µ Lame parameter, shear modulus; Section 3.1

ν ray variable that increases monotonically along the ray; possible variables are s, T , andσ; equation (3.4.6)

ξ configuration parameter; 2-D parameter, describing the seismic measurement configura-tion; horizontal coordinate vector of the data space; Section 2.2

ξ∗ stationary point of the Kirchhoff migration integral in equation (7.1.4); Figure 7.2, equa-tion (7.1.9)

% density of the ambient medium at the current position

ρ horizontal components of global Cartesian coordinates in the depth domain after remi-gration; Section 2.4.3

σ ray variable that increases monotonically along the ray, “optical length” of the ray; equa-tion (3.4.8a)

τ output time coordinate after configuration transform; Section 2.4.3

ϕ in-plane rotation angle between the projected slowness vector of the central ray p0 andthe x1-axis; Section 3.11.3

ϕx in-plane rotation angle between the x1-axis and the vertical plane; Section 3.11.4

ϕr in-plane rotation angle between the r1-axis and the vertical plane; Section 3.11.4

ω angular frequency

Greek capital letters

ΓR reflection time surface; Figure 2.4

ΓM diffraction time surface; Figure 2.4

ΓCOR common-offset reflection time surface; Section 2.4.3

ΓZOR zero-offset reflection time surface; Section 2.4.3

Γ˜S 2 × 2 configuration matrix, describing the source position subject to the chosen seismic

experiment; equation (2.2.14)

XX LIST OF SYMBOLS

Γ˜G 2 × 2 configuration matrix, describing the receiver position subject to the chosen seismic

experiment; equation (2.2.14)

Γ˜M projection matrix; equation (4.5.16)

Γ˜

3 × 3-matrix,‘Christoffel matrix’; page 62

Θ±R angles the incident and reflected rays at MR make with the vertical axis; Figure 8.5

Θ˜

3 × 3 rotation matrix from ray-centered coordinates q to the local Cartesian coordinatesx; equation (3.11.6)

Θ˜

upper left 2×2 submatrix of Θ˜

; projection matrix from ray-centered coordinates q to thelocal Cartesian coordinates x; equation (3.11.8)

Λ˜

traveltime derivative matrix in local Cartesian coordinates x; equation (4.5.15)

Λ˜

(r) traveltime derivative matrix in global Cartesian coordinates r; equation (5.3.10)

ˆΠ˜

ray-centered 4 × 4 propagator matrix of a paraxial ray in notation of Cerveny (Cervenyand Ravindra, 1971; Cerveny et al., 1977; Cerveny, 1987), it is made up by the 2 × 2-matrices Q

˜1, Q

˜2, P

˜1 and P

˜2; matrices P

˜1,Q

˜1 (plane-wave matrices) are obtained by

dynamic ray tracing along the central ray with the initial conditions of a plane waveat the starting point (P

˜1 = O

˜, Q

˜1 = I

˜); correspondingly, one will get the matrix pair

P˜

1,Q˜

1 (point-source matrices) by dynamic ray tracing along the central ray with theinitial conditions of a point source at the starting point (P

˜2 = I

˜, Q

˜2 = O

˜); equation

(3.10.24)

ΣN isochron surface associated with point N ; page 143, Figure 2.4, and Section 2.3

Σ surface of the volume V ; Section 3.6 and Chapter 6

Σ1 top surface of ray tube; Section 3.6

Σ2 bottom surface of ray tube; Section 3.6

ΣR reflecting interface, target reflector; Figure 2.4 Section 2.4.3

ΣR more accurate target reflector image after remigration; Section 2.4.3

ΣM measurement surface; Figure 2.4

ΥDS output amplitude factor of Kirchhoff (diffraction-stack) migration; equations (7.3.1)

ΥIS output amplitude factor of Kirchhoff (isochron-stack) demigration; equations (9.1.5)

ΥCC output amplitude factor of cascaded configuration transform; Section H.1

ΥCT output amplitude factor of single-step configuration transform; equation (H-9)

ΥCR output amplitude factor of cascaded remigration; Section H.2

ΥRM output amplitude factor of single-step remigration; equation (H-24a)

Φ migration output; equation (7.1.4)

Φb time-dependent migration output; equation (7.1.5)

ΦTA analytic true-amplitude signal, desired result of true-amplitude migration; equation (7.1.3)

Φ0(r) migrated trace amplitude as input for demigration; equation (9.1.3)

Φ multiple migration output vector; equation (8.4.9)

Φ˜

3 × 3 in-plane rotation matrix of the ray coordinates q in direction of the Cartesiancoordinates x; equation (3.11.6)

INDICES AND ACCENTS XXI

Φ˜

2 × 2 in-plane rotation matrix of the ray coordinates q in direction of the Cartesiancoordinates x; upper left 2 × 2 submatrix of Φ

˜; equation (3.11.8)

Φ˜x 3 × 3 in-plane rotation matrix between the x1-axis and the vertical plane; Section 3.11.4

Φ˜r 3 × 3 in-plane rotation matrix between the r1-axis and the vertical plane; Section 3.11.4

Ψ demigration output; equation (9.1.1)

Ψ vector function; equation (E-8); equation (E-18)

Ω plane perpendicular to the ray; Section 3.11.3

Ω0 tangent plane to the considered surface at P ; Section 3.11.1

ΩT tangent plane to the considered surface at P ; Section 3.11.1

ΩR tangent plane to the target reflector at MR; Figure 7.3

ΩM coordinate plane at point M ; its normal halves the angle between the up- and downgo-ing ray segments; thus, ΩM is tangential to the isochron and a possible reflector at M ;Figure 7.3

Other symbols

0 2-D zero vector; Section 3.10.2

0 3-D zero vector; equations (3.2.3)

∇ 2-D differential operator nabla with respect to horizontal global Cartesian coordinates;∇ = (∂/∂r1, ∂/∂r2); an index indicates derivatives are taken with respect to other coor-dinates; Section 3.1

∇ 3-D differential operator nabla; ∇ = (∂/∂r1, ∂/∂r2, ∂/∂r3); the symbols ∇·, ∇×, and ∇

signify the divergence, curl, and gradient operations, respectively; equation (3.1.1)

Indices and accents

In this sublist, we explain those indices that may vary, indicating that the variable they specify isthe same, just taken at a different location or ray segment. Indices that distinguish variables fromother, unrelated variables that use the same letter are not explained here.

Subscripts

i, j, k,

l,m, n

numbering indices, taking on values from 1 to 3 for three-dimensional quantities, 1 or 2for two-dimensional quantities, or 1 to n when numbering the interfaces in a system ofseismic layers

G specifies quantities belonging to the geophone position G

M specifies quantities belonging to an arbitrary depth point M

R specifies quantities belonging to the reflection point MR or its dual point NR

S specifies quantities belonging to the source position S

XXII LIST OF SYMBOLS

0 specifies quantities belonging the one-way normal ray and its paraxial vicinity

1 specifies quantities belonging to the descending ray segment of the central ray and itsparaxial vicinity

2 specifies quantities belonging to the ascending ray segment of the central ray and itsparaxial vicinity

∆ specifies a difference quantity.

Γ specifies quantities belonging to the reflection-time surface ΓR

Σ specifies quantities belonging to the reflector ΣR

Superscripts

s specifies quantities belonging to the scattered field

ref specifies quantities belonging to the reflected field

(q) specifies quantities in ray-centered coordinates

(r) specifies quantities in global Cartesian coordinates

(x) specifies quantities in local Cartesian coordinates

− specifies ray quantities taken at an interface immediately before reflection or transmission,i.e., on the incidence side of the interface; nomenclature in accordance with Ursin (1990)

+ specifies ray quantities taken at an interface immediately after reflection or transmission,i.e., on the outgoing side of the interface

T denotes the transpose of a vector or of a matrix

−1 denotes the inverse of a matrix

−T denotes the inverse of the transpose (transpose of the inverse) of a matrix

Mathematical accents

a (dot over symbol) time derivative: a =da

dt

a (check over symbol) denotes quantities in the Fourier domain

a (tilde above symbol) marks quantities belonging to the output space of a configurationtransform or a remigration

a (bold symbol) 2-D vector

a (bold symbol with hat) 3-D vector

a (bold symbol with check) 3-D vector in the Fourier domain

a∗ (asterisk after vector symbol) marks the stationary points of the migration and demigra-tion integrals

a′ (prime after vector symbol) denotes quantities at the endpoint of a transmitted ray, or atthe reflection point of a reflected ray

a′′ (double prime after vector symbol) denotes quantities at the endpoint of a reflected ray

OPERATIONAL SYMBOLS XXIII

A (small bar over symbol) distinguishes the reciprocal (i.e., energy-normalized) reflectionand transmission coefficients (Rc, T ) from the standard (amplitude-normalized) ones (Rc,T )

A (large bar over symbol) denotes points in the paraxial vicinity of a corresponding pointwithout bar

A˜

(tilde below bold symbol) 2 × 2 matrices

A˜

(tilde below bold symbol with hat) 3 × 3 matrices

ˆA˜

(tilde below bold symbol with double hat) 4 × 4 matrices

A˜∗ (asterisk after matrix symbol) marks propagator matrices that pertain to the reverse ray

Operational symbols

· symbolizes the scalar (or inner) product of two vectors (∇ · a indicates the divergenceoperation), i.e., the sum of the products of the corresponding components of these vectors(in matrix notation: a · b = aTb)

× symbolizes the vector product of two vectors (∇×a indicates the curl operation) or anykind of multiplication at a line break

XXIV LIST OF SYMBOLS

Contents

List of symbols IX

Variables and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX

Latin lowercase letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX

Latin capital letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII

Calligraphic capital letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVI

Greek lowercase letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVIII

Greek capital letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX

Other symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXI

Indices and accents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXI

Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXI

Superscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXII

Mathematical accents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXII

Operational symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXIII

1 Introduction 1

1.1 True-amplitude Kirchhoff migration . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 True-amplitude Kirchhoff demigration . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 True-amplitude Kirchhoff imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Additional remarks on true amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Description of the problem 19

2.1 Earth model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

XXV

XXVI CONTENTS

2.2 Measurement configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Hagedoorn’s imaging surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Mapping versus imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.1 Migration and demigration: mapping . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.2 Generalized Hagedoorn’s imaging surfaces . . . . . . . . . . . . . . . . . . . . 32

2.4.3 Unified approach: mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.4 Seismic reflection imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Zero-order ray theory 53

3.1 Wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Ray ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Homogeneous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.2 Inhomogeneous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.3 Time-harmonic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.4 Time-domain expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.5 Validity conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Eikonal and transport equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.1 Acoustic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.2 Elastodynamic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 Rays as characteristics of the eikonal equation . . . . . . . . . . . . . . . . . . . . . . 65

3.4.1 Slowness vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4.2 Characteristic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Ray fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5.1 Ray coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5.2 Transformation from ray to global Cartesian coordinates . . . . . . . . . . . . 69

3.5.3 Ray Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Solution of the transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6.1 Solution in terms of the ray Jacobian . . . . . . . . . . . . . . . . . . . . . . . 69

3.6.2 Point-source solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

CONTENTS XXVII

3.7 Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.8 Computation of the point source solution . . . . . . . . . . . . . . . . . . . . . . . . 74

3.8.1 Homogeneous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.8.2 Inhomogeneous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.9 Ray-centered coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.9.1 Transformation from ray-centered to global Cartesian coordinates . . . . . . 76

3.9.2 Transformation from ray to ray-centered coordinates . . . . . . . . . . . . . . 77

3.9.3 Ray Jacobian in ray-centered coordinates . . . . . . . . . . . . . . . . . . . . 78

3.9.4 Ray-tracing system in ray-centered coordinates . . . . . . . . . . . . . . . . . 78

3.10 Paraxial and dynamic ray-tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.10.1 Paraxial ray-tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.10.2 Dynamic ray-tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.10.3 Paraxial approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.10.4 Initial conditions for dynamic ray tracing . . . . . . . . . . . . . . . . . . . . 83

3.10.5 Ray-centered propagator matrix ˆΠ˜

. . . . . . . . . . . . . . . . . . . . . . . . 84

3.11 Rays at a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.11.1 Vector representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.11.2 Surface representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.11.3 Transformation from local Cartesian to ray-centered coordinates . . . . . . . 87

3.11.4 Transformation from local to global Cartesian coordinates . . . . . . . . . . . 89

3.11.5 Relationship between the slowness vector representations . . . . . . . . . . . 89

3.11.6 Surface-to-surface propagator matrix ˆT˜

. . . . . . . . . . . . . . . . . . . . . 92

3.12 Rays across an interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.12.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.12.2 Dynamic-ray-tracing matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.12.3 Ray Jacobian across an interface . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.13 Primary reflected wave at the geophone . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.13.1 Ray amplitude at the geophone . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.13.2 Complete transient solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

XXVIII CONTENTS

4 Surface-to-surface paraxial ray theory 103

4.1 Paraxial rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2 Traveltime of a paraxial ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.2.1 Infinitesimal traveltime differences . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2.2 Surface-to-surface propagator matrix . . . . . . . . . . . . . . . . . . . . . . . 109

4.2.3 Paraxial traveltime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.3 Ray-segment decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3.1 Chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3.2 Ray-segment traveltimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.4 Meaning of the propagator submatrices . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.4.1 Propagation from point source to wavefront . . . . . . . . . . . . . . . . . . . 120

4.4.2 Propagation from wavefront to wavefront . . . . . . . . . . . . . . . . . . . . 120

4.5 Fresnel zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.5.2 Time-domain Fresnel zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.5.3 Projected Fresnel zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.5.4 Time-domain projected Fresnel zone . . . . . . . . . . . . . . . . . . . . . . . 131

4.5.5 Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.6 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.6.1 Geometrical-spreading decomposition . . . . . . . . . . . . . . . . . . . . . . 132

4.6.2 Extended NIP-wave theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5 Duality 139

5.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.2 Duality of reflector and reflection-time surface . . . . . . . . . . . . . . . . . . . . . . 141

5.2.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.2.2 One-to-one correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.2.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.3 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

CONTENTS XXIX

5.3.1 Diffraction and isochron surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.3.2 Useful definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.3.3 Expressions in terms of paraxial-ray quantities . . . . . . . . . . . . . . . . . 147

5.4 Duality theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.4.1 First duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.4.2 Second duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.5 Proofs of the duality theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.5.1 First duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.5.2 Second duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.6 Fresnel geometrical-spreading factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.6.1 Curvature duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.6.2 Beylkin determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6 Kirchhoff-Helmholtz theory 159

6.1 Kirchhoff-Helmholtz integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.1.1 Kirchhoff-Helmholtz approximation . . . . . . . . . . . . . . . . . . . . . . . . 164

6.2 Asymptotic evaluation of the KHI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.2.1 Geometrical-spreading decomposition . . . . . . . . . . . . . . . . . . . . . . 170

6.3 Phase shift due to caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7 True-amplitude Kirchhoff migration 175

7.1 True-amplitude migration theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.1.1 Underlying assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.1.2 Diffraction stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.1.3 Evaluation at a stationary point . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.1.4 Evaluation elsewhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.1.5 Evaluation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.2 True-amplitude weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.2.1 Traveltime functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

XXX CONTENTS

7.2.2 Traveltime difference and Hessian matrix . . . . . . . . . . . . . . . . . . . . 186

7.2.3 Geometrical-spreading factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.2.4 Final weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

7.2.5 Alternative expressions for the weight function . . . . . . . . . . . . . . . . . 191

7.3 True-amplitude migration result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

7.4 Comparison with Bleistein’s weight function . . . . . . . . . . . . . . . . . . . . . . . 194

7.5 Free surface, vertical displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.6 Particular configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.6.1 Zero-offset (ZO) configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.6.2 Common-offset (CO) configuration . . . . . . . . . . . . . . . . . . . . . . . . 196

7.6.3 Common-midpoint offset (CMPO) configuration . . . . . . . . . . . . . . . . 196

7.6.4 Common-shot (CS) configuration . . . . . . . . . . . . . . . . . . . . . . . . . 196

7.6.5 Common-receiver (CR) configuration . . . . . . . . . . . . . . . . . . . . . . . 197

7.6.6 Cross-profile (XP) configuration . . . . . . . . . . . . . . . . . . . . . . . . . 197

7.6.7 Cross-spread (XS) configuration . . . . . . . . . . . . . . . . . . . . . . . . . 197

7.7 True-amplitude migration procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

8 Further aspects of Kirchhoff migration 203

8.1 Migration aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

8.1.1 Minimum aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

8.1.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8.2 Pulse distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

8.2.1 Geometrical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

8.2.2 Mathematical derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

8.2.3 Geometrical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8.2.4 Synthetic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

8.3 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

8.3.1 Mathematical derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

8.3.2 Synthetic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

CONTENTS XXXI

8.4 Multiple weights in Kirchhoff imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 224

8.4.1 Multiple diffraction-stack migration . . . . . . . . . . . . . . . . . . . . . . . 225

8.4.2 Three fundamental weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

8.4.3 Synthetic example in 2-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

9 Seismic imaging 239

9.1 Isochron stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

9.1.1 Asymptotic evaluation at the reflection-time surface . . . . . . . . . . . . . . 242

9.1.2 Isochron stack in the vicinity of the reflection-time surface . . . . . . . . . . . 243

9.1.3 Isochron stack elsewhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

9.1.4 True-amplitude kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

9.2 Diffraction- and isochron-stack chaining . . . . . . . . . . . . . . . . . . . . . . . . . 247

9.2.1 Chained solutions of Problem #1 . . . . . . . . . . . . . . . . . . . . . . . . . 248

9.2.2 Chained solutions of Problem #2 . . . . . . . . . . . . . . . . . . . . . . . . . 256

9.2.3 Some general remarks on image transformations . . . . . . . . . . . . . . . . 260

9.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

References 265

Appendices

A Reflection and transmission coefficients 277

A.1 Reflection coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

A.1.1 P-P reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

A.1.2 SV-SV reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

A.1.3 SH-SH reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

A.1.4 P-SV reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

A.1.5 SV-P reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

A.2 Transmission coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

A.2.1 P-P transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

A.2.2 SV-SV transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

XXXII CONTENTS

A.2.3 SH-SH transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

A.2.4 P-SV transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

A.2.5 SV-P transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

B Waves at a free surface 283

B.1 P-waves at a free surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

B.2 S-waves at a free surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

B.2.1 SV-waves at a free surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

B.2.2 SH-Waves at a free surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

B.3 Acoustic waves at a free surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

C Curvature matrices 289

D Relationships to Beylkin’s determinant 293

E Derivation of the scalar elastic Kirchhoff integral 295

E.1 A scalar wave equation for elastic elementary waves . . . . . . . . . . . . . . . . . . 295

E.2 Direct waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

E.3 Transmitted waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

E.4 Reflected waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

F Kirchhoff-Helmholtz Approximation 303

F.1 Plane wave considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

F.2 Local plane-wave approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

G The scalar elastic Kirchhoff-Helmholtz integral 307

G.1 The anisotropic elastic Kirchhoff integral . . . . . . . . . . . . . . . . . . . . . . . . . 307

G.2 Anisotropic Kirchhoff-Helmholtz approximation . . . . . . . . . . . . . . . . . . . . . 309

G.3 The Kirchhoff-Helmholtz integral for an isotropic medium . . . . . . . . . . . . . . . 310

H Evaluation of chained integrals 313

H.1 Cascaded configuration transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

H.2 Cascaded remigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

CONTENTS XXXIII

H.3 Single-stack remigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

I Hessian matrices 321

I.1 Configuration transform Hessian matrix . . . . . . . . . . . . . . . . . . . . . . . . . 321

I.2 Remigration Hessian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

XXXIV CONTENTS

Chapter 1

Introduction

In this book, we present a unified theory of three-dimensional (3-D) true-amplitude seismic reflectionimaging that can be applied to seismic records under general measurement configurations. Theprecise formulation of the true-amplitude concept, which depends on the specific imaging taskunder consideration, will be given below.

The theory relies on the ray-theoretical description of the seismic wave propagation involvedand assumes an a priori given macrovelocity model. This is an initial or reference velocity modelthat incorporates the basic information we have from the subsurface to be imaged.

The imaging theory consists of (a) a weighted true-amplitude diffraction stack to migrate theseismic reflection data from the time-trace domain into the depth domain, and (b) a weighted true-amplitude isochron stack to demigrate the migrated seismic image from the depth domain back intothe time-trace domain. Both the diffraction and isochron stacks are explained in connection withtrue-amplitude migration and demigration. The stacking operations can be cascaded or chained fordifferent measurement configurations, velocity models, or elementary waves to permit a variety oftrue-amplitude image transformations.

Many of the described ideas and results comprise the contents of a series of research articlesthe authors have published over the last few years. It is our aim to provide an updated, hopefullydidactic, tutorial of the subject, so as to have it accessible to a broader audience that wish tounderstand it and, above all, use it.

1.1 True-amplitude Kirchhoff migration

Much of the seismic literature is devoted to the imaging of seismic reflectors under the assumption ofa given macrovelocity model. The most widely investigated imaging processes are seismic prestackand post-stack depth migration. Since the early days of seismic migration, attempts have been madeto formulate depth migration, as well as a variety of other seismic imaging procedures in terms ofwave theory. Each exploration geophysicist familiar with the history of migration has probably hisown view on what were the most significant steps in the development of the theory. In our opinion,a list of important contributions to the subject should include the works of Hagedoorn (1954),Rockwell (1971), Claerbout (1971), Newman (1975), Loewenthal et al. (1976), Hubral (1977), Stolt

1

2 CHAPTER 1. INTRODUCTION

(1978), Schneider (1978), Bleistein and Cohen (1979), Berkhout (1981, 1982), Bortfeld (1982), Ursin(1984), Beylkin (1985a,b), Stolt and Weglein (1985), Lee and Wade (1986), Stolt and Benson (1986),Langenberg (1986), Nolet (1987), Bleistein (1987), Miller et al. (1987), Kanasewich and Phadke(1988), Keho and Beydoun (1988), Wenzel (1988), Larner and Hatton (1990), Goldin (1990), Kiehn(1990), Hubral et al. (1991), Docherty (1991), Bortfeld and Kiehn (1992), Cerveny and Castro(1993), Schleicher et al., (1993a), Hanitzsch (1995), and Sun and Gajewski (1997; 1998). A verycomprehensive list of important papers on seismic migration up to the year 1984 is contained inGardner (1985). A collection of recent articles on true-amplitude reflection imaging can be foundin Hubral (1998) and Tygel (2002).

Because of its fundamental role in seismic processing, migration has enjoyed a wide exposurein the literature. In this sense, we feel that a description and discussion on the various migrationmethods, including their practical advantages and disadvantages is redundant and out of the scopeof this book. For that information, the reader is referred to the classic textbook on seismic migrationof Stolt and Benson (1986). For a survey on the various migration methods in the context of itsapplication on seismic processing, the equally classic book of Yilmaz (1987) or its more recentversion (Yilmaz, 2000) are recommended.

In this book we shall concentrate on the development of a unified theory of true-amplitudeseismic reflection imaging. In this context, the process of Kirchhoff true-amplitude migration playsa fundamental role, thus deserving substantial attention. As our principal focus, we shall addressthe concept of true amplitudes in migration. In particular, we hope to show why this is a verynatural concept and why a true-amplitude migrated section is so useful for practical applications.

Our point of departure is a brief consideration of the overall properties of seismic reflectiondata that are used for imaging purposes. We suppose that the multicoverage field data have beenalready decomposed (sorted) into seismic sections, each of them corresponding to a specific distri-bution (configuration) of source and receiver pairs along the measurement surface. In other words,each seismic section is the result of the “illumination” provided by the specific source-receiverconfiguration involved. Let us consider the problem of obtaining an image of the subsurface fromone particular seismic section. What is then the information in this seismic section that is valu-able for imaging purposes? We believe that all geophysicists involved in reflection seismic imagingwould agree that traveltimes, amplitudes and shapes of some “key primary reflections” (or primaryevents) are the most valuable information from the data. In this way, an acceptable subsurfaceimage should contain a set of “key reflectors” that would “explain” the corresponding observedevents. This means that the traveltimes of the primary reflection events should be compatible withthe depth location of the reflectors and, moreover, the magnitude and shape of these events shouldbe in agreement with the variation of the medium seismic parameters across the reflectors.

The above simple and intuitive description of what should be an acceptable image of thesubsurface contains some fundamental, underlying, and tacitly accepted, assumptions on the earthmodels to be recovered, as well as of wave propagation involved in the seismic reflection method.At first, the earth model to be imaged has a layered structure. It roughly consists of an ensembleof inhomogeneous layers (or blocks) separated by interfaces (reflectors). The inhomogeneous mediawithin the layers are fairly arbitrary (e.g., acoustic, elastic, isotropic, anisotropic, with or withoutattenuation). Moreover, the primary reflections from the interfaces constitute separated events inthe data, each of them being individually characterized by its traveltime, amplitude and shape.

Seismic traveltimes are referred to as the kinematic part of the seismic data. They carry theinformation about the geologic structure of the medium under investigation, i.e., the depth location

1.1. TRUE-AMPLITUDE KIRCHHOFF MIGRATION 3

and shape of the layers and interfaces. Further influences on the traveltimes include those of themeasurement geometry and the medium velocities. Any migration scheme has, as its main task, toproperly position the key reflectors in depth. For this purpose, all that is needed are the primaryreflection traveltimes contained in the data.

Seismic amplitudes, called the dynamic part of the seismic data, carry the information onthe lithology of the geological structures that is contained in the seismic data. In a rather intricateway, they inform about the interface contrasts of the medium parameters. The amplitudes of thereflections that are observed at the receivers result from a combination of factors. Among these,we are only interested in those effects that are the result of wave propagation. Therefore, we willnot discuss any additional amplitude influences due to measurement instrumentation, topographicand surface conditions, etc. For a more detailed treatment of these effects, the reader is referred tothe comprehensive paper of Sheriff (1975).

According to the underlying physical processes, the wave propagation effects can be furthersubdivided. There are direct reductions of the amplitude due to energy loss of the propagating wave.These are caused by attenuation as well as reflections and transmissions at interfaces. There are alsoamplitude effects due to the focusing and defocusing of the wave during propagation. These have apurely geometrical character, accounting for the energy spreading on the wavefront. For instance,for a point source in a 3-D homogeneous medium that produces an expanding wavefront, theamplitude of the wave is inversely proportional to the distance between the observation point andthe source location. An opposite effect would result, for example, from the reflection from an upwardconcave (syncline) reflector. The contracting wavefront would increase the amplitude measured atthe receiver. The effect due to expansion or contraction of the wavefront during propagation isknown as geometrical spreading.

Parameter contrasts at seismic interfaces are generally weak, so that a reflection usuallyresults in a major amplitude loss. This explains why primary reflections, i.e., seismic events thatexperience only one reflection at a subsurface interface, are the most important events in seismicreflection data. For that reason, following the general practice, we will concentrate on the treatmentof primary reflected waves, considering other events in the data as noise. Of principal interest is thenthe reflection coefficient, that is, the factor that describes the reflected fraction of the amplitude ofthe incident wave. This coefficient carries direct information about the interface contrast, which canbe extracted by an amplitude-variations-with-offset or -angle (AVO or AVA) analysis. In such ananalysis, the reflection coefficient is investigated as a function of source-receiver offset or reflectionangle (see, e.g., Castagna and Backus, 1993).

Still under the assumption of weak contrasts in the reflector overburden, the only other majorinfluence on the amplitude of a seismic wave is the geometrical spreading. Thus, if the reflectioncoefficient is to be determined from the seismic amplitudes, this influence has to be accounted foror eliminated. This is the objective of a true-amplitude migration.

A common goal to all migration procedures is to correctly position the seismic reflectorsin depth. This aim has, thus, a kinematic or geometrical character. It is the purpose of a true-amplitude migration to aggregate a dynamic character on the migration output, namely that themigration amplitudes are free from geometrical-spreading losses. In this way, they provide a directmeasure of the reflection coefficient of primary reflections and become a valuable tool for inversion.

The definition of a true-amplitude migration as one that removes the geometrical-spreadingloss from seismic reflection amplitudes is in the tradition of Bortfeld (1982), Newman (1985),

4 CHAPTER 1. INTRODUCTION

Bleistein (1987), Bortfeld and Kiehn (1992), Schleicher et al. (1993a), Sun and Gajewski (1997),and several related papers. As a consequence, true-amplitude migration outputs can be used as ameasure of the local (angle-dependent) reflection coefficients.

Permitting three-dimensional (3-D) smoothly curved subsurface reflectors with locally vary-ing reflection coefficients to fall below a laterally inhomogeneous, layered overburden, we show inChapter 7 that both objectives, i.e., the (time or depth) migration and the estimation of angle-dependent reflection coefficients can be obtained in one step. The principal issue in the attempt torecover angle-dependent reflection coefficients becomes the removal of the geometrical spreading ofthe primary reflections.

For that aim, weighted Kirchhoff-type diffraction-stack procedures are applied to the seis-mic field data. This means that the images are obtained as a result of summations of the inputdata along auxiliary (stacking) surfaces suitably multiplied by weights. The stacking surfaces andweight functions required for the migration process are constructed in an a priori given macrovel-ocity model. The true-amplitude Kirchhoff migration technique discussed here handles wavefieldamplitudes and correctly recovers source pulses in the migrated image. The migrated primary re-flections are free from geometrical-spreading losses. Seismic primary reflections events that havebeen corrected for the geometrical-spreading factor are referred to as true-amplitude events.

The determination of the geometrical-spreading factor of primary reflections using only theirmeasured traveltimes is an important problem of seismic exploration. In the case of a horizontallystratified earth and short offsets, a solution to the problem was given by Newman (1973). Heprovided a simple relationship between the geometrical-spreading factor and the RMS-velocity,since then widely used in true-amplitude processing. Further extensions of Newman’s results aredescribed in Ursin (1978, 1986, 1990) and Hubral (1978). The determination of the geometrical-spreading factor for a zero-offset reflection was presented by Hubral (1983) and Krey (1983) for anearth model of homogeneous isotropic layers separated by smoothly curved interfaces using zero-offset and common midpoint data. Further generalizations to inhomogeneous layers and arbitraryoffsets were provided by Cerveny (1987) and Tygel et al. (1992).

It is, however, to be remarked that it is not adequate to simply remove the geometricalspreading from the input traces first and subsequently apply an unweighted Kirchhoff migration.The reason is that the Kirchhoff migration process automatically introduces a partial geometrical-spreading correction in the migrated output. As we will see in Chapter 7, an unweighted Kirchhoffmigration corrects for the part of the geometrical spreading that is due to the curvature of thereflector, called the Fresnel geometrical-spreading factor in Tygel et al. (1994a). Therefore, a fullgeometrical-spreading correction before migration will introduce an amplitude error that will affecta subsequent AVO analysis.

In spite of the increasing value given to amplitude attributes, current processing techniquesand the subsequent structural interpretation are still mainly based on traveltime measurements.This is easy to understand as traveltimes possess the robustness and stability attributes requiredto the implementation of most seismic data manipulations. However, since the work of Newman(1975), attempts have been made to incorporate amplitudes in inversion schemes. Amplitudes ofprimary reflection arrivals are strongly related to angular dependent reflection coefficients and, ifproperly processed, may be of great interpretational value.

The importance of preserving seismic reflection amplitudes in seismic processing, imaging,and inversion is widely recognized. As a result, amplitude-preserving imaging methods have been

1.1. TRUE-AMPLITUDE KIRCHHOFF MIGRATION 5

developed that encompass a whole spectrum of seismic imaging procedures. One of the aims ofthe efforts to preserve amplitudes is the extraction and inversion of angle-dependent reflectioncoefficients at selected points on a target reflector. The best domain to extract this informationappears to be the seismic image after prestack migration (Beydoun et al., 1993), e.g., from common-offset sections.

That a seismic migration can be done such that the migrated wavefield amplitudes becomea measure of angle-dependent reflection coefficients was previously shown, e.g., by de Bruin etal. (1990) using a wavefield-extrapolation concept. In this connection, also the works of Claytonand Stolt (1981), Cohen et al. (1986), Bleistein et al. (1987), and Goldin (1987) are to be cited.The papers of Bleistein (1987, 1989), Docherty (1991), and Dong et al. (1991) are devoted toa method that combines the acoustic Kirchhoff theory with the WKBJ approximation and themultidimensional stationary-phase method. Cerveny and Castro (1993) show how the weights ofBleistein can be computed using dynamic ray tracing. On the other hand, the developments ofBeylkin (1985a), Miller et al. (1987), and Beylkin and Burridge (1990) based on generalized Radontransform combined with the Born approximation also result in an equivalent migration scheme.Ikelle et al. (1992) show that the latter approach cannot correctly describe far-offset amplitudevariations due to velocity perturbations because of the Born approximation.

In this book, a 3-D dynamic imaging scheme is presented, which can be applied to fairlygeneral source-receiver configurations, including the ones most commonly used in practical dataacquisition. It can image acoustic and elastic monotypic (P-P and S-S) as well as converted (S-Pand P-S) reflections. The method in its present form uses primary reflection arrivals from the datavolume. However, in principle, multiple (monotypic and converted) reflections could also be imaged.

The true-amplitude imaging method is a natural extension of the P-P wave true-amplitudemigration proposed by Hubral et al. (1991) for zero-offset data and the work of Goldin (1992). Thepresent work is entirely based on ray-theoretical considerations and also addresses carefully theproblem of source-pulse recovery in the presence of wavefield caustics. In this way, an expression forthe weight function involved in the diffraction stack integral is obtained that is completely defined inray-theoretical quantities. Moreover, it allows, in most situations, for the existence of caustics alongthe ray paths, as long as these are not located at source and receiver points and at interfaces. Sincethe expressions for the weight functions are derived upon the use of the stationary-phase methodfor the asymptotic evaluation of the stacking integral, there are also some minor limitations on thegeometry of the reflectors.

Let us briefly comment on the question why the extensive use of ray theory in our approach isnot a fundamental restriction, in spite of the known limitations on the validity of the approximationsinvolved. The first argument is the very nature of seismic data, which, as indicated earlier, is, forpractical purposes, interpreted as a superposition of seismic events (mainly primary reflections),each of them characterized by a certain pulse shape, a certain arrival time and a certain amplitude.Under the separation of the total wavefield into individual events, the ray approximation has alreadybeen tacitly assumed.

The second argument is also based on practical observations. From years of experience, ithas been widely recognized that Kirchhoff-type migration methods work well even where the ray-theoretical approximations, upon which their theoretical description is based, should supposedlybreak down. This apparent paradox is based on the common allegation that ray theory, as originatedfrom geometrical optics, is a “high-frequency” approach. As such, imaging methods based on it arenot expected to provide realistic results when applied to “low-frequency,” seismic wavefields. As

6 CHAPTER 1. INTRODUCTION

explained by Bleistein et al. (2001), the applicability conditions of the ray method must contemplatenot only the frequencies involved, but also the so-called typical dimensions of the medium in whichthe wave propagation takes place. In the seismic situation, these are, for instance, the depths andradii of curvature of the reflectors, typically large quantities. The controlling factor is the ratiobetween the spatial wavelength and the smallest of the typical dimensions, a non-dimensionalquantity that has to be small.

The third argument is, in fact an extension of the second one. It helps explain why Kirchhoff-type migration methods may still provide interpretable results even beyond the range of validity ofthe ray-theoretical approximations. The reason is the restriction to high frequencies is “artificially”introduced in order to allow for mathematical proof. There is no implication whatsoever on whetheror not Kirchhoff methods work when the ray-theoretical conditions are invalid but just that it hasnot been proved for those situations. There is an indication why the validity of the concepts mayextend beyond the limits of ray theory. It lies in the fact that the major wavefield effects that arenot described by standard zero-order ray theory are diffraction events, head waves, and distortionsof the source pulse. These carry much less information about the searched-for subsurface structurethan the reflection events that are described by ray theory. These observations should help toexplain why ray-theoretical based procedures frequently work well in “low-frequency” cases.

The description of the seismic data as a superposition of “key reflection events” has animportant implication on the type of subsurface images that are expected to be constructed. Theseare, namely, the reconstruction of the “key reflectors” that originated the corresponding reflections.An acceptable image (migrated output) should, then, fulfill at least two requirements, namely (a)it should contain the reflectors that “explain” the reflection events observed in the data and (b)it should be consistent with the geological knowledge that is appropriate to the subsurface regionunder consideration.

The layered structures, prototypes of the subsurface images to be obtained, can be regardedas effective medium reconstructions to be derived from the seismic data. In these effective earthmodels, certainly much simpler than the actual earth, wave propagation is well described by raytheory, as required by our imaging methods. This provides one more justification that ray theoryis an adequate tool for our procedures.

The effective earth model that results from our imaging process, is assumed to consist of asystem of laterally smoothly varying isotropic elastic layers separated by many unknown smoothlycurved subsurface reflectors along which the reflection coefficients may vary as a function of reflectorposition and incidence angle. Within this earth model the elastodynamic wave equation holds. Theparticle motion of a primary reflection at the receiver location G resulting from a point source atS can be well described in terms of the zero-order approximation of ray theory. Furthermore, it isassumed that rays in the vicinity of the central ray connecting S with G can be well described bymeans of paraxial ray theory. The conditions of validity of ray-theory assumptions are extensivelydiscussed in the works of Ben-Menachem and Beydoun (1985) or Kravtsov and Orlov (1990).

As in any Kirchhoff-type migration scheme, the method uses an initially prescribed mac-rovelocity model, usually a smooth reference model specifically designed for the imaging purposeunder investigation. The macrovelocity model can be either a P- or a S-velocity model, or both,if converted waves are considered. Within the macrovelocity model, ray tracing is certainly jus-tified and used to construct the stacking surfaces and weights along which the actual data willbe summed to produce the migrated output (depth image). The macrovelocity model, namely anauxiliary construction that is required to perform the migration process, is not to be confused with

1.1. TRUE-AMPLITUDE KIRCHHOFF MIGRATION 7

the actual image that results from applying the migration scheme to the data.

The stacking surfaces along which the migration operation is performed are called diffractionor Huygens traveltime surfaces. As explained first in a simple and appealing geometrical way inChapter 2 and later in a rigorous form in Chapter 7, for every depth point M in the macroveloc-ity model, the Huygens traveltime surface is calculated and the weighted sum—called “weighteddiffraction stack” in accordance to Newman (1975, republished 1990)—is performed. The migrationscheme itself decides whether M is a reflection point or not. If it is one, the stacking procedure willyield the value of the reflection coefficient at M , if not, the sum will nearly be zero.

It is to be observed that the particular choice of the weight function does not affect thekinematic properties of Kirchhoff migration. Its stacking (Huygens or diffraction) surfaces are alwaysthe same, independently of whether one wants to realize a simple structural migration of the keyhorizons or a complete true-amplitude migration with full weights. As a consequence, although ourattention is focused on true amplitudes, all kinematic considerations about Kirchhoff migration inthis book remain valid when realizing the diffraction stack without weights.

True-amplitude migration can be considerably simplified by the multiple-weights approachdescribed in Section 8.4. It is shown there that the computation of the true-amplitude weightfunction is needed only for reflection rays. This may reduce the effort to be spent in order to obtaintrue amplitude by a substantial amount (Bleistein, 1987; Tygel et al., 1993).

We close this section with some overall observations on the nature of Kirchhoff migration,bringing about its most intuitive and physically plausible aspects. There are two conceptually dif-ferent approaches to derive the Kirchhoff migration integral and its true-amplitude weight function.One approach is based on a clever modification of the forward modeling Kirchhoff integral, so asto adapt it to the inverse task of migration. The idea is to propagate the reflected field recordedat the receivers back to the reflectors in depth. The approach is restricted to the common-sourceconfiguration, because this is the only situation where all gathered seismic data pertain to oneand the same physical experiment. The input wavefield is emitted by a single source point, thecorresponding reflected response being recorded by several receivers.

In this situation, the backpropagation is realized by means of a trick, namely by replacingin the Kirchhoff integral the retarded Green’s functions by advanced ones (Porter, 1970; Bojarski,1982) which leads to the Porter-Bojarski integral (Langenberg, 1986). In other words, the recordedreflected wave is restarted with Huygens waves at the measurement surface and propagates backinto the medium towards the secondary sources, i.e., towards the reflector. If considered in conjunc-tion with the forward propagated field from the common source and a suitable imaging condition(Claerbout, 1971), the reflector can be imaged in true amplitude (Miller et al., 1987; Bleistein,1987).

There is another approach to image reflected waves, based on the geometrically motivateddiffraction stack, which is valid for almost all arbitrary measurement configurations, the CMPgather excluded. According to Vermeer (2002), only so-called minimal data sets can be migrated(i.e., common-shot, common-receiver, cross-spread, zero-offset, common-offset records, etc.). In thismigration approach, the seismic traces are summed up along the surfaces of maximum convexity(Hagedoorn, 1954) or diffraction-traveltime surfaces, and the obtained value is assigned to therelated diffraction point specified in the given macrovelocity model (Rockwell, 1971). The mathe-matical formulation of this latter procedure in the high-frequency approximation (Schneider, 1978)leads to the weighted “diffraction-stack integral.” True-amplitude weights for this integral were de-

8 CHAPTER 1. INTRODUCTION

rived by Schleicher et al. (1993a). Its result is the image of the subsurface reflector giving a measureof the reflection coefficient at any reflector location.

In this book we try to combine both approaches. We give a physical meaning to the heuris-tic ansatz chosen for the diffraction-stack integral by revealing its relationship to the forwardKirchhoff-Helmholtz integral. As we will see, both integrals, although they are not strictly math-ematical inverse operations, can be understood as being “physically inverse” to each other. Thefact that the Kirchhoff-Helmholtz and diffraction-stack integrals have this relationship might ap-pear to be intuitively obvious. However, taking into account that the Kirchhoff-Helmholtz integralis traditionally only formulated for a shot record whereas the diffraction-stack integral allows foralmost all arbitrary measurement configurations, it still requires a sound mathematical analysis(that involves the application of the stationary phase method) to show their relationship.

Our analysis leads us to a physical and intuitive interpretation of both integrals. For didacticreasons, these will be described for the simple case of a single smooth reflector below a laterallyinhomogeneous overburden. The Kirchhoff-Helmholtz integral can be understood as the superposi-tion of Huygens elementary waves emitted by secondary point sources (called Huygens or diffractionsources) distributed along the reflector. The secondary sources are excited by the incident wave,their intensity being proportional to local plane-wave reflection coefficients.

Each Huygens source would, if exploding on its own, generate seismic energy distributed alongthe “diffraction-traveltime surface” (therefore also called “Huygens surface”) in the seismic recordthat results from the selected measurement configuration. The envelope of these Huygens surfacesis the reflection-time surface. In other words, the two reflector attributes “location” and “reflectioncoefficient” are mapped (or transformed) by way of the Huygens sources onto the recorded reflectionin the seismic record section.

In a reciprocal way, stacking the seismic trace amplitudes in the seismic record section alongthe diffraction-time surface that pertains to a Huygens secondary-source point involves summingup all contributions that come from this particular Huygens wave center. This operation, whichis done by the diffraction-stack integral with certain weights, recovers from the recorded reflectionboth the reflector location and the reflection coefficient, i.e., the two attributes that characterizethe Huygens source. It is in this way that the diffraction-stack integral can be interpreted as beingthe “physical inverse” to the Kirchhoff-Helmholtz integral.

Because of the above observations, the diffraction-stack or Kirchhoff migration integral is of-ten understood, even in a more mathematical, although approximate, sense, as the inverse operationto forward modeling with the classical Kirchhoff integral (Frazer and Sen, 1985). The Kirchhoff in-tegral can be used to propagate a given incident wavefield (e.g., an elementary compressional wave)from the reflector to the receiver point by superposing “Huygens secondary sources.” In the sameway, the Kirchhoff migration integral serves to reconstruct the same Huygens’ secondary sourcesalong the reflector (in position and strength) from the measured elementary-wavefield reflectionsat several receiver positions along the seismic line. Note, however, that the Kirchhoff migrationintegral only inverts the propagation effects of the Kirchhoff integral (Tygel et al., 1994a). Its finalresult is a migrated depth section. To reconstruct the physical model, i.e., the position of the re-flector and the values of the reflection coefficients, an additional process (usually called inversion)is needed. Therefore, from a strict mathematical point of view, the Kirchhoff migration integralcannot be considered an inverse to the Kirchhoff-Helmholtz integral. However, Kirchhoff migrationcan be interpreted as the adjoint operation to Kirchhoff modeling (Tarantola, 1984).

1.2. TRUE-AMPLITUDE KIRCHHOFF DEMIGRATION 9

1.2 True-amplitude Kirchhoff demigration

As discussed by Hubral et al. (1996a) and rigorously shown by Tygel et al. (1996), there existsa mathematical inverse to the Kirchhoff migration integral (in an asymptotic sense). This inversehas the same integral structure as Kirchhoff migration. It is given by a stacking process similar tothat of Kirchhoff migration, which is applied to the depth-migrated section. To better understandthe process, we first recall that the Kirchhoff migration operation is performed on seismic timesections by means of weighted stacks to produce a depth-migrated section, that is, the image.The stacking surfaces are the diffraction-traveltime surfaces, which are constructed in a givenmacrovelocity model without the need to determine (nor to identify) the location of the reflectiontraveltime surfaces in the seismic section. In Chapter 9, we prove that, in a completely analogousand complementary way, the inverse operation, called “demigration,” is given by a similar weightedstack performed on the depth-migrated image along related surfaces. These are also constructedon the given macrovelocity model without knowing the location of the reflectors in the migratedsection.

The stacking surfaces are simply the surfaces of equal reflection traveltime between a givensource and a given receiver, called “isochrons.” These isochrons (ellipsoids in the constant-velocitycase) are defined by the same traveltimes as the Kirchhoff-type diffraction traveltime surfaces

(hyperboloids in the constant-velocity case) that define the stacking surfaces for migration. Theseare the traveltimes from all source receiver pairs on the measurement surface to all points in thedepth region that is being imaged. As explained first in a simple and appealing geometrical way inChapter 2 and later in a rigorous form in Chapter 9, for every point N in the time section to besimulated, the isochron is calculated and the weighted sum – called “weighted isochron stack” inaccordance to the corresponding migration process – is performed. The demigration scheme itselfdecides whether N is a point on the reflection traveltime surface or not. If it is one, the stackingprocedure will yield the value of the reflection coefficient divided by the geometrical-spreading factorat N , if not, the sum will nearly be zero. Thus, because of its fundamental similarity to Kirchhoffmigration, the inverse process can be called an isochron stack or simply Kirchhoff demigration.

The concept of true amplitude in connection with Kirchhoff demigration follows immediatelyfrom its definition for Kirchhoff migration. Since demigration is the inverse process to migration,it has to undo whatever a previous migration process has done to the data. This implies that atrue-amplitude demigration must move the key reflectors back to the primary reflection traveltimesurfaces. Moreover, it must reintroduce the geometrical spreading to the data amplitudes that hasbeen removed by a true-amplitude migration.

The weights in the isochron stack can be tailored to achieve this desired true-amplitude be-havior, i.e., to reconstruct the correct elementary reflections in the time-trace domain, from whichthe depth-migrated image was originally obtained by a weighted diffraction stack. The macro-velocity model, the measurement configuration, and the selected wave mode for the ray tracingconstructions need, of course, to be the same in both steps in order to enable a complete recovery.In this case, the two operations can be shown to be asymptotic inverses to each other. If one ormore of these attributes is changed before the demigration is carried out, a transformed seismicsection will be obtained. These image transformations are the topic of the second part of Chapter 9.

A particularly attractive feature of the described demigration operation is that any soft-ware developed for true-amplitude Kirchhoff migration can be easily modified to perform a true-amplitude demigration. In fact, the structural similarity of the migration and demigration concepts

10 CHAPTER 1. INTRODUCTION

constitutes a significant part of the unified approach to seismic reflection imaging pursued in thisbook.

Although being a relatively recent development, seismic demigration is not a process unknownto the seismic world (Whitcombe, 1991; Kaculini, 1994). It has already found several different typesof practical applications.

One of the first seismic methods to be suggested that are based on the cascaded applicationof migration and demigration is the so-called “seismic migration aided reflection tomography” orbriefly SMART (Faye and Jeannot, 1986; Lailly and Sinoquet, 1996). Here, seismic reflection dataare migrated to depth using a simple, albeit probably wrong, macrovelocity model. In the depthdomain, the migrated primary reflection events, although not correctly located, are usually morecoherent and can, thus, be better identified and picked. The resulting picked “reflector maps”are then kinematically demigrated back into the time-trace domain using the same macrovelocitymodel. The demigrated reflector images can then help to identify and pick the traveltime surfacesin the original seismic data. A similar concept was independently described by Fagin (1994).

The same cascade of migration and demigration is also used in a non-layer-stripping approachfor depth-conversion purposes. As described by Whitcombe (1991; 1994), the combination of dem-igration with single-step ray migration can be used to improve a layered macrovelocity model. Inthis procedure, demigration is used to back out the effect of time migration prior to a ray-baseddepth migration (i.e., map migration) and to enable the lateral shift between the time migratedimage and a depth-migrated image. The needed velocity model is obtained from a conventionalvertical depth conversion of time-migrated data.

Another important field, where demigration has already found a practical application isvelocity analysis (Ferber, 1994). The procedure is similar to a conventional migration velocityanalysis. Conventionally, image gathers are formed after prestack depth migration in the migrateddomain. Of course, all so-obtained migrated seismic image gathers to be compared depend on themacrovelocity model, which will be generally incorrect. Thus, an interpretation of the image gathersmay be difficult. Demigration can be used to avoid this problem by enabling a comparison directlybetween seismic time sections. All that has to be done is to demigrate the migrated sections obtainedfrom different common-offset sections using the original macrovelocity model. However, instead ofdemigrating them back to their original offset, demigration is applied to all of them using a given,fixed offset that was actually used in the data acquisition geometry. After demigration, all so-obtained sections can be compared with a real common-offset section that was actually measuredin the field. The advantage is that the latter obviously does not depend on the macrovelocitymodel. Of course, if the macrovelocity model was correct, all constructed offset sections should beidentical to each other and to the section actually measured. Deviations between the constructedand measured sections can therefore be directly attributed to errors in the macrovelocity model.These deviations can be determined, interpreted, and used to update the velocity model in thesame way as in migration velocity analysis.

Moreover, the processing sequence of migration and demigration has the potential of beingutilized in data regularization. Seismic reflection data that were acquired on an irregular grid canbe migrated to depth (using a macrovelocity model as accurate as possible) and then demigratedwith the same model back into the time-trace domain onto a regular grid. Although expensive, thisis the best data interpolation (and even extrapolation) technique as it correctly accounts for thepropagation effects in the reflector overburden.

1.2. TRUE-AMPLITUDE KIRCHHOFF DEMIGRATION 11

Although the two integrals describing Kirchhoff forward modeling and Kirchhoff demigrationboth appear to be inverses to Kirchhoff migration in an approximate sense, they do not exactlycoincide. Their relationship was recently investigated by Jaramillo and Bleistein (1997). Consideringonly the leading order contributions, they have shown that the Kirchhoff modeling integral can bemodified in such a way that the Kirchhoff demigration integral results. As the main contributionsto the integration stem from the specular reflection point, this modification should not cause majordifferences. We may, thus, interpret the demigration integral as a “reorganized Kirchhoff modelingintegral,” which should give very similar results. The physical interpretation of this new integral is,however, different. Unlike the Huygens’ secondary source contributions in the Kirchhoff integral, itis now the individual Fresnel zone contributions to each primary reflection that are summed up bythe integration (Schleicher et al., 1997b).

The close relationship of the Kirchhoff demigration and modeling integrals, however, impliesthat it should be possible to use Kirchhoff demigration to achieve the goals of forward Kirchhoffmodeling. This has been shown by Santos et al. (2000). The missing fourth operation, the asymptoticinverse to the Kirchhoff-Helmholtz integral, is described in Tygel et al. (2000).

We stress that, in a very strict sense, demigration is conceptually different from modeling.Modeling, as we understand it, implies the analytical or numerical simulation of a physical processgiven all the equations and parameters for its complete description. In our case, the physical processto be simulated is seismic wave propagation. It is described, e.g., by the elastic or acoustic waveequation and the parameters are the velocity and density distributions within the medium, thesource and receiver locations, and the source wavelet together with appropriate boundary andinitial conditions. Seismic modeling is, then, realized by an implementation of the wave equation(e.g., using finite differences or the Born or Kirchhoff representation integrals) or its approximatesolutions (like asymptotic ray theory) to obtain a synthetic-seismogram equivalent of the seismicdata that would have been recorded if the same experiment had been actually carried out in thefield. For the meaningful case of a layered Earth model, we need, in particular, the precise locationand description of the interfaces, as well as the appropriate boundary conditions on them.

Demigration, on the other hand, although it envisages to provide very similar results, uses aconceptually different approach. Its aim is to reconstruct a seismic time section from a correspondingdepth-migrated section. In other words, demigration aims to invert the imaging process of migration.Of course, as migration aims at inverting the wave propagation effects, it is related in some wayto the wave equation. Correspondingly, also demigration, as the inverse process to migration, musthave some relationship to that equation. As opposed to direct forward modeling, however, we donot have to implement or even know this wave equation. Moreover, we do not have to preciselyknow all the true model parameters to actually perform the demigration process. Neither the truevelocity distribution in the earth, nor the source wavelet nor, above all, the position of the reflectinginterfaces have to be known in order to apply a demigration. All that is needed, apart from theseismic depth-migrated image section to be demigrated, is the macrovelocity model that has beenused for the migration process which produced this section. In fact, a table with all the Green’sfunctions as used in migration (i.e., from all sources and receivers to all subsurface points on anappropriate grid) would already be sufficient. Of course, the better the macrovelocity model is, thebetter will be the corresponding migrated section. This is, however, a problem of migration andnot of demigration. Even if the velocity model used for the original migration is erroneous, so thatthe depth-migrated image is wrong, a subsequent demigration is expected to correctly reconstructthe original time section. The only condition is that the same macrovelocity model has to beused for demigration as previously used for migration. In other words, the chain of migration and

12 CHAPTER 1. INTRODUCTION

demigration is a process that recovers the original time-domain data (in a high-frequency sense)with little sensitivity to the macrovelocity model.

It is to be observed that, in analogy to Kirchhoff migration, the particular choice of the weightfunction does not affect the kinematic properties of Kirchhoff demigration. Its stacking surfaces arealways the same, independently of whether one wants to realize a simple structural demigrationwith no weights or a complete true-amplitude demigration with full weights.

1.3 True-amplitude Kirchhoff imaging

The technological know-how exists already today to depth migrate a seismic record for a specificmeasurement configuration (e.g., a common-shot or common-offset section), a specific ray code(compressional or shear-wave, primary or multiple), and/or macrovelocity model, as well as todemigrate it with another configuration, ray code, and/or macrovelocity model. Various theoriesand numerical algorithms exist to achieve this goal, so that, at first sight, a book on this topicmay seem superfluous. However, the present work does not aim at rederiving the seismic migrationand demigration theory in a different or new form. Its main goal is to provide one single unifiedtheory to solve a variety of seismic reflection imaging problems in true amplitude with a suitablecombination of true-amplitude migration and demigration operations. A particular attraction inthe proposed approach is that, for each specific problem, the migration and demigration operationscan be merged into a single weighted Kirchhoff-type imaging process that solves the problem. Thenew operator will have a specific traveltime stacking surface and specific weights. Moreover, thetrue-amplitude character of the new operator will also have a specific meaning that corresponds tothe problem under investigation.

Apart from depth migration, many other imaging techniques exist. These include, amongothers, the processes of migration-to-zero-offset (MZO) or its close relative, dip moveout (DMO),redatuming, etc. These operations can be termed “configuration transform,” since they transformthe data acquired with a certain measurement configuration into some simulated data as if acquiredwith another configuration. Further imaging processes include remigration (i.e., the updating of adepth migrated image of a seismic record or section for different or improved macrovelocity models),and elementary-wave transformation (e.g., changing P-P reflections into P-S reflections or multiplesinto primaries in a specific or different seismic record), etc.

It is not difficult to observe that all imaging techniques addressed above can be realized bythe cascaded use of a depth migration and a subsequent demigration or vice versa. This was themain message of our articles “A unified approach to seismic reflection imaging”, parts I and II, inGeophysics (Hubral et al., 1996a; Tygel et al., 1996). In this book, the fundamental ideas expressedin compact form in those articles are described in a thorough, updated, and more didactic way. Oneof the main objectives of this book is, thus, to show how the cascaded or chained use of the twoKirchhoff operations of true-amplitude migration and demigration can be used to solve a numberof reflection seismic problems in a simple and elegant unified way.

By the terminology unified approach to seismic reflection imaging, we shall refer to anycombined use of Kirchhoff true-amplitude migration and demigration. These operations can beapplied in any order, in separate steps or merged into one step. Surely, there are a number ofimaging procedures that cannot be described by the approach outlined above. These include, forexample, normal-moveout (NMO) correction, the CMP stack, homeomorphic imaging (Gelchinsky,

1.3. TRUE-AMPLITUDE KIRCHHOFF IMAGING 13

1988), and the change of a time-migrated section into a depth-migrated section. Such imagingprocedures do not make part of the unified approach.

Of course, the concept of true amplitudes must be extended to cover all possible imagetransformation problems included in the unified approach. This is, however, as straightforward asfor demigration. Consider, for example, the cascaded application of a common-offset migration and asubsequent zero-offset demigration. By definition of true amplitudes for migration and demigration,the former eliminates the original geometrical spreading of the wave that travels from the source toan offset receiver, and the latter introduces the corresponding zero-offset spreading. In other words,the two-step true-amplitude imaging process consisting of true-amplitude Kirchhoff migration anddemigration automatically corrects the geometrical spreading correspondingly to the desired datatransformation. Consequently, it is exactly this amplitude change that a one-step true-amplitudemigration to zero offset must achieve. This is readily generalized to any other kind of seismicreflection imaging. A true-amplitude imaging method replaces the original geometrical spreadingof the seismic reflections in the input section by the one pertaining to the corresponding outputreflections. All other amplitude effects (in particular the reflection and transmission coefficients)remain unaltered.

A particular attraction of the seismic true-amplitude imaging methods as described in thisbook is that the geometrical spreading factor is automatically taken care of during the imagingprocess. This is done without any knowledge of the reflector position or its shape but only of amacrovelocity model, as is the case in all Kirchhoff-type imaging techniques. The adequate care onthe geometrical spreading of primary reflections is an essential requirement to a reliable access toreflection coefficients, as well as other useful seismic attributes.

By treating all imaging problems that can be handled by the unified approach in a true-amplitude way, we will construct the “best possible solutions” in terms of the wavefield amplitudes.Note that the kinematic part of operator obtained by the use of the unified approach is determinedsolely by the resulting stacking surface and not from the weights. In this sense, even without weightsthe obtained operators can provide useful imaging results.

The proposed imaging theory has as its two key transformations the weighted diffractionstack and the weighted isochron stack. Both are based on two different but closely related stackingintegrals. They are described in Chapters 7 and 9. They permit the mathematical description ofthe true-amplitude migration and demigration procedures for specific elementary reflections (e.g.,primary P-P reflections or P-S reflections) recorded by arbitrary measurement configurations. Bothintegrals can be analytically chained, so as to construct the seismic image in either the time-traceor depth domain by only one weighted stack. For instance, a 3-D (true-amplitude) MZO correctioncan be performed entirely with stacking surfaces in the time-trace domain (see Tygel et al., 1998)or, correspondingly, a (true-amplitude) remigration can be achieved using only stacking surfaces inthe depth domain. Moreover, the theory that we present is entirely target-oriented, i.e., images canbe constructed in confined windows in either domain. For example, it enables the 3-D remigrationof a depth-migrated image in a selected depth-domain target window by a stack on that imageperformed exclusively on specific depth-domain stacking surfaces. These are confined to a well-defined (limited) target window. A remigration of a certain target zone as proposed here, thus,requires neither the whole original seismic record or section nor the repeated depth migration witha new or improved macrovelocity model.

Since the famous work of Hagedoorn (1954), concepts like the “surface of maximum convex-ity” (which has also become known as the “diffraction-time surface” or “Huygens surface”), and

14 CHAPTER 1. INTRODUCTION

concepts like the “isochron surface” (also called “aplanatic surface” or simply “aplanat”) are well-known in the world of seismic-reflection imaging. The Huygens surface is the “kinematic image”in the time-trace domain of a point in the depth domain. The isochron, on the other hand, is the“kinematic image” in the depth domain of a point in the time-trace domain. Both the Huygensand isochron surfaces thus represent the most basic kinematic concepts upon which a migrationand demigration are based. We will see that corresponding kinematic concepts can be employed forall other imaging problems. They are the “bones” of the theory for which we want to provide the“flesh”. Our attempt is to extend well-known kinematic mapping techniques (i.e., those exclusivelybased on traveltime, ray or wavefront concepts) to dynamic ones (i.e., to image transformationsusing traveltime and amplitude information simultaneously).

For that purpose, the true-amplitude diffraction-stack and isochron-stack integrals are quan-titatively described. They constitute an asymptotic transform pair which is well interlinked by theduality theorems derived in Chapter 5. The pair can be used to solve a multitude of amplitude-preserving target-oriented seismic imaging (or image-transformation) problems. These include, forinstance, the dynamic counterparts of the kinematic map-transformation examples given in Chap-ter 2. All image-transformation problems can be addressed by applying both stacking integrals insequence, whereby the macrovelocity model, the measurement configuration or the ray-code of theconsidered elementary reflections may change from step to step. This leads to weighted (Kirchhoff-type) summations along certain stacking surfaces (or inplanats) for which we provide true-amplitudeweights. To demonstrate the value of the proposed imaging theory which is based on analyticallychaining the two stacking integrals, we have solved the amplitude-preserving DMO-correction andremigration problem for the case of a 3-D laterally inhomogeneous velocity medium.

There exists a large number of seismic image transformation problems that may be solvedwith the proposed theory. For instance, one important image transformation procedure, requiringboth the diffraction-stack and isochron-stack integrals applied to an identical macrovelocity model,identical elementary wave, but different input and output measurement configurations is the 3-D true-amplitude DMO correction (Black et al., 1993; Oliveira et al., 1997; Tygel et al., 1999)or the related migration to zero offset (MZO) (Tygel et al., 1998; Bleistein et al., 1999). Thesetransformations are special cases of what we will refer to in Chapter 2 as Problem #1.

Another very similar task that falls in the same category of problems is the 3-D shot-continuation operation (SCO) that transforms the seismic primary reflections of one 3-D seismicshot record into those of another one for a displaced source location. A comparison of the thussimulated SCO shot record with that of the actually acquired field record for the very same dis-placed shot location allows for a validation and updating of the macrovelocity model employed forthe SCO (Bagaini and Spagnolini, 1993). Both the DMO correction and the SCO can be describedby the same general image-transformation approach (Problem #1 in Chapter 2). This general im-age transformation is also referred to as the “configuration transform” (CT). Other CT problemsinclude Azimuth MoveOut (AMO) (Biondi et al., 1998) Common-Shot DMO (Bagaini and Schlei-cher, 1997), Offset Continuation (OCO) (Santos et al., 1997), Redatuming (i.e., the change of themeasurement surface) (Sheaffer and Bleistein, 1998), data regularization (i.e., the change from anirregular to a regular measurement grid), the transformation of a seismic record (e.g., a shot or acommon-midpoint record) into a vertical seismic profile (VSP) record, etc., and their correspondinginverse transformations. Further time-domain imaging processes include wave-mode transformation(e.g., transforming P-P reflection data into corresponding P-S reflection data) and simulation ofmultiples out of primary reflections for multiple suppression (Filpo and Tygel, 1999).

1.4. ADDITIONAL REMARKS ON TRUE AMPLITUDE 15

A somewhat different problem, where both stacks are applied to different macrovelocitymodels but identical measurement configurations and elementary waves in the input and outputspace, is 3-D true-amplitude remigration (Hubral et al., 1996b; Schleicher et al., 1997a; Adler,2002). This is treated as Problem #2 in Chapter 2. In all cases, no more than one single weighted(Kirchhoff-type) summation is required along the problem-specific inplanats. The proposed theorycan also be used for elementary-wave transformations (e.g., a P-S shot record could be changed intoa P-P shot record). Also, transformations can be conceived whereby the elementary wave and themeasurement configuration may change at the same time (e.g., a P-S shot record could be changedinto a P-P zero-offset record). In addition, it should be mentioned that the proposed theory canbe used to correct images from previous inaccurate transformations. For instance, a correction forlateral velocity changes could be applied to depth-migrated images (or to sections after a DMOcorrection) that were very efficiently computed for simple velocity laws.

1.4 Additional remarks on true amplitude

The concept and terminology of true amplitudes as introduced in this book, although very adequatefor the present unified approach of seismic reflection, are not universal. In fact, the discussion ofwhat a true amplitude should mean is being thoroughly carried out in the literature and meetings(see, e.g., Berkhout, 1994; Hubral, 1998; Gray, 1999). Moreover, as the role of amplitudes, in seismicprocessing as a whole and in imaging in particular, is definitely growing, these discussions are boundto persist.

Depending on the framework or context under consideration, it is easy to expect that onecan understand the term “true amplitude” in various ways. Of course, the term carries such anoverwhelming and absolute character that it must not be taken literally. A tacit translation intoa more modest equivalent such as “best possible amplitude” or “most important amplitude” isalways implied. For example, in seismic acquisition, the attribute true-amplitude can be associatedwith field data of the best quality and accuracy one can actually achieve. In seismic processing,a true-amplitude data set can be understood as one to which none of the various procedures hasbeen applied that clearly do not preserve amplitudes (e.g., Common Midpoint (CMP) stacking,automatic gain control (AGC)).

As explained in the previous sections, our meaning of true amplitudes is attached to imagingand inversion. Our starting point is migration. We adopt the point of view of Newman (1975) whorecognized that migrated primary-reflection outputs can provide a useful measure of the reflec-tion coefficients if the geometrical-spreading losses are removed. The recognition that geometricalspreading is the main cause of distortion of reflection coefficients is the historical reason for callingtrue-amplitude any migration procedure that automatically removes these effects in the migratedamplitudes. In this context, it is important to observe that there is no conceptual difference betweenthe recovery of reflection coefficients, as proposed here, or reflectivity as in the works of Bleisteinand co-workers (see, e.g., Bleistein, 1987; Bleistein et al., 1987). Both results can be readily trans-formed into each other by multiplication with a simple scale factor depending on the reflection angleand the velocity (Dellinger et al., 2000). The main emphasis of geometrical-spreading correction isidentical in both approaches.

The definition of a true-amplitude demigration is a logical consequence of the true-amplitudemigration concept. True-amplitude demigration should undo the amplitude effects true-amplitude

16 CHAPTER 1. INTRODUCTION

migration had done to primary reflections. This means demigrated primary reflections should havetheir original geometrical spreading back as before the migration operation.

In the spirit of our unified approach of seismic reflection imaging, we define as true amplitudeany imaging procedure that is obtained by means of a cascaded use of true-amplitude migration anddemigration. This means that each of these imaging operations contains a specific true-amplitudecharacter that is defined by its specific use of the migration and demigration transformations.

We recognize, of course, that geometrical spreading is not the only factor affecting seismicamplitudes. For example, sources and receivers have certain characteristics that may vary from shotto shot and influence the observed amplitudes. Moreover, amplitudes are also expected to changedue to transmission and attenuation during propagation in the reflector overburden. For a detaileddiscussion of amplitude effects, see the comprehensive paper of Sheriff (1975). We stress, however,that these additional amplitude effects do not pose any restriction to the concept of true amplitudesas introduced above. A true-amplitude imaging process will achieve its goal of correcting for thegeometrical spreading independently of other amplitude factors that may be present in the data.Experience has shown that true-amplitude imaging is valuable and can improve the data quality(Hanitzsch et al., 1993; Beydoun et al., 1994; Winbow 1995; Tura et al., 1998). This happensindependently of whether effects other than geometrical spreading can be simply neglected (asis often the case in practice) or have been compensated for using some pre- or post-processingmethods. How transmission losses can be successfully removed from seismic data has been shownrecently by Hatchell (2000).

As a final remark, let us mention that other uses of the term true amplitude exist in theseismic community that do not match the above definition. For that reason, we prefer to employdifferent names for such schemes. For example, from our point of view, the term true amplitudeseems inappropriate for field amplitudes that have already been corrected for geometrical-spreadingeffects. Such data require imaging processes that do not alter the incoming amplitudes with respectto geometrical spreading. To distinguish these processes from the ones defined above, we prefer tolabel them with the term “amplitude preserving.” An example for such an amplitude-preservingimaging process is the DMO scheme proposed in Black et al. (1993).

1.5 Overview

This book is organized as follows. After this Introduction, Chapter 2 contains a non-mathematicaland pictorial description of the reflection seismic problem to be solved, as well as some basicconcepts that will be needed throughout.

In Chapter 3, the mathematical basics of zero-order ray theory are presented that are relevantfor the understanding of the later chapters. This chapter, by no means a complete introduction tothe theory, tries to present the needed results in an organized and self-contained manner.

The fundamentals of paraxial ray theory, especially the relation between paraxial traveltimesand the dynamic quantities of the central ray, are discussed in Chapter 4. The concept of surface-to-surface propagator matrix (Bortfeld, 1989; Hubral et al., 1992a,b; Cerveny, 2001) is used as amain tool to derive the required paraxial traveltime expressions. These expressions are very muchimportant for the derivation of the weight functions in the true-amplitude imaging operationsdiscussed later.

1.5. OVERVIEW 17

Chapter 5 examines the kinematic or geometric properties that relate primary reflections inthe time and depth domains. These involve the basic kinematic concepts of all seismic imaging pro-cedures, namely the diffraction-time surface and the isochron (Hagedoorn, 1954). How both surfacesare related, i.e., their fundamental duality (Tygel et al., 1995), is investigated and mathematicallyquantified in that chapter.

The Kirchhoff modeling integral, to which Kirchhoff migration is historically related, is de-rived and discussed in Chapter 6. Our derivation is focussed on the case of elastic isotropic media,from which the analogous expressions for acoustic media can be readily seen as a simple particularcase. We also comment on the extension of the results to generally elastic anisotropic media. Abetter understanding of the geometrical spreading of a primary reflection event can be obtained bythe asymptotic (stationary-phase) analysis of the Kirchhoff modeling integral. This is quantifiedby a decomposition formula that can be derived from the analysis. This formula will play an im-portant role in the derivation, later, of the weight functions in the various true-amplitude imagingalgorithms to be constructed.

The method of true-amplitude Kirchhoff or diffraction-stack migration, the first fundamentaloperation of the unified approach to seismic reflection imaging (Hubral et al., 1996a; Tygel et al.,1996), and upon which all other imaging methods are based, is presented in Chapter 7. It is shownthat true-amplitude migration outputs (i.e., the elimination of the geometrical spreading fromprimary reflections) can be obtained by incorporating a suitable weight function to the conventional(unweighted) Kirchhoff migration algorithm (Schleicher et al., 1993). The required true-amplitudeweights are derived using the machinery and results described in the previous chapters. In particular,the duality theorems and the geometrical-spreading decomposition formula play a most significantrole in the derivations. Expressions of the weight functions for the mostly used seismic measuringconfigurations are explicitly obtained.

Further important aspects of the Kirchhoff migration technique that also influence the ap-pearance of the migration results are considered and quantified in Chapter 8. These include thequantification of the pulse stretch that is observed in the migration outputs (Tygel et al., 1994b),the relationship between migration apertures and the Fresnel zones (Schleicher et al., 1997b) andthe influence of the migration on vertical and horizontal resolution.

Chapter 8 also considers the use of two or more simultaneous Kirchhoff-type stacks alongthe same stacking surfaces, but with different weights. It is shown that, in this situation, theobtained results can be combined to yield useful seismic attributes. For didactic reasons, the methodis explained for diffraction-stack migration (Bleistein, 1987; Tygel et al., 1993). Its natural andstraightforward extension for any other Kirchhoff-type imaging method is also briefly indicated.Applications of this technique include the determination of incidence angles of primary reflectionsleading to more reliable AVO/AVA analysis (Bleistein, 1987; Hanitzsch, 1995; Tygel et al., 1999;Bleistein et al., 1999), as well as the derivation of simpler and less expensive computation of true-amplitude weights (Tygel et al., 1993; Hanitzsch, 1995).

The very heart of this book is its final Chapter 9. It presents in its first part the isochron-stackdemigration, the second fundamental transformation of the unified approach to seismic reflectionimaging. This is a stacking procedure that is fully analogous to diffraction-stack migration towhich it is the (asymptotic) inverse operation (Hubral et al., 1996a; Tygel et al., 1996). All featuresdiscussed above for diffraction-stack migration can be transferred, in a simple and direct manner,to isochron-stack demigration.

18 CHAPTER 1. INTRODUCTION

The operations of diffraction-stack migration and isochron-stack demigration, are the fun-damental tools, on which a number of other true-amplitude imaging procedures are based. Thesecond part of Chapter 9 is devoted to the imaging operations that are derived by chaining thediffraction-stack and isochron-stack integrals. There, we will derive the rules for constructing therespective stacking surfaces and true-amplitude weight functions for the various true-amplitudeimage transformations that can be described by the unified approach to seismic reflection imaging.Using these rules, true-amplitude Kirchhoff-type stacking operations have been developed for offsetcontinuation (Santos et al., 1997) and migration to zero offset (Tygel et al., 1998).

To facilitate the exposition, a number of appendices are included containing the technicaldetails and mathematical derivations that would otherwise interrupt the flux of presentation.

Chapter 2

Description of the problem

In this chapter, we discuss in more detail and from a mainly geometrical point of view the principleson which the two fundamental seismic processes of true-amplitude migration and demigration arebased. We will see how they form the basis for a unified theory of Kirchhoff-type seismic reflectionimaging. By applying these two operations in sequence (i.e., chaining them), a wide class of seismicimaging problems can be solved. These include

(a) the transformation of a seismic data section in the time-trace domain, recorded with a givenmeasurement configuration into a section as if recorded with another configuration, exceptfor the reflection and transmission coefficients. This imaging process is generally referred toas a configuration transform (CT). As particular CT transforms, we can cite the operationsof dip moveout (DMO), azimuth moveout (AMO), migration to zero offset (MZO), shot oroffset continuation, etc. In this chapter, we concentrate our discussion on the MZO operation,i.e., the transformation of a common-offset section into a zero-offset section.

(b) the transformation of a 3-D migrated image in the depth domain into another one for adifferent (improved) macrovelocity model. This imaging process is referred to as remigration.

Other possible image transforms that can be solved by chaining the migration and demi-gration operations include redatuming, wavemode transformations, transformation of surface datainto VSP data, etc. These possible applications will not be discussed in detail in this book. Notethat in this context, imaging does not only imply going from the time-trace domain to the depthdomain or vice versa. In the framework of the unified approach, it can also imply going from onetime-trace domain to another or from one depth domain to another.

We start by giving a brief description of the earth and macrovelocity models to be considered,as well as of the seismic measurement configurations that are commonly used. Thereafter, wesummarize the basic (mainly kinematic) aspects of the theory in order to gain a good geometricalunderstanding of all imaging operations involved.

19

20 CHAPTER 2. DESCRIPTION OF THE PROBLEM

2.1 Earth model

Throughout this book, we shall consider that seismic waves propagate within 3-D layered models.The medium within each layer is smoothly inhomogeneous, acoustic or elastic isotropic. The layersare bounded by smooth interfaces. Across these interfaces, the medium parameters may present upto first-order (jump) discontinuities. The above conditions are chosen such that primary reflections,the wave propagation events of our main interest, are adequately described by zero-order ray theory(see, e.g., Cerveny, 1987, 1995, 2001). It is to be noted that the conditions of “smoothness” of theinterfaces, as well as of the medium parameters within the layers, are not absolute requirements.They depend, in particular, on the main frequencies of the input signals and on the dimensions(depth, curvature) of the reflectors. Experience has shown that these requirements are, to a largeextent, quite reasonably met for the purposes of seismic reflection imaging.

Macrovelocity model.—Even though certain general, “ray-theoretical,” assumptions have beenmade about the earth model, this model remains nevertheless unknown with respect to the positionsof the reflectors that are to be determined by seismic imaging procedures. As explained below, theseprocedures require the construction of auxiliary (stacking) surfaces along which the seismic datawill be summed (stacked). The stacking surfaces, as well as the corresponding weighting factors thatwill be applied in the stacking process, will be constructed using an a priori given macrovelocitymodel. This is typically a smooth time or depth velocity model that incorporates in the “best way”all geophysical and geological information available about the region to be imaged. How to establishthe macrovelocity model is a complicated problem on its own, which is not discussed in this book.A vast literature is devoted to the construction of a macrovelocity model that is most adequateto the imaging purpose under consideration. The topic is also subject of active ongoing research.For example, recent papers on the subject of migration velocity analysis include those of Berkhout(1997), Liu (1997), Jones et al. (1998), Zhu et al. (1998), and Sacchi (1998).

Wavemode selection.—All true-amplitude Kirchhoff-type operations described by the unifiedapproach to seismic reflection imaging are single wavemode methods. This means that they aredesigned to achieve, at a time, the correct imaging of events with only one particular preselectedwavemode (as, for example, pure P-P reflections, simple P-S conversions, or acoustic propagationplus conversion to S wave at the sea bottom plus conversion to P at the target reflector, etc.). Allother events with different wavemodes are considered as noise or well suppressed in one particularapplication of the method. They may, however, be imaged independently by further applicationsof the same imaging scheme with other preselections of the wavemode.

In this context, let us remark that all formulas derived in this book are valid for all types ofisotropic elementary reflections (namely, acoustic, elastic non-converted and converted reflections)with straightforward changes or interpretations of the formulas involved (Beydoun and Keho, 1987;Schleicher, 1993). The imaging theory presented here can be generalized to multi-component dataand anisotropic media (see, e.g., de Godoy, 1994; Ikelle, 1994; Hokstad, 2000).

Coordinate system.—We describe the domain of the seismic earth and macrovelocity models,briefly called the depth domain, by means of a global Cartesian coordinate system. The 3-D Carte-sian coordinate vector r consists of the 2-D horizontal coordinate vector r = (r1, r2) and thevertical coordinate z, i.e., r = (r, z) (see Figure 2.1). Surfaces in this domain will be assumed to be

2.2. MEASUREMENT CONFIGURATIONS 21

Fig. 2.1. Parameterization of the depth domain. The seismic reflector ΣR below a laterally inho-mogeneous overburden is supposed to be given by z = ZR(r). Correspondingly, the measurementsurface ΣM is described by z = ZM (r).

parametrized in the form z = Z(r). In this way, the 3-D global Cartesian coordinate vector of anarbitrary point P on a given surface is given by rP = (rP , z = Z(rP )). Due to above considerations,we will adopt the natural identification between a surface point P and its 2-D horizontal coordinatevector rP . For example, expressions such as “point P at rP ” will be used freely.

2.2 Measurement configurations

Measurement surface.—Seismic reflection data are gathered by exciting the propagation ofwaves into the earth by a given distribution of sources (explosions, vibrators, air guns, etc.) andrecording the emitted energy at another given distribution of receivers (geo- or hydrophones, oceanbottom seismographs, etc.). We assume that all sources and receivers employed for the seismic dataacquisition are distributed along some surface ΣM , usually coinciding with the earth’s surface. Wewill refer to ΣM as the measurement surface (see Figure 2.1). Like the interfaces, we also assume ΣM

to be smoothly curved. For simplicity of presentation, we shall sometimes depict the measurementsurface ΣM as a horizontal plane. The formalism, however, can take smooth curvatures into account.

Nonetheless, one might wish to think of the measurement surface as being planar. This is nota real restriction since, for the marine environment, this simple assumption is automatically met.For land seismics, it is true once the usual preprocessing routines for redatuming have been applied.

22 CHAPTER 2. DESCRIPTION OF THE PROBLEM

These are designed to eliminate the undesirable effects of topography, ground roll, weathering zone,so as to simulate the data as being acquired at a new (planar) datum in depth, where these effectsdo not occur. Further discussion on the above-described preprocessing issues are beyond the scopeof the present work, the interested reader being referred, for instance, to the classic text book ofYilmaz (1987,2000).

We consider a distribution of sources and receivers on the measurement surface above thesubsurface region to be imaged. We say that these source and receivers provide an illumination ofthat target region. The recording at a fixed receiver point G due to a fixed source at point S forvarying recording time t is called a seismic trace, denoted by U(S,G, t). The seismic traces mayrepresent, for instance, pressure or displacement, for marine or land data, respectively. The dataacquisition is realized by a sequence of seismic experiments. Each of them consists of one shot ata specific source location, the reflection wavefield being recorded at a number of specific receivers.The collection of seismic experiments designed to illuminate a given region is called a seismic survey.

Measurement configuration.—The seismic data, to which our imaging methods will be ap-plied, consists of an ensemble of seismic traces, generally obtained from many overlapping seismicexperiments. This ensemble of traces is called the “multi-coverage data.” Especially in the 3-Dsituation as envisaged here, this ensemble of traces is huge. To be able to handle the data, a verysuccessful strategy is to separate (sort) the data into well designed subsets, called seismic sections.Each seismic section consists of seismic traces in which the sources and receivers are grouped intowell defined pairs. Within a seismic section, we use the terminology seismic measurement configu-ration to designate the rule upon which each source uniquely corresponds to one receiver. In thisway, one divides the multi-coverage data into several “single-coverage data” sections. Only thesesingle-coverage sections, also called minimal data sets (Vermeer, 2002), are suited for a Kirchhoffmigration. It is to be noted that although the CMP configuration provides a unique one-to-onerelationship between sources and receivers, it generally does not give rise to a single-coverage orminimal data set.

Please note that our imaging methods will be applied independently to each seismic section.The corresponding seismic configuration involved provides a specific illumination of the targetsubsurface region. The various images of the same target region that originate from different single-coverage seismic sections can be combined in a variety of strategies so as to come up with the bestsolution of the imaging task under consideration that the original multi-coverage data allow for.

A natural way to specify the source-receiver pairs (S,G) within a given seismic configurationis in parametric form, namely S = S(ξ) and G = G(ξ). Here, ξ = (ξ1, ξ2) is a 2-D vector parameterreferred to as the configuration parameter. Seismic traces U(S(ξ), G(ξ), t) that belong to a givenseismic configuration will be denoted simply by U(ξ, t). The planar region A where the parameterξ varies is called the seismic aperture of the configuration.

We will now describe how a seismic configuration can be specified with the help of theconfiguration parameter ξ. For that purpose, let rS and rG denote the 2-D coordinate vectors of anarbitrary shot point S and an arbitrary receiver point G, respectively, on the measurement surfaceΣM: z = ZM(r). In other words, the 2-D vectors rS and rG point simply to the projections of S andG on the plane z = 0. We assume throughout that all sources and receivers are uniquely specified inthis way with respect to the global Cartesian (r, z) coordinate system. As mentioned above, for anyspecified measurement configuration, sources and receivers are systematically grouped into pairs(rS , rG). These pairs can be described by the configuration parameter ξ according to the simple,

2.2. MEASUREMENT CONFIGURATIONS 23

but general, relations

rS(ξ) = aS + Γ˜Sξ , rG(ξ) = aG + Γ

˜Gξ . (2.2.1)

Here, aS and aG are the 2-D global coordinate vectors of two fixed reference points that depend onlyon the arbitrary choice of the origin for ξ. Moreover, Γ

˜S and Γ

˜G are constant 2×2-matrices, which

we will call configuration matrices. The meaning of these vectors and matrices will become clearfrom the following examples for some standard seismic configurations typically used in practice.

For the most frequently used seismic measurement configurations (see Figure 2.2), equations(2.2.1) can be readily recast into a more comprehensive form.

(a) Common-shot (CS) configuration: The shot location S is fixed at rS0 and the receiver positionG falls upon a 2-D measurement grid (Figure 2.2a). The CS configuration is described as

rS(ξ) = rS0 and rG(ξ) = rG0 + ξ . (2.2.2)

This is equation (2.2.1) with aS = rS0, aG = rG0, Γ˜S = O

˜and Γ

˜G = I

˜. Here, O

˜and I

˜are the 2 × 2 zero and identity matrices, respectively. Moreover, rG0 is an arbitrary initial orreference receiver coordinate. This is the most basic measurement configuration. The meaningof the parameter vector ξ = rG− rG0 is here that of a dislocation vector between the varyingreceivers and the reference one.

(b) Common-receiver (CR) configuration: The receiver location G is fixed at rG0 and the sourceposition S falls upon a 2-D measurement grid (Figure 2.2b). We have, in this case,

rS(ξ) = rS0 + ξ and rG(ξ) = rG0 . (2.2.3)

In terms of the configuration equations (2.2.1), we now have aS = rS0, aG = rG0, Γ˜S = I

˜and

Γ˜G = O

˜. The parameter vector ξ = rS − rS0 now represents the dislocation vector between

the varying sources and the reference one.

(c) Common-offset (CO) configuration: Source and receiver coordinate pairs (rS, rG) are movedby the same dislocation vector from a fixed, reference source-receiver pair (rS0, rG0). Thus, apossible way to express rS and rG in the form of equations (2.2.1) is

rS(ξ) = rS0 + ξ and rG(ξ) = rG0 + ξ . (2.2.4)

In this form, the parameter vector ξ = rS − rS0 = rG− rG0 is the common dislocation of thesource-receiver pairs. The above specification of the CO-configuration has the general form(2.2.1) with aS = rS0, aG = rG0 and Γ

˜S = Γ

˜G = I

˜.

There is another, more useful representation of the source and receiver coordinates in a COconfiguration. Taking the difference between equations (2.2.4), we recognize the half-offsetvector

1

2

(rG(ξ) − rS(ξ)

)=

1

2

(rG0 − rS0

)= h , (2.2.5)

which is constant for this configuration. Using this vector h, the source-receiver pairs can bealternatively represented as

rS(ξ) = − h+ ξ and rG(ξ) = h+ ξ . (2.2.6)

24 CHAPTER 2. DESCRIPTION OF THE PROBLEM

(a)

(f)(e)

(d)(c)

(b)

1

3

3

1 2

2

3

21

3

2 1

1 2

3

1 2

3

1 2

3

1

2

3

G

S

SS

S S

GG

GG

S

S S

S

GG

G G

SS

S SG

G

GG

S

G

S

G

G G

S S

S

G

Fig. 2.2. Displacement of source S and receiver G for typical seismic measurement configurations:(a) Common Shot, (b) Common Receiver, (c) Common Offset, (d) Common Midpoint, (e) CrossProfile Experiment. (f) Cross Spread Experiment.

2.2. MEASUREMENT CONFIGURATIONS 25

Observe that the ξ in equations (2.2.6) is not the same as the one in equations (2.2.4).In fact, ξ = 1

2(rS + rG) has now the meaning of the midpoint coordinate between rS andrG. Only if the initial midpoint is set to zero, equations (2.2.6) and (2.2.4) are consistent.Nonetheless, equations (2.2.6) also have the general form (2.2.1) with aS = −h, aG = h andΓ˜S = Γ

˜G = I

˜. In the case that rS0 = rG0 we obtain h = 0, which describes the zero-offset

(ZO) configuration.

(d) Common-midpoint offset (CMPO) configuration: Source and receiver coordinate pairs (rS , rG)are dislocated by a fixed amount but opposite directions, from a fixed, reference source-receiverpair (rS0, rG0). Thus, a possible way to express rS and rG in the form of equations (2.2.1) is

rS(ξ) = rS0 + ξ and rG(ξ) = rG0 − ξ . (2.2.7)

In this form, the parameter vector ξ relates to the source and receiver dislocations as ξ =rS − rS0 = − (rG − rG0). The CMPO configuration is then given in terms of the generalequations (2.2.1) by using aS = rS0, aG = rG, Γ

˜S = I

˜and Γ

˜G = −I

˜.

As for the CO configuration, there is another possibility of representing the source-receiverpair in the form of equations (2.2.1). Taking the sum of equations (2.2.7) we recognize thatthe midpoint vector

1

2(rS(ξ) + rG(ξ)) =

1

2(rS0 + rG0) = m (2.2.8)

is fixed for this configuration. Using the constant midpoint coordinatem, we may alternativelyspecify the CMPO configuration as

rS(ξ) = m− ξ and rG(ξ) = m+ ξ . (2.2.9)

where ξ = 12(rG − rS) has now the meaning of the half-offset coordinate vector. The new

CMPO configuration equation has still the general form (2.2.1) with aS = aG = m, Γ˜S = −I

˜and Γ˜G = I

˜.

In the case that rS0 = rG0 we obtain m = rS0 = rG0 which describes the mostly usedcommon-midpoint (CMP) configuration. Note that the distinction between the CMP andCMPO configurations is rather artificial since the choice of rS0 and rG0 is arbitrary. We willneed this distinction, however, for didactic reasons at a later stage.

(e) Cross-profile (XP) configuration: The source-receiver coordinate pairs (rS , rG) are dislocatedby a fixed amount, but in orthogonal directions, from a reference source and receiver coordi-nate pair (rS0, rG0) (Figure 2.2e). In symbols,

rG − rG0 = R˜

(rS − rS0) , (2.2.10)

where R˜

is the 2 × 2-matrix

R˜

=

(

cos π2 sin π2

− sin π2 cos π2

)

=

(

0 1−1 0

)

. (2.2.11)

describing a clockwise rotation of 90. Introducing the configuration parameter as the shotdislocation ξ = rS − rS0, this configuration can be described by equations (2.2.1) with aS =rS0, aG = rG0, Γ

˜S = I

˜and Γ

˜G = R

˜. Similar configurations with other angles α between the

source and receiver dislocations are described in a corresponding way using the respectiverotation matrix R

˜(α).

26 CHAPTER 2. DESCRIPTION OF THE PROBLEM

(f) Cross spread (XS) configuration: All receivers are located on a single line perpendicularlyoriented to one single line of sources (see Vermeer, 1995). The situation is shown in Figure 2.2f.Each shot is recorded by all receivers. This configuration is subdivided into single-coveragesubsets described by equations (2.2.1) with aS = rS0, aG = rG0,

Γ˜S =

(

1 00 0

)

, and Γ˜G =

(

0 00 1

)

. (2.2.12)

Here, the first and second components of the configuration parameter vector describe thesource and receiver dislocations, respectively.

The parameterization (2.2.1) can also be useful to describe sources and receivers S and G in therespective vicinities of a given pair (S,G). Let us consider two local Cartesian coordinate systemsxS and xG with origins at S and G, respectively. The coordinates of S and G in these systems canthen be expressed by the vectors

xS = Γ˜Sξ , xG = Γ

˜Gξ , (2.2.13)

where ξ is now the value of the configuration parameter relative to the chosen coordinate originsat S and G. In other words, equations (2.2.13) describe the coordinates of sources S and receiversG independently of where the local Cartesian coordinate systems are centered.

Above we have assumed Γ˜S and Γ

˜G to be constant matrices, i.e., strictly speaking, equations

(2.2.13) are only valid for regular profiles. For other, less common configurations, equations (2.2.13)may still represent a reasonable first-order approximation for small distances between S and S andbetween G and G. However, in that case, Γ

˜S and Γ

˜G have to be determined for each source-receiver

pair (S,G) independently. Then, Γ˜S and Γ

˜G may be determined from

ΓSij =∂xSi∂ξj

S

and ΓGij =∂xGi∂ξj

G

. (2.2.14)

For irregular profiles, the assumption of constant configuration matrices is valid only in some vicinityof the chosen coordinate centers S and G. Moreover, the matrices may depend on the initial choiceof the vectors aS and aG. However, even in that case, equations (2.2.13) may represent usefulapproximations describing the measurement configuration in some region.

Data space description.—The configuration parameter ξ = (ξ1 , ξ2 ), together with the recordingtime t, provide us with the Cartesian coordinate system for the description of the data space (seeFigure 2.3). A generic point with coordinates (ξ, t) within the seismic section will be denoted byN . Among the reflection traveltime surfaces within the seismic section, one pertains to the chosentarget reflector. It will be denoted by ΓR: t = TR(ξ) and is described by points NR with thecoordinates (ξ, t = TR(ξ)), where ξ is varying on the seismic aperture A.

We close this section with some remarks concerning the traces U(ξ, t) within a given seismicconfiguration. We assume that all these traces come from reproducible sources and receivers. Bythis we mean that all sources and all receivers have identical characteristics at every time they areused. Moreover, as indicated above, all possible effects due to positioning of sources and receivershave already been removed or accounted for. We also suppose that in each seismic section thedistribution of sources and receivers on the measurement surface is sufficiently dense to guarantee

2.3. HAGEDOORN’S IMAGING SURFACES 27

Fig. 2.3. Parameterization of the data space. The reflection traveltime surface ΓR is supposed tobe given by t = TR(ξ).

an adequate illumination of the target reflector, so that the imaging procedures to be appliedare possible. To theoretically describe these procedures, we will use the general tools of Calculus,in particular integrals, which considers a continuous distribution of traces. In the real world, allprocedures are implemented with discrete data, requiring in many instances adequate interpolationof missing data, as well as a variety of procedures to handle the actual field data. In this book, weshall be concerned with the description and understanding of a unified theory of seismic reflectionimaging that has already shown to yield good results on real problems. The actual specifics of itsimplementation, albeit of prime practical importance, falls beyond the scope of the present work.

2.3 Hagedoorn’s imaging surfaces

For all purposes of the unified approach, we assume a fixed seismic configuration of sources andreceivers grouped into pairs, (S(ξ), G(ξ)), specified by the configuration parameter ξ. These intro-duce a 3-D Cartesian system, the (ξ, t)-domain, that is simply referred to as time-trace domain (seeupper part of Figure 2.4).

A counterpart spatial 3-D Cartesian system, the (r, z)-domain is also assumed to be definedin the subsurface region illuminated by the sources and receivers in the given seismic configuration.It is called the depth domain (see lower part of Figure 2.4).

28 CHAPTER 2. DESCRIPTION OF THE PROBLEM

reflection-signal strip

t

R

( ;M )

( )( )

D

RR

z

rr

MR

M

NRR

depth-migrated strip

R

R

R

R

R

( )r

( ; N )r

R

RR

I

N

S( )G( )

ξ R

ξ

R

ΓM

ξ

ξ

ξ

ϑβ

Σ

ξ Σξ

ΣR

( )r

ΓR

Fig. 2.4. 2-D sketch of 3-D seismic time and depth sections. Upper part: Data space or time-tracedomain. A seismic reflection event is confined to a “reflection-signal strip” attached to the reflec-tion-time surface ΓR given by t = TR(ξ). Also shown is the Huygens surface ΓM : t = TD(ξ;MR)associated with point MR on the reflector ΣR. Lower part: Image space or depth domain. Thereflector image is confined to a “depth-migrated strip” attached to the reflector ΣR given byz = ZR(r). Also shown is the specular reflection ray connecting the source point S to the receiverposition G. It is reflected at the point MR on the reflector ΣR under an angle ϑR. The depictedisochron ΣN : z = ZI(r;NR) is associated with the specified point NR on the reflection-time surfaceΓR : t = TR(ξ).

2.3. HAGEDOORN’S IMAGING SURFACES 29

Moreover, we consider a macrovelocity model that incorporates the general geologic featuresof the region to be imaged to be given a priori. Finally, we suppose that the wavemode of theprimary reflections to be imaged has been selected. We will refer to this wavemode as the “imagingwavemode.”

The imaging transformations described in this book will be realized upon stacking (summing)or smearing (broadcasting) selected parts of the input data along problem-specific imaging surfacesthat are constructed to solve the given imaging task. The imaging surfaces that correspond tothe migration and demigration operations are the diffraction traveltime or Huygens surface (inthe time-trace domain) and the isochron surface (in the depth domain). These are of paramountimportance, since all imaging surfaces used in the unified approach are obtained upon the systematiccombination of them (Hagedoorn, 1954; Bleistein, 1999). The definition of the imaging surfaces ofmigration and demigration is provided below.

The diffraction traveltime or Huygens surface.—For any fixed depth point M and vary-ing source-receiver pairs (S(ξ), G(ξ)), the diffraction-traveltime or Huygens surface is denoted byΓM : t = TD(ξ;M). Figure 2.4 shows the Huygens surface ΓM of a point MR on the reflector ΣR.For each value ξ of the configuration parameter, TD(ξ;M) represents the sum of traveltimes alongthe two rays that link the source point S(ξ) to the subsurface point M and point M to the receiverpoint G(ξ), respectively. Here, the wavemodes for rays SM and MG are the same as the downgoingand upgoing parts of the imaging wavemode, respectively. This is the surface of maximum convexityof Hagedoorn (1954).

A physical interpretation of the above construction is that of a point diffractor at M thatis illuminated by the seismic configuration according to the imaging wavemode. The resultingtraveltime surface in the seismic section would exactly be ΓM .

The isochron surface.—For any fixed point N with coordinates (ξ, t) in the time-trace domain,i.e., a fixed source-receiver pair (S,G) together with a given time t, and varying r, the isochronsurface is denoted by ΣN : z = ZI(r;N). Figure 2.4 shows the isochron ΣN of a point NR on thereflection-time surface ΓR. For each value, r, of the horizontal depth coordinate, the correspondingpoint MI = (r,ZI(r;N)) on the isochron ΣN is implicitly defined by the following condition. Thesum of traveltimes from the fixed source at S(ξ), defined by the given coordinate ξ of N , to thedepth point MI and from there to G(ξ) has to be constant and equal to the time coordinate t of N .Again, the wavemodes for rays SMI and MIG are the same as the downgoing and upgoing partsof the imaging wavemode, respectively. This is the surface of equal reflection time of Hagedoorn(1954).

The isochron can be constructed by placing point sources both into S(ξ) and G(ξ), computingfrom both points the traveltimes to all points M in the (r, z)-domain, and selecting thereafter thosesubsurface points MI for which the sum of the traveltimes from S(ξ) to MI and from G(ξ) to MI

equals t. The points MI fall then onto the isochron z = ZI(r;N). The physical interpretation isthat of a caustic mirror at ΣN that has all its reflections arrive at the same point at the same time.

Note that both the Huygens and isochron surfaces ΓM and ΣN are defined by the very sametraveltime function TD. To obtain the Huygens surface ΓM , one has to keep the subsurface point M(i.e., the coordinates r and z) fixed and let ξ and t vary. On the other hand, to obtain the isochronΣN , one has to keep point N (i.e., the coordinates ξ and t) fixed and let r and z vary.

30 CHAPTER 2. DESCRIPTION OF THE PROBLEM

Hagedoorn’s imaging conditions.—As shown in the pioneering work of Hagedoorn (1954),under the correct macrovelocity model, the Huygens or diffraction traveltime surface, as well as theisochron surface satisfy the following tangency properties (see also Figure 2.4):

(H1) The Huygens or diffraction traveltime surface ΓM pertaining to a reflection point M = MR

on a target reflector ΣR and the primary-reflection traveltime surface ΓR of ΣR are tangentsurfaces in the time-trace domain.

(H2) In the same way, for any point NR on ΓR, the corresponding isochron ΣN is tangent to thereflector ΣR in the depth domain.

We will refer to the above statements as Hagedoorn’s imaging conditions1. Below, they will be seento play a decisive role in the imaging methods that are dealt with in this book.

Hagedoorn’s imaging conditions can be alternatively formulated in a more imaging-orientedmanner, namely

(H1’) Any reflector at depth can be understood as the envelope of all isochrons issued from pointsalong the corresponding primary-reflection traveltime surface.

(H2’) Any primary-reflection traveltime surface in the time-trace domain can be understood as theenvelope of all diffraction traveltime surfaces issued from the points along the correspondingreflector.

A complete account of these and other geometrical relationships that link the two fundamentalsurfaces to a given reflector and its corresponding reflection traveltime surface will be describedin more rigorous detail in Chapter 5. Chapters 7 and 9 describe how the obtained results canbe applied to derive the weights designed to achieve the required true-amplitude character in alltransformations.

In the next section, we will show, in a pictorial and qualitative manner, how Hagedoorn’simaging conditions can be effectively used to produce seismic images by suitably stacking thedata. We start by considering the basic processes of seismic migration and demigration. Thereafter,we show how the underlying principles can be generalized to encompass all transformations thatcomprise our unified approach to seismic reflection imaging.

2.4 Mapping versus imaging

Before entering into the details of the true-amplitude imaging theory, we first want to addressthe question of what is meant in this book by the words “mapping” or “imaging.” We refer toall migration and demigration schemes that only use and provide the kinematic and geometricalinformation of elementary reflections (i.e., the reflection times and reflection points) as mappingprocedures. Those migration, demigration and transformation schemes, however, that also accountfor seismic pulse forms and amplitudes are referred to as imaging procedures. Let us now furtherelaborate on the introduced terminology of mapping and imaging and give some examples.

1In the seismic literature, the term “imaging condition” is traditionally used as defined in Claerbout (1971). Thiscondition is the following: a target reflector can be imaged (depth migrated) if we (forward) propagate the wavefieldof the source and (backward) propagate the field of a receiver until the full propagating time has elapsed (t = 0).

2.4. MAPPING VERSUS IMAGING 31

2.4.1 Migration and demigration: mapping

The kinematic and geometric objectives of migration and demigration consist of determining, froma selected reflection-time surface ΓR, the corresponding location of the subsurface reflector ΣR andvice versa. Figure 2.4 shows a 2-D sketch of a primary reflection in the (ξ, t)-domain. Due to thefinite length of the source pulse, the reflection event is confined to the shaded 2-D reflection-signalstrip that extends in 3-D space to a full 3-D reflection-signal sheet. A map migration requiresthe identification of the target (primary) reflections of interest (possibly in the presence of manyother, non target, events). This means the determination of the reflection time surface by a pickingprocedure. After the traveltime surface and a suitable macrovelocity model are available, we have atour disposal the following two (kinematic) map-migration procedures. These will find their equivalentformulations in seismic reflection imaging.

(M1) Map migration with isochrons

(a) Select for each point NR on the identified and picked reflection-time surface ΓR inthe time-trace domain the associated source-receiver pair (S,G) and determine its theisochron ΣN : z = ZI(r;NR) in the depth domain as defined above. For instance, forpoint NR with coordinates (ξR, TR(ξ)) in Figure 2.4, the isochron is z = ZI(r;NR).

(b) Now find the envelope to all isochrons associated with all points NR(ξ, t) on ΓR. Thisenvelope determines the depth-migrated reflector ΣR.

A useful generalization of this scheme, which will be of importance below, consists of theconstruction of the isochrons ΣN for all points N on a dense grid in the time domain and of asubsequent selection for the map migration of only those points that fall upon the reflection-time surface ΓR: t = TR(ξ).

(M2) Map migration with Huygens surfaces

(a) Specify a dense grid of points M in the (r, z)-domain (i.e., the depth domain) belowthe measurement surface ΣM . Then compute the diffraction traveltime function ΓM :t = TD(ξ;M) as defined above. In this way, an ensemble of Huygens surfaces ΓM for allpoints M is constructed in the (ξ, t)-domain.

(b) Now select, from this ensemble, all Huygens surfaces ΓM that are tangent to the targetreflection-time surface ΓR. Note that the latter is also a function of the source-receiverconfiguration, ΓR : t = TR(ξ). In Figure 2.4, for instance, the Huygens surface ΓM : t =TD(ξ;MR) of a point MR is depicted. It is tangent to the reflection-time surface ΓR atpoint NR. As a consequence of the tangency of both surfaces, one knows that point MR

falls upon the searched-for reflector ΣR. The reflector ΣR can thus be constructed as theset of all points M = MR the Huygens surfaces ΓM of which are tangent to ΓR.

The two kinematic migration methods reviewed above describe the transformation of thereflection-time surface ΓR of a primary reflection in the (ξ, t)-domain onto the reflector ΣR in the(r, z)-domain provided the correct macrovelocity model was used. Both methods are closely relatedto the following two methods that solve the kinematic demigration problem.

(D1) Map demigration with Huygens surfaces

32 CHAPTER 2. DESCRIPTION OF THE PROBLEM

(a) Construct for all points MR on the given reflector ΣR the Huygens surfaces ΓM : t =TD(ξ;MR) in the (ξ, t)-domain.

(b) Observe that they have as their common envelope the searched-for reflection-time surfaceΓR.

As a generalization of this scheme, which is to be used below when we talk about imaging, wemay construct the Huygens surfaces ΓM for all points M on a dense grid in the (r, z)-domainand then select only those points MR that fall upon the known reflector ΣR.

(D2) Map demigration with isochrons

(a) Define a dense grid of points N in the (ξ, t)-domain and construct the correspondingisochrons ΣN in the (r, z)-domain.

(b) Now search for those isochrons ΣN that are tangent to the reflector ΣR. All points N =NR in the (ξ, t)-domain that specify isochrons ΣN tangent to ΣR define the searched-forreflection-time surface ΓR.

Let us now see how other mapping problems can be described in analogy to these migration anddemigration techniques. As a first step, we need to generalize Hagedoorn’s imaging surfaces.

2.4.2 Generalized Hagedoorn’s imaging surfaces

The above map-migration and map-demigration procedures may to some readers appear not tohave a direct practical application2. However, the above kinematic mapping schemes will gain afundamental significance as soon as we put flesh to the bones, i.e., incorporate them into the unifiedtrue-amplitude imaging theory presented below. For that reason, it makes good sense to commentnot only on these mapping schemes but also on the many different ways how a map migration andmap demigration can be combined to solve a variety of other useful (now only mapping but laterimaging) problems. Below, we outline only two of many possible combinations.

Prior to doing this, however, it is instructive to comment on a common feature of all mappingand imaging problems treated below. For that matter, we introduce two new concepts, namely thatof an “inplanat” and that of an “outplanat.” These are generalizations of the classical isochron andHuygens-surface concepts. For any mapping (or imaging) procedure that can be handled by theunified approach and which includes the map migration and demigration described above, we canintroduce the following definitions.

Definition #1:An inplanat is the surface in the input space that corresponds to a point in the output space.

Definition #2:An outplanat is the surface in the output space that corresponds to a point in the inputspace.

2In fact, routine map-migration schemes (particularly for zero-offset sections as, e.g., simulated by the CMP-stackor a NMO/DMO/stack) may follow yet a different strategy. Map demigration is not a real practical problem.

2.4. MAPPING VERSUS IMAGING 33

In other words, the outplanat is the image of a point and the inplanat is the surface the image ofwhich is a point. Observe that the “input space” is the space (either the (ξ, t)-domain or (r, z)-domain) in which the seismic data are found that are to be mapped or imaged, while the “outputspace” is the one in which the result obtained by the mapping or imaging procedure is to be placed.

To explain both definitions, let us give two examples. We consider at first the map migration.In this case, the time domain is obviously the input space of the seismic map migration and the depthspace correspondingly specifies the output space. A point in the output space of a map migrationis, therefore, the depth point M used in method (M2). After Definition #1, the Huygens surfaceis therefore the inplanat for a migration, or briefly the “migration inplanat.” Thus, we could callmethod (M2) the “map migration with inplanats.” On the other hand, let us consider point N inthe time domain used in the map migration method (M1). After Definition #2, the isochron is theoutplanat for the migration or briefly the “migration outplanat.” Method (M1) therefore describesthe “map migration with outplanats.” Now consider the demigration. As the roles of input spaceand output space are now interchanged in comparison to the migration, so are the roles of theinplanat and outplanat. We find now that the Huygens surface is the “demigration outplanat”and the isochron is the “demigration inplanat.” Methods (D1) and (D2) describe then the “mapdemigration with outplanats” and the “map demigration with inplanats,” respectively.

At this stage, the above two definitions may seem to confuse rather than to enlighten thesituation. However, in general, we will encounter map transformations or imaging procedures whereboth the input and output spaces may coincide with either the time-trace domain or the depthdomain. In such cases, it no longer makes sense to speak about Huygens surfaces and/or isochrons.But it still makes sense to speak about inplanats and outplanats according to the above definitions.It will turn out, in fact, that for all mapping and imaging problems (such as DMO, MZO, AMO,redatuming, remigration, etc.), there always exists one solution based on inplanats and one onoutplanats.

Generalized Hagedoorn’s imaging conditions.—As we will see below, the general inplanatsand outplanats satisfy corresponding tangency properties to the ones formulated above for theHuygens surface and the isochron. The general statements corresponding to Hagedoorn’s imagingconditions (H1) and (H2) are

(H3) The outplanat pertaining to a point on a given surface in the input domain is tangent to thatsurface’s image in the output domain.

(H4) The inplanat pertaining to a point on a searched-for image surface in the output domain istangent to the original surface in the input domain.

Correspondingly to the alternative formulations (H1’) and (H2’), also the generalized Hagedoorn’simaging conditions (H3) and (H4) can be recast as

(H3’) Any searched-for image surface in the output domain can be understood as the envelope ofall outplanats belonging to points along the original surface in the input domain.

(H4’) Any given surface in the input domain can be understood as the envelope of all inplanatsbelonging to points along that surface’s image in the output domain.

34 CHAPTER 2. DESCRIPTION OF THE PROBLEM

It is these fundamental properties that permit a Kirchhoff-type treatment of seismic imaging prob-lems as described in this book. Let us now see how these unified imaging conditions give rise toother mapping procedures that are completely analogous to the above described map migrationand demigration transformations.

2.4.3 Unified approach: mapping

In view of the above, we are ready to deal with many different map-transformation problems thatcan be solved by the unified approach, that is, by applying migration and demigration in sequence.We recognize that there are two fundamental ways of realizing this sequence. The more intuitiveone is to first apply a migration to the data and then demigrate the resulting depth image. Thisorder describes processes such as the transformation of data from one measurement configurationto another (including MZO, AMO, SCO, etc.), wavemode transformation (e.g., simulating P-Sreflections from P-P data), redatuming, wavefield continuation, data regularization, etc. In otherwords, it describes data transformations that take place entirely in the time-trace domain. Below, werestrict ourselves to the description of the general configuration transform problem, with an MZOas an application example. However, we keep in mind that the other operations are conceptuallyidentical and can thus be discussed in a completely analogous way.

To avoid confusion with the nomenclature, we stress that Bleistein and Jaramillo (1998; seealso Bleistein et al., 2001) refer to the complete set of these operations as “data mapping.” Note,however, that their concept of “mapping” is different from the one used in this book. In Bleisteinet al., (2001), “data mapping” includes the full true-amplitude operations that we describe inSection 2.4.4 and refer to as “imaging.”

The other possibility of chaining migration and demigration is to first apply a demigration toan already available depth image and afterwards migrate the resulting simulated data. This ordergives rise to a process called remigration, that is, the updating of a depth image due to an improvedmacrovelocity model. Of course, other image transformations in the depth domain can be thoughtof, most notably a wavemode transformation (e.g., transformation of a P-P migrated image into aP-S migrated one). Here, we will focus on the remigration problem.

It is worthwhile to note that all problems described now as purely kinematic map transfor-mations will have their imaging counterparts in the true-amplitude imaging theory indicated belowand fully developed in Chapter 9.

Let us now proceed with the consideration of the configuration transform and remigration asillustrations of mapping transformations.

• Problem #1: Configuration transformCompute from a given traveltime surface for a primary reflection in one (input) seismic

configuration the corresponding primary-reflection time surface in a different (output) config-uration.

Solution: Let the input and output configurations be parameterized by the vector parametersξ and η, respectively. Moreover, let the time coordinates in the input and output sections bedenoted by t and τ , respectively. Also, let ΓR: t = TR(ξ) and ΓR: τ = TR(η) denote the inputand output primary-reflection time surfaces, respectively.

2.4. MAPPING VERSUS IMAGING 35

1. Two-step solution:The problem can be solved by a direct two-step procedure. Apply as a first step one ofthe above two map-migration procedures (M1) or (M2) to the input primary-reflectiontime surface ΓR in order to construct the reflector ΣR in the (r, z)-domain. As a secondstep, the reflector ΣR is then again demigrated by one of the two above map-demigrationschemes (D1) or (D2) using now the output configuration. The result is the searched-foroutput primary-reflection traveltime surface ΓR.

2. One-step solution:We now observe that, besides the previous two-step solution, there exists two alternativeone-step solutions.(i) Solution with outplanatsFor each point NR on an input reflection-time surface, ΓR, construct its outplanat in

the output time-trace domain. This surface is obtained by the following procedure. Firstdetermine the isochron ΣN in depth that corresponds to NR using the input configura-tion. Thereafter, demigrate that isochron (by treating it as a reflector) with the outputconfiguration into the output domain. As stated in Hagedoorn’s imaging condition (H3’),the envelope of all outplanats pertaining to all points NR on ΓR is then the searched-foroutput reflection-time surface ΓR. We call this mapping procedure “configuration trans-form with outplanats.”(ii) Solution with inplanatsFor each point N on a dense grid in the output time-trace domain, construct its in-

planat in the input domain. This is now obtained as follows. First determine the isochronΣN in depth that corresponds to N using the output configuration. Thereafter, demi-grate that isochron (by treating it as a reflector) with the input configuration into theinput domain. According to Hagedoorn’s imaging condition (H4), all points N = NR

in the output space that have inplanats tangent to the known surface, ΓR, define thenthe searched-for (simulated) output reflection-time surface, ΓR. We call this mappingprocedure “configuration transform with inplanats.”

Note that, although based on the cascaded solution, the two solutions (i) and (ii) are confinedto transformations performed exclusively in the time-trace domain once the inplanats andoutplanats are known. We will later extend these solutions to realize the true-amplitudeimaging configuration transformation.

Example: map migration to zero offset (MZO).—A useful example of the configura-tion transform procedure is provided by migration to zero offset (MZO)3. MZO means thetransformation of primary reflection surfaces, ΓCOR : t = T co

R (ξ), from an input constant-offsetsection into their corresponding primary-reflection counterparts ΓZOR : τ = T zo

R (η) in a simu-lated zero-offset output section. For simplicity, we illustrate the procedure with figures froma 2-D MZO, where all surfaces cited above are just curves.

1. Two-step solution:First, migrate the input primary-reflection time surface ΓCOR to the reflector ΣR in depth

3In practice, the MZO procedure is generally broken up into two steps. The first step consists of an application ofan NMO correction to the common-offset reflections. The resulting section is then subjected to a DMO correction. TheMZO process is a special case of the more general offset-continuation (OCO) transformation, where primary reflectionsin one (input) common-offset section are transformed into the corresponding ones under a different common-offsetconfiguration.

36 CHAPTER 2. DESCRIPTION OF THE PROBLEM

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Fig. 2.5. Two-step solution for MZO, first step: CO map migration (a) Migration with outplanats(isochrons). Top: A CO reflection-time curve ΓCOR is indicated as a sequence of points (diamonds) inthe input space, i.e., the CO time-trace domain. Bottom: Shown are the isochrons for the indicatedpoints on the CO reflection (dashed lines). The envelope of these isochrons defines the migratedtarget reflector ΣR in the depth domain (bold line). (b) Migration with inplanats (Huygens curves).Top: A CO reflection-time curve ΓCOR is indicated in the input space, i.e., the CO time-trace domain.Also shown are some of the CO Huygens curves (dashed lines) that are constructed for all pointson a grid in the depth domain (bottom). Certain Huygens curves (bold lines) are tangent to theinput CO reflection-time curve ΓCOR . Bottom: Grid of points (diamonds) in the depth domain forwhich Huygens curves are computed. Depth points pertaining to those Huygens curves that aretangent to the input CO reflection-time curve ΓCOR define the migrated target reflector ΣR.

with method (M1) or (M2) using the constant-offset configuration. This step is illustratedin Figure 2.5 for a single dome-like reflector in a 2-D constant velocity medium. Thevelocity above the reflector is 2000 m/s and the offset is 1000 m. After this depthmigration, demigrate the so-obtained reflector image ΣR with method (D1) or (D2)using the zero-offset configuration. This step is illustrated in Figure 2.6 for the samemodel as before. Note that in Figure 2.6, the process flow is inverted with respect toFigure 2.5, i.e., the input space is the depth domain in the bottom part of the figure andthe output domain is the ZO time-trace domain in the top part of the figure. The resultof this two-step procedure is the searched-for zero-offset primary-reflection traveltimesurface ΓZOR . In other words, the MZO has been successfully performed.

2. One-step solution:(i) MZO with outplanats

2.4. MAPPING VERSUS IMAGING 37

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Fig. 2.6. Two-step solution for MZO, second step: ZO map demigration. (a) Demigration withoutplanats (Huygens curves). Top: ZO Huygens curves computed for points along the migratedtarget reflector (bottom). The envelope of all Huygens curves defines the true ZO reflection-timecurve ΓZOR . Bottom: String of points (diamonds) along the target reflector ΣR as constructed inthe first step of the two-step MZO. (b) Demigration with inplanats (isochrons). Top: Given is agrid of points (diamonds) in the output space (ZO time-trace domain). To each point belongs aZO isochron in the depth domain (bottom). Bottom: Some of the isochrons constructed for all gridpoints in the ZO time-trace domain (dashed lines). Those isochrons (bold lines) that are tangent tothe migrated target reflector define grid points on the searched for ZO reflection-time curve ΓZOR .

Construct an MZO outplanat for each point NR = (ξ, T coR (ξ)) on the constant-offset

reflection-time surface ΓCOR . The MZO outplanat is the envelope of all zero-offset Huygenssurfaces that correspond to points on the common-offset isochron defined by point NR. Inother words, the MZO outplanat is the reflection-traveltime surface of the CO isochronwhen treated as a reflector in an experiment with the ZO configuration. The constructionof an MZO outplanat is illustrated in Figure 2.7a. The envelope of all MZO outplanatspertaining to the points NR on ΓCOR is the zero-offset reflection-time surface ΓZOR . Thismapping procedure is graphically explained in Figure 2.8a for the same earth model as inthe previous figures. Note that the surface that we have called here the “MZO outplanat”(as in Definition #2) coincides with the “MZO smear-stack surface” or “smile operator”as defined in Deregowski and Rocca (1981). The ”DMO smile” is obtained from this”MZO smile” by an NMO correction.(ii) MZO with inplanatsLet a point N be given in the zero-offset (output) section to be constructed. Its MZO

38 CHAPTER 2. DESCRIPTION OF THE PROBLEM

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Fig. 2.7. One-step solution for MZO. (a) Construction of an MZO outplanat. Top: Indicated is apoint N (diamond) in the CO time-trace domain (chosen on the CO reflection-time curve ΓCOR ) andits respective MZO outplanat (bold line) that is tangent to the ZO reflection-time curve ΓZOR inthe ZO time-trace domain. Bottom: Depicted are the CO isochron (bold line) for point N and theZO (normal-incidence) rays from this isochron to the measurement surface. The emergence pointsof these ZO rays define the lateral position of the MZO outplanat and their traveltimes define itsshape. (b) Construction of an MZO inplanat. Top: Indicated is a point N (diamond) in the ZOtime-trace domain (chosen on the ZO reflection-time curve ΓZOR ) and its respective MZO inplanat(bold line) that is tangent to the CO reflection-time curve ΓCOR in the CO time-trace domain.Bottom: ZO isochron (bold line) for point N and three CO reflection rays from this isochron to themeasurement surface. The midpoint coordinates of all such CO rays define the lateral position ofthe MZO inplanat and their traveltimes define its shape.

2.4. MAPPING VERSUS IMAGING 39

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Fig. 2.8. One-step solution for MZO. (a) Direct one-step MZO with outplanats. A CO reflec-tion-time curve ΓCOR is indicated by a string of points (diamonds) in the input space, i.e., theCO record. To all points, outplanats are constructed that provide as their envelope the ZO reflec-tion-time curve ΓZOR in the output space, i.e., the ZO time-trace domain, which will constitute theZO record to be simulated. (b) Direct one-step MZO with inplanats. To each grid point in the ZOtime domain belongs an inplanat in the CO time domain. Inplanats (displayed as bold lines) forgrid points on the searched-for ZO reflection-time curve ΓZOR are tangent to the CO reflection-timecurve ΓCOR . In other words, the grid points of these bold inplanats define the searched-for ZOreflection-time curve ΓZOR .

inplanat is the envelope of all constant-offset Huygens surfaces that correspond to pointson the zero-offset isochron defined by the given point N . In other words, the MZOinplanat is the reflection traveltime curve of the ZO isochron when treated as a reflectorin an experiment with the CO configuration. The construction is graphically explainedin Figure 2.7b. All points in the zero-offset section that have MZO inplanats tangentto the known surface ΓCOR in the ZO time-trace domain define the unknown zero-offsetreflection-time surface ΓZOR in the CO time-trace domain. The situation is visualized inFigure 2.8b for the same earth model as before.

• Problem #2: RemigrationLet an inaccurate reflector location be given, e.g., as originally obtained from a map migrationunder an initial (input) macrovelocity model. The task of a remigration is, then, to relocatethe original reflector to a more accurate location using an updated (output) macrovelocitymodel4.

Solution: Let the input and output depth sections be parametrized by global Cartesiancoordinate systems (r, z) and (ρ, ζ), respectively. Also, let ΣR : z = ZR(r) and ΣR : ζ = ZR(ρ)denote the input and output reflector locations and v(r, z) and v(ρ, ζ) the initial and updatedmacrovelocity models, respectively.

1. Two-step solution:This problem can again be solved by a two-step procedure. Apply at first a map dem-

4If the two macrovelocity models do not differ too much, one generally calls the imaging procedure that correctsthe image a “residual migration” (Rothman et al., 1985). Here, we allow significant differences between both and,therefore, refer to the process as remigration. In the seismic literature, it is also known as velocity continuation(Fomel, 1994). Again, only the kinematic aspects are treated here. The corresponding imaging problem is discussedbelow.

40 CHAPTER 2. DESCRIPTION OF THE PROBLEM

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Fig. 2.9. Two-step solution for map remigration, first step: Map demigration. (a) Demigrationwith outplanats (Huygens curves). Bottom: A wrongly migrated reflector image ΣR is indicated asa sequence of points (diamonds) in the input space, i.e., the wrongly migrated depth domain. Top:Shown are the Huygens curves for the indicated points on the reflector image ΣR (dashed lines),calculated using the incorrect migration velocity v(r, z). The envelope of these Huygens curvesdefines the demigrated Reflection-time curve ΓR in the time-trace domain (bold line). (b). Demi-gration with inplanats (isochrons). Bottom: A wrongly migrated reflector image ΣR is indicated inthe input space, i.e., the wrongly migrated depth domain. Also shown are some of the isochrons(dashed lines) that are constructed with the wrong velocity v(r, z) for all points on a grid in thetime-trace domain (top). Certain isochrons (bold lines) are tangent to the input reflector image ΣR,Top: Grid of points (diamonds) in the time-trace domain for which the isochrons are computed.Grid points pertaining to those isochrons that are tangent to the input reflector image ΣR definethe demigrated reflection-time curve ΓR.

igration to the (inaccurately mapped) subsurface reflector ΣR with v(r, z) as the mac-rovelocity model. This provides the (reconstructed) reflection-time surface ΓR in the(ξ, t)-domain. The map demigration can be realized, of course, either by method (D1) or(D2). This first step is graphically explained in Figure 2.9 for the same 2-D earth modelas to illustrate the MZO. The input for this example, i.e., the wrongly positioned reflec-tor image has been generated by a map migration of the (originally picked) zero-offsettraveltime curve with a wrong migration velocity of v(r, z) = 1500 m/s. As a secondstep, use method (M1) or (M2) to depth-migrate the reconstructed traveltime curve ΓRresulting from step one with the (updated) output velocity field v(ρ, ζ). This second stepis illustrated in Figure 2.10. Its result is the searched-for improved reflector ΣR. In other

2.4. MAPPING VERSUS IMAGING 41

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Fig. 2.10. Two-step solution for map remigration, second step: Map migration. (a) Migrationwith outplanats (isochrons). Bottom: Isochrons computed with the updated velocity v(ρ, ζ) forpoints along the demigrated traveltime curve ΓR (top). The envelope of all isochrons defines theremigrated reflector image ΣR. Top: String of points (diamonds) along the reflection-time curve ΓRas constructed in the first step of the two-step remigration. (b) Migration with inplanats (Huygenscurves). Bottom: Given is a grid of points (diamonds) in the output space (updated depth domain).To each point belongs a Huygens curve in the time-trace domain (top), calculated with the updatedvelocity v(ρ, ζ). Top: Some of the Huygens curves constructed for all grid points in the depth domain(dashed lines). Those Huygens curves (bold lines) that are tangent to the demigrated reflection-timecurve ΓR define grid points on the searched-for remigrated reflector image ΣR.

words, the remigration has been successfully completed.

2. One-step solution:Correspondingly to Problem #1, there also exist two alternative one-step solutions:(i) Solution with outplanatsFor each point MR on the inaccurately mapped reflector ΣR in the input space, i.e.,

the (r, z)-domain, construct its “remigration outplanat” in the output space, i.e., the(ρ, ζ)-domain. This surface is obtained by the following procedure. First, determine theHuygens surface ΓM of MR using the input velocity field v(r, z). Thereafter, migratethat Huygens surface (by treating it as a reflection-time surface) with the output mac-rovelocity model v(ρ, ζ) into the output domain (the (ρ, ζ)-domain). The constructionis graphically explained in Figure 2.11a. According to Hagedoorn’s imaging condition(H3’), the envelope of all outplanats that correspond to points MR on ΣR is then thesearched-for reflector image ΣR. We call this mapping procedure a “remigration with

42 CHAPTER 2. DESCRIPTION OF THE PROBLEM

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Fig. 2.11. One-step solution for map remigration. (a) Construction of a remigration outplanat.Bottom: Indicated is a point M (diamond) in the (wrongly migrated) input depth domain (chosenon the reflector image ΣR) and its respective remigration outplanat (bold line) that is tangent to theupdated reflector image ΣR in the (remigrated) output depth domain. Top: Depicted is the Huygenscurve for point M (bottom) as calculated with the input velocity v(r, z). A map migration of thistraveltime curve to depth with the improved velocity v(ρ, ζ) provides the remigration outplanat. (b)Construction of a remigration inplanat. Bottom: Indicated is a point M (diamond) in the outputdepth domain (chosen on the updated reflector image ΣR) and its respective remigration inplanat(bold line) that is tangent to the wrongly migrated reflector image ΣR in the input depth domain.Top: Huygens curve for point M as calculated with the updated velocity v(ρ, ζ). A map migrationof this traveltime curve to depth with the wrong velocity v(r, z) provides the remigration inplanat(bottom).

2.4. MAPPING VERSUS IMAGING 43

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Fig. 2.12. One-step solution for map remigration. (a) Remigration with outplanats. A wronglymigrated reflector image ΣR is indicated by a string of points (diamonds) in the input space,i.e., the wrongly migrated depth domain. To all points, remigration outplanats are constructed(dashed lines) that provide as their envelope the improved reflector image ΣR in the output space,i.e., the remigrated depth domain. (b) Remigration with inplanats. To each grid point in the(remigrated) output depth domain (diamonds) belongs an inplanat in the (wrongly migrated) inputdepth domain. Inplanat (bold lines) that for grid points on the searched-for improved reflector imageΣR are tangent to the wrongly migrated reflector image ΣR. In other words, the grid points thatpertain to these bold inplanats define the searched-for improved reflector image ΣR.

outplanats.” Figure 2.12a graphically explains the remigration with outplanats for a 2-D constant velocity model and a zero-offset configuration. The updated (output) velocityis the true medium velocity of v(ρ, ζ)=2000 m/s and the initial (input) velocity is theincorrect migration velocity of v(r, z)=1500 m/s.(ii) Solution with inplanatsFor each point M on a dense grid in the output depth domain, construct its inplanat

in the input domain. This is now obtained as follows. First determine the Huygens sur-face that corresponds to M using the output velocity field v(ρ, ζ). Thereafter, migratethat Huygens surface (by treating it as a reflection-time surface) with the input velocityfield v(r, z) into the input depth domain. The construction of an inplanat is graphicallyexplained in Figure 2.11b. According to Hagedoorn’s imaging condition (H4), all pointsM = MR in the output space that have inplanats tangent to the known surface ΣR de-fine then the searched-for (simulated) output reflector image, ΣR. We call this mappingprocedure a “remigration with inplanats.” The process is illustrated in Figure 2.12b forthe same 2-D earth model as before.

Correspondingly to Problem #1, the two solutions (i) and (ii) are now confined to the depthdomain. We will later extend these solutions to solve the true-amplitude remigration imagingproblem. It will become clear, then, why it is better to develop a theory that starts fromthe wrongly migrated depth image rather than from the original time-trace domain reflectionseismograms to obtain an improved depth-migrated image with v(ρ, ζ).

Note that it is fairly simple to invert any of the map transformations described above, i.e.,to perform the inverse transform operation. As we have already indicated in the case of a mapmigration and demigration, one only needs to exchange the input space by the output space andthe roles played by the inplanats and outplanats. This generally applies to all seismic map trans-formations and later also to the image transformations involving amplitudes. In other words, for

44 CHAPTER 2. DESCRIPTION OF THE PROBLEM

each imaging procedure (e.g., true-amplitude MZO, true-amplitude remigration, true-amplituderedatuming, etc.) there exists an inverse operation that is described by the same means.

The above-discussed kinematic migration and demigration mapping schemes (i.e., methods(M1) through (D2) or any combination of them) are simple to conceive in terms of the kinematicsand the geometry involved. However, when we try to apply them in practice, their simplicity hasto be paid for with a tremendous interpretational effort which requires identification of selectedreflections and the picking of their arrival times. These events are generally difficult to identify,especially in the time-trace domain and in the presence of complex velocity fields and severalreflectors. In addition, the inherent property of map transformations of not being able to useseismic amplitudes has more and more turned out to be disadvantageous. Nevertheless, as alreadyindicated, the same geometrical concepts as discussed above remain useful in connection with thetrue-amplitude migration, demigration, and all other image transformation procedures describedin Chapter 9. True-amplitude migration based on a weighted diffraction stack, and true-amplitudedemigration based on a weighted isochron stack, will turn out to be the key operations for allimage transformations. They incorporate the same simple kinematic and geometrical concepts asthe map-migration, map-demigration and map-transformation methods presented above.

At this point, let us mention that the above defined inplanats and outplanats are not onlyuseful concepts for Kirchhoff-type imaging as discussed in this book, but also for a solution of thesame imaging problems using the concept of image waves (Hubral et al., 1996b). In that approach,the inplanats and outplanats assume the role of Huygens image waves similar to conventionalHuygens secondary waves excited by incident wavefields.

2.4.4 Seismic reflection imaging

All the above kinematic map-migration, map-demigration, and map-transformation procedureshave their counterparts in the more general framework of the Kirchhoff-type seismic-reflection-imaging procedures. For this, an identification of events and picking of arrival times is no longerrequired. A true-amplitude image migration, for example, is applied to a seismic section as a wholeand migrates all present reflection events of the selected imaging wavemode into a depth section. Asin the kinematic mapping procedures, we have at our disposal two Kirchhoff-type migration methods(subsequently called image migrations) that are very similar to the above kinematic methods (M1)and (M2). These are

(M3) Image migration with Huygens surfaces (inplanats)This image migration can be looked upon as the “dynamic counterpart” to the kinematic

map migration with Huygens surfaces explained in method (M1). Step (a) remains the sameas described there. Step (b), however, consists now of stacking (i.e., summing) the seismictrace amplitudes, modified by a suitable true-amplitude migration weight. For each pointM in the migrated section to be constructed, the stack is carried out along the Huygenssurface ΓM (i.e., along the migration inplanat) and the obtained stack value is assigned tothat point. For this reason, this Kirchhoff-type image-migration method is sometimes alsocalled “diffraction-stack migration” or briefly “diffraction stack.” According to Hagedoorn’simaging condition (H1), those diffraction traveltime surfaces ΓM that pertain to points MR onan actual—although unknown—reflector are tangent to actual reflection traveltime surfaces.When performing the diffraction stack along these tangent Huygens surfaces it gives rise toconstructive interference. Thus, it will produce a significantly higher value than along another

2.4. MAPPING VERSUS IMAGING 45

diffraction traveltime surface that does not belong to a point on any reflector in depth. Inother words, those points M = MR, to which a significant amplitude is assigned by thediffraction stack, provide the searched-for depth migrated reflector-image strip attached toΣR. Moreover, due to the migration weights applied in the stacking process, true-amplitudemigration outputs, in the sense defined earlier, are obtained.

(M4) Image migration with isochrons (outplanats)This image migration is the dynamic counterpart to the above described kinematic map

migration with isochrons explained in method (M2). Step (a) remains essentially the sameas before. However, the isochrons ΣN are constructed not only for points on t = TR(ξ) butfor all points N on a dense grid in the (ξ, t)-domain. Step (b) now consists of distributing(i.e., smear-stacking or smearing out), the seismic trace amplitude found at N along thecorresponding isochron ΣN , i.e., the amplitude value at N , multiplied with suitable true-amplitude migration weights, is assigned to each depth point MI on the isochron (i.e., themigration outplanat). According to Hagedoorn’s imaging condition (H2), the isochrons ΣN

pertaining to points N = NR on ΓR are tangent to the searched-for reflector, thus giving riseto constructive interference at the reflector. Therefore, after the “weighted smearing” has beenperformed for all points N of the (ξ, t)-domain, those depth points M , for which a significantamplitude is obtained as a sum of all smeared-out values, provide the searched-for depth-migrated reflector-image strip attached to ΣR. The use of appropriate weights guaranteesthat the true-amplitude migration outputs, in the sense defined earlier, are achieved.

Note that both image-migration methods (M3) and (M4) are completely equivalent. Only theorder of summation is changed from one method to the other. This can be explained as follows.Suppose that the data consists of just one data sample located at point N in the time-trace domain.Perform the diffraction stack for all subsurface points M , i.e., perform the same image migrationas described in (M3). Note that the record space has zero values at all points, except N . Thus,the stack will provide results different from zero only at those depth points M whose Huygenssurfaces pass through N . According to the definition of the isochron, these are exactly the pointsMI on ΣN . Thus, the amplitude value given at N , multiplied with a true-amplitude weight, isdistributed along the isochron ΣN . In the case of an arbitrary data set, we can conclude thatsmearing out the seismic amplitudes at all points N in the (ξ, t)-domain along their respectiveisochrons ΣN can be looked upon as individually performing the diffraction stack for each datasample in the (ξ, t)-domain. Summing thereafter all the obtained diffraction-stack migration outputsthat result from all individual data samples, provides then the complete migrated image. Within theframework of seismic image migration, both schemes (M3) and (M4) are, thus, identical, i.e., theycan mathematically be described by one and the same diffraction-stack integral. Consequently,the diffraction-stack migration can be viewed as being the image-migration counterpart of bothmap-migration methods (M1) and (M2).

The migrated image of a reflection-signal strip (Figure 2.4) is a depth-migrated strip of acertain width that depends on the length of the source pulse, the reflector dip, the incidence angle,and the local velocity in the macrovelocity model. Therefore, the thickness of this depth-migratedstrip may vary along the reflector (Tygel et al., 1994b).

The analogous inverse operation to the above procedure is Kirchhoff image demigration. It isapplied to a given migrated section in the depth domain. Image demigration transforms all presentdepth-migrated reflector images in the depth section back into their counterpart time-trace domainreflections. Correspondingly to the methods (M3) and (M4), we have for the image demigration

46 CHAPTER 2. DESCRIPTION OF THE PROBLEM

the following two imaging methods (D3) and (D4). They are the dynamic counterparts of the mapdemigration methods (D1) and (D2), respectively.

(D3) Image demigration with Huygens surfaces (outplanats)This image demigration is the dynamic counterpart to the above described kinematic map

demigration with Huygens surfaces explained in method (D1). Step (a) remains essentiallythe same as before. However, the Huygens surfaces are constructed, not only for points MR

on ΣR but for all points M on a dense grid in the (r, z)-domain. Step (b) now consists ofdistributing (i.e., smear-stacking or smearing out) the seismic trace amplitude at M along thecorresponding Huygens surface ΓM . In other words, the amplitude value atM is assigned (withcertain weights) to each point N on the Huygens surface ΓM (the demigration outplanat).Again according to Hagedoorn’s imaging condition (H1), the Huygens surfaces ΓM pertainingto points M = MR on ΣR are tangent to the searched-for reflection-time surface ΓR, thusgiving rise to constructive interference at ΓR. Therefore, after the “weighted smearing” hasbeen performed for all points M of the (r, z)-domain, those points N = NR, for which asignificant amplitude is obtained as a sum of all smeared-out values, provide the searched-for demigrated reflection-signal strip attached to the reflection-time surface ΓR. The use ofappropriate weights in the smearing procedure guarantees the true-amplitude character ofthe resulting demigration output.

(D4) Image demigration with isochrons (inplanats)As in method (D2), step (a) consists of constructing the isochrons ΣN (i.e., the demigrationinplanats) for all points N on a dense grid in the (ξ, t)-domain. Step (b) is analogous to itscounterpart in the diffraction stack, i.e., to step (b) of method (M3). However, the Huygenssurfaces ΓM (i.e., the migration inplanats) are now replaced by isochrons ΣN (the demigrationinplanats). In other words, for each point N on a dense grid in the trace-time domain, Step (b)consists of weighted stacking (i.e., summing) the data along the respective isochron ΣN andassigning the result to that point. Correspondingly, this method is called the “isochron-stackdemigration” or briefly “isochron stack.” Hagedoorn’s imaging condition (H2) guaranteesthat those isochrons ΣN that belong to points NR on actual—although unknown—reflection-traveltime surfaces are tangent to reflectors in the migrated section. Due to the resultingconstructive interference, the stack along these will then produce a significantly higher valuethan along isochron surfaces not attached to a point on any reflection traveltime surface inthe time-trace domain. Thus, all points N = NR in the (ξ, t)-domain for which the isochronstack provides a significant amplitude value pertain to the searched-for reflection-signal stripattached to the reflection-time surface ΓR. Again, the stacking weights are necessary to obtaintrue demigration amplitudes in the sense discussed in the Introduction (Chapter 1).

By arguments very similar to the ones above in connection with the image migration, one can seethat both image-demigration methods (D3) and (D4) are also equivalent. Therefore, we can saythat both map-demigration methods (D1) and (D2) find also their dynamic counterpart in onlyone seismic image-demigration scheme, namely the isochron stack.

By looking back once more at the migration and demigration methods (M1) through (D4),we observe the following interesting property. For all methods discussed above, the inplanat of amapping procedure becomes the stacking surface for the corresponding (stacking-type) imagingprocedure. In the same way, the outplanat for a mapping method becomes the smearing surfacefor the corresponding imaging method. This observation is not confined to true-amplitude image

2.4. MAPPING VERSUS IMAGING 47

migration and demigration. It holds for all true-amplitude image-transformation procedures de-scribed by the unified approach, such as the configuration transform and remigration describedabove. The two methods of either stacking or smearing out seismic amplitudes turn always out tobe fully equivalent. For that reason, we will restrict ourselves only to the discussion of performingweighted stacks over problem-specific inplanats.

We are now ready to describe the unified approach to true-amplitude seismic-reflection imag-ing. It is based on nothing more than a weighted diffraction stack in the time-trace domain and aweighted isochron stack in the depth domain that can both be cascaded or chained. Both opera-tions are mathematically represented by certain weighted surface integrals described in Chapters 7and 9. There, the weights will be suitably defined. After a proper definition, both weighted stack-operations can be applied in sequence in order to solve a variety of true-amplitude imaging problems.These include, of course, the imaging counterparts of Problem #1 (Configuration Transform) andProblem #2 (Remigration). Let us stress, once more, that all image transformation problems un-der the unified approach admit both a two-step solution, as well as, more attractively one singlestacking procedure in the input space. In the latter case, there is no need of explicitly performingthe sequence of image migration and demigration or vice versa. All that is essentially involved isthe weighted summation of trace amplitudes along the corresponding problem-specific inplanats.In case of Problem #1, a single stack in the time-trace domain and in case of Problem #2, a singlestack in the depth domain is all that is needed. Of course, the single-stack approach does make useof both the diffraction-stack migration and the isochron-stack demigration integrals. However, asshown in Chapter 9, these two integrals can be suitably merged into one single stacking operationthat characterizes the individual image transformation. The analysis will include the developmentof adequate true-amplitude weights for the chained imaging operations.

At this stage, let us readdress Problem #1 and Problem #2 in the more general frameworkof seismic reflection imaging.

• Problem #1: Configuration transformTransform a given seismic section recorded under a certain (input) measurement config-

uration into a corresponding section as if recorded under a different (output) measurementconfiguration. All primary reflections in the simulated output section have not only the correcttraveltimes, but also true amplitudes in the sense described in the Introduction (Chapter 1).

Solution:

1. All one needs to solve Problem #1 is to apply as a first step a weighted true-amplitudediffraction-stack migration to the input section in order to obtain the depth-migratedimage. As a second step, this image is then demigrated by a weighted true-amplitudeisochron stack, using, however, the output instead of the input configuration. The resultis the searched-for simulated section.

2. There exists, however, a single-stack solution using inplanats. For each point N on adense grid in the output space, its inplanat is constructed as previously explained for themap configuration transform. A single stack along that inplanat, with suitably computedtrue-amplitude weights, yields the output of the configuration transform at N . Thosepoints N = NR in the output space to which a significant stack result is assigned arethe ones that determine the simulated output primary reflections as required by thetransformation.

48 CHAPTER 2. DESCRIPTION OF THE PROBLEM

Example: image MZO.—As an important example of the above described configurationtransform, we consider image MZO. In the framework of seismic imaging, its task can be for-mulated as follows. Transform all (unidentified) primary reflections in a given common-offsetrecord into their corresponding zero-offset reflections. The simulated zero-offset reflectionshave, besides the correct traveltimes, also true amplitudes5

1. Two-step solutionFirst, migrate the constant-offset record by a weighted true-amplitude diffraction stack.Then, demigrate this image by a weighted true-amplitude isochron stack, using, however,the zero-offset instead of the constant-offset configuration. The result is the searched-forsimulated zero-offset record.

2. One-step solutionThere exists, however, a single-stack solution using MZO inplanats. For each point N inthe common-offset record, construct the MZO inplanat as explained for map MZO. Asingle stack along the MZO inplanat yields the MZO output at N . All points N = NR inthe output space to which a significant stack result is assigned, define then the simulatedzero-offset primary reflections.

As already indicated above, when addressing the kinematic aspects, the MZO procedure is inseismic processing implemented in two steps, first by performing an NMO correction followedby a DMO correction. The first operation represents an image transformation that cannot bedescribed by a chained migration and demigration (upon which the unified approach is based)because it is a point-to-point rather than a point-to-surface transformation. In fact, the NMOcorrection is just a sophisticated coordinate change. However, the DMO correction, as wellas the overall transformation, i.e., the MZO procedure, can be described by this sequence. Asimple way to obtain the DMO inplanat is to apply the NMO correction to the MZO inplanat(Tygel et al., 1998).

• Problem #2: RemigrationTransform a given (input) seismic depth section, obtained by means of a true-amplitude imagedepth migration with an initial macrovelocity field v(r, z), into a corresponding output true-amplitude depth-migrated section, that would be obtained using an updated macrovelocitymodel v(ρ, ζ).

Solution:

1. Problem #2 can be solved by applying at first an isochron-stack true-amplitude demigra-tion to the input depth section using the initial velocity field v(r, z). The resulting timesection is then true-amplitude migrated with the new velocity field v(ρ, ζ) to providethe searched-for updated depth-migrated section.

2. Correspondingly to Problem #1, there also exists an alternative one-step solution usinginplanats. For each depth point M in the output space (the updated depth-migratedsection to be obtained), a “remigration inplanat” is constructed in the input space (theinitial, given depth-migrated section). The inplanat is determined by constructing theHuygens surface ΓM of M using the new macrovelocity field v(ρ, ζ) and thereafter map-migrating this Huygens surface using the old velocity field v(r, z). The image remigration

5As explained in the Introduction (Chapter 1), this means that the original common-offset geometrical-spreadingfactors are automatically transformed into their corresponding ones that would be observed under a zero-offsetmeasurement configuration. The angle-dependent reflection coefficients are, however, preserved.

2.4. MAPPING VERSUS IMAGING 49

is then performed by a single stack along the remigration inplanats for all points M inthe output space. All points M = MR, to which a significant amplitude is assigned (as aresult of constructive interference in the remigration inplanat stack), pertain to updatedreflector images in the remigrated depth section. The specific remigration weights thatare used in the process guarantee true-amplitude results in the sense defined earlier.

Above we have indicated that image transformations, as, e.g., dealt with in Problems #1 and #2can be performed in either two steps (i.e., by separately performing an explicit image migrationand a subsequent demigration) or one step (i.e., by one single stacking procedure). Involved in theone-step procedure are summations of trace amplitudes along inplanats in the input space.

It is interesting to observe (see Figure 2.7) that outplanats or inplanats in the one-stepapproach can be restricted to a smaller spatial extent than the apertures that correspond to thestacking surfaces involved in the two-step approach. The smaller the offset for the configurationtransform, or the less the difference between the velocity fields v(r, z) and v(ρ, ζ) for the remigration,the more limited is the spatial extent of the corresponding inplanats, along which the stack is tobe applied. As a consequence, the one-step approach leads to less stacking operations. Moreover,the computational effort can be further reduced by performing the implementation in a target-oriented way, i.e., in a selected record- or image-space window. This property makes the single-stack solutions of image transformations attractive. This even applies to Problem #2 which oneusually solves by computing the depth-migrated section for v(ρ, ζ) from scratch, i.e., by directlydepth-migrating the original record. It should, however, be kept in mind that this reduction of theoperator size also means a reduction of traces that will be summed together by the process. In thisway, the constructive interference implicitly assumed in Kirchhoff-type methods can be reduced,thus leading to a poorer image quality. There is, thus, a trade off between computational efficiencyand image quality that has to be taken into account in the actual implementation choices involved.

Let us stress, once more, that all Kirchhoff-type imaging methods described by the unifiedapproach can be realized, not only by a stack along the problem-specific inplanats, but also bya smear-stack along the corresponding outplanats. Both descriptions of the process are basicallyequivalent as they involve only a different order of summation. The discrepancies that can occur aredue to numerical problems associated with the implementation. These include, e.g., different finitesampling intervals, different interpolation, etc. For definiteness (and without loss of generality),we restrict ourselves to the discussion of stacking and do not further comment on smear-stacking.The mathematical description in Chapters 7 and 9, nevertheless encompasses both procedures. Theresulting integrals can be directly implemented either as stacking or smear-stacking schemes.

We have provided an overall, pictorial and nonmathematical view of the unified approachto seismic reflection imaging that is the subject of this book. It is now natural that we say a fewwords about the present status and future potential of the application of this theory to real practicalproblems. The best known and most commonly applied transformation of the theory is Kirchhoffprestack depth migration (PreSDM). Its kinematic use in 3-D is nowadays a practical reality. Forother imaging schemes, the effort to calculate traveltime tables does not exceed that of Kirchhoffmigration by a significant amount. A configuration transform can essentially be realized making useof the very same traveltime tables twice, combining ray segments differently. For remigration witha slightly modified macrovelocity model, one can also use, on account of the linearization principle(Nolet, 1987), the same rays to compute the traveltimes in both models. Thus, available computersshould pose no restriction to the kinematic use of Kirchhoff imaging as described here.

50 CHAPTER 2. DESCRIPTION OF THE PROBLEM

Concerning amplitudes, an active debate on the pros and cons of using full true-amplitudeweight functions for Kirchhoff migration is being carried out within the Geophysical community.Workshops on true amplitudes have been organized and sponsored by the SEG and EAGE on aregular basis for several years. On the more theoretical side, one of the main topics of discussion ishow to make full use of amplitudes and other related attributes as obtained from true-amplitudeimaging. On the computational level, the discussions involve the determination and storage of thevarious quantities that make up the weights, these being required at a huge number of grid points.

There is, however, a very fortunate property common to all Kirchhoff imaging processes suchas, e.g., true-amplitude migration or MZO. They can be easily applied to selected small, targetregions using a fraction of the whole data sets. Utilizing this property, which allows for an efficientapplication of the methods, 3-D true-amplitude migration has already been carried out in practice(see, e.g., Tura et al., 1997, 1998). Another way of efficiently implementing true-amplitude imagingmethods is using simpler macrovelocity models, where analytic formulas for traveltimes and weightfunctions are available (see, e.g., Martins et al., 1997). Dellinger et al. (2000) have demonstratedfor true-amplitude migration that this can lead to depth images that are satisfactory to the eye ofthe interpreter. Similar behavior can be expected for other Kirchhoff-type imaging methods.

We feel, nevertheless, that computer limitations should not be an argument in the long run.A good example is full 3-D kinematic Kirchhoff migration. This procedure was impossible a fewyears ago due to prohibitive costs of the necessary traveltime computations. Today, it has becomea routine practice. It is to be expected that computer hardware and software will further evolve,thus enabling true-amplitude imaging as a feasible option. It is to be kept in mind that providingsatisfactory depth images is not the only objective of true-amplitude imaging methods. Their maingoal is to determine high-quality amplitudes with a great potential to be useful for the inversionof geologic, especially reservoir, attributes. In this respect, the already available results on thecombined use of true-amplitude migration and MZO for AVO/AVA inversion are most encouraging(Beydoun et al., 1993; Hanitzsch, 1995; Oliveira et al., 1997; Tura et al., 1998; Gray, 1999).

2.5 Summary

The aim of this book is to introduce and thoroughly describe a Unified Approach to Seismic Re-flection Imaging. In this chapter, we have discussed the main assumptions, fundamental conceptsand principles that constitute its basis.

We have defined the earth model for which the proposed theory is valid, and we have com-mented on the assumptions about the macrovelocity model that has to be given a priori in orderto apply the imaging methods. Note that only one selected “imaging wavemode” can be imagedat a time. Moreover, we have specified an adequate parameterization of the seismic measurementconfigurations that are used to acquire and organize the seismic reflection data that are the inputto the various imaging transformations comprised by the Unified Approach.

To distinguish between purely kinematic and fully dynamic seismic operations, we have de-noted these as mapping and imaging, respectively. Procedures that kinematically transform pickedsurfaces (traveltimes or reflectors) into other, corresponding surfaces are called mapping procedures.On the other hand, techniques that achieve a dynamic transformation of a complete seismic section,including pulse shapes and amplitudes, are referred to as imaging techniques. If the amplitudes aretreated in the way specified in the Introduction (Chapter 1), the attribute “true-amplitude” is used.

2.5. SUMMARY 51

We have next introduced the concepts of a diffraction-time or Huygens surface in the time-trace domain and of an isochron in depth domain. These surfaces, which we have called “Hage-doorn’s imaging surfaces,” play a fundamental role in the imaging theory to be developed in thisbook. They are distinguished by important tangency properties that we have termed “Hagedoorn’simaging conditions.” These conditions will be rigorously addressed and quantified in Chapter 5.Hagedoorn’s imaging surfaces and conditions can be generalized to encompass other imaging prob-lems. This gives rise to surfaces we call “inplanats” and “outplanats.” An inplanat is a surface inthe input space of a certain image transformation that is mapped into a single point in the outputspace. For instance, inplanats for migration are the Huygens surfaces and inplanats for demigrationare the isochron surfaces. Correspondingly, an outplanat is the surface in the output space thatresults from the application of a certain seismic image transformation to a single point in the inputspace. In other words, a point in the input space is mapped into the outplanat in the output spaceof the given image transformation. For example, outplanats for migration are the isochrons andoutplanats for demigration are the Huygens surfaces.

With these definitions in mind, we have outlined the key operations or “building blocks”for the unified approach. These are the migration and demigration procedures. We have shownby simple geometric arguments how Hagedoorn’s imaging conditions work to realize the migrationand demigration operations. By cascading or chaining a migration and a demigration for differentmacrovelocity models, measurement configurations and/or imaging wavemodes, it is possible toformulate a wealth of arbitrary seismic image-transformation problems. As illustrative examples,we have discussed the configuration transform and remigration. Since the configuration transformis, on its own, a generalization of various problems, we have selected for its illustration the importantexample of migration to zero offset (MZO). For MZO and for remigration, we have demonstratedhow inplanats and outplanats are constructed. In this way, we have laid the foundations for theresulting true-amplitude imaging theory, to be fully elaborated in Chapter 9. There exist, however,many more imaging problems for which problem-specific inplanats and outplanats can be specifiedand which can therefore be solved with the present approach.

Our intention in this Chapter was divided into two main aspects. First, we wanted to providethe reader with an appreciation of the geometrical or kinematic (i.e., mapping) aspects that formthe basis of and will be incorporated into the theory described in this book. Second, we wanted tooutline the idea of how the computational kinematic aspects come into play when implementing anyof the proposed image-transformation methods. In fact, if no attention is to be paid to amplitudes,and all the kinematic and geometrical aspects to construct inplanats are understood, all that hasbeen said in this Chapter is already sufficient to practically implement any image transformation.However, whoever desires to better understand how seismic trace amplitudes are accounted for intrue-amplitude image transformations, such as image migration, demigration, redatuming, MZO,remigration, etc., should proceed to the mathematical details presented in the following chapters.

52 CHAPTER 2. DESCRIPTION OF THE PROBLEM

Chapter 3

Zero-order ray theory

In this chapter, we give a brief introduction to high-frequency wave propagation in isotropic, in-homogeneous, layered 3-D media as is described by zero-order ray theory. We will go as far asneeded for the development and understanding of the imaging theory described in the later chap-ters. The main purpose of this chapter is therefore twofold. One aim is to formulate ray-theoreticalexpressions for the elementary seismic waves by which seismic reflections are described in seismicrecords (as, e.g., common-shot, common-receiver, common-midpoint, or common-offset gathers). Itis these elementary waves from which the seismic images (e.g., depth-migrated images) are to beconstructed by the imaging processes that are studied below. The second aim is to provide ray-theoretical expressions for all quantities that will appear in the imaging theory to be developed.For a more detailed treatment of most of the topics dealt with in this chapter, see, e.g., the variousextensive publications of Cerveny (1985, 1987, 1995, 2001) on the topic. Those readers whose maininterests lie in true-amplitude imaging in the spirit of the previous chapter, without going intothe underlying details of forward wave propagation, are suggested to continue their lecture withChapter 7.

3.1 Wave equations

Wave propagation in isotropic, inhomogeneous media is described by the well-known elastodynamicwave equation. Its derivation from linear elastodynamics is well explained in many text books (e.g.,Aki and Richards, 1980; Cerveny, 2001) and need not be repeated here. Let us denote the three-dimensional displacement vector (that describes the vector displacement of a medium particle fromits original position r = (r1, r2, r3 = z) at time t in global Cartesian coordinates) by u = u(r, t).Here and in the following, 3-D vectors are characterized by a hat above the symbol to distinguishthem from 2-D vectors. In particular, the three-dimensional position vector in global Cartesiancoordinates r = (r1, r2, r3 = z) is to be distinguished from the two-dimensional vector r (withouta hat) that will be used later. This 2-D vector consists of the first two components of r, i.e.,r = (r1, r2). Correspondingly, ∇ = (∂/∂r1, ∂/∂r2, ∂/∂r3) in distinction to ∇ = (∂/∂r1, ∂/∂r2).The symbols ∇·, ∇×, and ∇ signify the divergence, curl, and gradient operations, respectively.

53

54 CHAPTER 3. ZERO-ORDER RAY THEORY

Elastic Parameter SymbolFunction ofλ and µ

Function ofMp and µ

Function ofk and µ

Function ofEY and σ

1st Lame parameter λ λ Mp − 2µ k − 2µ

3

σEY(1 − 2σ)(1 + σ)

2nd Lame parameter(Shear Modulus)(Rigidity)

µ µ µ µEY

2(1 + σ)

P-Modulus Mp λ+ 2µ Mp k +4µ

3

EY (1 − σ)

(1 − 2σ)(1 + σ)

Bulk Modulus k λ+2µ

3Mp −

4µ

3k

EY3(1 − 2σ)

Compressibility κ3

3λ+ 2µ

3

3Mp − 4µ

1

k

3(1 − 2σ)

EY

Young Modulus EYµ(3λ+ 2µ)

λ+ µ

µ(3Mp − 4µ)

Mp − µ

9µk

3k + µEY

Poisson Ratio σλ

2(λ+ µ)

Mp − 2µ

2Mp − 2µ

3k − 2µ

2(3k + µ)σ

P-wave velocity α

√

λ+ 2µ

%

√

Mp

%

√

3k + 4µ

3%

√

EY (1 − σ)

%(1 − 2σ)(1 + σ)

S-wave velocity β

õ

%

õ

%

õ

%

√

EY2%(1 + σ)

Acoustic velocity c

√

λ

%

√

Mp

%

√

k

%–

Table 3.1. Relationships between different elastic parameters

Provided one neglects body forces (i.e., the source term), the elastodynamic wave equationcan be written as

(λ+ µ)∇(∇ · u) + µ(∇ · ∇)u+ ∇λ(∇ · u) + ∇µ× (∇ × u) + 2(∇µ · ∇)u = %∂2

∂t2u . (3.1.1)

Here, λ and µ are the (spatially varying) Lame parameters. They are related to other, more physicalelastic parameters that are given in Table 3.1. Parameter % denotes the (spatially varying) densityof the medium.

A medium with µ = 0 is usually called a fluid or an acoustic medium. We immediately observefrom Table 3.1 that in this case also EY = 0, Mp = k = λ = κ−1, and σ = 1

2 . In such a medium,the elastodynamic wave equation (3.1.1) reduces to

λ∇(∇ · u) + ∇λ(∇ · u) = %∂2

∂t2u (3.1.2a)

or

∇(λ∇ · u) = %∂2

∂t2u . (3.1.2b)

3.2. RAY ANSATZ 55

Dividing equations (3.1.2) by %, taking the divergence, and replacing the displacement vector u bythe pressure field p = −λ∇ · u, commonly used to describe waves in acoustic media, one arrives at

∇ ·(

1

%∇p

)

=1

λp , (3.1.3)

where the dot denotes the time derivative. Equation (3.1.3) is known as the acoustic wave equation.For a medium with constant density %, this equation reduces to the well-known form

∇2p =

1

c2p , (3.1.4)

where c =√

λ/% is the acoustic wave velocity. Note that even in constant-density media, c may bespatially varying due to a nonconstant λ.

3.2 Ray ansatz

3.2.1 Homogeneous medium

It is well-known (see, e.g., Aki and Richards, 1980) that in a homogeneous, isotropic elastic medium(with λ, µ, and % constant and µ different from zero) two types of elastic elementary waves of avectorial character may exist: the compressional, or so-called primary (P) wave and the shear, orso-called secondary (S) wave, which are completely decoupled. If the wavefield is generated by anomnidirectional compressional point source, i.e., a source that is concentrated in one single pointand emits an equal amount of compressional energy in all directions, only a P-wave exists.

Provided the medium is homogeneous and described by the constant parameters λ, µ, and%, the elastic wave equation with a omnidirectional compressional point source at rS, emiting asource wavelet f [t], reads

%∂2

∂t2u− (λ+ 2µ)∇(∇ · u) + µ∇ × (∇ × u) = f = −αF [t] ∇δ(r − rS) , (3.2.1)

where F [t] = f [t] and where α =√

(λ+ 2µ)/% is the P-wave propagation velocity (see also Ta-ble 3.1). The source term on the right-hand side generally may contain an additional factor thatdepends on the direction of r − rS and describes the source directivity (also known as radiationpattern). That factor is constant here because of the spherical symmetry due to our assumption ofan omnidirectional point source.

The solution of the elastic wave equation (3.2.1) can be obtained by Lame’s Theorem (see,e.g., Aki and Richards, 1980). This theorem states that upon substitution of u = ∇Φ + ∇ × Ψand f = ∇φ+ ∇ × ψ, equation (3.2.1) decouples into

%d2Φ

dt2= φ+ %α2

∇2Φ , (3.2.2a)

%d2Ψ

dt2= ψ + %β2

∇2Ψ , (3.2.2b)

where β =√

µ/% is the S-wave propagation velocity (see also Table 3.1). Due to the particularform of the source term f in equation (3.2.1), we immediately see that φ = −αF [t] δ(r − rS) and

56 CHAPTER 3. ZERO-ORDER RAY THEORY

ψ = 0, where 0 the 3-D zero vector. With homogeneous initial conditions for the displacementfield, i.e., u(r, 0) = 0 and ˙u(r, 0) = 0, which translate into Φ(r, 0) = 0 and Ψ(r, 0) = 0, thesewave equations have the well-known solutions

Φ = − 1

4π%α |r − rS |F

[

t− |r − rS |α

]

, (3.2.3a)

Ψ = 0 . (3.2.3b)

The particle displacement is then given bu u = ∇Φ. Thus, in far-field approximation, i.e., neglectingthe term of order |rG − rS |−2, the analytic solution at a receiver position rG reads

u(rG, t) =1

4π%α2

f [t− |rG − rS | /α]

|rG − rS |rG − rS|rG − rS |

, (3.2.4)

where the polarization vector, (rG − rS)/|rG − rS | is the unit vector in propagation direction atthe receiver location G.

3.2.2 Inhomogeneous medium

In inhomogeneous media, the solution of the elastodynamic wave equation (3.1.1) is considerablymore difficult than in a homogeneous medium. In general, the wavefield is not separable intoseveral independent elementary waves, because the propagation of the compressional and shearportions of the wavefield are no longer independent. Although the earth appears often to be wellapproximated by an inhomogeneous medium, independent compressional and shear waves havenevertheless been observed in almost every seismic record. Actually, in a smooth inhomogeneousmedium, the complete high-frequency elastic wavefield is indeed approximately separable into manyelementary compressional and shear-wave contributions. To be precise, we have to remark that inan inhomogeneous layered medium, the term “primary wave” is also used to denote a particular(not necessarily compressional) body wave, which causes the primary reflections. In the context ofthe ray method these are easily explained. They are introduced to distinguish them from “multiplereflections.”

For the following considerations, we assume a 3-D model of the earth. This consists of a stackof isotropic, laterally and vertically inhomogeneous layers separated by continuous and smoothfirst-order discontinuities of almost arbitrary shape. At a first order discontinuity at least one ofthe three medium parameters changes abruptly. Higher-order discontinuities (Cerveny, 2001) willnot be considered. The radii of curvature of interfaces are assumed to be large in comparison toseismic wavelengths and the lateral and vertical velocity and density variations within a layer arealso expected to be smooth within a typical seismic wavelength. A ray-theoretical ansatz for anelementary wave to describe the solution of the elastodynamic wave equation will thus be justified.In the following, we will refer for brevity to the model that is implicitly assumed in ray theory as an“inhomogeneous medium.” The properties of the resulting high-frequency elementary compressionaland shear waves are very similar to those of P- and S-waves propagating in a homogeneous elasticmedium. Therefore, they are also called P- and S-waves.

3.2.3 Time-harmonic approximation

The most common way to represent the total wavefield in an inhomogeneous medium by elementarybody waves is by expressing the solution of the wave equation for each elementary body wave in

3.2. RAY ANSATZ 57

form of a so-called ray series. In the frequency domain, this is a series in inverse powers of thecircular frequency ω (see, e.g., Babich, 1956; Karal and Keller, 1959; Luneburg, 1964: Cerveny,2001). In the present work, we rely, as is usually done in most practical applications in seismologyand seismics, only on the leading term of the ray series which is of the order ω0. Therefore, thefollowing description of high-frequency elementary body wave propagation is called the zero-orderray-theory description. This description also implies that in all derivations below, we always haveto consider only terms of the highest order in ω.

The fundamental idea upon which zero-order ray theory is based is to approximate an “ele-mentary time-harmonic body wave” with frequency ω at a position r in the form of a time-harmonicplane wave, however with a spatially varying amplitude and phase. In symbols, the ansatz for thesolution of the elastodynamic wave equation (3.1.1) reads

u(r, ω) = U(r)e−iωT (r)f [ω] , (3.2.5)

where the factor expiωt has been ignored on both sides of equation (3.2.5). In other words,equation (3.2.5) represents the Fourier transform of u(r, t). The function f [ω] denotes the complexFourier spectrum of the seismic source signal f [t]. The direction of the (real) vectorial amplitudefactor U(r) determines the polarization direction of the considered elementary seismic wave. Itstraveltime is given by the function T (r) and, thus, its “propagating wavefront” is described byt = T (r). The traveltime T (r) is closely related to the so-called eikonal S(r) = T (r)v(r) (Bornand Wolf, 1987), where v(r) is the local wave velocity at r. Therefore, also T (r) is often referredto as the eikonal. For arbitrary inhomogeneous media t = T (r) is an arbitrarily curved and oftenmulti-valued surface. Note that the vectorial amplitude factor U , the traveltime T , and also thepropagation direction available from ∇T of the time-harmonic wave (3.2.5) generally depend on rbut are independent of ω.

In correspondence to equation (3.2.5), the zero-order high-frequency ray ansatz to find thesolution of the acoustic wave equation (3.1.3) in the frequency domain reads

p(r, ω) = P(r)e−iωT (r)f [ω] (3.2.6)

with the obvious meaning of the quantities involved. Before we show how to find the two unknownfunctions P(r) and T (r) for a specific problem, we show how to reformulate the time-harmonicansatzes (3.2.5) and (3.2.6) in the time domain.

3.2.4 Time-domain expressions

The time-domain or transient counterpart u(r, t) of approximation (3.2.5) can be obtained from asuperposition of time-harmonic solutions u(r, ω) expiωt by means of an inverse Fourier transform.

The Fourier-transform pair is defined in this book by

u(r, ω) =

∞∫

−∞

u(r, t)e−iωtdt (3.2.7a)

and

u(r, t) =1

2π

∞∫

−∞

u(r, ω)eiωtdω . (3.2.7b)

58 CHAPTER 3. ZERO-ORDER RAY THEORY

Given a (real) source wavelet, source pulse or source signal f [t] (which in reality is causalbut may be defined as a high-frequency signal for positive and negative t), its Fourier transformf [ω] according to equation (3.2.7a) satisfies the well-known relationship f [ω]∗ = f [−ω], where theasterisk denotes the complex conjugate. Using the inverse Fourier transform (3.2.7b), this permitsus to represent f [t] in the form

f [t] =1

πRe

∞∫

0

f [ω]eiωtdω

. (3.2.8)

In this way we can avoid an integration over negative frequencies.

It is to be stressed that one of the basic assumptions of the ray approximation is that f [ω] ' 0for small frequencies 0 < ω < ω0, where ω0 is sufficiently large so that possible variations of thethe medium can be considered low frequency.

Applying the inverse Fourier transform as given by equations (3.2.7b) or (3.2.8) to expression(3.2.5), we obtain (for real U) its time domain form

u(r, t) = U(r)f [t− T (r)] . (3.2.9)

In many cases, it turns out to be very useful to leave aside the symbol Re in equation(3.2.8) and to work with the full complex source signal

F [t] =1

π

∞∫

0

f [ω]eiωtdω . (3.2.10)

This is called the “analytic source signal” assigned to f [t]. As is well known (see, e.g., Tygel andHubral, 1987), its real part is the original source wavelet, i.e., Re F [t] = f [t], and its imaginarypart is related to f [t] by means of the Hilbert transform, i.e.,

Im F [t] = HT f [t] =1

πPV

∞∫

−∞

f [t′]

t− t′dt′ , (3.2.11)

where PV denotes the Cauchy principal value of the integral. We have thus

F [t] = f [t] + iHT f [t] . (3.2.12)

In that way, one is led to the so-called “transient analytic solution” of an elementary body wave.This results from a frequency-integration of formula (3.2.5), leading to the analytic source waveletF [t] multiplied by the amplitude factor U(r). The transient analytic elastodynamic ray ansatz(3.2.5) and consequently also the resulting solution reads then in the time domain

U(r, t) = U(r)F [t− T (r)] . (3.2.13)

As we will see below, it will be useful in this description to allow for a complex amplitude factorU(r) to simplify the description of caustics and overcritical reflections. Of course, the real signalu(r, t) as recorded in a seismic survey will now be described by the real part of the analytic signalof equation (3.2.13), i.e.,

u(r, t) = Re

U(r, t)

= Re

U(r)F [t− T (r)]

. (3.2.14)

3.2. RAY ANSATZ 59

The very same reasoning as above applies to the acoustic case, where the real ray ansatzp(r, t) is analogously replaced by the transient analytic acoustic ray ansatz given by

P (r, t) = P(r)F [t− T (r)] . (3.2.15)

Here, P(r) is the amplitude factor of the acoustic pressure at r, which is also allowed to be complex.From now on, we will use these analytic quantities for all our considerations. However, we wantto emphasize that only the real parts of the resulting final transient analytic solutions remain thephysically meaningful quantities.

Note, however, that in contrast to the source signals and amplitude factors, we assume thetraveltime T in equations (3.2.13) and (3.2.15) [or (3.2.5) and (3.2.6)] always to be real. In this way,we neglect evanescent waves and complex rays, which have little importance in seismic reflectionimaging. We also neglect beams and packages (Popov, 1982), of which we make no use.

3.2.5 Validity conditions

Ray theory is a very widely developed tool to describe high-frequency seismic body wave propa-gation in inhomogeneous layered media. Its validity conditions were extensively discussed, e.g., byBen-Menachem and Beydoun (1985), Kravtsov and Orlov (1990), Cerveny (1995, 2001), or Popovand Camerlynck (1996). Therefore we will only briefly address the question of validity. In spite ofthe many investigations on the subject, only heuristic criteria exist to determine whether zero-orderray theory is a good approximation to true wave propagation. One of the most frequently givenconditions is the following one. To guarantee the approximate validity of the above ansatzes (3.2.5)and (3.2.6), the Fourier spectrum f [ω] of the seismic source wavelet f [t] is required to effectivelyvanish for frequencies ω < ω0 = v(r)/`0, where `0 is a length scale of the inhomogeneities of themedium. It can, for example, be determined as the smallest of all distances `ψ, where ψ stands forany of the involved quantities like, e.g., the medium parameters λ, µ, or %, the wave velocity, or thereflector curvature, etc. For each of these quantities characterizing the medium, `ψ is the maximaldistance for which

`ψψ

∣∣∣∇ψ

∣∣∣ 1 . (3.2.16)

In other words, the approximate high-frequency solutions (3.2.5) and (3.2.6) require that the ma-terial parameters of the medium should not significantly vary within distances of the order of theprevailing wavelength of the wavefield to be described. An alternative formulation of the aboveconditions would claim that within a so-called Fresnel volume around the ray, the velocity anddensity variations are smooth and of small amount.

Let us illustrate condition (3.2.16) by means of an example. Consider a seismic velocity thatvaries over a distance of 100 m from 3 km/s to 3.3 km/s. This is equivalent to a velocity gradientof 3/s. Thus, upon division of this value by the velocity of 3 km/s, condition (3.2.16) for thecorresponding length scale of the velocity requires that `ψ 1 km. Accepting that 300 m is muchless than 1 km, we find that ray theory can be expected to well-describe waves with frequencieshigher than 10 Hz in such a medium. By the same considerations, the minimum frequency for avelocity variation from 3 km/s to 3.1 km/s is 3 Hz. Of course, to determine whether ray theory canprovide an adequate description of wave propagation in a given medium, all its parameters have tobe studied in the same way.

Of course, the very existence of reflections tells us that earth properties often vary muchmore rapidly than this. What we actually do is to partition changes into into “slow” and “fast”,

60 CHAPTER 3. ZERO-ORDER RAY THEORY

and then use the slow changes to construct the propagation model and the fast changes to describethe reflections.

It is to be observed, however, that even where condition (3.2.16) is satisfied, the zero-orderray approximation may fail. This was shown by Popov and Camerlynck (1996) for an acousticmedium using a counterexample involving a fairly simple model. They suggest the following, morerigorous condition that is based on the zero- and first-order terms of the ray series. It reads

∣∣∣U (1)/ωU (0)

∣∣∣ 1 , (3.2.17)

where the zero-order amplitude coefficient U (0) =∣∣∣U

∣∣∣ in the elastic case and U (0) = P in the

acoustic case. Also, U (1) is the corresponding first-order amplitude coefficient of the ray series.Since we consider only zero-order ray theory, we drop the upper index from U (0) from now on.

Condition (3.2.17) is, of course, much more difficult to check than (3.2.16), since it involvesthe computation of U (1). Popov and Oliveira (1997) discuss this condition for elastic media. Prac-tical observations confirm that once the above condition (3.2.16) is satisfied, the ray-theoreticaldescription of a wavefield in form of the above product ansatz (3.2.6) or (3.2.5) is usually justified.In any case, the fact that ray theory describes actually observed reflections reasonably well in manypractical situations is its final justification.

3.3 Eikonal and transport equations

We now insert the ray ansatz (3.2.5) into the elastodynamic wave equation (3.1.1) and ansatz (3.2.6)into the acoustic wave equation (3.1.3). In this way, we obtain separate partial differential equations(the eikonal and transport equations) for the traveltime or eikonal T and amplitude U of therespective elementary wave to be described. Considering the transient solutions (3.2.13) and (3.2.15)for a high-frequency source signal would lead to the same equations.

3.3.1 Acoustic case

Let us start for didactic reasons with the acoustic wave equation (3.1.3) in the frequency domain.Differentiating ansatz (3.2.6) we obtain

∇P =[

∇Pe−iωT −Pe−iωT iω∇T]

f [ω] . (3.3.1)

With the identity

∇ ·(

1

%∇P

)

= − 1

%2∇% · ∇P +

1

%∇

2P , (3.3.2)

the substitution of expression (3.3.1) in equation (3.1.3) leads to

1

λ(iω)2Pe−iωT = − 1

%2∇% · ∇Pe−iωT − 1

%2∇% · ∇T Pe−iωT (−iω) +

P%e−iωT (−iω)∇

2T

+1

%∇

2Pe−iωT +2

%∇P · ∇T e−iωT (−iω) +

P%e−iωT (−iω∇T )2 . (3.3.3)

3.3. EIKONAL AND TRANSPORT EQUATIONS 61

Here, we have introduced the notation ∇2 = ∇ ·∇. Multiplication with %, division by Pe−iωT , andreordering the result in powers of ω yields

−ω2[

∇T 2 − %

λ

]

− iω

P

[

2∇P · ∇T + P∇2T − P

%∇% · ∇T

]

+%

P ∇ ·(

1

%∇P

)

= 0 . (3.3.4)

For ansatz (3.2.6) to be a solution of equation (3.1.3), this equation must be satisfied for all highfrequencies. This implies that the coefficients of ω2, ω1, and ω0 must vanish independently. However,with only two quantities, P and T , to be determined, this can, in general, only be achieved for thecoefficients of ω2 and ω1. From these, we obtain two key equations for the searched-for quantitiesP and T in the ansatzes (3.2.6) or (3.2.15) for the elementary acoustic wave.

Eikonal equation

The coefficient of ω2 provides the eikonal equation

∇T 2= 1/c2 (3.3.5)

that describes the wavefront t = T (r) of the considered elementary wave. Here, we have used againthe acoustic wave velocity c =

√

λ/%.

Transport equation

The coefficient of ω1 provides the transport equation

2∇P · ∇T + P∇2T − P

%∇% · ∇T = 0 (3.3.6)

that describes the amplitude P(r) once the traveltime function T (r) is known, i.e., once the eikonalequation (3.3.5) is solved. Both key equations (3.3.5) and (3.3.6) need of course to be subjectedto certain initial conditions to determine a specific elementary wave that propagates from a well-specified source into the homogeneous medium and to one or many receivers. More details onspecifying solutions are described below.

After a multiplication of equation (3.3.6) with P/%, we recognize that it may be rewritten inthe more convenient form

∇ ·(

P2

%∇T

)

= 0 . (3.3.7)

This is the most compact expression for the transport equation that will be solved in Section 3.6.

It is interesting to observe that the transport equation is directly related to the energyflux of the elementary wave under consideration. By multiplying equation (3.3.6) with the complexconjugate P∗ of P, then taking the complex conjugate of the whole equation, adding both equations,and dividing by 4

√%, one obtains

∇ ·(

1

2

P∗P%

∇T)

= 0 , (3.3.8)

which is the law of conservation of energy. It states that the divergence of the energy flux vanisheswithin a propagating acoustic wave. Below we will introduce the concept of a ray tube. The law of

62 CHAPTER 3. ZERO-ORDER RAY THEORY

conservation of energy is often associated with the statement that in the ray method, the energyflux is confined to a ray tube. Note that for real P, equations (3.3.7) and (3.3.8) coincide, i.e., thecompact form (3.3.7) of the acoustic transport equation is the law of conservation of energy.

Additional condition

The coefficient of ω0 in equation (3.3.4) provides the condition

%

P ∇ ·(

1

%∇P

)

=∇2PP − ∇%

%· ∇P

P = 0 , (3.3.9)

which can only be fulfilled exactly in homogeneous media. It implies that the derivatives of themedium density and of the elementary wave amplitude must be negligible. This condition canbe understood as an additional criterion on whether the obtained ray expressions are a goodapproximation to true wave propagation. For further details, see Cerveny (2001).

3.3.2 Elastodynamic case

Along similar lines (but in a more tedious way) as indicated in detail for the acoustic case above,the insertion of ansatz (3.2.5) into the elastodynamic wave equation (3.1.1) yields three equationsfor the coefficients of ω2, ω1, and ω0, respectively. These are

−%U + (λ+ µ)(∇T · U)∇T + µ ∇T 2U = 0 , (3.3.10a)

(λ+ µ)[

(∇ · U)∇T + ∇(U · ∇T )]

+ µ[

2(∇T · ∇)U + ∇2T U

]

+(U · ∇T )∇λ+ (∇µ · ∇T )U + (∇µ · U)∇T = 0 , (3.3.10b)

(λ+ µ)(∇ · ∇)U + µ∇2U + (∇ · U)∇λ

+2(∇µ · ∇)U + ∇µ× (∇ × U) = 0 , (3.3.10c)

where the third equation cannot be fulfilled exactly but provides a criterion on the validity of theapproximation.

Eikonal equations

Equation (3.3.10a) can be rewritten in the form

(Γ˜− I

˜)U = 0 or Γ

˜U = U , (3.3.11)

where I˜

is the 3× 3-unit matrix. Equation (3.3.11) formally represents an eigenvector equation fora unit eigenvalue. The 3 × 3-matrix Γ

˜, which has the elements

Γij =λ+ µ

%

∂T∂xi

∂T∂xj

+µ

%δij ∇T 2

, (3.3.12)

is commonly called the Christoffel matrix. Its eigenvalues can be readily determined as

G1 =λ+ 2µ

%∇T 2

, G2 = G3 =µ

%∇T 2

. (3.3.13)

3.3. EIKONAL AND TRANSPORT EQUATIONS 63

Besides the trivial solution U = 0, equation (3.3.11) can obviously only be fulfilled, if eitherG1 orG2

and G3 equal one. This fact can be interpreted in the following way: For high-frequency elementarybody waves as implied in the ray method, there exist two types of body waves that propagateindependently in an inhomogeneous, isotropic, elastic medium. The body wave corresponding toG1 = 1 is the elementary compressional wave or P-wave. Its traveltime is described by the eikonalequation

∇T 2= 1/α2 , (3.3.14)

where α =√

(λ+ 2µ)/% is the P-wave velocity. The body wave corresponding to G2 = G3 = 1 isthe elementary shear wave or S-wave. Its traveltime is described by the eikonal equation

∇T 2= 1/β2 , (3.3.15)

where β =√

µ/% is the S-wave velocity. Equations (3.3.14) and (3.3.15) express the fact that P-and S-waves are decoupled asymptotically (i.e., for high frequencies) in (slightly) inhomogeneousmedia.

Transport equation for the P-wave

The solution of equation (3.3.11) shows that the eigenvector of Γ˜

corresponding to G1 is parallel to∇T and, thus, perpendicular to the propagating wavefront t = T (r). In other words, the elemen-tary compressional body wave satisfying equation (3.3.14) is linearly polarized in the propagation

direction ∇T / ∇T . This is in fact the justification to call a P-wave also a longitudinal wave.

Therefore, the vectorial amplitude U(P )

of this compressional wave can be expressed as

U(P )

= U (P ) ∇T∇T

= U (P )α∇T , (3.3.16)

where we have used the fact that for G1 = 1 the modulus ∇T = 1/α due to equation (3.3.14).The scalar amplitude factor U (P ), yet to be determined, is often simply called the amplitude of theP-wave.

Note that the linear polarization in propagation direction of the P-wave (that correspondsto the first eigenvalue G1 of Γ

˜) is only guaranteed in the zero-order ray approximation. If higher-

order terms in the ray-series are considered, U (P ) becomes the amplitude of the so-called “principalcomponent” of what is then referred to as the ray series solution of the P-wave. The higher-orderray-series terms in (1/ω)n lead to the so-called “additional components” that are perpendicularto ∇T . Since we stick in this book to the zero-order ray theory, we consider only the “principalcomponent” of the more general ray-series solution.

Insertion of the expression (3.3.16) into equation (3.3.10b) and multiplying it with ∇T yieldsafter several algebraic operations

2%α2∇U (P ) · ∇T + %α2U (P )

∇2T + U (P )

∇(%α2) · ∇T = 0 (3.3.17)

where we have used the expression λ + 2µ = α2%. After a multiplication by√

%α2U (P ), equation(3.3.17) can be written even more compactly as

∇ · (%α2U (P )2∇T ) = 0 . (3.3.18)

64 CHAPTER 3. ZERO-ORDER RAY THEORY

This is the form of the transport equation that will be solved in Section 3.6. We observe that theamplitude expression

√

%α2U (P ) plays the same role for elastic P-waves as P/√% does for acousticwaves. This may look surprising but in the high-frequency approximation and for µ = 0, we haveP = %cU (P ). Thus, the above transport equation (3.3.17) reduces to the acoustic one, equation(3.3.6).

The law of conservation of energy for the P-wave is obtained by multiplying equation (3.3.17)with the complex conjugate amplitude U (P )∗ , taking the complex conjugate of the resulting equationand adding both equations. After a multiplication by

√

%α2/4, this yields

∇ · (12%α2U (P )U (P )∗

∇T ) = 0 . (3.3.19)

As in the acoustic case, for real U (P ), equations (3.3.18) and (3.3.19) coincide.

Transport equation for the S-wave

The eigenvectors of Γ˜, corresponding to G2 and G3, fall both into the plane tangent to the wavefront

t = T (r) at the point r and perpendicular to ∇T . Equation (3.3.11), however, is not sufficient touniquely determine them. We have, thus, any freedom to choose any pair of mutually perpendicularunit vectors e1 and e2 within that plane such that the triplet e1, e2, t = ∇T / ∇T = β∇Tgenerates a right-handed Cartesian coordinate system. A useful choice for e1 and e2 results fromconsidering the so-called ray-centered coordinates described in Section 3.9 [see also Popov andPsencık (1976; 1978) and Cerveny (1987) for details].

Irrespective of the particular choice of e1 and e2, we can express the zero-order vectorial

amplitude U(S)

of the shear wave as

U(S)

= U (S)1 e1 + U (S)

2 e2 , (3.3.20)

where U (S)1 and U (S)

2 are (generally complex) amplitude factors yet to be determined for any specific

solution. Note that in general, it is not possible to find a real polarization vector eS such that U(S)

=U (S)eS , because S-waves are not linearly (as P-waves) but elliptically polarized. An expression of

this type can only be found if both U (S)1 and U (S)

2 are real or at least have the same phase. We will,

however, see that even for complex U (S)1 and U (S)

2 , the factor U (S) =

√(

U (S)1

)2+(

U (S)2

)2plays

the role of the scalar amplitude of the S-wave. Just like the amplitude U (P ) of the P-wave, thequantity U (S) becomes the amplitude of the principal component in higher-order ray theory. Thehigher-order terms of the ray-series solution for the S-wave provide additional components in thepropagation direction.

Insertion of expression (3.3.20) into equation (3.3.10b) and multiplication with e1 and e2,

respectively, yields two coupled equations for U (S)1 and U (S)

2 . They may be written as

2µ∇T · ∇U (S)1 + U (S)

1 ∇µ · ∇T + U (S)1 µ∇

2T + 2U (S)2 µ∇T · E

˜2e1 = 0 , (3.3.21a)

2µ∇T · ∇U (S)2 + U (S)

2 ∇µ · ∇T + U (S)2 µ∇

2T + 2U (S)1 µ∇T · E

˜1e2 = 0 , (3.3.21b)

where E˜k(k = 1, 2) are 3 × 3-matrices with the elements Ekij = ∂ekj/∂xi. Here, Ekij denotes the

element in the ith row and jth column of matrix E˜k. Correspondingly, ekj denotes the jth element

of vector ek.

3.4. RAYS AS CHARACTERISTICS OF THE EIKONAL EQUATION 65

By multiplying equation (3.3.21a) with U (S)1 , equation (3.3.21b) with U (S)

2 , and adding bothequations, one obtains

∇

[

µ(

U (S))2]

· ∇T + µ(

U (S))2

∇2T + 2U (S)

1 U (S)2 µ∇T · ∇(e1 · e2) = 0 . (3.3.22)

As e1 and e2 are always perpendicular to each other, their dot product vanishes everywhere and

thus the last term in equation (3.3.22) equals zero. Dividing the remaining terms by√

µ(U (S)

)2,

we may write

2∇T · ∇√

%β2(U (S)

)2+√

%β2(U (S)

)2∇

2T = 0 , (3.3.23)

where we have used the relation µ = %β2. Again, we also may write equation (3.3.23) in the moreconvenient form

∇ · (%β2(

U (S))2

∇T ) = 0 . (3.3.24)

The law of conservation of energy for the S-wave is obtained from multiplying equation

(3.3.21a) with U (S)∗

1 and taking its complex conjugate equation, as well as multiplying equation

(3.3.21b) with U (S)∗

2 and taking its complex conjugate equation. By adding the resulting fourequations and multiplying with

√

%/β2/4, one obtains

∇ · (12%β2(U (S)∗

1 U (S)1 + U (S)∗

2 U (S)2 )∇T ) = 0 . (3.3.25)

As before, for real U (S)1 and U (S)

2 , equations (3.3.24) and (3.3.25) coincide.

Generic transport equation

Comparing equations (3.3.6), (3.3.17) and (3.3.23), we see that√

%β2U (S) plays the same rolefor S-waves as

√

%α2U (P ) does for P-waves and P/√% for acoustic waves. Hence, we will confineourselves below to an explicit treatment of a generic situation with velocity v and amplitude factorU , however keeping in mind that by a proper substitution of v and U by α and U (P ), β and U (S),or c and P/%c, P-waves, S-waves, or acoustic waves can be described by the same formalism. Interms of the generic quantites v and U , the generic transport equations reads

∇ · (%v2U2∇T ) = 0 . (3.3.26)

We will solve this equation in Section 3.6.

3.4 Rays as characteristics of the eikonal equation

It is now our aim to derive the solution of the equations that govern the zero-order ray approxi-mation, that is, the eikonal and transport equations. Since the latter involves the solution of theformer, it is natural to start with the eikonal equation.

We have observed that acoustic as well as elastic P- and S-wave propagation in the high-frequency range is described by the same type of eikonal equation. Therefore, there is no need tosolve the eikonal equations (3.3.5), (3.3.14), and (3.3.15) independently, but only the general form

∇T 2= 1/v2 , (3.4.1)

66 CHAPTER 3. ZERO-ORDER RAY THEORY

where v may be replaced by any of the (spatially varying) wave velocities c, α, or β, correspondingto the type of elementary wave to be considered.

3.4.1 Slowness vector

To find solutions to equation (3.4.1), it is useful and convenient to introduce the slowness vector pof a wave. In isotropic media as considered here, it coincides with the propagation direction. It isdefined as

p = ∇T , (3.4.2)

where the gradient may be taken in any arbitrary coordinate system. For the moment, we stickfor simplicity to global Cartesian coordinates. Together with the definition (3.4.2), equation (3.4.1)reads now

p · p = |p|2 = 1/v2 , (3.4.3)

from which we immediately observe that |p| = ∇T = 1/v. Equation (3.4.3) is a nonlinear partialdifferential equation of the first order. It is a particular case of the class of Hamilton-Jacobi equations

H(r, p) = 0 , (3.4.4)

which can be solved by the method of characteristics (see, e.g. Herzberger, 1958).

Even in the case of equation (3.4.3), there are several possible ways to specify a so-calledHamiltonian H. From these possibilities, we will consider

H =1

2

(

p · p− 1/v2)

, (3.4.5a)

H = |p| − 1/v , (3.4.5b)

H = ln(v |p|) , (3.4.5c)

as the most useful ones. The so-called characteristics of equations of the type (3.4.4) are 3-D spatialtrajectories along which equation (3.4.4) is satisfied. In the particular case of the eikonal equation,these trajectories are called rays. Each of these rays is described by a function r = r(ν), where ν isa variable that monotonically increases along the ray. It cannot be chosen arbitrarily but dependson the particular choice of H. It must satisfy (Herzberger, 1958)

p · ∇pH = ∇T · drdν

=dTdν

, (3.4.6)

and thus

dν =dT

p · ∇pH, (3.4.7)

where ∇p = (∂/∂p1, ∂/∂p2, ∂/∂p3). For the above choices of H in equations (3.4.5), we have

dν = v2dT = dσ , (3.4.8a)

dν = vdT = ds , (3.4.8b)

dν = dT , (3.4.8c)

where s is the arclength along the ray and σ is another variable varying monotonically along theray that is often very useful in certain applications of ray theory (see, e.g., Bleistein, 1986; Cerveny,2001). In optical contexts, it is often referred to as the “optical length” (Sommerfeld, 1964).

3.5. RAY FIELDS 67

The traveltime T as a function of the Cartesian coordinates r is given by an integration alongall possible rays. Due to equations (3.4.8), the form of the integral depends on the particularlychosen variable. In symbols,

T =

∫

dT =

∫1

vds =

∫1

v2dσ . (3.4.9)

Note that in order to actually calculate the traveltime field T (r), one first needs to trace all raysinvolved.

3.4.2 Characteristic equations

With a characteristic (or ray) parameterized in this way, the Hamilton-Jacobi equation (3.4.4) canbe replaced by the so-called characteristic equations (Herzberger, 1958)

dr

dν= ∇pH ,

dp

dν= −∇H . (3.4.10)

For the above choices of H, these read

dr

dσ= p ,

dp

dσ=

1

2∇

(1

v2

)

, (3.4.11a)

dr

ds= vp ,

dp

ds= ∇

(1

v

)

, (3.4.11b)

dr

dT = v2p ,dp

dT = ∇

(

ln1

v

)

. (3.4.11c)

To arrive at equations (3.4.11), equation (3.4.3) has been used, i.e., |p|2 has been replaced by1/v2. Any pair in equations (3.4.11) is also called a ray-tracing system. It consists of six ordinarydifferential equations of the first order in r and p. The location vector r(ν) describes the spatialtrajectory of the ray as a function of ν and p(ν) describes the slowness vector tangent to the rayat r(ν).

To trace a single ray through an arbitrary 3-D inhomogeneous medium, one needs to solveone of the above ray-tracing systems (3.4.11) numerically. The initial conditions to be specifiedare the point O = (x0, y0, z0) where the ray starts and its initial direction, specified either by thespatial angles or by the initial slowness vector at O. The modulus of the initial slowness vectormust equal the inverse of the local velocity at O. Of course, it depends on the particular problem tobe solved which of the above ray-tracing systems (3.4.11) is the most useful one. In the following,we will restrict ourselves to the system (3.4.11b), where ν = s is the arclength of the ray. This willlead to convenient expressions. Note that

t =dr

ds= vp = p/ |p| (3.4.12)

is the unit tangent vector to the ray at r(ν).

3.5 Ray fields

Up to now, we have only considered one single ray. However, to study ray amplitudes, i.e., to solvethe transport equation along a chosen ray, we need to consider the complete wavefront t = T (r)

68 CHAPTER 3. ZERO-ORDER RAY THEORY

Fig. 3.1. A possible choice of ray coordinates at a point source.

of an elementary wave propagating through the medium along this ray. The wavefront can berepresented by a system of rays in the vicinity of the ray under investigation. Of course, all raysof this system are (in an isotropic medium) always orthogonal to the wavefront at all instants oftime. This system of rays is called a ray field. Thus, the necessity exists to uniquely identify eachray within this field.

3.5.1 Ray coordinates

Let O(x0, y0, z0) denote the initial point of one selected ray of the ray field to be studied. To uniquelyspecify each ray within the ray field, we need two ray coordinates γ1 and γ2 . A possible choice ofray coordinates for any arbitrary ray in the field could be the coordinates of the intersection pointof that ray with an arbitrary surface crossed by the selected ray at O(x0, y0, z0). More specifically,the surface can be chosen as the wavefront at O(x0, y0, z0). However, this specification will not beunique if more than one ray of the field crosses the wavefront at O. In particular, this choice for theray coordinates cannot be used to uniquely describe a wave emanating from a point source, whereall rays start from the very same point O. In this situation, another pair of ray coordinates is needed.The most common choice for the ray coordinates at a point source are two angular coordinates ofthe outgoing rays (see Figure 3.1). However, as we will see below, it is more convenient to use asthe ray coordinates γ1 and γ2 two independent components of the slowness vector at the source.These implicitly determine its angular coordinates.

Note that together with the third coordinate γ3 = s, i.e., the arclength along the selectedray, the pair of ray coordinates γ1 and γ2 provides a valid coordinate description for any pointin the medium reached by the wavefront (as, for example, demonstrated in Figure 3.1). Thus, wehave a 3-D curvilinear system of ray coordinates γ = (γ1 , γ2 , γ3 ) for any point in the ray field.Medium points that are not reached by any ray of the field under consideration have no valid raycoordinate description. This is, however, no restriction as these points need not be described whentreating that ray field.

3.6. SOLUTION OF THE TRANSPORT EQUATION 69

3.5.2 Transformation from ray to global Cartesian coordinates

Since the triplet γ = (γ1 , γ2 , γ3 ) with γ3 = s is a valid coordinate description for any point in theray field there exists a relationship r = r(γ) that can be expressed by

dr = Q˜

(r)dγ , (3.5.1)

where Q˜

(r) is the transformation matrix from ray coordinates γ to global Cartesian coordinates r.It has the elements

Q(r)ij =

∂ri∂γj

. (3.5.2)

The matrix Q˜

(r) will be of use later on when describing ray amplitudes. Note again that in corre-spondence to the Cartesian coordinate vector r, we have introduced the 3-D vector γ = (γ1 , γ2 , γ3 )to distinguish it from the 2-D vector γ = (γ1 , γ2 ) that will be used later.

3.5.3 Ray Jacobian

Another quantity of fundamental importance to describe ray amplitudes is the Jacobian determi-nant of the above transformation (3.5.1), often also called the ray Jacobian. It is expressed by

J = det Q˜

(r) =

(∂r

∂γ1× ∂r

∂γ2

)

· drds

. (3.5.3)

If the ray Jacobian is well defined and does not vanish at an arbitrary point in the medium, theray field of the considered elementary wave is called regular at that point. Points, where J eithervanishes or is not defined are called singular or caustic points.

3.6 Solution of the transport equation

With the above coordinate transformation, we are now ready to solve the transport equation. Forsimplicity, we will treat here the generic transport equation (3.3.26), where we have introduced thenotations v for the generic wave velocity and U for the generic amplitude factor. We remind thereader that the acoustic, P-, or S-wave transport equations (3.3.7), (3.3.18), or (3.3.24) are obtained

by replacing√

%v2 U with P/√%,√

%α2 U (P ), or with√

%β2U (S) =

√

%β2

((

U (S)1

)2+(

U (S)2

)2)

, re-

spectively, in equation (3.3.26). Thus, the following considerations describe acoustic, compressional,or shear waves upon the adequate substitution.

3.6.1 Solution in terms of the ray Jacobian

Using the ray Jacobian (3.5.3), the transport equation (3.3.26) simplifies considerably and can besolved analytically along the known ray. Equation (3.3.26) states that the divergence of the vectorfield %v2U2∇T vanishes. It is well known that the divergence of a vector field can also be expressed

70 CHAPTER 3. ZERO-ORDER RAY THEORY

(γ ,γ +∆γ )21 2

∆

(γ +∆γ ,γ )(γ ,γ )

1

Σs+ s

1 2

1 2

n

s

1

(γ +∆γ ,γ +∆γ )21 1 2

V

Σ2

Fig. 3.2. Choice of an arbitrary volume V over which equation (3.3.26) is integrated.

in integral form. Using Green’s theorem in an arbitrary volume V , the transport equation (3.3.26)can be rewritten as

∫∫

V

∫

∇ · (%v2U2∇T )dV =

∫

©∫

Σ

%v2U2∇T · n dΣ = 0 . (3.6.1)

Here, Σ is the surface of the volume V and n the outward-pointing normal vector to Σ. We chooseV to be the volume of an elementary ray tube between the arclength values s and s + ∆s (seeFigure 3.2). The ray tube itself is defined by the four rays specified by the ray coordinate pairs(γ1 , γ2 ), (γ1 + ∆γ1 , γ2 ), (γ1 , γ2 + ∆γ2 ), (γ1 + ∆γ1 , γ2 + ∆γ2 ). The ray tube volume V has thetotal surface Σ and dΣ is a surface element. Note that the integral representation (3.6.1) of thetransport equation (3.3.26) is obviously invalid where V is not defined or vanishes. Therefore, thesolution for U obtained in this way will not be valid at singular points or in their close vicinity.

As the side walls of the volume V are formed by rays, the scalar product ∇T · n vanishesalong these walls. Let us denote the remaining surface elements of the ray tube, which are crossedby the rays, by Σ1 (at s+ ∆s) and Σ2 (at s). At these surfaces elements, which are perpendicularto the rays, the outward-pointing normal vector n is given by

ndΣ = ±(∂r

∂γ1× ∂r

∂γ2

)

dγ1 dγ2 . (3.6.2)

where the upper sign holds for the surface Σ1 and the lower one for Σ2. The different signs are dueto the different orientation of n with respect to ∇T .

Using equation (3.4.12), the slowness vector p = ∇T may be represented as

p = ∇T =1

vt =

1

v

dr

ds, (3.6.3)

and, thus, the scalar product in integral (3.6.1) is

∇T · ndΣ = ±(∂r

∂γ1× ∂r

∂γ2

)

· drds

1

vdγ1 dγ2

= ±(J/v)dγ1 dγ2 . (3.6.4)

3.6. SOLUTION OF THE TRANSPORT EQUATION 71

Insertion of equation (3.6.4) into formula (3.6.1) yields thus∫

Σ1

∫

%vJU2 dγ1 dγ2 =

∫

Σ2

∫

%vJU2 dγ1 dγ2 . (3.6.5)

Since this is true for any arbitrarily chosen ray tube, it follows that the integrands of both sidesof equation (3.6.5) must be equal. As a consequence, the expression %vU 2J is constant along a rayand equals its initial value at the starting point P0 of the ray. Expressing the latter in the form%0v0J0U2

0 , the solution for U along the ray reads

U =

[%0v0J0

%vJ

] 1

2

U0 . (3.6.6)

We observe that the ray Jacobian J defined in equation (3.5.3) is the main factor that determines

the variation of the wave amplitude along the ray. The expression (J0/J)1

2 describes the ampli-tude loss due to the geometrical divergence of the wavefront. It is commonly referred to as thegeometrical-spreading factor. Note that although the ray Jacobian, according to equation (3.5.3),depends on the choice of the ray coordinate system γ, the ratio J0/J and thus the ray amplitudedo not.

The square root in equation (3.6.6) remains to be defined. At this stage, let us only mentionthat U may become real or imaginary as J (and J0) may take on positive or negative values. Inother words, we must find a physically meaningful definition of

√J for negative values of J . Note

also that equation (3.6.6) is obviously invalid at singular points. For finite frequencies, it providesnoncorrect values even in a close vicinity of singular points, so that one can speak about singularregions. The size of a singular region depends on the frequency content of the seismic signal f [t].

As a final comment in this section, let us remark that the solution of the transport equation(3.3.7) of the acoustic wave equation (3.1.3) can be obtained analogously or, alternatively, it canbe inferred form equation (3.6.6) by replacing

√

%v2U with P/√%. The resulting expression reads

P =

[%cJ0

%0c0J

] 1

2

P0 . (3.6.7)

3.6.2 Point-source solutions

Equation (3.6.6) is the solution of the transport equation (i.e. for the ray amplitude) at any non-singular point P on the ray where J is defined and not zero. The initial values %0, v0, J0 need tobe known at a likewise nonsingular point P0 where J0 is also defined and not zero. However, inthis book we only consider elementary body waves that have their origin in an omnidirectionalpoint source. For a ray emanating from such a point source, say, at S, the latter assumption isobviously violated. At a point source, the initial value JS of the ray Jacobian vanishes. To obtainnonvanishing amplitudes along the ray, we would have to require that the initial amplitude US isinfinite at the point source. If we assume that equation (3.6.6) is valid for any point P0 on the raynear the point source at S, then the amplitude US at S must be infinite in such a way that

limP0→S

√

%0v0J0(P0)U0(P0) = gS (3.6.8)

is a constant. In general, this constant will contain a source directivity, i.e., a dependence on theray coordinates γ.

72 CHAPTER 3. ZERO-ORDER RAY THEORY

To evaluate this limit, we suppose that there is a small sphere of radius ε around the sourcepoint S where the medium is homogeneous with density %S and velocity vS. According to equation(3.2.4), the wave amplitude at a point P0 on this sphere is given by

U0 =1

4π%Sv2Sε

. (3.6.9)

Therefore, in this situation the above limit can be written as

limε→0

√J0

4πε√

%Sv3S

= gS . (3.6.10)

It is convenient to replace the constant gS by

gS =g√%SvS

, (3.6.11)

where g is called the source strength. If the ray coordinates γ are chosen to be represented by twocomponents of the slowness vector at S, g is an adimensional quantity. Using g instead of gS , theabove limit reads now

limε→0

√J0

εvS= 4πg . (3.6.12)

On the other hand, by application of equation (3.6.6) to amplitude (3.6.9), the amplitudeU at any point P on the same ray through P0 further away from the source, where density andpropagation velocity are % and v, respectively, is given by

U =1

4π

1√%SvS%v

1√J

√J0

vSε. (3.6.13)

Because of equation (3.6.12), this reduces, in the limit of ε tending to zero, to

U =g√

%SvS%v

1√J. (3.6.14)

In the case of an omnidirectional point source as assumed here, amplitude U0 is independentof the initial propagation direction at the source. Thus, it will be possible to find a particularchoice for the ray coordinates γ such that g is independent of them. One such choice is to let thecoordinates γ of any ray in the ray field be equal to its first two components of its slowness vector,represented in a coordinate system in which the third axis points into the propagation directionof the ray under investigation. This is the ray-centered coordinate system that will be discussed indetail in Section 3.9. Therefore, we will assume from now on that a transformation to such a choiceof γ has been already performed and that g is independent of γ.

In equation (3.6.14), the geometrical divergence of a wave emanating from a point source isdescribed by

√J alone. The quantity

L =1√vSv

√J (3.6.15)

is referred to as the normalized geometrical-spreading factor for a point source. For the indicatedchoice of γ, it can be shown that in a homogeneous medium, L equals the distance between the point

3.7. CAUSTICS 73

source and the observation point. In a horizontally layered medium, it reduces to the expression ofNewman (1973).

With this definition (3.6.15) of the normalized geometrical-spreading factor L, equation(3.6.14) becomes

U =g

√

%Sv2S%v

2

1

L . (3.6.16)

Note that equation (3.6.16) describes the amplitude of an elementary wave that emanates from anomnidirectional point source at S in a slightly inhomogeneous medium with smooth variations ofthe velocity field. How this expression must be modified to remain valid in a layered, inhomogeneousmedium will be discussed in Section 3.13.

Observe that the amplitude at a point P of a wave with an arbitrary initial wavefront cur-vature may be expressed using the ratio of two point-source geometrical-spreading factors. Thereason is that J0 in equation (3.6.6) can be interpreted as describing the point-source geometricalspreading of the wave at the initial point P0 assuming a fictitious point source at a location earlieron the ray. Equation (3.6.6) may then be rewritten as

U =

√

%0v20

%v2

L0

L U0 , (3.6.17)

where L0 denotes the normalized geometrical-spreading factor from the fictious point source to P0

and L that from that same point source to P .

Since the ray Jacobian J in the above equations can be positive or negative, its squareroot remains to be defined. This must be done honoring the physical observations as well as themathematical conditions that govern seismic wave propagation. How L is actually computed alonga ray is shown in Sections 3.9 and 3.10.

3.7 Caustics

Caustic points are defined as such points of the ray, at which the ray Jacobian vanishes (J = 0).At these points, the cross-sectional area of the ray tube shrinks to zero, and the description ofwavefield amplitudes by means of equations (3.6.6) or (3.6.14) breaks down. Even in regions awayfrom caustic points, those caustic points which the elementary wave already passed still have aninfluence on the phase of the seismic wave. If this effect is not explicitly taken into account, thezero-order ray solution for the seismic wave will remain incorrect.

We must distinguish between two types of caustic points. A caustic point of the first order isa point of a line caustic. At such a point, only one dimension of the cross-sectional area shrinks tozero, i.e., the area shrinks to a line. Mathematically, this means that the rank of the transformationmatrix Q

˜

(r) given by equation (3.5.2) reduces to 2. A caustic point of the second order is a pointcaustic or a focus point. At such a point, both dimensions of the cross-sectional area shrink to zero,i.e., the area shrinks to one point. The rank of matrix Q

˜

(r) then reduces to 1. Talking in terms ofthe curvatures of the wavefront along a ray—not yet discussed—one could distinguish between afirst- and a second-order caustic by whether only one or both principal radii of curvature of thewavefront change their sign at the caustic point. If only one of them does, also the ray JacobianJ = det Q

˜

(r) will. However, if both of them do, J will keep its sign.

74 CHAPTER 3. ZERO-ORDER RAY THEORY

It is well known from many seismic observations that the phase spectrum of a wavelet changesby the amount of π

2 when the elementary wave travels through a line caustic and by the amountof π when it propagates through a point caustic. In equation (3.6.6) the ray Jacobian J is theonly quantity to describe a possible phase change along the ray (% and v are real quantities unlessabsorption is taken into account and J0 is a fixed, constant, initial value). Thus, the square rootin equations (3.6.6) and (3.6.14) must be defined in such a way that this physical observation ishonored.

Note that the amplitude and phase shift of a seismic wave in the close vicinity of a causticcan also be differently described in form of a more accurate high-frequency approximation. Theappropriate foundation is Maslov theory (see, e.g., Chapman 1978, 1985). Unlike ray theory, itprovides a continuous transition of the phase of an elementary wave crossing a caustic point. Thefull amount of the phase shift can then also be mathematically proven to be π

2 for a line causticand π for a point caustic.

With this understanding, we are now ready to define the square root in equation (3.6.6) as

√J = |J | 12 · e−iπ

2κ (3.7.1)

where κ is the so-called KMAH index of the considered ray. It counts the number of causticsalong the ray from its starting point to its end point. Note that κ increases along the ray by theorder of the caustic points encountered, i.e., by 1 for a line caustic and by 2 for a point caustic.In this way, the actually continuous transition of the elementary-wave phase across a caustic pointis approximated in the ray method by a discontinuous phase jump. By squaring equation (3.7.1),we readily see that J = |J | e−πκ honors the conditions on its sign, i.e., definition (3.7.1) is alsoconsistent with the mathematical requirements as discussed above.

3.8 Computation of the point source solution

In order to summarize what has been said so far, let us address the computation of the zero-orderray solution that would result from an omnidirectional compressional point source, located at aposition S with global coordinates rS, and emiting a source pulse f [t].

3.8.1 Homogeneous medium

Provided the medium is homogeneous and described by the constant parameters % and v, the raysolution for the P-wave reduces to

u(rG, t) =g

%v2

f [t− |rG − rS | /v]|rG − rS |

rG − rS|rG − rS |

, (3.8.1)

where (rG − rS)/|rG − rS | is the unit tangent vector, tG, to the ray at the receiver location G.Note that this expression coincides with the analytic solution for the particle displacement at thereceiver location G with coordinates rG in the far-field approximation [see equation (3.2.4)] if thesource strength is g = 1/4π.

3.9. RAY-CENTERED COORDINATES 75

3.8.2 Inhomogeneous medium

Given this very same point source in a slightly inhomogeneous medium (without first-order in-

terfaces) the displacement u(rG, t) = Re

U(rG)F [t− T (rG)]

at the receiver location rG has

to be computed. The traveltime T (rG) is determined with the help of equation (3.4.9). By equa-tion (3.6.14), the problem of computing of the ray amplitude has been shown to be solved by thedetermination of the Jacobian J along the ray.

How to compute the ray Jacobian J has yet to be described. This can be achieved bestwith the help of the ray-centered coordinate system that can be defined at any point on the rayconnecting rS with rG. This will be described in Section 3.9. The ray-centered coordinate system isvery much suited to formulate a dynamic ray-tracing system, with which J can then be continuouslycomputed at any point along the ray. For details see Section 3.10.

In the above introduction to the zero-order ray method, we have up to now ignored interfaces.However, generally, a ray connecting a source with a receiver is traced through a system of layersor blocks separated by zero or first-order reflecting or transmitting interfaces. The question of howto determine J in such a situation is addressed in Section 3.12. At the end of this chapter we willestablish the formula for displacement vector of an elementary wave at an arbitrary point rG of aray that may have crossed a number of interfaces.

3.9 Ray-centered coordinates

Above, we have seen that the main factor determining the ray amplitude is the ray Jacobian Jdefined in equation (3.5.3). Let us now see how J can be calculated for a given ray.

So far, all calculations have been performed in global Cartesian coordinates or (arbitrary)ray coordinates. We will now introduce a new, ray-centered coordinate system along a known ray(Popov and Psencık, 1978). This will lead to simplified expressions not only for the ray Jacobian,but also for various useful ray quantities, as, e.g., a paraxial ray or the wavefront curvature that canbe computed along an already established ray. One main advantage of the ray-centered coordinatesystem consists in the fact that any unknown ray close to the known ray can be traced with anew ray tracing system—the dynamic ray tracing system—in which the number of independentequations in the ray-tracing system (3.4.11) reduces from six to four.

Consider one chosen ray of the ray field under consideration to be known one. At a pointP0(s) with a given arclength s on that ray, the three unit vectors of the ray-centered coordinatesystem are e1(s), e2(s), e3(s) = t(s), where t(s) is the unit tangent vector that is tangent to thechosen ray at P0(s) as introduced in equation (3.4.12). The vectors e1 and e2 are defined as thosemutually normal unit vectors perpendicular to the chosen ray, the changes of which fall completelyinto the direction of the ray. In other words, at any arbitrary point on the chosen ray, they have tosatisfy the conditions

e1 · e2 = 0 , (3.9.1a)

ei · p = 0 , (3.9.1b)

and∂ei∂s

= a(s)p , (3.9.1c)

76 CHAPTER 3. ZERO-ORDER RAY THEORY

where a(s) is a proportionality factor that can be determined from combining the above conditions.Multiplication of condition (3.9.1c) with p provides

a(s) = v2p · ∂ei∂s

, (3.9.2)

and differentiation of condition (3.9.1b) with respect to s yields

∂ei∂s

· p = −∂p∂s

· ei . (3.9.3)

By substituting equation (3.9.3) in expression (3.9.2) we obtain

a(s) = −v2 ∂p

∂s· ei = ∇v · ei , (3.9.4)

where the second equation in (3.4.11b) has been used. Thus, the ray-centered coordinate vectorscan be computed from the system of ordinary differential equations (Cerveny, 2001)

deids

= (ei · ∇v)p . (3.9.5)

Note that the orientation of e1 and e2 at the initial point of the ray can be chosen arbitrarily.Further along the ray, however, they rotate around it as described by equation (3.9.5). In this way,they accompany the torsion of the ray. A more detailed discussion of the ray-centered coordinatesystem can be found in Cerveny (1987, 2001).

In the ray-centered coordinate system defined by the unit vectors e1, e2, and t, we denote thecoordinates of a point P off the ray by qi, where q3 = s is the arclength of that point P (s) on theray with the shortest distance to P . Thus, a point P (s) on the ray has the ray-centered coordinates(0, 0, s). We will denote its position vector r in global Cartesian coordinates by r(0, 0, s). It sat-isfies the above ray-tracing equations (3.4.11b). A point off the ray with ray-centered coordinates(q1, q2, s) has the global Cartesian position vector r(q1, q2, s) = r(0, 0, s) + q1e1 + q2e2. In otherwords (q1, q2) are the coordinates of a point off the ray within the plane perpendicular to the rayat arclength s. As before for the ray coordinates and global Cartesian coordinates, we will use thenotations q = (q1, q2, q3) for the 3-D vector and q = (q1, q2) for the 2-D vector of the ray-centeredcoordinates.

One of the many nice features of the ray-centered coordinate system e1, e2, t it the factthat it has a diagonal metric tensor. Its diagonal elements, the so-called scale factors, have rathersimple expressions that reduce to unity for points on the ray. Thus, on the ray, the ray-centeredcoordinate system coincides with a local ray Cartesian coordinate system. The transformation fromray-centered coordinates to global Cartesian coordinates at a point on the ray involves thus simplya translation plus a rotation, i.e., the transformation matrix is simply a rotation matrix.

The unit vectors e1 and e2 turn out to be particularly advantageous for the description ofshear waves. If one of these vectors is chosen such that it coincides with an S-wave polarizationvector at an arbitrary point on the ray, it keeps this property along the whole ray. Note, however,that at interfaces this must be guaranteed by choosing the correct boundary conditions.

3.9.1 Transformation from ray-centered to global Cartesian coordinates

After introducing a new coordinate system, the first question is again: How do elementary-wavequantities transform from one set to another set of coordinates? The transformation from ray-

3.9. RAY-CENTERED COORDINATES 77

centered to global Cartesian coordinates can be represented, as before in differential form, as

dr = H˜dq , (3.9.6)

where the elements of the transformation matrix H˜

are given by

Hij =∂ri∂qj

. (3.9.7)

As mentioned above, for a fixed point on the chosen ray, the ray-centered coordinate system coin-cides with a local Cartesian coordinate system. Thus, we observe that H

˜is a rotation matrix with

the known propertiesH˜T = H

˜−1 and det H

˜= 1 , (3.9.8)

where H˜T and H

˜−1 denote the transposed and inverse matrix, respectively, of H

˜. It should be

noted that H˜

varies along the chosen ray because of the variation of the orientation of the ray-centered coordinate system along the ray, i.e., H

˜= H

˜(s).

When leaving the chosen ray, there exist two possibilities to define H˜

. One consists of calculat-ing H

˜independently for the neighboring ray. This is exact, but leads to non-Cartesian coordinates

q. In this book, we prefer to remain with the same H˜

for all points in the vicinity of the chosenray, which is only approximately correct. In fact, this is consistent with the paraxial approximationas discussed in Section 3.10. This approximation makes the ray-centered coordinate system a trulyCartesian one. It implies that the matrix H

˜is always calculated on the chosen ray, i.e.,

Hij =∂ri∂qj

∣∣∣∣∣q1=q2=0

. (3.9.9)

Note that the columns of H˜T constitute the ray-centered coordinate vectors e1, e2, and t as

represented in global Cartesian coordinates.

3.9.2 Transformation from ray to ray-centered coordinates

We have already introduced the ray coordinates γ in Section 3.5.2. Therefore, we also need thetransformation from ray-centered to ray coordinates. We write it as

dq = Q˜dγ , (3.9.10)

where the elements of the transformation matrix Q˜

are given by

Qij =∂qi∂γj

. (3.9.11)

Note that we have chosen γ3 = q3 = s, so that Q33 = ∂s/∂s = 1. Along the ray, both coordinatesq1 and q2 vanish identically. Therefore Q13 = ∂q1/∂s and Q23 = ∂q2/∂s vanish at all points on theray. At any arbitrary point on the ray, we have, thus

det Q˜

= detQ˜, (3.9.12)

where Q˜

is the upper left 2 × 2 submatrix of the 3 × 3-matrix Q˜

. In other words, Q˜

is the trans-formation matrix from γ = (γ1 , γ2 ) to q = (q1, q2).

78 CHAPTER 3. ZERO-ORDER RAY THEORY

3.9.3 Ray Jacobian in ray-centered coordinates

The ray Jacobian J was defined in equation (3.5.3) as the determinant of the transformationmatrix Q

˜

(r) from ray coordinates γ to global Cartesian coordinates r. Inserting equation (3.9.10)into equation (3.9.6), we obtain a new expression for this transformation, namely

dr = H˜Q˜dγ . (3.9.13)

A comparison with equation (3.5.1) immediately shows that

Q˜

(r) = H˜Q˜. (3.9.14)

Insertion of equation (3.9.14) into equation (3.5.3) for the ray Jacobian J leads to

J = det(H˜Q˜

) = det H˜

det Q˜. (3.9.15)

For points on the ray, one obtains thus with equations (3.9.8) and (3.9.12)

J = detQ˜. (3.9.16)

We observe that once the matrix Q˜

is known, the main factor that determines the ray amplitudecan be computed. The ray Jacobian is obviously entirely determined by the geometrical behaviorof the ray tube that surrounds the ray.

3.9.4 Ray-tracing system in ray-centered coordinates

We now return to the equations (3.4.11) that define the three different kinematic ray-tracing sys-tems. However, as we have decided to choose the parameter ν along the ray to be the arclength s,we will confine ourselves to the ray-tracing system (3.4.11b). Equations (3.4.11b) are expressed inglobal Cartesian coordinates. To represent them in ray-centered coordinates, we start by multiply-ing the left one of equations (3.4.11b) by H

˜T = H

˜−1. Using equation (3.9.6), we obtain

dq

ds= vp(q) . (3.9.17)

The vector H˜Tp = p(q) = (p

(q)1 , p

(q)2 , p

(q)3 ) denotes the 3-D slowness vector in ray-centered coordi-

nates. Note that equation (3.9.17) is strictly valid only for points on the ray, because only thereH˜

is the true transformation matrix between global Cartesian and ray-centered coordinates. InSection 3.10, we will assume it to remain approximately valid in some vicinity of the chosen ray.

Let us first address the third component of equation (3.9.17). As q3 = s, the correspondingequation reads

ds

ds= vp

(q)3 , (3.9.18)

from which we immediately conclude that p(q)3 = 1/v(s) along the ray. Moreover, as q1 = q2 = 0

on the ray, we observe from the first two components of equation (3.9.17) that also p(q)1 = p

(q)2 = 0

along the ray. Thus, along the ray,

p(q)(s) = (0, 0, 1/v(s)) . (3.9.19)

3.9. RAY-CENTERED COORDINATES 79

The transformation of the right one of equations (3.4.11b) into ray-centered coordinates isa little trickier, because H

˜depends on s, and thus, H

˜Tdp/ds 6= d(H

˜Tp)/ds = dp(q)/ds. By the

product rule, the derivative of p(q) with respect to to s can be written as

dp(q)

ds=dH

˜Tp

ds=dH

˜T

dsp+ H

˜T dp

ds. (3.9.20)

With the second of equations (3.4.11b), this becomes

dp(q)

ds=dH

˜T

dsp+ H

˜T∇

(1

v

)

=dH

˜T

dsp+ ∇q

(1

v

)

, (3.9.21)

where ∇q = (∂/∂q1, ∂/∂q2, ∂/∂q3) is the 3-D gradient in ray-centered coordinates. Since the columns

of the transformation matrix H˜T are formed by the ray-centered coordinate vectors e1, e2, and t,

the third component of the first term in the above sum can be recast into the form

(

dH˜T

dsp

)

3

=dt

ds· p = 0 , (3.9.22)

which vanishes, because the changes of the tangent vector t are perpendicular to the ray. Corre-spondingly, the first two components of the first term in the above sum satisfy

(

dH˜T

dsp

)

i

=deids

· p (i = 1, 2) . (3.9.23)

With equation (3.9.5), we obtain

(

dH˜T

dsp

)

i

=(

ei · ∇v)

p · p =∂v

∂qi

∣∣∣∣q=0

1

v2= − ∂

∂qi

(1

v

)∣∣∣∣q=0

. (3.9.24)

Therefore, we can write the second of equations (3.4.11b) in ray-centered coordinates as

dp(q)

ds= ∇q

(1

v

)

− Dq , (3.9.25)

where

Dq =

∂v−1

∂q1q=0

,∂v−1

∂q2q=0

, 0

. (3.9.26)

The third component of equation (3.9.25) reduces to

dp(q)3

ds=

d

ds

(1

v

)

, (3.9.27)

which is nothing more than a restatement of equation (3.9.19). In ray-centered coordinates, we thussee that the ray-tracing system consisting of equations (3.9.17) and (3.9.25) consists of only fourindependent equations for the first two components of q and p(q). We will make use of this fact inthe next section.

80 CHAPTER 3. ZERO-ORDER RAY THEORY

3.10 Paraxial and dynamic ray-tracing

In this and the following sections, we will study the above ray-tracing system consisting of equations(3.9.17) and (3.9.25) in more detail. Our purpose is to investigate the behavior of so-called paraxialrays in the near vicinity of a ray that is now assumed to be known. This will provide us with thesystem of differential equations that are actually used to calculate the ray Jacobian and thus thegeometrical spreading of that known ray. This latter ray will therefore be called from now on thecentral ray.

For the described aim, we interpret system (3.9.17,3.9.25) to be approximately valid alsofor any neighboring ray in the close vicinity of the central ray (where it is exact), i.e., for smalldistances q. This is the so-called paraxial approximation, and the vicinity of the central ray where itapproximately holds, is called the paraxial vicinity. Note that many of the derivations below oftendo not follow the lines of Cerveny (1985, 1987, 1995, 2001). However, the main intermediate andall final results are identical.

3.10.1 Paraxial ray-tracing

Assuming that equations (3.9.17) and (3.9.25) are approximately valid for paraxial rays in thevicinity of the central ray, there remains the following system of four equations for the first twocomponents of q and p(q), denoted by q and p(q):

dq

ds= vp(q) ,

dp(q)

ds= ∇q

(1

v

)

−Dq , (3.10.1)

where q = (q1, q2), p(q) = (p

(q)1 , p

(q)2 ), ∇q = (∂/∂q1, ∂/∂q2), and Dq is the vector of the first two

components of Dq, i,e.,

Dq = ∇q

(1

v

)∣∣∣∣q=0

. (3.10.2)

We stress once more that we assume the validity of system (3.10.1) on each paraxial ray in thevicinity of the central ray, i.e., for small q, with the same Dq as on the central ray. Equations(3.10.1) describe how q and p(q) change along one paraxial ray. Under the assumption of small q,we can expand the gradient of the slowness into a Taylor series up to the first order in q,

∇q

(1

v

)

= ∇q

(1

v

)∣∣∣∣q=0

− 1

v2V˜q = Dq −

1

v2V˜q , (3.10.3)

where v denotes the velocity at s on the central ray, and where V˜

is a 2×2 matrix with the elements

Vij =∂2v

∂qi∂qj

∣∣∣∣∣q=0

(i, j = 1, 2) . (3.10.4)

Again, approximation (3.10.3) is consistent with the paraxial approximation as discussed below.

Inserting the above expression in equation (3.10.1), we thus obtain the ray-tracing system inits paraxial approximation

dq

ds= vp(q) ,

dp(q)

ds= − 1

v2V˜q . (3.10.5)

3.10. PARAXIAL AND DYNAMIC RAY-TRACING 81

Note that in this approximation, the third component of the slowness vector p(q)3 equals 1/v(s) not

only on the ray but also in its vicinity. However, off the ray the first two components p(q)1 and p

(q)2

of p(q) are generally different from zero.

3.10.2 Dynamic ray-tracing

We are now interested in finding an approximation for the dynamic properties along the centralray. As we have seen above, the ray Jacobian J is directly related to the geometrical behaviorof the neighboring rays. We can thus expect the above ray-tracing systems (3.10.1) or (3.10.5)to contain some dynamic information. This will become obvious when we take the derivatives ofthe ray-tracing system (3.10.5) with respect to ray coordinates γ. In other words, we study thevariation of system (3.10.5) when moving from one ray to another. In component notation, we havefrom the left equation

∂

∂γj

dqids

=∂

∂γj(v p

(q)i ) (i, j = 1, 2) . (3.10.6)

Interchanging the order of differentiation on the left-hand side and applying the chain rule to theright-hand side, we arrive at

d

ds

∂qi∂γj

=∂v

∂γjp(q)i + v

∂p(q)i

∂γj. (3.10.7)

Note that i, j = 1, 2 as we consider the first two components only. As equations (3.10.5) are onlyapproximately valid in a small vicinity of the central ray, we will consider equations (3.10.7) only at

points on the central ray itself, i.e., for q = 0. We have seen that on the ray p(q)1 = p

(q)2 = 0. Hence,

the first term on the right-hand side of the equations (3.10.7) vanishes. The remaining equationreads

d

ds

∂qi∂γj

= v∂p

(q)i

∂γj. (3.10.8)

Correspondingly, we also take the derivatives of the right equations of system (3.10.5) withrespect to γ to obtain in component notation

∂

∂γj

dp(q)i

ds=

∂

∂γj

(

− 1

v2

2∑

k=1

∂2v

∂qi∂qkqk

)

= −2∑

k=1

[

∂

∂γj

(

1

v2

∂2v

∂qi∂qk

)

qk +

(

1

v2

∂2v

∂qi∂qk

)

∂qk∂γj

]

. (3.10.9)

Since the derivatives are taken again at one constant s on the central ray, i.e., at q = 0, the first termin the last equation vanishes. Therefore, we obtain after interchanging the order of differentiationon the left-hand side,

d

ds

∂p(q)i

∂γj= − 1

v2

2∑

k=1

∂2v

∂qi∂qk

∂qk∂γj

. (3.10.10)

At this point, we identify in equations (3.10.8) and (3.10.10) the factor ∂qk/∂γj with the elementsQkj of the matrix Q

˜introduced in equation (3.9.11). Moreover, we introduce the corresponding

2 × 2 matrix P˜

with the elements

Pij =∂p

(q)i

∂γj. (3.10.11)

82 CHAPTER 3. ZERO-ORDER RAY THEORY

Then, equations (3.10.8) and (3.10.10) can be finally rewritten in matrix notation as

d

dsQ˜

= vP˜,

d

dsP˜

= − 1

v2V˜Q˜, (3.10.12)

where the elements of V˜

are again given by equation (3.10.4). Note again that all derivatives insystem (3.10.12) are taken at points on the ray. By writing P

˜and Q

˜in the form of a 4× 2-matrix

ˆW˜

=

(

Q

P˜

)

, (3.10.13)

the system (3.10.12) can be written in the compact form

d

dsˆW˜

= ˆS˜

ˆW˜. (3.10.14)

Here, ˆS˜

is the 4 × 4-matrix given by

ˆS˜

=

(

O˜

v I˜− 1

v2V˜

O˜

)

, (3.10.15)

where O˜

is the 2 × 2 zero matrix and I˜

is the 2 × 2 unit matrix. Note that system (3.10.5) takes

the same form of equation (3.10.14) if ˆW˜

is defined as the 4 × 1-column matrix

ˆW˜

=

q1q2

p(q)1

p(q)2

. (3.10.16)

Even though we have derived equations (3.10.12) from considering paraxial rays, they arecommonly not referred to as the paraxial ray-tracing system but as the dynamic ray-tracing system.The reason is that for appropriate initial values (apart from many useful quantities such as, e.g.,the two principal wavefront curvatures and the corresponding principal planes), they provide thematrix Q

˜and thus J = detQ

˜along the central ray. The ray Jacobian in turn determines the

complex ray amplitude U (i.e., the dynamic quantities |U| and argU) of the elementary wave onaccount of equations (3.6.6), (3.7.1), and (3.9.16).

3.10.3 Paraxial approximation

Prior to investigating how to solve the above dynamic ray-tracing system (3.10.12), let us addressin more detail the following two questions, namely (a) how can paraxial ray tracing be achievedwith equations (3.10.12) and (b) what approximation has really been done by assuming the validityof system (3.10.1) in the paraxial vicinity of the ray, i.e., where the linear approximation (3.10.3)holds.

To answer the first question, we consider a small difference ∆γ between the ray coordinatesof a paraxial ray (to be determined) and of the central ray (already computed). We observe fromthe definitions (3.9.11) of Q

˜and (3.10.11) of P

˜(assumed to be also already known along the central

ray) that we may approximately write up to first order in ∆γ

q(s) = Q˜

(s)∆γ , p(q)(s) = P˜

(s)∆γ , (3.10.17)

3.10. PARAXIAL AND DYNAMIC RAY-TRACING 83

i.e., the location and the slowness vector of a paraxial ray in the vicinity of the known central rayare entirely determined by the given quantity ∆γ, once the matrices P

˜and Q

˜are known. In this

way, any paraxial ray can be traced in the vicinity of a known central ray once system (3.10.12)has been solved.

The second question can be answered after eliminating ∆γ from equations (3.10.17). Then,these equations can be combined to yield

p(q) = P˜

(s)Q˜

−1(s)q(s) . (3.10.18)

We see that equation (3.10.17) is a linear approximation in ∆γ, and thus equation (3.10.18) isone in q, which exactly corresponds to approximation (3.10.3). Therefore, after system (3.10.12) issolved, the slowness vector of any paraxial ray can be determined up to first order in q. Because ofequation (3.4.2), this will lead to an approximation for the traveltime along the paraxial ray whichwill be correct up to the second order in q.

This first-order slowness or second-order travaltime approximation is the so-called “paraxialapproximation” on which all following derivations rely. We thus shall always neglect terms of higherorder in q as we have already done in equation (3.10.3).

3.10.4 Initial conditions for dynamic ray tracing

For a specific solution of equations (3.10.12), we will need initial values for P˜

and Q˜

at an initialpoint on the central ray with coordinate s = s0 Obviously, these initial values, which we denoteas as P

˜0 and Q

˜0, depend on the particular choice of the ray coordinates γ. Here, we discuss two

particular choices for γ which are adequate for plane-wave and point-source initial conditions.

Plane-wave initial conditions

If the wavefield in the vicinity of the initial point of the central ray is that of a plane wave, each rayin the ray field can be uniquely specified by the two in-plane coordinates of its intersection pointwith the wave front. Therefore, it is useful to choose γ to be represented by these coordinates.Then, at the initial point s = s0, the difference between the ray coordinates of a paraxial and thecentral ray coincides with the ray-centered coordinates, i.e., ∆γ = q(s=s0) = q0. Inserting thiscondition into equations (3.10.17), taken at s = s0, we observe that the initial values for P

˜and Q

˜for plane-wave initial conditions, P

˜

(pw)0 and Q

˜

(pw)0 , must satisfy

q0 = Q˜

(pw)0 q0 , p0

(q) = 0 = P˜

(pw)0 q0 . (3.10.19)

Here we have used the fact that the slowness vectors of a plane wave are all parallel and thusp0

(q) = 0. The plane-wave initial conditions for the dynamic ray-tracing system (3.10.12) are thus

P˜

(pw)0 = O

˜, Q

˜

(pw)0 = I

˜. (3.10.20)

Point-source initial conditions

On the other hand, for a point source at the initial point s = s0, a different choice for γ leadsto convenient initial conditions. In this case, unique specification of all rays is achieved by their

84 CHAPTER 3. ZERO-ORDER RAY THEORY

slowness vectors. We therefore choose γ to be represented by the first two components of the initialslowness vector of the paraxial ray in ray-centered coordinates, i.e., such that ∆γ = p(q)(s=s0) =p0

(q). Inserting this into equations (3.10.17), again taken at s = s0, leads to the condition

q0 = 0 = Q˜

(ps)0 p0

(q) , p0(q) = P

˜

(ps)0 p0

(q) . (3.10.21)

Here we have used the fact that at a point source, all rays emanate from the same point and thusq0 = 0. The point-source initial conditions for the dynamic ray-tracing system (3.10.12) are thus

P˜

(ps)0 = I

˜, Q

˜

(ps)0 = O

˜. (3.10.22)

General initial conditions

As equations (3.10.12) constitute a system of two linear differential equations of the first order in P˜andQ

˜, we know that two solutions with different initial conditions are sufficient to solve the general

problem. Each solution for arbitrary initial conditions can be described by a linear combinationof two independent solutions. In agreement with Cerveny (2001), we denote the solution of thedynamic ray-tracing system with plane-wave initial conditions by P

˜1,Q

˜1 and with point-source

initial conditions by P˜

2,Q˜

2. Then, we may write the general solution P˜,Q˜

with arbitrary initialconditions P

˜0,Q

˜0 as

Q˜

= Q˜

1Q˜

0 +Q˜

2P˜

0 ,

P˜

= P˜

1Q˜

0 + P˜

2P˜

0 .(3.10.23)

This is a general property of systems like (3.10.14) that does not depend on the particular choiceof the initial conditions. This can be readily verified. Insertion of the plane-wave initial conditions(3.10.20) leads again to P

˜= P

˜1 andQ

˜= Q

˜1, point-source initial conditions (3.10.22) yield P

˜= P

˜2

and Q˜

= Q˜

2.

3.10.5 Ray-centered propagator matrix ˆΠ˜

The 4 × 4-matrix ˆΠ˜

that is constituted by the four 2 × 2 matrices Q˜

1,Q˜

2,P˜

1,P˜

2 as

ˆΠ˜

=

(

Q˜

1 Q˜

2

P˜

1 P˜

2

)

(3.10.24)

assumes the role of a propagator matrix. The particular simple form of system (3.10.23) results ofcourse from the convenient choice of the initial conditions (3.10.20) and (3.10.22) for P

˜1, P

˜2, Q

˜1,

and Q˜

2 which leads to the following initial condition for ˆΠ˜:

ˆΠ˜0 = ˆI

˜, (3.10.25)

where ˆI˜

is the 4 × 4-unit matrix. Equations (3.10.23) are a consequence of the fact that ˆΠ˜

can bethought of as being composed of two fundamental solutions of equations (3.10.14) for the 4 × 2

matrix ˆW˜

as defined in equation (3.10.13). In the same way, however, ˆΠ˜

can be thought of as being

composed of four fundamental solutions of equations (3.10.14) for the 4 × 1-column matrix ˆW˜

as

defined in equation (3.10.16). Thus, it is obvious that ˆΠ˜

also defines the equations

q(s) = Q˜

1q(s0) +Q˜

2p(q)(s0) ,

p(q)(s) = P˜

1q(s0) + P˜

2p(q)(s0) ,

(3.10.26)

3.11. RAYS AT A SURFACE 85

i.e., ˆΠ˜

is not only a propagator matrix for the general solution of the dynamic ray-tracing system(3.10.12) but also for the coordinates and slowness vector of a paraxial ray. In one single matrixequation, we may write

ˆW˜

(s) = ˆΠ˜

ˆW˜

(s0) , (3.10.27)

where ˆW˜

may be interpreted in either of the two ways defined above.

System (3.10.26) is a particularly useful tool to study the properties of paraxial ray tracing.Much information can be extracted from its manipulation. For example, by solving the top equationof system (3.10.26) for p(q)(s0), one can immediately find

p(q)(s0) = Q˜

−12

[

q(s) −Q˜

1q(s0)]

. (3.10.28)

This equation provides an expression for the initial slowness vector of a paraxial ray at q(s0) thathas its final point at q(s). In other words, equation (3.10.28) can be directly used for paraxialtwo-point ray tracing.

3.11 Rays at a surface

In most seismic applications, the sources and receivers are placed along the earth surface. Thismeans that the initial and end points of the searched-for paraxial rays often do not lie within theplanes perpendicular to the central ray at its initial and end points but in arbitrarily oriented andcurved surfaces. To be able to describe paraxial rays emanating from a source at one surface andemerging at another surface, we have to generalize the above paraxial ray theory accordingly. Forthat purpose, we will need an additional, local Cartesian coordinate system within the tangentplane to each of the surfaces under consideration. The formulas derived in this section will also beof need in Section 3.12, where we will see how a ray behaves on crossing an interface.

3.11.1 Vector representations

Consider a central ray impinging on (or emanating from) an arbitrary surface ΣA at a givenintersection point P (Figure 3.3a). Let its slowness vector at P in global 3-D Cartesian coordinatesbe represented by p0. We refer to the plane tangent to the surface ΣA at P as the plane Ω0. Followingthe lines of Bortfeld (1989), we define a local 3-D Cartesian coordinate system x = (x1, x2, x3) withits origin at P . The x3-axis of this coordinate system is normal to the plane Ω0 and is chosen suchthat it makes an acute angle with the slowness vector p0 of the central ray. Let us denote the

representation of p0 in these local coordinates by p(x)0 = (p01, p02, p03).

It will be convenient to let first two components of x define a local 2-D Cartesian coordinatesystem within the plane Ω0 and are denoted by the 2-D vector x= (x1, x2). Correspondingly, the

vector formed by first two components p01 and p02 of p(x)0 within the plane Ω0 is denoted in local

Cartesian coordinates by p0, i.e., p0 = (p01, p02).

In the same way, we consider a paraxial ray impinging on (or emanating from) the samesurface ΣA at a generic point P in the vicinity of P (Figure 3.3b). The projection of the distancevector from P to P into the plane Ω0 has the components xP = (xP1 , x

P2 ). To avoid an overloading

of the notation, we omit below the superscript P and refer to the local 2-D Cartesian coordinates

86 CHAPTER 3. ZERO-ORDER RAY THEORY

P

P

pp

0

P

p0

(a)

x2 x1

0

0

(b)

P

p0

p

pT

p0 p

p

p

A

A

T

Ω

Σ

Σ

Ω

Ω

central ray

central ray

paraxial ray

paraxial ray

x

Fig. 3.3. A central and a paraxial ray impinge on (or emanate from) a surface ΣA at points P andP , respectively. (a) Definition of the local 2-D Cartesian coordinate system (x1, x2) within planeΩ0 with origin at P . (b) 2-D view of part (a) to visualize the cascaded slowness vector projection.

3.11. RAYS AT A SURFACE 87

of P as x= (x1, x2). Also, the paraxial ray at P has a slowness vector pp which will be represented

in local Cartesian coordinates as p(x)p = (pp1, pp2, pp3).

To find a 2-D representation of pp within the plane Ω0 that will be of use later on, we performthe following cascaded projection (Bortfeld 1989). We construct the plane ΩT that is tangent tothe surface ΣA at P (see again Figure 3.3b). The projection pT of pp at P into plane ΩT is denoted

in local Cartesian coordinates by p(x)T = (pT1, pT2, pT3). By projecting pT a second time, now into

the plane Ω0, we find its 2-D representation in local Cartesian coordinates, formed by the first two

components of p(x)T , i.e., pT = (pT1, pT2). Again for notational simplicity, we will drop the index T

from the 2-D vector after this cascaded projection, denoting it simply by p = (p1, p2). It should bekept in mind that in general, p1 6= pp1 and p2 6= pp2.

Note that, for a given surface ΣA and velocity v at P , the relationship between the 3-Dslowness vector pp and its 2-D representation p within plane Ω0 is unique. In other words, thefull 3-D slowness vector pp can be reconstructed from its projection p. The reason is that theorientation and magnitude of pp are known. How this can be actually done is discussed in Section3.11.5.

3.11.2 Surface representation

In second-order approximation, the surface ΣA can be expressed as a paraboloid. In the local 3-Dcoordinate system x, this paraboloid is representable as

x3 =1

2x · F

˜x , (3.11.1)

where F˜

is the matrix of second derivatives of x3 with respect to x1 and x2. In other words,F˜

represents the surface’s curvature. Note that F˜

is symmetric, i.e., F12 = F21, because it isindependent of the order of differentiations. The surface normal n(x) to this paraboloid at point Pis then given by

n(x) = (F11 x1 + F12 x2, F21 x1 + F22 x2,−1) =

(

F˜x

−1

)

. (3.11.2)

It is not difficult to see that n(x) is indeed normal to the surface ΣA at P , since it is perpendicularto two basic tangent vectors,

(1, 0, dx3/dx1) = (1, 0, F11 x1 + F12 x2) and (0, 1, dx3/dx2) = (0, 1, F21 x1 + F22 x2) .(3.11.3)

In paraxial approximation, n(x) is the unit normal vector to the surface ΣA because its modulusdiffers from unity only in second order.

3.11.3 Transformation from local Cartesian to ray-centered coordinates

To be able to express the above formulas (3.10.26) and (3.10.27) in the local 2-D Cartesian coordi-nate system x at the surface ΣA, we need to find the transformation from this system to the 2-Dray-centered coordinate system q with the same origin P . For that purpose, we consider at first

88 CHAPTER 3. ZERO-ORDER RAY THEORY

the corresponding 3-D transformation from x to q, where, as before, q3 = s is the arclength alongthe ray.

Both coordinate systems x and q are Cartesian ones on the central ray at P . Under theassumption that the x2-axis is perpendicular to the plane of incidence, i.e., the plane defined byp0 and n(x), the 3-D transformation can be represented as a sequence of two elementary rotations(Cerveny, 1987, 2001). Firstly, the plane Ω0 is rotated around the x2-axis by the angle ϑP (which theslowness vector p0 makes with the interface normal at P ) into the plane Ω (not shown in Figure 3.3)perpendicular to the central ray at P . Secondly, the resulting coordinate system is rotated withinplane Ω, i.e., around the q3-axis, by an angle ϕ until it coincides with the ray-centered coordinatesystem q at P . In symbols

dq = G˜dx = Φ

˜Θ˜dx , (3.11.4)

where the 3 × 3-matrix G˜

is given by its elements

Gij =∂qi∂xj

(i, j = 1, 2, 3) (3.11.5)

and

Φ˜

=

cosϕ − sinϕ 0sinϕ cosϕ 0

0 0 1

, Θ

˜=

cosϑP 0 − sinϑP0 1 0

sinϑP 0 cosϑP

. (3.11.6)

For an arbitrarily oriented system x, prior to the two rotations described above, an additional oneis necessary within plane Ω0, i.e., around the x3-axis, which brings the x2-axis into the desiredposition.

In paraxial approximation, the above transformation (3.11.4) can be written in terms of thesmall vectors x and q instead of the infinitesimally small vectors dx and dq. Moreover, for pointsP on the surface ΣA, the terms containing x3 can be neglected as they are of second order in x dueto equation (3.11.1). Thus, the transformation from local 2-D Cartesian coordinates x to local 2-Dray-centered coordinates q is described by the upper left 2 × 2 submatrix G

˜of G

˜, i.e.,

q = G˜x= Φ

˜Θ˜x , (3.11.7)

where

Φ˜

=

(

cosϕ − sinϕsinϕ cosϕ

)

and Θ˜

=

(

cosϑP 00 1

)

. (3.11.8)

In other words, the 2-D transformation (3.11.7) amounts to a rotated projection of the vector xfrom the plane Ω0 tangent to the surface ΣA at P into the plane Ω that is perpendicular to the rayat P .

Note that, whereas the inverse of the 3-D transformation matrix G˜

equals the transposedmatrix, i.e., G

˜−1 = G

˜T (because both describe just the same rotations by negative angles), this is

no longer true for the 2 × 2 submatrix G˜

, i.e., G˜

−1 6= G˜T . The reason is that G

˜−1 describes the

inverse projection, i.e., the reconstruction of a vector that was projected from plane Ω0 into planeΩ, whereas G

˜T describes the projection in opposite direction, i.e., from the plane Ω into the plane

Ω0. In symbols, we have

G˜−1 = Θ

˜−1Φ

˜−1 = Θ

˜−TΦ

˜T , G

˜T = Θ

˜TΦ˜T = Θ

˜Φ˜−1 , (3.11.9)

where Θ˜

−T is the inverse of the transposed (or transposed of the inverse) matrix of Θ˜

. Here, wehave used that Φ

˜−1 = Φ

˜T and Θ

˜T = Θ

˜.

3.11. RAYS AT A SURFACE 89

3.11.4 Transformation from local to global Cartesian coordinates

To find the 2-D transformation from x to r, let us first look at the related 3-D transformation fromx to r. The transformation

dr = G˜

(r) dx (3.11.10)

defines the 3 × 3-matrix G˜

(r) with the elements

G(r)ij =

∂ri∂xj

. (3.11.11)

As both coordinate systems r and x are Cartesian ones, G˜

(r) is a rotation matrix that can bedecomposed into a sequence of three elementary rotations. The first one is the rotation around thex3-axis by an angle ϕx until the rotated x1-axis lies within the vertical plane that includes the localdip direction (i.e., the normal vector) of the considered surface ΣA at P . The second rotation isperformed around the rotated x2-axis by the angle βP , where βP is the local dip angle of the surfaceΣA at P . This rotates the x3-axis onto the global z-axis. Finally, the third rotation is performedaround the z-axis by an angle ϕr that defines the particular orientation of the first two axes of theglobal coordinate system. In symbols, we have

G˜

(r) = Φ˜r B˜

Φ˜x , (3.11.12)

where

Φ˜x =

cosϕx − sinϕx 0sinϕx cosϕx 0

0 0 1

, B

˜=

cos βP 0 − sinβP0 1 0

sinβP 0 cos βP

, (3.11.13)

and the matrix Φ˜r is of the same form as Φ

˜x where only the index x is replaced by r.

In paraxial approximation, the infinitesimally small vectors dr and dx can again be replacedby the small vectors ∆r and x. Also, for points P on ΣA, the terms containing x3 may again beneglected due to equation (3.11.1). Thus, the 2-D transformation from x to ∆r corresponding toequation (3.11.10) is then given by

∆r = r − r0 = G˜

(r)x , (3.11.14)

where r and r0 are the 2-D global Cartesian coordinate vectors of P and P , respectively. We remindthat the origin of the x-coordinate system is located at P . The 2 × 2 matrix G

˜(r) is the upper left

2 × 2 submatrix of G˜

(r). Note that

detG˜

(r) = cos βP , (3.11.15)

i.e., like G˜

, also the 2 × 2 matrix G˜

(r) is a projection rather than a rotation matrix and thusG˜

(r)T 6= G˜

(r)−1.

3.11.5 Relationship between the slowness vector representations

In this section, we are looking for the relationship between the slowness vector representations pin local Cartesian and p(q) in ray-centered coordinates.

90 CHAPTER 3. ZERO-ORDER RAY THEORY

Let us assume in the following that p(x)0 , p(x)

p , and p(x)T are the representations of p0, pp,

and pT , respectively, in the local 3-D coordinate system x (see again Figure 3.3). In that case, the

first two components p(x)01 = p01 and p

(x)02 = p02 of p

(x)0 constitute the 2-D projected slowness vector

p0 introduced above, i.e., p0 = (p01, p02). Correspondingly, the first two components p(x)T1 = p1

and p(x)T2 = p2 of p

(x)T constitute the 2-D vector p = (p1, p2). Additionally, we will denote the 2-D

vector constructed by the first two components of p(x)p , i.e., its direct projection into plane Ω0, by

pp = (pp1, pp2).

Recall that p(q) is the slowness vector of the paraxial ray in ray-centered coordinates, i.e., theprojection of pp−p0 into plane Ω. Thus, in order to find the desired relationship between p and p(q),we have to derive the relationship between p and pp, i.e., we have to perform the above described

cascaded projection analytically. As p(x)T is the projection of p(x)

p into the plane ΩT tangent to the

surface ΣA at P , we may write

p(x)T = p(x)

p − (p(x)p · n(x))n(x) , (3.11.16)

where n(x) is the unit surface normal at P .

Inserting the surface normal determined in equation (3.11.2) into equation (3.11.16) andneglecting again the terms of second order in x leads to

p(x)T = p(x)

p +

(

p(x)p3 F

˜x

pp · F˜x− p

(x)p3

)

. (3.11.17)

We now use the fact that the difference between p(x)p3 and p

(x)03 is linear in x and thus leads to

quadratic terms when multiplied with F˜x. Moreover, we have

p(x)03 =

cosϑPV3

, (3.11.18)

with V3 being the velocity at P and ϑP being the angle the ray makes with the surface normal at

P . Thus, the relationship between p(x)T and pp reads

p(x)T =

(

pp + cosϑP

V3F˜x

pp · F˜x

)

. (3.11.19)

As we are only interested in the first two components of equation (3.11.19), we may also write

p = pp +cosϑPV3

F˜x . (3.11.20)

In order to find a suitable expression for pp in equation (3.11.20), we expand p(x)p in a Taylor

series up to the first order in x, and obtain

p(x)p = p

(x)0 + M

˜(x) x . (3.11.21)

Here, we have introduced the 3 × 3-matrix M˜

(x) given by its elements

M (x)ij =

∂p(x)i

∂xj

∣∣∣∣∣P0

=∂2T∂xi∂xj

∣∣∣∣∣P0

(i, j = 1, 2, 3) , (3.11.22)

3.11. RAYS AT A SURFACE 91

where the second equality is a consequence of equation (3.4.2). From equation (3.11.22) it is obviousthat M

˜(x) is symmetric. For convenience, we will, however, not use M

˜(x) but its transformation

into ray-centered coordinates given by

M˜

= G˜M˜

(x) G˜T , (3.11.23)

where the transformation matrix G˜

is defined in equation (3.11.5) and where

Mij =∂p

(q)i

∂qj

∣∣∣∣∣P0

=∂2T∂qi∂qj

∣∣∣∣∣P0

(i, j = 1, 2, 3) . (3.11.24)

Note that due to equations (3.4.8b),∂T∂q3

=dTds

=1

v(3.11.25)

and thus,

Mi3 = M3i =∂ 1v

∂qiP0

= − 1

V 23

∂v

∂qiP0

(i = 1, 2, 3) . (3.11.26)

For the upper left 2 × 2 submatrix M˜

of M˜

, we observe from equation (3.11.24) together withexpressions (3.9.11) and (3.10.11)

Mij =∂p

(q)i

∂qj=

2∑

k=1

∂p(q)i

∂γk

∂γk∂qj

=2∑

k=1

Pik Q−1

kj (i, j = 1, 2) , (3.11.27)

where the derivatives are taken at P on the central ray. Here, we have used the fact that at any

point on the central ray, p(q) = 0 and thus ∂p(q)i /∂γ3 = 0.

Inserting equation (3.11.23) into formula (3.11.21), one obtains for the ith component of p(x)p

p(x)pi = p

(x)0i +Gli Mlk Gkj xj , (3.11.28)

where i, j, k, l = 1, 2, 3. Note that Gli = GT il. For clarity of notation, the Einstein sum conventionhas been used in equation (3.11.28), i.e., a sum over repeated indices j, k, and l is implicit. Wecontinue to use this convention below.

By insertion of equation (3.11.1), we immediately observe that at points P on ΣA, terms withj = 3 can be neglected as they are of second order in x. Thus, we may rewrite equation (3.11.28)in the following form:

p(x)pi = p0i +Gli Mlk Gkj xj +Gli Ml3 G3j xj +G3i M3l Gljxj +G3i M33 G3jxj , (3.11.29)

where now i, j, k, l = 1, 2. Subsituting equation (3.11.20) in (3.11.29), we arrive at

pi = p0i +cosϑP

V3Fijxj +GliMlkGkjxj +GliM3lG3jxj

+G3iM3lGljxj +G3iM33G3jxj .(3.11.30)

92 CHAPTER 3. ZERO-ORDER RAY THEORY

Upon the use of equation (3.11.7), we recognize that in paraxial, i.e., linear approximation Gkjxj =qk (k, j = 1, 2) and together with equation (3.11.24), we have

MlkGkjxj =∂p

(q)l

∂qkqk = p

(q)l (j, k, l = 1, 2) . (3.11.31)

Thus, equation (3.11.30) reduces to

pi = p0i +Gli p(q)l +Xijxj , (3.11.32)

where the 2 × 2 matrix X˜

is given by its elements

Xij =cosϑPV3

Fij +M33G3iG3j +M3l(G3iGlj +GliG3j) . (3.11.33)

In vector notation, we may write

p = p0 +G˜Tp(q) +X

˜x , (3.11.34a)

or vice versa,

p(q) = G˜

−T (p− p0 −X˜x) , (3.11.34b)

where, as before, G˜

−T = (G˜T )−1. Finally, we have obtained the relationship between p and p(q).

Together with equation (3.11.7), we have now found a useful representation of the paraxial-rayquantities at a surface.

3.11.6 Surface-to-surface propagator matrix ˆT˜

In seismics, we mostly have to deal with rays that emanate from sources that are located on asurface and impinge at receivers that are located on another surface. Here, we will refer to thesource surface as the anterior surface and to the receiver surface as the posterior surface. Notethat these surfaces may coincide, as is usually the case in reflection seismics. For the followingconsiderations, we assume a point source at a point S with the global Cartesian coordinate vectorrS on the anterior surface. In the same way, we consider a receiver pointG with the global coordinatevector rG on the posterior surface. The wave from S to G propagates through the inhomogeneousmedium without internal interfaces along a central ray connecting S to G. We refer to this ray,which is assumed to be already known, as the “ray SG.” In the paraxial vicinity of the ray SG,where the medium parameters change only gradually, we have a second source point S on theanterior surface and a second receiver point G on the posterior surface. This source-receiver pairis linked by a paraxial ray S G, which we want to characterize. It is always assumed that boththe central and paraxial rays under consideration make part of one ray family, i.e., there exists acommon wavefront.

Thus, we have to consider equations of the form (3.11.7) and (3.11.34) for both the initialand end points of the paraxial ray S G. Denoting quantities at the ray’s initial points by unprimedand at their end points by primed symbols, we can write

q = G˜x , p(q) = G

˜−T (p− p0 −X

˜x) ,

q′ = G˜

′ x′ , p(q)′ = G˜′−T (p′ − p′0 −X

˜′x′) .

(3.11.35)

3.12. RAYS ACROSS AN INTERFACE 93

Inserting equations (3.11.35) into equations (3.10.26) leads to a system of equations establishing apropagator matrix for the quantities x and p− p0 in the plane Ω0 between the initial and the endpoints S and G of the paraxial ray. This system is expressed as

x′ = A˜x+B

˜(p− p0) , (3.11.36a)

p′ − p′0 = C˜x+D

˜(p− p0) . (3.11.36b)

The 2 × 2 matrices A˜,B˜,C˜,D˜

are then given by

A˜

= G˜′−1 Q

˜1 G

˜−G

˜′−1 Q

˜2 G

˜−TX

˜, (3.11.37a)

B˜

= G˜′−1 Q

˜2 G

˜−T , (3.11.37b)

C˜

= G˜′TP

˜1 G

˜−G

˜′T P

˜2 G

˜−TX

˜+X

˜′ G˜′−1Q

˜1 G

˜−X

˜′ G˜

′−1Q˜

2 G˜−TX

˜, (3.11.37c)

D˜

= G˜′TP

˜2 G

˜−T +X

˜′G˜′−1Q

˜2 G

˜−T . (3.11.37d)

They constitute the so-called 4 × 4-surface-to-surface propagator matrix ˆT˜

given by

ˆT˜

=

(

A˜

B˜C

˜D˜

)

. (3.11.38)

With the propagator matrix ˆT˜

we may rewrite equations (3.11.36) in a form similar to equations(3.10.27)

(

x′

p′ − p′0

)

= ˆT˜

(

x

p− p0

)

. (3.11.39)

The propagator matrix ˆT˜

describes how the quantities p and x characterizing the paraxialray change as a result of the wavefront propagation in the vicinity of the central ray startingat an anterior and ending at a posterior surface. The idea of describing paraxial-ray quantitiesby a propagator matrix of this type was previously expressed by Deschamps (1972). As statedbefore, the linear relationship (3.11.39) between primed and unprimed quantities is the so-calledparaxial approximation. In contrast to Bortfeld (1989), who postulated the existence of equation(3.11.39) and derived expressions for the paraxial-ray traveltimes from it, we do not assume apiecewise homogeneous medium, but smoothly varying inhomogeneous layers for which the rayvalidity conditions (3.2.16) are fulfilled. Moreover, as rules will be derived in the next sectionon how the matrices P

˜and Q

˜are transformed when paraxial rays are to be traced across an

interface, equations (3.11.37) (and thus equation (3.11.39), too) remain valid in the presence ofinternal interfaces in the medium. As we will see later, equations (3.11.39) turn out to be extremelyvaluable to solve inversion problems, where the medium between the anterior and posterior surfaceis considered unspecified. In such situations, the matrix ˆT

˜can be considered like a “black box.”

In the next chapter we will express the traveltime of paraxial rays and the geometrical-spreading factor of the central ray in terms of the 2 × 2 submatrices of ˆT

˜.

3.12 Rays across an interface

So far, we have only addressed the question how a paraxial ray in the vicinity of a known central raypropagates through a medium encompassed between an anterior and a posterior surface without

94 CHAPTER 3. ZERO-ORDER RAY THEORY

considering internal interfaces. However, at a zero-order or first-order internal interface, those rayswill reflect or transmit. Thus, to come up with the final expression of an elementary wave asrecorded at a geophone in Section 3.13, we must describe these reflections and transmissions. To doso, we have to specify physically meaningful boundary conditions at the interface. In the following,we will say that a ray “crosses an interface” implying that it hits an interface and is either reflectedor transmitted there. The ray segments on both sides of the interface may belong to different wavemodes, that is, P- or S-waves.

3.12.1 Boundary conditions

Together with the above analysis of rays at a (anterior or posterior) surface, we are now readyto consider a known central ray traced across a reflecting or transmitting curved interface. Asthis situation is not covered by equations (3.10.26), we have to find out how the matrices P

˜and

Q˜

change when the central ray crosses the interface. For that purpose we need to specify someboundary conditions for the two matrices. These are provided by the physics of wave propagationacross an interface.

Of course, any ray field needs to be continuous across an interface. As any given ray in thefield under consideration is specified by its ray coordinates γ1 and γ2 , we require that at the pointof incidence P , the same value of γ specifies the same ray on both sides of the interface, and thatthe change of γi with respect to the local coordinates xj at P also remains the same across theinterface. In mathematical terms, we require that

γi |− = γi |+ , (3.12.1)

where − denotes the side of the interface at P where the wave impinges and + denotes the side ofthe outgoing wave. The latter may be either a reflected or a transmitted wave. As we assume thiscondition to be fulfilled for all paraxial rays in the vicinity of the central ray that make part of aray family, this implies also that

∂γi∂xj −

=∂γi∂xj

+

(i, j = 1, 2) , (3.12.2a)

where x denotes the local Cartesian coordinate vector as defined above.

The second boundary condition is Snell’s law. It requires that the projected slowness vectorpT , i.e., the component of the slowness vector in the plane ΩT tangent to the interface ΣA (see Figure3.3), remains constant across the interface. Note that for a first-order paraxial-ray approximation, adiscontinuous velocity across the interface ΣA implies the modulus and direction of p—and thus alsothe ray direction—change discontinuously, because the modulus of p is 1/v [see equation (3.4.3)].As p is the projection of pT , also p is required to remain constant across the interface, i.e.,

pi|− = pi|+ . (3.12.2b)

This is again true for all paraxial rays, and thus, also the variation of p within the tangent planeΩ0 at P must be continuous. This condition can be expressed as

∂pi∂xj −

=∂pi∂xj

+

(i, j = 1, 2) , (3.12.2c)

i.e., the change of p along the interface is the same on both sides of the interface.

3.12. RAYS ACROSS AN INTERFACE 95

3.12.2 Dynamic-ray-tracing matrices

Matrix Q across the interface

How the matrix Q˜

transforms for transmission or reflection at an interface point P can be derivedfrom the boundary condition (3.12.2a). Since at the central ray, the components of γ do not changewith q3 = s, this can be rewritten as

∂γi∂qk −

∂qk∂xj −

=∂γi∂qk +

∂qk∂xj

+

(i, j, k = 1, 2) . (3.12.3)

Here we recognize the elements of the 2× 2 matrices Q˜

and G˜

as defined in equations (3.9.11) and(3.11.5), respectively. In matrix notation, we have

Q˜

−1− G

˜− = Q

˜

−1+ G

˜+ . (3.12.4)

In other words, the matrix Q˜

transforms across the interface at P according to

Q˜

+ = G˜

+ G˜

−1− Q

˜− . (3.12.5)

Note that G˜−1− describes (a) a rotation within plane Ω− (perpendicular to the central ray at P

before crossing the interface) and (b) an inverse projection from the plane Ω− into the plane Ω0

tangent to the interface at P . Correspondingly, G˜

+ describes (c) a projection from plane Ω0 intothe plane Ω+ (perpendicular to the central ray at P after crossing the interface) and (d) a rotationwithin plane Ω+. The rotations within the planes Ω− and Ω+ must be inverse to each other inorder to correctly recover the ray-centered coordinate system after crossing the interface.

Matrix P across the interface

From the second boundary condition (3.12.2c), we can derive the rule for the transformation of P˜across an interface. Taking the xj-derivative of equation (3.11.30) leads, using equation (3.11.33)

in matrix notation, toG˜T− M

˜− G

˜− +X

˜− = G

˜T+ M

˜+ G

˜+ +X

˜+ . (3.12.6)

Because of equation (3.11.27), this is equivalent to

G˜T− P

˜− Q

˜

−1− G

˜− +X

˜− = G

˜T+ P

˜+ Q

˜

−1+ G

˜+ +X

˜+ . (3.12.7)

Multiplying from the right with (Q˜

−1− G

˜−)−1, we find because of equation (3.12.4)

G˜T− P

˜− +X

˜− G

˜−1− Q

˜− = G

˜T+ P

˜+ +X

˜+ G

˜−1− Q

˜− . (3.12.8)

This can be solved for P˜

+ to lead to the final expression

P˜

+ = G˜

−T+

[

G˜T− P

˜− + (X

˜− −X

˜+)G

˜−1− Q

˜−

]

. (3.12.9)

Together with equations (3.12.5) and (3.12.9), the system of equations (3.10.26), or on account ofequations (3.11.37), also system (3.11.39) enables us to perform paraxial and dynamic ray-tracingthrough a piecewise continuous, inhomogeneous medium separated by zero- or first-order interfaces.Note that Q

˜is not affected by a first-order interface as the ray direction does not change there and

thusG˜

+ = G˜−. However, P

˜is discontinuous across a first-order interface as the velocity derivatives

appearing in X˜

are discontinuous at such an interface. At zero-order interfaces, both matrices arediscontinuous.

96 CHAPTER 3. ZERO-ORDER RAY THEORY

3.12.3 Ray Jacobian across an interface

The determinant of equation (3.12.5) is

detQ˜

+ =detG

˜+

detG˜

−detQ

˜− . (3.12.10)

As any matrix G˜

can be represented in the form indicated in equation (3.11.7) by two matrices Θ˜and Φ

˜given in equations (3.11.8), its determinant is given by

detG˜

= detΦ˜

detΘ˜

= cosϑP . (3.12.11)

We recall that ϑP is the angle the slowness vector of the ray segment under consideration makeswith the interface normal. Inserting equation (3.12.11) into expression (3.12.10), we find for thetransformation of the ray Jacobian across the interface on account of equation (3.9.16)

J+ = detQ˜

+ =detG

˜+

detG˜−

detQ˜− =

cosϑ+

cosϑ−J− . (3.12.12)

We observe that J is discontinuous across the interface. The amount of the change is given by theratio of the cosines of the angles ϑ− and ϑ+ which the incident and the outgoing ray segmentsmake with the interface normal at P .

3.13 Primary reflected wave at the geophone

The most widely used 3-D earth model for the construction of synthetic seismograms with zero-order ray theory consists of inhomogeneous layers or blocks with smoothly varying velocities andseparated by smoothly curved interfaces. Given a concentrated source at some point S in thatmedium, a multitude of elementary waves will generally constitute the complete wavefield at onesingle or a string of receivers G. In the framework of seismic imaging, the objective is not toconstruct the whole wavefield of that source for a given earth model, but to give a description ofelementary waves that appear as strong reflections in seismic records. It is these strong reflectionsthat are used to image the searched-for subsurface reflectors without the need to have the reflectionsidentified in the records. The best candidates to identify key reflectors are their elementary primaryP-P or P-S reflections. The P-P wave commonly originates at a compressional point source andtravels to the target reflector and from there to the receiver as a P-wave. A P-S wave differs fromthe P-P wave by the fact that it returns from the target reflector as an S-wave. Both waves havein common that they follow direct (i.e., no multiple) ray paths from the source to the reflectorand from the reflector to the receiver. Both elementary waves thus do not suffer from internalreverberations in the layers or blocks. Though our ultimate aim in this chapter is to analyticallyexpress (but not to actually compute) primary P-P or P-S reflections, we proceed to describearbitrary elementary waves that propagate through piecewise continuous, inhomogeneous media.The resulting ray formulas for elementary waves could, of course, be used in case of a given earthmodel to construct synthetic seismograms as a superposition of a multitude of elementary-wavecontributions.

3.13. PRIMARY REFLECTED WAVE AT THE GEOPHONE 97

S

G

S=O0G=On+1

O1

Oj

On

interface j

interface 1 interface n

anterior surface posterior surface

Oj+1

Ok

interface k k

k

n

n

ϑ

ϑ

-ϑ

ϑ

interface j+1

+

ϑ+

ϑ-

Fig. 3.4. Ray path of an arbitrary primary reflected elementary wave traveling through a piecewisecontinuous, inhomogeneous medium from a point source at S to a receiver at G.

3.13.1 Ray amplitude at the geophone

We are now ready to derive an expression for the ray amplitude and displacement vector of anelementary seismic wave at any ray point, e.g., at a geophone position G. However, let us stressonce more that this expression will be invalid, if the chosen ray point falls onto, or into the nearvicinity of, a singular point where the ray Jacobian J vanishes or tends to infinity.

We consider now an arbitrary primary-reflected elementary wave that travels along a raythrough a piecewise continuous, inhomogeneous medium after having originated at a point sourceat a position S (see Figure 3.4). Of course, multiple reflections can be treated in a completelyanalogous way. The elementary wave may or may not be subjected to a mode conversion at theinternal interfaces which it hits along its propagation path from S to G.

98 CHAPTER 3. ZERO-ORDER RAY THEORY

We denote the total number of interfaces encountered by the ray by n. At the interface withnumber k, we denote the velocity, density, angle to the surface normal, ray Jacobian, and amplitudeof the incident wave by v−k , %−k , ϑ−k , J−

k , and U−k , respectively. Correspondingly, we denote the same

quantities pertaining to the outgoing wave by v+k , %+

k , ϑ+k , J+

k , and U+k , respectively. Moreover, we

denote the amplitude-normalized displacement transmission coefficient at interface k by Tk. Let ussuppose that the reflection takes place at the interface with the number j. Then, at that interface,we need the amplitude-normalized displacement reflection coefficient Rc.

Using this notation, equation (3.6.6) expresses the amplitude UG = U(rG) at the geophoneposition G as

UG =

[

%+n v

+n J

+n

%GvGJG

] 1

2

U+n , (3.13.1)

where vG, %G and JG are the values of v, %, and J at G (see Figure 3.4) on the posterior surfaceafter propagation through the whole stack of layers.

Let us now see what happens to the above formula when the amplitude change due to thetransmission at interface n is taken into account. Recall that the wave may have been transmittedmonotypically or mode-convertedly at interface n. Under the assumption that the amplitude lossdue to transmission is well described by a corresponding (real) transmission coefficient Tn, we mayreplace U+

n by TnU−n . Usually, this assumption is satisfied with Tn being the plane-wave transmission

coefficient. Note that an incident S-wave has to be separated into its SV and SH components withrespect to the local coordinate system at interface n before the chosen component can be multipliedby the corresponding transmission coefficient.

Exact expressions for the plane-wave reflection and transmission coefficients can also befound in Cerveny et al. (1977) or Cerveny (2001). However, linearized formulas for the reflectioncoefficients are frequently used, too. For completeness, the most important linearized formulas forthese coefficients are stated in Appendix A.

Moreover, to relate UG to U−n , the incident ray amplitude at interface n, the ray Jacobian,

J+n , must be replaced with the help of equation (3.12.12) by cosϑ+

n

cosϑ−nJ−n . The next step consists of

using again equation (3.6.6) to relate U−n to the amplitude U+

n−1 of the ray after transmission atinterface n− 1. We have, after a little rearrangement of the factors,

UG =

[

%+n v

+n cosϑ+

n

%−n v−n cosϑ−n

] 1

2

Tn ·[

%+n−1v

+n−1J

+n−1

%GvGJG

] 1

2

U+n−1 . (3.13.2)

In each layer and at each interface encountered by the central ray from S to G the very sameconsiderations apply. Therefore, additional factors of the same form appear in the above equation.The only difference is to be made at interface j, where the amplitude loss is described by thereflection coefficient Rc. Thus, we can trace the amplitude factor recursively back from G to thefirst interface. Under the slight abuse of notation Rc = Tj, we may write

UG =

n∏

k=2

[

%+k v

+k cosϑ+

k

%−k v−k cosϑ−k

] 1

2

Tk

[

%+1 v

+1 J

+1

%GvGJG

] 1

2

U+1 . (3.13.3)

At the first interface, we use again equation (3.12.12) to replace J+1 by

cosϑ+

1

cosϑ−1

J−1 and account for

the transmission loss by setting U+1 = T1U−

1 . Since we have assumed that at the initial point S

3.13. PRIMARY REFLECTED WAVE AT THE GEOPHONE 99

of the ray a point source is situated, we must now use equation (3.6.14) to find the point sourcesolution of the transport equation for U−

1 , namely

U−1 =

g√

%−1 v−1 %SvS

1√

J−1

, (3.13.4)

where vS and %S are the velocity and density at S. Using this expression, we finally obtain for UG

UG =g√

vS%SvG%G

n∏

k=1

(

%+k v

+k cosϑ+

k

%−k v−k cosϑ−k

) 1

2

Tk

1√JG

. (3.13.5)

Observe that this is just the amplitude factor of the smooth medium of equation (3.6.14) multipliedby a factor accounting for the amplitude loss at transmitting and reflecting interfaces.

The ray Jacobian JG at G for a point source at S is related to the corresponding normalizedgeometrical-spreading factor L according to its definition (3.6.15) as

L =1√vGvS

√

JG =1√vGvS

√

detQ˜

2 (3.13.6a)

or, taking into account the definition of the square root of the ray Jacobian in equation (3.7.1),

L =1√vGvS

∣∣∣detQ

˜2

∣∣∣

1

2 e−iπ

2κ , (3.13.6b)

where κ is the number of caustics encountered along the ray. For a ray traced from an anterior toa posterior surface, we may also write due to equations (3.11.37b) and (3.11.15)

L =

√

cosϑS cosϑGvGvS

|detB˜| 12 e−iπ

2κ , (3.13.6c)

where ϑS and ϑG are the angles the ray makes with the surface normals at S and G, respectively.

Note that equations (3.13.6) exclusively define the geometrical-spreading factor for a pointsource with the chosen parametrization γ as only in this case, the matrix Q

˜equals its particular

value Q˜

2 for point-source initial conditions [see equations (3.10.23)]. In a homogeneous medium, orfor a reflection at a planar reflector below a homogeneous overburden, the so-defined L equals thedistance between S and G along the ray. For a normal ray in a vertically inhomogeneous medium,it reduces to the well-known formula of Newman (1973), that is, L = v2

RMSt0/v0.

As another abbreviating notation, we introduce the “total transmission loss” A along the raySG due to all discontinuities in the 3-D medium caused by velocity and density changes. We alsoinclude the source strength g and the velocities and densities at the source and the receiver into itsdefinition, but exclude the particular reflection coefficient Rc at the target reflector (interface j inFigure 3.4), because to recover the latter is the aim of our true-amplitude imaging. We then writethis real quantity as

A =g

√

%Gv2G%Sv

2S

n∏

k = 1k 6=j

(

%+k v

+k cosϑ+

k

%−k v−k cosϑ−k

) 1

2

Tk , (3.13.7)

100 CHAPTER 3. ZERO-ORDER RAY THEORY

where j signifies the interface number of the target reflector. Note that the factor

Tk =

(

%+k v

+k cosϑ+

k

%−k v−k cosϑ−k

) 1

2

Tk (3.13.8)

is the so-called reciprocal (or energy-normalized) transmission coefficient at interface k (Cerveny,2001). Note that for a monotypic (i.e., P-P or S-S) reflection, the correspondingly defined reciprocalreflection coefficient

Rc =

(

%+k v

+k cosϑ+

k

%−k v−k cosϑ−k

) 1

2

Rc (3.13.9)

is equal to the standard (amplitude-normalized) reflection coefficient Rc, because in that case thevalues of velocity, density and propagation angle before and after reflection are equal to each other.With definition (3.13.8), we can also write

A =g

√

%Gv2G%Sv

2S

n∏

k = 1k 6=j

Tk . (3.13.10)

Inserting definitions (3.13.6a) and (3.13.7) into equation (3.13.5) leads to our final expressionfor the ray amplitude of the direct reflected elementary wave of Figure 3.4 at the receiver G,

UG =AL Rc . (3.13.11)

We remark once more that this amplitude may assume complex values in case the consideredelementary wave encounters caustics or overcritical reflections along its ray path through the layeredor blocked inhomogeneous medium.

Note that equation (3.13.11) describes the amplitude factor U (P ) or U (S), according to whetherthe wave emerging at G is a P- or an S-wave. For an acoustic wave, due to the correspondencebetween

√

%v2U and P/√%, a corresponding equation holds for the pressure amplitude P at G,where the transmission-loss factor A is given by

A = g√%G%S

n∏

k = 1k 6=j

(

%−k c−k cosϑ+

k

%+k c

+k cosϑ−k

) 1

2

Tk . (3.13.12)

The difference to the elastic version, equation (3.13.10), lies in the different dependence on the

source and receiver velocities and densities, and in the inversion of the factors%+

kc+k

%−kc−k

.

3.13.2 Complete transient solution

Collecting results, we find the final expression for the ray-theoretical high-frequency approxima-tion of the displacement vector of a selected elementary wave traveling through an inhomogeneousmedium from a point source at S to a geophone at G. We now only state the particular form thatresults for a reflected P-wave emerging at G. However, we remind the reader that by simple substi-tutions as indicated above, the corresponding formulas for an S-wave or an acoustic wave emerging

3.14. SUMMARY 101

at G are obtained. Thus, mode-converted elementary waves present no principal difficulties. Theycan also be built up by hopping from interface to interface. All one has to do is to account for therespective mode-conversions in the above formula (3.13.7). Using equations (3.3.16), (3.4.12), and(3.13.11), we may express the displacement vector of any elementary transient P-wave reflection(3.2.13) at the geophone G as

u(rG, t) = Re U(rG, t) tG , (3.13.13)

where tG is the unit vector in ray direction at G and U is the analytic principal component of theparticle displacement.

For an emerging S-wave, an equation of the simple form (3.13.13) cannot be found as S-wavesare in general elliptically polarized. This means that the general form of the particle displacementis

u(rG, t) = Re U1(rG, t) e1 + Re U2(rG, t) e2 , (3.13.14a)

where U1 and U2 are the analytic contributions to the S-wave displacement vector in the directionsof the ray-centered coordinate vectors e1 and e2, respectively. However, as we have seen before, thescalar function

U(rG, t) =√

U21 (rG, t) + U2

2 (rG, t) (3.13.14b)

describes the analytic principal component of the particle displacement of the S-wave at G.

In both cases, i.e., for P- and S-waves, the analytic principal component of the particledisplacement is given by

U(rG, t) = U(rG) F [t− TR(rG)]

=AL Rc F [t− TR(rG)] , (3.13.15)

where we have substituted equation (3.13.11). The factors A and L are given in equations (3.13.6)and (3.13.7), respectively. The factor Rc is the reciprocal reflection coefficient at the target reflector.If the reflection at the target reflector is an acoustic or monotypic (P-P or S-S) reflection, thereciprocal reflection coefficient Rc equals the plane-wave reflection coefficient Rc. The functionF [t] denotes the analytic seismic source pulse, and TR(rG) denotes the reflection traveltime at thegeophone position G. How the amplitude is affected if the wave is recorded at a free surface isdiscussed in Appendix B.

For an acoustic wave, the analytic pressure field is again described by equation (3.13.15),where A is, however, given in equation (3.13.12).

Note that the factor A assumes the fairly simple form given in equation (3.13.7) (or (3.13.12))due to the simplifying assumptions about the medium. However, if other factors affect the seismicdisplacement vector like, for instance, scattering attenuation due to a thinly layered overburden,intrinsic absorption, geophone coupling, wave-conversion at a free surface, etc., the factor A andthe dislocation vector must be modified accordingly. How, e.g., a thinly layered overburden can betaken into account is shown in Shapiro et al. (1994).

3.14 Summary

In this chapter, we gave an introduction to zero-order ray theory as far as it is needed for theunderstanding of the following chapters. We started from the elastodynamic wave equation (3.1.1)

102 CHAPTER 3. ZERO-ORDER RAY THEORY

and derived by means of the ray ansatz (3.2.5) for elementary-wave solutions the eikonal equation(3.4.1) and the transport equation (3.3.26). Rays turned out to be the characteristics of the eikonalequation. They can be traced in a medium free of interfaces by means of so-called ray-tracingsystems (3.4.11) which constitute a system of first-order partial differential equations. Later, wesaw that interfaces can be incorporated by specifying certain physical boundaries (3.12.2). Bysolving the transport equation (3.3.26) the expression (3.6.6) for the ray amplitude in terms of theray Jacobian (3.5.3) was obtained.

A transformation of the ray-tracing system (3.4.11b) into ray-centered coordinates and ap-plication of the paraxial approximation lead to the approximate ray-tracing system (3.10.1). Usingthis system in the paraxial vicinity of a known central ray, we also derived the dynamic ray-tracingsystem (3.10.12). Although it was derived from studying paraxial rays, the latter system turnedout to describe also the ray Jacobian and thus the amplitude of the ray. Two fundamental solutionsof the dynamic ray-tracing system (3.10.12) (or four fundamental solutions of the approximate

ray-tracing system (3.10.1)) constitute the so-called propagator matrix ˆΠ˜

(equation (3.10.24)) thatpropagates the dynamic quantities by equations (3.10.23) and the paraxial-ray parameters by equa-tion (3.10.26) from the initial point S to the end point G of the central ray.

Thereafter, paraxial ray theory was extended to wave propagation from an anterior surfaceto a posterior surface. Cartesian coordinates and slowness vector representations within the planetangent to the considered surface were introduced as paraxial ray parameters. They helped to sim-plify the expressions. Besides the propagator matrix ˆΠ

˜in ray-centered coordinates, the alternative

surface-to-surface propagator matrix ˆT˜

[see equation (3.11.38)] was introduced that propagates theparaxial parameters from the anterior to the posterior surface [see equation (3.11.39)]. With thehelp of the slowness vector projection (3.11.34a), it was simple to specify physically meaningfulboundary conditions (3.12.2) at interfaces. In this way, interfaces can be incorporated into paraxialray theory. Taking these boundary conditions into account, it was shown that not only the raydirection but also its propagator submatrices (3.12.5) and (3.12.9) are discontinuous across an in-terface. However, with these modified propagator submatrices, the formulas for paraxial ray theoryderived for interface-free media remain valid in the presence of internal interfaces.

The ray Jacobian J defines the geometrical-spreading factor of a wave emanating from a pointsource by equation (3.6.15). It was constructed in terms of the propagator submatrices Q

˜2 of ˆΠ

˜or B

˜of ˆT

˜in equations (3.13.6). Together with equation (3.6.6) that propagates the ray amplitude

through a medium without interfaces and with equation (3.12.12) that transforms J across aninterface, we derived an expression for the ray amplitude at a geophone G of an elementary wavewith a given ray code that starts at a point source S.

This finally leads to a high-frequency approximation of the transient single elementary re-flected wave as it would be measured at an arbitrary receiver point G [given in equation (3.13.15)].Note, however, that a seismic record always consists of a superposition of many elementary reflectedwaves all of which can be described by the above theory.

Chapter 4

Surface-to-surface paraxial ray theory

This chapter contains certain paraxial-ray theoretical foundations. These are necessary for the un-derstanding of most of the derivations and analyses presented later on in connection with amplitude-preserving seismic reflection imaging. Additionally, it will provide the links between the factors thatwill appear later in the imaging formulas and dynamic ray tracing, thus indicating how to prac-tically compute the quantities involved in the Kirchhoff-type imaging methods. The definitions ofquantities and variables are, wherever possible, chosen in agreement to the conventions previouslyused by us and other authors. However, some differences in notation to previous publications couldnot be avoided.

4.1 Paraxial rays

In the vicinity of a known ray SG (the central ray, see Figure 4.1), other rays can approximately becalculated by the well-known paraxial ray theory (Cerveny, 1985, 1987, 2001; Ursin, 1986; Bortfeld,1989; Virieux and Farra, 1991). As we have already seen in Chapter 3, the parameters describing aparaxial ray S G are its distance to the central ray SG and the deviation of its slowness vector fromthat of the central ray. Paraxial ray theory implies that the values of these parameters at any pointof a paraxial ray are linearly dependent on those at its initial point. This dependency is describedby either the popular propagator matrix ˆΠ

˜(Cerveny, 1985, 1987, 2001) in ray-centered coordinates

or the surface-to-surface propagator matrix ˆT˜

(Bortfeld, 1989), both introduced in Chapter 3. Inthis chapter, we will see how the two-point traveltime T (S,G) along the paraxial ray S G andits second-order derivative matrices, as well as the geometrical-spreading factor L(S,G) and theFresnel volume (Kravtsov and Orlov, 1990; Cerveny and Soares, 1992) along the central ray can beobtained by paraxial ray theory.

In three-dimensional (3-D) forward and inverse seismic modeling and imaging investigations,paraxial ray tracing plays a key role because of its appealing geometrical features and its mathe-matical simplicity. All paraxial ray parameters including its propagation direction and distance toa central ray, and even the amplitude along the central ray are described by the propagator matrix.Various propagator matrix approaches to paraxial ray tracing have been introduced in recent years,based on the work of Deschamps (1972), Cerveny (1987), Bortfeld (1989), or Virieux (1991), to citea few. The propagator matrix of a ray is a very useful quantity to solve a number of problems. Themain difference between different approaches to paraxial ray tracing consists in the dimensionality

103

104 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

G

S S

pp0 p

p

p0

0

A

x

G

pp

0

p0

0

P

x

central rayparaxial ray

’

’’

’

’

anterior surface

posterior surface

interface

Ω

Σ

Ω

Σ

Fig. 4.1. A central ray SG is assumed to be known in a layered, inhomogeneous medium. Theneighboring ray S G is to be described by paraxial ray theory. Note that ray SG symbolicallyrepresents any arbitrary ray that may reflect and transmit an arbitrary number of times at allpossible interfaces encountered along its path. It is implicitely understood that all paraxial rays toSG will reflect and transmit at the same interfaces the same number of times.

4.2. TRAVELTIME OF A PARAXIAL RAY 105

of the propagator matrix. As a 4 × 4 propagator matrix reflects the four degrees of freedom of theparaxial ray tracing problem, we consider it advantageous e.g. over a 6 × 6-propagator matrix asused by Virieux (1991). One popular 4× 4 propagator matrix is, for example, the matrix ˆΠ

˜that is

computed in a ray-centered coordinate system by dynamic ray tracing (Cerveny, 1987, 2001). Bort-feld (1989) formulated without dynamic ray tracing a 4 × 4 surface-to-surface propagator-matrixfor a bundle (or pencil) of rays passing through a system (i.e. model) of constant-velocity layersseparated by smoothly curved interfaces (a so-called seismic system).

Paraxial ray theory is not only of use for solving forward problems (Popov and Psencık,1978; Cerveny et al., 1977; Cerveny, 1985, 1987, 2001; see also references there). Particularly inthe seismic reflection method it has been used to solve a number of kinematic and dynamic (non-iterative) inversion problems as addressed, e.g., by Hubral and Krey (1980), Bortfeld (1982), Ursin(1982b), Hubral (1983), Goldin (1986) and many others. Entirely based on the propagator-matrixconcepts presented here, we developed in Tygel et al. (1992) and Schleicher et al. (1993a) a methodto perform a true-amplitude correction of seismic reflections purely from (picked) traveltimes (i.e.,without any knowledge about the earth model) and a theory of prestack seismic true-amplitudemigration. Further applications of paraxial ray theory can be found in two workshop proceedings ontrue-amplitude imaging (Hubral, 1998) and macro-model-independent imaging methods (Hubral,1999).

In Chapter 3, we have generalized Bortfeld’s results to inhomogeneous isotropic layers orblocks by relating the surface-to-surface propagator matrix ˆT

˜to the well-known ˆΠ

˜propagator

matrix (Hubral et al., 1992a). In this chapter, we present all of the paraxial ray theory that isneeded to geometrically understand the later derivations of the theory of seismic imaging. With thehelp of the formulas derived in this chapter, it will also become possible to compute by dynamicray tracing the quantities needed for the weight functions in the amplitude-preserving imagingalgorithms.

Finally, we also give some examples that prove the usefulness of the developed theory whenapplied to seismic problems (see also Hubral et al., 1992b). We derive a formula for the computationof Fresnel zones along a central ray and its projection into the earth’s surface. This particularexpression will then help us to find a decomposition formula (i.e. the so-called chain rule) of ˆT

˜and particularly of the important 2 × 2 submatrix B

˜of ˆT

˜. We also show in Section 4.5 that the

so-called NIP-wave theorem (where NIP stands for normal-incidence point), which is fundamentalfor generalized Dix-type traveltime inversion schemes (Hubral and Krey, 1980; Hubral, 1983), doesin general not hold if the central ray is no longer a (two-way) normal incidence ray. All resultingformulas in this chapter will be of use when addressing general problems of seismic imaging.

4.2 Traveltime of a paraxial ray

In this section, we will derive an expression for the traveltime T = T (S,G) along any givenparaxial ray S G in the vicinity of the central ray SG. This expression then describes a wavefrontas T (S,G) =constant. However, we are not interested in an exact expression for T but only in anasymptotic one. More explicitly, we look for an expression for T of the second order in the 2-Ddislocation coordinates x between S and S and x′ between G and G (see Figure 4.1). The two pairsof points S and S as well as G and G fall onto the anterior and posterior surfaces, respectively.

106 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

G

GGw

S

SS

2

1

w 1

2wavefront

wavefront

ray 2

ray 1

Fig. 4.2. Arbitrary ray 2 from S2 to G2 in the vicinity of the given ray 1 from S1 to G1.

There is one basic assumption for establishing a paraxial approximation for T (S,G): For anyarbitrary paraxial ray in the vicinity of the known central ray, there is a source-receiver distributionsuch that a physical wave would actually propagate along these two rays with a continuous wavefrontconnecting them. Only if such a wavefront can exist, a description of the paraxial ray in dependenceon the central ray makes sense. Of course, if we want to describe a second paraxial ray, we mayhave to consider a different physical wave with a different wavefront. In other words, we want todescribe any arbitrary possible paraxial ray and not only a set that pertains to a single physicalsituation.

Under these assumptions, we can make use of Hamiltonian theory. On the basis of this theory,Bortfeld (1989) derived the traveltime for a paraxial ray S G in the vicinity of a normal ray, i.e., aray SG reflected in itself with its source and receiver location being coincident i.e., G = S. Here,we generalize Bortfeld’s derivation to an arbitrary central ray where S and G may be separated onthe anterior surface by a certain offset or even be located on different surfaces.

4.2.1 Infinitesimal traveltime differences

Before entering into paraxial approximation, let us now study the infinitesimal traveltime differencebetween two neighboring rays that pertain to the same ray family (Lagrangian manifold) froma physical point of view (see Figure 4.2). The resulting expression can be derived on a strictlymathematical basis by Hamiltonian or Lagrangian theory as it can be found in many textbooks ontheoretical optics or mechanics (see, for instance, Herzberger, 1958; Luneburg, 1964; Kline and Kay,1965; Born, 1965; Sommerfeld, 1964; Buchdahl, 1970; Born and Wolf, 1987; Landau and Lifshits,1998).

As derived in Chapter 3, the traveltime of an elementary seismic wave pertaining to thecentral and one arbitrary neighboring ray pertaining to the same family must be a solution of theeikonal equation (3.4.1). However, let us stress again that for any two given neighboring rays, theirtraveltimes do not necessarily fulfill the eikonal equation with the same initial conditions, i.e., theydo not belong to the same physical experiment. In fact, we are considering here the whole set ofall possible initial conditions for the eikonal equation, i.e., the whole set of possible physical waves

4.2. TRAVELTIME OF A PARAXIAL RAY 107

p

P

P

drdr

rr

w

P2

w

1

2

rw

w

O

wavefront

ray 2

ray 1

Fig. 4.3. Relationship between the slowness vectors and traveltimes of the central and paraxialrays. For details, see text.

traveling in the vicinity of the given central ray.

Let us consider the situation in Figure 4.2. Assume that we know the traveltime T1 for ray 1from S1 to G1 and we would like to determine the traveltime T2 along the neighboring ray 2 from S2

to G2, where S2 and G2 are in infinitesimally small vicinities of S1 and G1, respectively. Accordingto our assumption, there exists a wavefront between both rays at S1 cutting ray 2 at Sw and asecond one at G1 cutting ray 2 at Gw. Thus, the traveltime along ray 2 from Sw to Gw is just givenby T1. Then, the traveltime difference between the ray segments S1G1 and S2G2 can be expressedas

dT = dTG − dTS (4.2.1)

where dTS and dTG are the traveltime differences because of the dislocations of S2 from Sw and G2

from Gw, respectively, along ray 2. In equation (4.2.1), the sign in front of dTS is negative becausethe traveltime T2 is smaller than T1 if S2 is displaced from Sw in the positive ray direction. Notethat we assume the distances from S2 to Sw and from G2 to Gw and the respective traveltimedifference to be infinitesimally small. Since the situations at S1 and G1 are conceptually identical,we discuss the traveltime differences referring to Figure 4.3 for a generic pair of points P1 and P2.

We want to determine the traveltime difference dTP due to a dislocation from a given pointP1 on a given ray 1 to an arbitrary point P2 located in its close vicinity on an arbitrary neighboringray 2. For that purpose, let us consider a point Pw on ray 2 and the wavefront defined by P1 andray 1. Let Pw be described in global coordinates by its position vector rw(s) in dependence ofthe arclength s. Moreover, let its slowness vector pw at Pw(s), with global coordinates rw(s) (seeFigure 4.3), be defined as in equation (3.4.2) by

pw = ∇T =1

vtw (4.2.2)

where tw is the unit vector in propagation direction at Pw, given by equation (3.4.12), and where vis the local velocity at Pw(s). Finally, T is the traveltime along the ray from some previous point,

108 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

at which by definition T = 0 and s = 0, to Pw(s).

As shown in Chapter 3, the eikonal equation (3.4.1) can be replaced, for instance, by theray-tracing system (3.4.11b), which reads at Pw

drwds

= vpw , (4.2.3a)

dpwds

= ∇

(1

v

)

. (4.2.3b)

Solving equation (4.2.3a) for drw yields

drw = vpw ds , (4.2.4)

which shows that an infinitesimal step drw along the ray is always directed parallel to the slownessvector pw. This means that, since we are considering infinitesimally small distances in Figure 4.3,the ray segment between Pw(s) and P2 and the wavefront segment between Pw(s) and P1 haveto be considered straight lines that are perpendicular to each other. The dot product of equation(4.2.4) with pw yields

pw · drw = (pw)2 v ds =1

vds = dT , (4.2.5)

where we have made use of equations (3.4.3) and (3.4.8b). Equation (4.2.5) holds for any arbitrarypoint on any ray and thus in particular for point Pw in Figure 4.3. Thus, according to equation(4.2.5) the traveltime difference between points Pw and P2 along ray 2 is given by

dTP = pw · drw , (4.2.6)

where drw is the dislocation vector from Pw to P2. Again, we make use of the fact that pw anddrw are parallel to each other. Since pw is perpendicular to the wavefront at Pw, we observe thatpw · drw = pw · dr, where dr is the dislocation from P1 to P2. From that fact that the differencebetween the slowness vectors p at P1 and pw at Pw is of the order of drw, we can moreover concludethat pw ·dr = p ·dr. Since we have already observed above that Pw and P1 have the same traveltimebecause they are located on the same wavefront, dTP is the desired traveltime difference betweenthe wavefronts at P1 and P2. Thus, we finally find

dTP = p · dr (4.2.7)

at any arbitrary point P1 along the central ray. Physically, this means that the change of thetraveltime due to a perturbation of point P1 with coordinates r to P2 with coordinates r+ dr onlydepends on the component drw of dr in the direction of the neighboring ray.

Equation (4.2.7) holds in the vicinity of any arbitrary point P1 of the central ray, in particularat points S1 and G1. By insertion of equations of the type (4.2.7) into identity (4.2.1), we obtainfor the searched-for traveltime difference in the inhomogeneous medium between the rays joiningpoints S1 and G1 and S2 and G2

dT =∂T∂r′j

dr′j +∂T∂rj

drj = p′ · dr′ − p · dr . (4.2.8)

This is Hamilton’s equation for two-point ray tracing.

As mentioned in connection with equation (4.2.1), the signs in equation (4.2.8) reflect thefact that the traveltime is diminished when the initial point S2 on ray 2 is displaced in the directionof the central ray but it is increased if the end point G2 is displaced in the ray direction.

4.2. TRAVELTIME OF A PARAXIAL RAY 109

It should be noted that equations (4.2.5) and (4.2.8) are nothing else but an alternativemathematical formulation of Fermat’s principle. They state that the first derivative of the traveltimein the direction vertical to the ray vanishes. Let us mention once more that the traveltime differencedepends on six and not on only three free parameters because it denotes the traveltime differencebetween the central and any arbitrary neighboring ray, i.e., the solution of any arbitrary possibleeikonal equation.

We stress once more that up to this point, there is no paraxial approximation involved in thederivation. However, for the purpose of integrating equation (4.2.8), i.e., to find an expression forthe traveltime along ray S G in dependence on that of ray SG, we will assume linear dependence ofthe ray quantities on the ray coordinates. This assumption, called paraxial approximation, is thebasis of the derivations presented in the next section.

4.2.2 Surface-to-surface propagator matrix

In this section, we study the traveltime of a paraxial ray S G in the vicinity of a known centralray SG following the lines of Chapter 3 and the original work of Bortfeld (1989) and Hubral etal., (1992a,b). We suppose the initial and end points of the central and paraxial rays to fall uponarbitrarily curved surfaces. The surface where S and S are located is referred to as the anteriorsurface, and the surface where G and G are located as the posterior surface. The situation isdepicted in Figure 4.1.

We start by considering small perturbations dr and dr′ of the paraxial ray’s initial and endpoints S and G along the anterior and posterior surfaces, respectively. Note, however, that toallow for the combination of the above Hamiltonian theory with the paraxial-ray formalism derivedin Chapter 3, we will not be able to use the 3-D perturbation vectors dr and dr′ nor the 3-Dslowness vectors pp and p′p, but we need their 2-D projections into the planes Ω0 and Ω′

0 thatare tangent to the anterior and posterior surfaces at S and G, respectively. Suppose the necessarycascaded projections have been carried out as described in Section 3.11.1 and actually carried out inSection 3.11.5. The resulting projected slowness vectors, expressed in local Cartesian coordinates,are p and p′. Moreover, let the perturbation vectors dr and dr′ be represented in local Cartesiancoordinates as dx and dx′ and have the 2D projections dx and dx′.

As discussed above, these projections are unique. Therefore, the 3-D vectors can be recon-structed from the 2-D projections using the respective surface. Thus, Hamilton’s equation (4.2.8)can be represented as a function of the 2-D projection vectors only. To understand this, let usconsider the situation at the anterior surface. The situation at the posterior surface is analogous.In local Cartesian coordinates, a small perturbation of the paraxial ray’s initial point S along theanterior surface can be described by a dx which is obtained from the perturbation of equation(3.11.1), viz.,

dx =

(

dxx · F

˜dx

)

. (4.2.9)

As before x denotes the local Cartesian coordinates of S within the plane Ω0 and F˜

is the curvaturematrix of the anterior surface as defined by means of equation (3.11.1). After multiplication ofequation (3.11.19) with dx from expression (4.2.9), we can neglect all terms of higher order in x toobtain in linear approximation

p(x)p · dx = p

(x)T · dx , (4.2.10)

110 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

where p(x)T is the projection of p(x)

p into the plane tangent to the anterior surface at S as definedin equation (3.11.16). In the same way, by multiplication of equation (3.11.19) with the above dx,we also have in linear approximation in x

p(x)T · dx = pp · dx+

cosϑPv0

x · F˜dx = p · dx , (4.2.11)

where we have used that F˜

is a symmetric matrix. Note that the last equality in the above relation-ship is due to equation (3.11.20). This discussion applies of course to both primed and unprimedquantities, i.e., to both the vector projections at the anterior and posterior surfaces. Thus, inparaxial approximation, Hamilton’s equation (4.2.8) may be rewritten as

dT = p′ · dx′ − p · dx . (4.2.12)

At this point, we consider the linear relationship between primed and unprimed quantitiesgiven by equations (3.11.36). By solving equation (3.11.36a) for p−p0 and inserting the result intoequation (3.11.36b), one obtains the following two equations for p and p′,

p = p0 + B˜

−1x′ − B˜

−1A˜x , (4.2.13a)

p′ = p′0 + D˜B˜

−1x′ − D˜B˜

−1A˜x + C

˜x . (4.2.13b)

To end up with a second-order approximation of the traveltime difference between the centraland paraxial rays, we now insert equations (4.2.13) into formula (4.2.12). However, before we canintegrate the resulting expression for dT , we have to check that it is indeed integrable. The firstcondition is that the second-derivative matrices of the traveltime T with respect to unprimedcoordinates x as well as those with respect to primed coordinates x′ must be independent of theorder of differentiations, i.e.,

∂2T∂xi∂xj

=∂pi∂xj

!=

∂pj∂xi

=∂2T∂xj∂xi

, (4.2.14a)

∂2T∂x′i∂x

′j

=∂p′i∂x′j

!=

∂p′j∂x′i

=∂2T∂x′j∂x

′i

, (4.2.14b)

where the exclamation mark above the equal sign symbolizes that these are physically requiredconditions. By taking the respective derivatives of equations (4.2.13), the above conditions yield

the following two identities for the submatrices of the propagator matrix ˆT˜:

A˜B˜T = B

˜A˜T , (4.2.15a)

D˜TB

˜= B

˜TD

˜. (4.2.15b)

Moreover, from elementary mathematical rules, we know that equation (4.2.12) is only integrableif the expression on the right-hand side is a total differential. This is the case if and only if themixed second derivatives of T with respect to primed and unprimed coordinates are independentof the order of differentiation. This can be checked by comparing ∂pi/∂x

′j with ∂p′j/∂xi, i.e.,

∂2T∂xi∂x

′j

=∂pi∂x′j

!= −

∂p′j∂xi

=∂2T∂x′j∂xi

. (4.2.16)

4.2. TRAVELTIME OF A PARAXIAL RAY 111

The negative sign on the right-hand side is due to the negative sign of the second term in equation(4.2.8). This last condition translates into the following identity for submatrices of the propagator

matrix ˆT˜:

B˜

−1 =(

D˜B˜

−1A˜−C

˜

)T

= A˜TB

˜−TD

˜T −C

˜T . (4.2.17)

Multiplication of equation (4.2.17) from the left and right, respectively, with B˜

leads, under con-sideration of equations (4.2.15), to

A˜D˜T − B

˜C˜T = I

˜, (4.2.18a)

A˜TD

˜− C

˜TB

˜= I

˜. (4.2.18b)

From the symmetry of the unit matrix I˜, we have that also

D˜A˜T − C

˜B˜T = I

˜, (4.2.19a)

D˜TA˜

− B˜TC˜

= I˜. (4.2.19b)

By multiplication of equation (4.2.17) from the left with D˜

and from the right with A˜

, respectively,D˜C˜T and C

˜TA˜

are seen to be represented by a difference of symmetric matrices and are, thus,symmetric themselves. In symbols

D˜C˜T = C

˜D˜T , (4.2.20a)

C˜TA˜

= A˜TC

˜. (4.2.20b)

The four equations (4.2.15), (4.2.18), (4.2.19), and (4.2.20) together describe the so-called symplec-

ticity of the propagator matrix ˆT˜. This property can be expressed in one single equation defining

the inverse of ˆT˜, i.e.,

ˆT˜−1 =

A˜

B˜

C˜

D˜

−1

=

D˜T −B

˜T

−C˜T A

˜T

. (4.2.21)

The identity ˆT˜

ˆT˜−1 = ˆI

˜then reproduces the above four equations (4.2.15a), (4.2.18a), (4.2.19a),

and (4.2.20a). The corresponding identity ˆT˜−1 ˆT

˜= ˆI

˜yields the other four equations, (4.2.15b),

(4.2.18b), (4.2.19b), and (4.2.20b).

From equation (3.11.39) together with the symplecticity (4.2.21) of ˆT˜, we conclude how the

initial values x and p of the paraxial-ray parameters for S can be computed if the end values x′

and p′ for G are known. From the avove analysis, we know that ˆT˜−1 exists. Thus, it can be applied

to equation (3.11.39) to yield(

x

p− p0

)

= ˆT˜−1

(

x′

p′ − p′0

)

. (4.2.22)

With the help of equation (4.2.22), we can now also determine the propagator matrix ˆT˜∗

of the reverse ray, i.e., the ray that starts at point G and ends at point S. In correspondence toequation (3.11.39), rays paraxial to the reverse ray must satisfy

(

x∗′

p∗′ − p∗0′)

= ˆT˜∗

(

x∗

p∗ − p∗0

)

, (4.2.23)

112 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

where x∗ and x∗′ describe the coordinates of the initial and end points of the reverse paraxial rays,i.e., of points G and S, respectively. Therefore, we have x∗ = x′ and x∗′ = x. In the same way, p∗

and p∗′ are the corresponding slowness vector projections at G and S, respectively.

To express ˆT˜∗ with the help of the submatrices of ˆT

˜, we make use of the kinematic reciprocity

of seismic rays. In this connection, reciprocity means that a wave which travels from G to S followsthe identical central ray trajectory as the one from S to G, but in opposite direction. This directionis mathematically expressed by the fact that the 3-D slowness vectors p∗ and p∗′ of the reverseparaxial rays and thus also their projections p∗ and p∗′ possess the same components as thecorresponding vectors of the original ray but with the opposite sign, i.e., p∗ = −p′ and p∗′ = −p.With the help of equations (4.2.21) and (4.2.22), one thus finds

ˆT˜∗ =

A˜∗ B

˜∗

C˜∗ D

˜∗

=

D˜T B

˜T

C˜T A

˜T

. (4.2.24)

Expressions (4.2.21) and (4.2.24) are very important for certain derivations in the following sections

and chapters. The possibility to express ˆT˜−1 and ˆT

˜∗ in terms of ˆT

˜provides the basis for the

derivation of some useful decomposition formulas for certain quantities to be computed along anarbitrary central ray that can be divided into two or more segments (see also Hubral et al., 1995).

4.2.3 Paraxial traveltime

We are now ready to integrate equation (4.2.12) in x (i.e., from S to S) and x′ (i.e., from G toG) to find the two-point traveltime T of a wave traveling along a paraxial ray S G in the vicinityof a known central ray SG that possesses the traveltime T0. For that purpose, we insert equations(4.2.13) into formula (4.2.12). Under consideration of the symplectic property of ˆT

˜, the so-called

“paraxial traveltime” T is found to be

T (x,x′) = T0 − p0 · x + p′0 · x′ − x ·B˜

−1x′

+1

2x ·B

˜−1A

˜x +

1

2x′ ·D

˜B˜

−1x′ . (4.2.25)

Equation (4.2.25) provides a neat and compact second-order approximation for the traveltime of atransmitted paraxial ray through a piecewise continuous, inhomogeneous medium with smoothlycurved interfaces. This equation is only valid in a certain neighboring region, i.e., the so-calledparaxial vicinity, of a (known) central ray. The extent of the paraxial vicinity strongly depends onthe inhomogeneity of the medium. In a homogeneous medium, for instance, it extends to infinity.

As they will become useful at a later stage, let us at this point introduce the second-derivative(or Hessian) matrices of traveltime (4.2.25), viz.,

N˜GS =

(

∂2T∂xi∂xj

)

i,j=i,2

= B˜

−1A˜, (4.2.26a)

N˜SG =

(

∂2T∂x′i∂x

′j

)

i,j=i,2

= D˜B˜

−1 , (4.2.26b)

N˜SG = −

(

∂2T∂xi∂x′j

)

i,j=i,2

= B˜

−1 , (4.2.26c)

4.2. TRAVELTIME OF A PARAXIAL RAY 113

where all derivatives are evaluated at the central ray, i.e., at x= x′ = 0. Matrix N˜GS is the second-

derivative matrix of T with respect to the coordinates of S keeping those of G fixed, and N˜SG the

corresponding matrix where the roles of S and G are exchanged. Matrix N˜SG is obtained by taking

the derivatives of T first with respect to source and then with respect to receiver coordinates. It isto be noted that N

˜SG and N

˜GS are symmetric matrices whereas N

˜SG generally is not. It satisfies

N˜SG = N

˜TGS, where N

˜GS is the corresponding matrix for the reverse propagation direction, i.e.,

with source at G and receiver at S. Therefore, det(N˜SG) = det(N

˜GS).

With these matrices, we can rewrite equation (4.2.25) as

T (x,x′) = T0 + p0 · x − p′0 · x′ − x ·N˜SGx

′ +1

2x ·N

˜GSx +

1

2x′ ·N

˜SGx

′ , (4.2.27)

which will be useful in Chapter 7. Various versions of equation (4.2.27) can be found in theliterature. Cerveny (1985; 1987; 2001; see also Cerveny et al., 1984) provides a similar expressionto formula (4.2.27) in terms of ray-centered coordinates. Bortfeld (1989) works with a very similarformula to (4.2.27) for so-called seismic systems (i.e., stacks of constant-velocity layers boundedby curved interfaces), and Ursin (1982a) considers another expression which also coincides withequation (4.2.27) up to the second order (Schleicher et al., 1993b). The relationship between thetraveltime equations of Cerveny (1985) and Bortfeld (1989) and the above Taylor polynomial inequation (4.2.27) can be found in Hubral et al. (1992a).

Matrix relationships

From equations (4.2.26) together with (4.2.21), we find the relationships

A˜

= N˜

−1SGN

˜GS , (4.2.28a)

B˜

= N˜

−1SG , (4.2.28b)

C˜

= − (N˜TSG −N

˜SGN

˜−1SGN˜

GS ) , (4.2.28c)

D˜

= N˜SGN

˜−1SG . (4.2.28d)

By means of equation (4.2.28), the surface-to-surface propagator matrix ˆT˜

can be constructed fromtraveltime derivatives. This has been used for traveltime interpolation by Vanelle and Gajewski(2002).

The second-derivative matrices N˜GS , N

˜SG, andN

˜SG of the traveltime can also be expressed in

terms of the submatrices Q˜

1, Q˜

2, P˜

1, and P˜

2 of the propagator matrix ˆΠ˜

using equations (3.11.37).One obtains

N˜GS = G

˜T Q

˜

−12 Q

˜2G˜− X

˜, (4.2.29a)

N˜SG = G

˜′TP

˜2Q˜

−12 G

˜′ + X

˜′ , (4.2.29b)

N˜SG = G

˜TQ˜

−12 G

˜′ . (4.2.29c)

In this way, the N˜

-matrices can also be computed by dynamic ray tracing.

Let us make a remark for the reader familiar with the book of Goldin (1986): The 2 × 2matrix M

˜x introduced by Goldin is closely related to B

˜−1. His matrix M

˜x denotes the matrix

of second mixed derivatives of the traveltime with respect to source and receiver coordinates fora two-dimensional problem. It is thus the 2-D equivalent to a 3 × 3 matrix M

˜x of which −B

˜−1

is the upper left 2 × 2 submatrix, as we recognize from equation (4.2.26c). In other words, −B˜

−1

corresponds to M˜x in the same way as Q

˜corresponds to Q

˜(see previous chapter).

114 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

Measurement configuration

In a seismic experiment, sources and receivers usually are grouped into pairs that vary jointly inaccordance with a pre-defined measurement configuration. How this can actually be described interms of a parameter vector ξ has been discussed in Section 2.2. As detailed there, the vector ξcan be understood as a position vector that defines the coordinates of a source and a receiver inlocal Cartesian coordinate systems (see equations (2.2.13)) centered at some reference source andreference receiver. Common seismic measurement configurations are depicted in Figure 2.2.

To be able to make use of the description of Section 2.2, we simply must identify the coordi-nates xS and xG of the seismic sources and receivers with the coordinates x and x′ of initial andend points S and G of paraxial rays on the anterior and posterior surfaces. Then, we can rewrite thetraveltime equation (4.2.25) under consideration of the configuration of the seismic experiment asdescribed by equations (2.2.13). We arrive at the paraxial approximation of the traveltime surfaceof a reflection event in a seismic section, which is given in dependence on the parameter vector ξas

T (ξ) = T0 − p0 · Γ˜S ξ + p′0 · Γ

˜G ξ

+1

2ξ ·(

Γ˜TSB

˜−1A

˜Γ˜S + Γ

˜TGD

˜B˜

−1Γ˜G − Γ

˜TSB

˜−1Γ

˜G − Γ

˜TGB

˜−TΓ

˜S

)

ξ . (4.2.30)

As before, notation B˜

−T stands for the inverse of the transpose of B˜

.

It is to be observed that a simple substitution of equation (2.2.13) in expression (4.2.25) doesnot directly lead to the form (4.2.30). To give the second-order term its symmetric form we haveused that

ξ · Γ˜TSB

˜−1Γ

˜G ξ = ξ · Γ

˜TGB

˜−TΓ

˜S ξ =

1

2ξ · (Γ

˜TSB

˜−1Γ

˜G + Γ

˜TGB

˜−TΓ

˜S) ξ . (4.2.31)

This fact becomes obvious when one notes that ξ · Γ˜TSB

˜−1Γ

˜Gξ is a scalar and thus equal to its

transpose. It is important to recognize that this does not mean that the antisymmetric part ofΓ˜TSB

˜−1Γ

˜G is equal to zero, but just that it does not influence the value of T (ξ). This observation

is in due agreement with the fact that the second-derivative (Hessian) matrix of T (ξ) with respectto ξ has to be a symmetric matrix. In fact, in the above form, equation (4.2.30) directly exhibitsthis Hessian matrix inside the parentheses.

Equation (4.2.30) represents the reflection traveltime surface as a function of the parametervector ξ in paraxial approximation in the vicinity of the central ray defined by ξ = 0 for any arbi-trary seismic experiment (see Section 2.2). Using theN

˜-matrices, this equation can be alternatively

expressed as

T (ξ) = T0 − pS · Γ˜Sξ + pG · Γ

˜Gξ

− 1

2ξ ·[

Γ˜TSN

˜SGΓ

˜G + Γ

˜TGN

˜GSΓ

˜S − Γ

˜TSN

˜GSΓ˜S − Γ

˜TGN

˜SGΓ˜G

]

ξ , (4.2.32)

where we have used that N˜TSG = N

˜GS . Expression (4.2.32) is the final expression for the traveltime

of a paraxial ray in the vicinity of a known central ray. It will be of further use in Chapter 7,particularly describing the traveltime of a primary reflected paraxial SMRG in the vicinity of aprimary reflected central ray SMRG.

4.3. RAY-SEGMENT DECOMPOSITION 115

4.3 Ray-segment decomposition

For the application of ray theory in seismic problems, it is often useful to think of a completepropagation trajectory of a seismic wave as a sequence of individual pieces, so-called ray segments.In this section, we study how the properties of the complete ray path can be composed from thoseof the segments.

4.3.1 Chain rule

M M M

MM

M

SS

S G

G G

S

SS

G

GG

Fig. 4.4. Ray-segment decomposition. Top left: Decomposition at an arbitrary ray point M . Topright: Decomposition at a point M on a transmitting interface. Bottom: Decomposition at a pointM on a reflecting interface. The curly brackets indicate arbitrary layered, inhomogeneous media,where the rays may multiply reflect or transmit.

The most basic decomposition of a complete ray path is one into two individual segments.Referring to Figure 4.4, we consider a point M , where an arbitrary (real or fictitious) transmitting

116 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

or reflecting interface cuts the central ray SG (now also referred to as ray SMG). For any such

point M on any such interface, the propagator matrix ˆT˜

satisfies the following equation

ˆT˜(G,S) = ˆT

˜(G,M) ˆT

˜(M,S) , (4.3.1)

which we refer to as the chain rule. Here, ˆT˜(M,S) and ˆT

˜(G,M) denote the surface-to-surface

propagator matrices for the ray segments SM and MG that build up the total ray SMG. Equation(4.3.1) holds for all situations of Figure 4.4, whetherM lies upon an actual reflecting or transmittingor even an arbitrarily introduced fictitious interface.

The aim of this section is to develop the chain rule [equation (4.3.1)] for the propagator

matrix ˆT˜. For this purpose we consider Figure 4.4, where the central ray SMG is decomposed into

two ray segments, namely

(a) segment 1: the ray connecting the initial point S to point M and

(b) segment 2: the ray connecting point M to the end point G of the total ray SMG.

As indicated by Figure 4.4, point M can be either an arbitrary point within a continuouslayer or a transmission or reflection point at a first- or second-order interface. Also shown in Figure4.4 is a paraxial ray S M G decomposed into two ray segments.

We denote the propagator matrix of segment i by

ˆT˜i =

A˜i B

˜i

C˜i D

˜i

(i = 1, 2) (4.3.2)

and have, similar to equation (3.11.36), the two linear relationships

x′M

p′M − p′M0

= ˆT

˜1

x

p− p0

(4.3.3a)

and

x′

p′ − p′0

= ˆT

˜2

xM

pM − pM0

. (4.3.3b)

Here, we have denoted by xM′ and pM

′ − pM0′ the end point parameters of ray segment 1 at M

and by xM , pM −pM0 the initial point parameters of ray segment 2 at M . Since xM as well as xM′

describe the coordinates of point M , we have

xM′ = xM . (4.3.4a)

Referring to Figure 4.4, we also have at point M the equation

pM′ − pM0

′ = pM − pM0 , (4.3.4b)

since both vectors represent the same slowness vector projection. Equation (4.3.4b) is also true forall the situations depicted in Figures 4.4. This is due to Snell’s law, which states that the slowness

4.3. RAY-SEGMENT DECOMPOSITION 117

vector component tangent to an interface is continuous across that interface. In fact, the simplicityof the relationship (4.3.4b) is the motivation for the rather complicated construction of the cascadedprojection that led to the 2-D representations of the slowness vector. It is the cascaded projectionthat makes p remain a faithful representative of the tangential components of the slowness vector.

By using equations (4.3.4a) and (4.3.4b) one can eliminate xM and pM −pM0 from equations(4.3.3a) and (4.3.3b) and find a composed expression for the propagator matrix from S via M to

G. A comparison of this result with the analogous formula for the propagator matrix ˆT˜

of the totalray SG [equation (3.11.39)] yields the relationship

ˆT˜

= ˆT˜2

ˆT˜1 . (4.3.5)

The above formula translates into the following chain-rule equations for the four 2× 2 submatricesof ˆT

˜A˜

= A˜

2A˜

1 +B˜

2C˜

1 , (4.3.6a)

B˜

= A˜

2B˜

1 +B˜

2D˜

1 , (4.3.6b)

C˜

= C˜

2A˜

1 +D˜

2C˜

1 , (4.3.6c)

D˜

= C˜

2B˜

1 +D˜

2D˜

1 . (4.3.6d)

Just as much as ˆT˜

can be decomposed into the product of ˆT˜2 and ˆT

˜1, the latter two ray-segmentpropagator matrices may be further decomposed. This means that ultimately the propagator matrixˆT˜

can be written as a product of many ray-segment propagator matrices. This general decomposition

is referred to as the chain rule of the ˆT˜

propagator matrix. Hubral et al. (1995) show how this canbe used for a quick and efficient computation of the total-ray propagator matrix.

Two important relationships that will be of use later follow directly from this chain rule:

B˜

−11 A

˜1 − B

˜−1A

˜= B

˜−1B

˜2B˜

−T1 (4.3.7a)

D˜

2B˜

−12 − D

˜B˜

−1 = B˜

−TB˜T1B

˜−12 . (4.3.7b)

Proof of equation (4.3.7a): Starting from expression (4.3.7a), we have using equation (4.3.6a):

B˜

−11 A

˜1 = B

˜−1(B

˜2B˜

−T1 +A

˜)

= B˜

−1(B˜

2B˜

−T1 + A

˜2A˜

1 + B˜

2C˜

1)

= B˜

−1(B˜

2(B˜

−T1 + C

˜1) + A

˜2A˜

1) . (4.3.8)

With the symplecticity equation (4.2.17), this can be written as

B˜

−11 A

˜1 = B

˜−1(B

˜2D˜

1B˜

−11 A

˜1 + A

˜2A˜

1)

= B˜

−1(B˜

2D˜

1 + A˜

2B˜

1︸ ︷︷ ︸

B˜

)B˜

−11 A

˜1. (4.3.9)

Since all operations here are invertible, the proof is complete.

Proof of equation (4.3.7b): Using symplecticity equation (4.2.15b), we recognize that the left-hand side of equation (4.3.7b) is symmetric and thus, if equation (4.3.7b) is correct, so has to bethe right-hand side, i.e.,

D˜

2B˜

−12 − D

˜B˜

−1 = (D˜

2B˜

−12 − D

˜B˜

−1)T = B˜

−T2 B

˜1B˜

−1 , (4.3.10)

D˜

2B˜

−12 = (B

˜−T2 B

˜1 + C

˜2B˜

1 + D˜

2D˜

1)B˜

−1 , (4.3.11)

118 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

where we have used equation (4.3.6d). Again using the symplecticity equation (4.2.17), we obtain.

D˜

2B˜

−12 = (D

˜2B˜

−12 A

˜2B˜

1 + D˜

2D˜

1)B˜

−1

= D˜

2B˜

−12 (A

˜2B˜

1 +B˜

2D˜

1︸ ︷︷ ︸

B˜

)B˜

−1 . (4.3.12)

Again, all operations applied are invertible and thus, the proof is complete.

Zero-offset ray

In the zero-offset situation (when the central ray of a primary reflection reduces to the two-waynormal ray), we can write equation (4.3.5) as

ˆT˜

= ˆT˜∗0

ˆT˜

0 , (4.3.13)

since in this case ˆT˜1 = ˆT

˜0 and ˆT˜2 = ˆT

˜∗0. Here,

ˆT˜0 =

A˜

0 B˜

0

C˜

0 D˜

0

(4.3.14)

is the propagator matrix of the one-way normal ray that starts at S (= G) and ends at the normal-

incidence-point NIP (see Figure 4.9a) and ˆT˜∗0 is the corresponding reverse-ray propagator matrix.

In particular, using the form of reverse ray propagator given in equation (4.2.24), equation (4.3.6breduces in this situation to

B˜

= D˜T0B

˜0 + B

˜T0D

˜0 = 2D

˜T0B

˜0 = B

˜T , (4.3.15)

where the second of the above equalities follows from the symplecticity equation (4.2.15b). Equation(4.3.15) is of significant importance for the computation of zero-offset true-amplitude reflections(Hubral, 1983; Tygel et al., 1992) and for true-amplitude zero-offset migration (Hubral et al., 1991).

4.3.2 Ray-segment traveltimes

For later constructions of traveltime surfaces that will define the inplanats and outplanats, theparaxial-ray traveltimes along the ray segments are needed (see Figure 4.4). These are easily foundfrom the basic quadratic traveltime formula (4.2.25) together with the continuity equations (4.3.4)that connect the ray segments. We have for ray segment 1

T1(x,xM ) = T01 − p0 · x + pM0 · xM − x ·B˜

−11 xM

+1

2x ·B

˜−11 A

˜1x +

1

2xM ·D

˜1B

˜−11 xM (4.3.16a)

and for ray segment 2

T2(xM ,x′) = T02 − pM0 · xM + p′0 · x′ − xM ·B

˜−12 x′

+1

2xM ·B

˜−12 A

˜2xM +

1

2x′ ·D

˜2B

˜−12 x′ . (4.3.16b)

4.3. RAY-SEGMENT DECOMPOSITION 119

Measurement configuration

Considering a seismic measurement configuration as described by equations (2.2.13), the paraxialtraveltimes of the ray segments can also be expressed in dependence on the parameter vector ξ andthe coordinates xM of point M . We have

T1(ξ,xM ) = T01 − p0 · Γ˜Sξ + pM0 · xM − ξ · Γ

˜TSB

˜−11 xM

+1

2ξ · Γ

˜TSB

˜−11 A

˜1Γ˜Sξ +

1

2xM ·D

˜1B

˜−11 xM (4.3.17a)

for ray segment 1 and

T2(xM , ξ) = T02 − pM0 · xM + p′0 · Γ˜Gξ − xM ·B

˜−12 Γ

˜Gξ

+1

2xM ·B

˜−12 A

˜2xM +

1

2ξ · Γ

˜TGD

˜2B

˜−12 Γ

˜Gξ (4.3.17b)

for ray segment 2. Using the corresponding N˜

-matrices N˜SM , N

˜MS , N

˜SM , and N

˜MG, N

˜MG , N

˜GM ,

for the two ray segments, defined analogously to the N˜

-matrices in equation (4.2.26), we may alsowrite

T1(ξ,xM ) = T01 − p0 · Γ˜Sξ + pM0 · xM − ξ · Γ

˜TSN

˜SMxM

+1

2ξΓ˜TSN

˜MS Γ

˜Sξ +

1

2xM ·N

˜SMxM (4.3.18a)

for ray segment 1 and

T2(xM , ξ) = T02 − pM0 · xM + p′0 · Γ˜Gξ − xM ·N

˜MGΓ

˜Gξ

+1

2xM ·N

˜GMxM +

1

2ξΓ˜TGN

˜MG Γ

˜Gξ (4.3.18b)

for ray segment 2.

Global coordinates

Let us now finally make also use of equation (3.11.10) to replace the local coordinates xM at Mwith the corresponding global coordinates rM . Note, however, that also the dislocation of the originbetween the global and the local coordinates must be taken into account. If we denote the differenceof the 2-D coordinate vectors of M and M by rM , we arrive at

T1(ξ, rM ) = T01 − p0 · Γ˜Sξ + pM0 ·G

˜(r)−1rM − ξ · Γ

˜TSB

˜−11 G

˜(r)−1rM

+1

2ξ · Γ

˜TSB

˜−11 A

˜1Γ˜Sξ +

1

2rM ·G

˜(r)−TD

˜1B

˜−11 G

˜(r)−1rM (4.3.19a)

for ray segment 1 and

T2(rM , ξ) = T02 − pM0 ·G˜

(r)−1rM + p′0 · Γ˜Gξ − rM ·G

˜(r)−TB

˜−12 Γ

˜Gξ

+1

2rM ·G

˜(r)−TB

˜−12 A

˜2G˜

(r)−1rM +1

2ξ · Γ

˜TGD

˜2B

˜−12 Γ

˜Gξ (4.3.19b)

for ray segment 2.

120 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

For future reference, it is useful to introduce the second-order mixed derivative matrices ofthese traveltimes in global coordinates of M . Corresponding to equation (4.2.26c), we have

N˜

(r)SM = −

(

∂2T1

∂xSi∂rMj

)

i,j=i,2

= N˜SMG

˜(r)−1 = B

˜−11 G

˜(r)−1 (4.3.20a)

and

N˜

(r)MG = −

(

∂2T2

∂rMi∂xGj

)

i,j=i,2

= G˜

(r)−TN˜MG = G

˜(r)−TB

˜−12 . (4.3.20b)

Of course, N˜

(r)SM and N

˜

(r)MG possess the same reverse-ray property as N

˜SG, i.e., N

˜

(r)MS = N

˜

(r)SM

T

and N˜

(r)GM = N

˜

(r)MG

T .

4.4 Meaning of the propagator submatrices

We have already seen in equation (3.13.6c) that the determinant of the submatrix B˜

of ˆT˜

is closelyrelated to the (point-source) geometrical spreading factor L along the ray SG. In this section, we

will interpret also the other submatrices of the surface-to-surface propagator matrix ˆT˜

by relatingthem to the physical situations depicted in Figure 4.5.

4.4.1 Propagation from point source to wavefront

At first, we study the propagation of a wave from a point source at S to a wavefront at G. In otherwords, we let the posterior surface at G (see Figure 4.5a) coincide with the actual wavefront ofthe propagating wave at G. Let us consider how this affects equations (3.11.36). At the anteriorsurface at S, all rays emanate from the very same point source S, i.e., x = 0. At the posteriorsurface, all slowness vectors are perpendicular to the wavefront and thus to the posterior surface,i.e., p′ = p′0 = 0. Insertion into equations (3.11.36) yields

x′ = B˜

(p− p0) , (4.4.1a)

0 = D˜

(p− p0) . (4.4.1b)

Since these equations have to be satisfied by all paraxial rays, we immediately observe from equation(4.4.1b) that in this case necessarily D

˜= O

˜. This equation thus describes the wavefront of a wave

emanating from a point source S. For instance, by insertion of equations (3.11.37d) and (3.11.33)and solving for F

˜, an expression is found how the wavefront curvature can be computed by dynamic

ray tracing.

Insertion of x = p′0 = 0 and D˜

= O˜

into traveltime equation (4.2.25) leads to T = T0

expressing the obvious fact that all rays joining a point source at S to a wave front at G haveidentical traveltime.

4.4.2 Propagation from wavefront to wavefront

As a second example, we study again the propagation of a wave from an arbitrarily specified initialwavefront at S to another wavefront at G (see Figure 4.5b). In other words, we let the anterior and

4.4. MEANING OF THE PROPAGATOR SUBMATRICES 121

central ray

central ray

p

p

pp

p

p

’

’

’

’

^

^

^^

^

^

paraxial ray

0

0

paraxial ray

0

p

p

p

source

S

G

S

G

point

(b)

(a)

wavefront

wavefront

wavefront

Fig. 4.5. Physical interpretation of the matrices A˜

, B˜

, C˜

, and D˜

(2-D sketch of a 3-D situation).(a) Propagation from a point source to a wavefront. In this situation, B

˜relates to the geometri-

cal-spreading factor and D˜

= O˜

. (b) Propagation from a wavefront to another wavefront. In thissituation, A

˜relates to the geometrical-spreading factor and C

˜= O

˜.

122 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

GS

S

central ray

wavefrontwavefront

paraxial ray

t

Fig. 4.6. 2-D sketch of a 3-D situation. An elementary 3-D wave with arbitrary initial wavefront atS travels from point S to point G. To understand the geometrical spreading between S and G, weconsider the unknown medium to the left of point S to be replaced by a suitable known one, e.g., ahomogeneous medium. In this region, we consider a fictitious point source at a point St such thatthe wave originating at St has the same wavefront curvature at S as the wave to be investigated.

the posterior surfaces coincide with the actual wavefronts. This implies that p = p0 = p′ = p′0 = 0,for now all slowness vectors are perpendicular to the anterior and posterior surfaces. Thus, equations(3.11.36) become

x′ = A˜x , (4.4.2a)

0 = C˜x . (4.4.2b)

In the same way as before, we observe from equation (4.4.2b) that in this case necessarily C˜

= O˜

.This equation thus describes the wavefront emanating from an arbitrary wavefront. As before, anequation on how to compute this wavefront by dynamic ray tracing can be derived with equations(3.11.37c) and (3.11.33).

Insertion of p0 = p′0 = 0 together with equation (4.4.2a) into the traveltime equation (4.2.25)leads to

T = T0 +1

2x ·A

˜T[

D˜B˜

−1A˜−B

˜−T]

x . (4.4.3)

Considering the symplecticity of the propagator matrix, we recognize the expression within thesquare brackets to be equal to C

˜, which equals zero in this case. Thus, also in this case, T = T0 as

required.

Geometrical spreading

The physical meaning ofA˜

is not as clearly visible as those ofB˜

, C˜

andD˜

. However, asB˜

relates tothe geometrical-spreading factor of a wave emanating from a point source (where D

˜= O

˜), the idea

that A˜

may determine the geometrical-spreading factor for a wave emanating from an arbitraryspatial wavefront (where C

˜= O

˜) seems not too farfetched. In fact, this is indeed the case and is

not difficult to prove. Consider the situation depicted in Figure 4.6. A wave with an arbitrarilyshaped wavefront travels along the central ray from S to G. In order to investigate this situationin detail, we consider a particular continuation of the medium to the left of the ray’s initial pointS. We assume it to be inhomogeneous in such a way that the wavefront at S can be thought of ashaving originated at the (fictitious) point source at St. We have already seen in equation (3.6.17)

4.5. FRESNEL ZONE 123

that in such a situation, the geometrical spreading of an arbitrary wave traveling from S to G canbe expressed as the ratio of two point-source geometrical-spreading factors for the rays StG andStS. Using equation (3.13.6c), we find

LtLS

=

√

vM cosϑGvG cosϑM

|detB˜t|

1

2

|detB˜

1|1

2

expiπ2

(κt − κ1) , (4.4.4)

where we have used the index t to identify the quantities pertaining to the total ray from thefictitious point source at St to the ray’s end point at G. Correspondingly, the index 1 identifies thequantities pertaining to the ray segment from the fictitious point source at St to the wavefront atthe ray’s initial point S. Together with the above chain rule (4.3.6b) for B

˜at the initial wavefront

where D˜

1 = O˜

as explained above, this yields

LtLS

=

√

vM cosϑGvG cosϑM

|detA˜| 12 expiπ

2(κt − κ1) , (4.4.5)

The matrix A˜

(without index) pertains to the ray under consideration, i.e., the segment from S toG of the total ray StG as used above. At a later stage of this book, we will see how the KMAHindex (κt − κ1) appearing in the above formula can be expressed in terms of the ray from M to Gonly.

4.5 Fresnel zone

We are now ready to provide some examples of applying the ˆT˜-propagator-matrix formalism to

problems of reflection seismics that will be useful later on. In this section, we derive an interestingexpression for the (first) Fresnel zone at the intersection point M of an arbitrary (real or fictitious)interface with the ray SG (see Figure 4.7). All (first) Fresnel zones encountered along the ray SGdefine its (first) Fresnel volume. This is the (frequency-dependent) volume surrounding the centralray that influences the (high-frequency) wave propagation at G for a point source at S. In fact, forthe validity of ray-theory, it is often sufficient if the ray validity conditions (3.2.16) are satisfiedwithin the (first) Fresnel volume.

Zero-order ray theory is a concept that is mathematically valid in the limit of infinite fre-quency. Therefore, the high-frequency range of seismic wave propagation is usually well-describedby ray theory. For lower frequencies, however, a mathematical ray can no longer be viewed upon asa valid physical concept. In fact, there is a (frequency-dependent) spatial region in the vicinity ofsuch a “mathematical ray” that influences the time-harmonic wavefield received at the end of theray. This region is the so-called (first) Fresnel volume of the ray or the “physical ray”. Any cross-cut of the Fresnel volume by an arbitrary curved surface intersecting the ray (not necessarily aninterface in the medium) is called a (first) Fresnel zone for that surface (Gelchinsky, 1985; Knapp,1991; Cerveny and Soares, 1992; see also references there). The Fresnel volume is consequently theenvelope of all possible Fresnel zones along the ray. Fresnel zones and volumes can be computedvery efficiently by forward dynamic ray tracing given a ray traced through a known velocity model(Cerveny and Soares, 1992). In this way, the validity of seismic events computed by ray theorycan be checked (see also Chapter 3). Only if no violation of the basic ray-theoretical assumptionsoccurs within the whole Fresnel volume of the ray, ray theory will correctly compute the seismicevents. In the application of ray theory to seismic stratigraphic modeling, usually the Fresnel zoneupon a key reflector is of interest, so as to know which part of it contributes to the reflected wave.

124 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

Fresnel zone

M

S

G

Centralray

Segment 1

Segment 2

Paraxialray

M

Fig. 4.7. All paraxial ray points M on the (real or fictitious) interface which the central rayintersects at M belong to the first Fresnel zone at M (defined with respect to a certain frequency).For these pointsM the sum of traveltimes along the ray segments SM andMGminus the traveltimealong the central ray SG is not greater than half the period of the mono-frequency (i.e., harmonic)wave traveling along the ray SG from a point source at S to a receiver at G. The ray SG symbolicallyrepresents any ray, e.g. that of a primary reflection.

4.5. FRESNEL ZONE 125

In addition to playing the above described role in forward modeling, Fresnel zones are alsowidely used in stratigraphic analyses (Sheriff, 1980, 1985; Lindsey, 1989; Knapp, 1991). However,seismic data are often more accurate in the time domain than in the depth domain, because anytime-to-depth conversion depends on a possibly wrong velocity model. For that reason, explorationgeophysicists involved in a stratigraphic analysis generally prefer to know already the Fresnel zoneupon a reflecting interface, or its projection onto the earth’s surface or into the identified reflectiontime surface within the seismic section without knowing the details of the reflector overburden,i.e., without having a depth model already at their disposal. In other words, they are interested insolving an inverse problem (Sheriff, 1980, 1985; Lindsey, 1989).

4.5.1 Definition

In this section, we consider the determination of the so-called (first) Fresnel zone (Kravtsov andOrlov, 1990) at a point M on a central ray SG upon an arbitrary smoothly curved (real or fictitious)surface that cuts the ray at M (see Figure 4.7). This surface may, e.g., be a reflecting/transmittinginterface or the tangent plane to it, the so-called image plane (Gelchinsky, 1985).

All points M that are confined to the intersecting interface are said to define the (first)Fresnel zone if a wave from S, when scattered at M , contibutes constructively to the reflected waveat G. This constructive interference happens, if the sum of traveltimes from S to M and from M toG does not differ from the traveltime along the reflected ray SMG by more than half a period T ofthe considered mono-frequency (with ω = 2π/T ) wave traveling along the ray SMG (Cerveny andSoares, 1992). In symbols,

|T (S,M ) + T (M,G) − T (S,G)| ≤ T/2 . (4.5.1)

Medium parameters in the total Fresnel volume, and not only those encountered along the centralray SG, influence the wavefield recorded at G due to a point source at S.

Let us now calculate the Fresnel zone in terms of paraxial quantities. The ray-segment trav-eltimes T (S,M ) and T (M,G) are known from equations (4.3.16). As S is the origin for all raysparaxial to segment SM that is under consideration, we have x= 0 in equation (4.3.16a). Corre-spondingly, as G is fixed for all rays paraxial to segment MG that is under consideration, we havex′ = 0 in equation (4.3.16b). Inserting the resulting ray-segment traveltimes into formula (4.5.1)leads to the following expression for the Fresnel zone at M ,

|xM ·H˜FxM | ≤ T , (4.5.2)

where xM denotes the Cartesian coordinates of the normal projection of point M onto the planetangent to the surface at M . The Fresnel zone matrix H

˜F is given by

H˜F =

∂2(

T (S,M ) + T (M,G))

∂xM i∂xM j

(i,j=1,2)

(4.5.3a)

= B˜

−12 A

˜2 + D

˜1B˜

−11 (4.5.3b)

= B˜

−12 B

˜B˜

−11 . (4.5.3c)

We remind that, while in equation (4.5.3a) the derivatives with respect to the horizontal coordi-nates xM of M are taken in the tangent plane to the intersecting interface, M actually varies on

126 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

the interface, the variation perpendicular to the tangent plane being implicitly considered by thedependence of xM 3 on xM . From expression (4.5.3b), we observe that, because of the properties(4.2.15) of the propagator matrix,H

˜F is a symmetric matrix, i.e., H

˜F = H

˜TF . The identity in equa-

tion (4.5.3c) is a consequence of the ray-segment decomposition formula (4.3.6b) for the submatrix

B˜

of ˆT˜.

Note that, as a consequence of the paraxial approximation, the condition (4.5.2) definesstrictly speaking only a paraxial Fresnel zone (Cerveny and Soares, 1992). This may slightly differfrom the exact Fresnel zone defined by condition (4.5.1). However, for high frequencies, for whichthe zero-order ray-theory assumptions are valid, these differences should be negligible. We observethat theB

˜-matrices play a fundamental role in result (4.5.3), which can be readily used to construct

the Fresnel zone at any point along a ray SG. Thus, the problem of determining the Fresnel zoneat a certain point M (e.g., on the reflector) reduces to that of computing H

˜F at M .

Equation (4.5.3) states how the paraxial Fresnel zone can be computed in forward problemsusing dynamic ray-tracing along the two ray segments SM and MG. Note, however, that for thispurpose the intersecting interface at M must be known, because its curvature enters into matricesD˜

1 and A˜

2 (see Section 3.11.6).

However, equation (4.5.3) is not directly applicable in seismic reflection imaging, where themodel is a priori not known. In spite of this fact, in a similar way as the factor |L| for a particularelementary wave (e.g. a primary reflection) at G (Figure 4.7) can be derived from a pure traveltimeanalysis involving a perturbation of points S and G (i.e., without any knowledge of the model; seeTygel et al., 1992), it is also possible for certain subsurface models to obtain the Fresnel zone upona reflector purely from a traveltime analysis. This is shown for the zero-offset case (i.e., S = G) inHubral et al. (1993b). In the framework of this book, we will observe that Fresnel zones are of greatimportance in seismic modeling (Santos et al., 2000) and true-amplitude seismic reflection imaging(Hubral et al., 1996a; Tygel et al., 1996). They also enter into the determination of optimummigration apertures (Schleicher et al., 1997b; see also Section 8.1).

4.5.2 Time-domain Fresnel zone

Although the Fresnel zone is obviously a frequency-domain concept, we will also use it in the timedomain. This can be done under the assumption of a causal source pulse F [t] with a constant lengthTε, i.e., F [t] vanishes outside an interval 0 ≤ t ≤ Tε. To carry the concept of a Fresnel zone to thetime domain, we replace the half period T/2 in equation (4.5.1) with the length Tε of the seismicsource pulse. In symbols,

|T (S,M ) + T (M,G) − T (S,G)| ≤ Tε . (4.5.4)

The so-defined time-domain Fresnel zone will be an important concept when investigating theoptimal migration aperture and the horizontal resolution of seismic depth migration.

In correspondence to equation (4.5.2), we obtain then as the paraxial approximation to thetime-domain Fresnel zone

1

2|xM ·H

˜FxM | < Tε , (4.5.5)

where H˜F is again given by equation (4.5.3).

4.5. FRESNEL ZONE 127

4.5.3 Projected Fresnel zone

In order to evaluate the resolution of a primary reflection for a stratigraphic analysis, its (first)Fresnel zone upon the reflector is required. Its projection into the measurement surface (in arbitrarymeasurement configurations like, e.g., common-shot or constant-offset records) provides the imagein the seismic section of that part of the subsurface reflector that influences the considered reflection.The concept of a projected Fresnel zone was introduced for zero-offset reflections (Hubral et al.,1993b) and has been extended to arbitrary offset rays by Schleicher et al. (1997b). The projectedFresnel zone can be determined from the data when knowing the velocity model above the reflectorbut not the reflector itself. It plays an important role in diffraction-stack (or Kirchhoff) migrationin connection with the migration aperture, i.e., the number of traces that are summed up alongdiffraction time surfaces. As we will see in Chapter 7, the projected Fresnel zone corresponds to theminimum migration aperture that is needed to guarantee correctly recovered migration amplitudestogether with the best signal-to-noise (S/N) ratio. This may provide a criterion on whether themeasurement aperture was sufficiently big to perform an amplitude-preserving migration or whetherthe migration aperture can be even restricted to avoid the summation of noisy traces containingno signal.

For the 3-D zero-offset case, where a primary reflection follows the trajectory of a normalray, it was shown by Hubral et al. (1993b) that it is possible to determine the projection intothe measurement surface of the actual Fresnel zone at the lower end of the normal ray upon thekey reflector. It is this projected Fresnel zone the seismic interpreter is interested in to study theinfluence of the change in the reflector properties on identified zero-offset reflections in the absenceof a known velocity model for the overburden. The computation is entirely based on a traveltimeanalysis of identified near-offset reflections at the key reflector that are routinely obtained by the3-D common-midpoint (CMP) profiling technique.

However, for a 3-D finite-offset primary reflection, the situation is different. The concept ofa projected Fresnel zone can, of course, be extended to this case, but its computation from a puretraveltime analysis is no longer possible. In this section, we also discuss what additional informationis needed for the construction of the projected Fresnel zone.

The projected Fresnel zone is defined as that region on the earth’s or measurement surfacethat contains the events reflected from the actual Fresnel zone on the reflector along the pertinentrays corresponding to the measurement configuration. In Figure 4.8, we have depicted a projectedFresnel zone for a common-shot experiment.

Starting from representation (4.5.2) of the paraxial Fresnel zone at a given point M on acertain (true or hypothetical) interface, it is not difficult to derive a suitable expression for itsprojection into the measurement surface. For simplicity, we assume this surface to be planar as isdepicted in Figure 4.8.

We now mathematically formulate the concept of the projected Fresnel zone for the case ofan arbitrary reflection ray SMRG (Figure 4.8) in an arbitrary measurement configuration. Theprojected Fresnel zone is obtained by projecting the true Fresnel zone on the reflector ΣR along thebundle of reflection rays S G paraxial to SG into the ξ-plane. In Figure 4.8 there are depicted four ofthese paraxial rays S G that reflect at the actual boundary of the Fresnel zone at R thus defining theboundary of the projected Fresnel zone in the ξ-plane. The bundle of rays required of the projectiondepends, of course, on the seismic measurement configuration involved. In Figure 4.8, the situationis depicted for a common-shot configuration for simplicity. Note that the considerations below are

128 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

S

reflection traveltime surface

G

M

MFresnel zone at

reflector

projectedFresnel zone

x

x

t

z

R

2

R

RΣ

ΓR

1

ξ

ξ2

1

Fig. 4.8. Projected Fresnel zone for common-source geometry. It is constructed by projection alongparaxial rays that reflect from the actual Fresnel zone at R.

valid for any arbitrary measurement configurations.

Consider now point M of Figure 4.7 (where the Fresnel zone has been computed) to be thereflection point MR of an arbitrary primary reflected ray SG, which is from now on also denotedas ray SMRG (Figure 4.8). Let MR be a point in the vicinity of MR upon the key reflector and letSMRG be a ray paraxial to SMRG and specularly reflected at MR. The 2-D Cartesian coordinatesxM = (xM 1, xM 2) of MR are obtained in the same way as before by a normal projection of MR

into the plane tangent to the reflector ΣR at MR. Then the projection of the Fresnel zone from ΣR

along paraxial rays into the seismic section, i.e., into the ξ-plane, is represented by the projectionmap from xM to ξ. This map can, in a first-order approximation, be expressed by

xM = Γ˜Mξ , (4.5.6a)

or

ξ = Γ˜−1M xM (4.5.6b)

4.5. FRESNEL ZONE 129

where Γ˜M is a projection matrix to be specified below. Inserting this expression for xM into the

Fresnel zone definition (4.5.2), we obtain the Fresnel zone projected onto the ξ-plane as

|ξ ·H˜Pξ| ≤ T , (4.5.7)

where we have denoted by

H˜P = Γ

˜TMH

˜F Γ

˜M (4.5.8)

the “projected Fresnel zone matrix.”

Projection matrix

To determine the projection matrix Γ˜M , we assume that the sources S and receivers G are dis-

tributed in the vicinity of the reference points S and G. In this case, their positions in the measure-ment plane are well-described by equations (2.2.13). To each source-receiver pair (S,G) there existsone reflection point MR (supposed to be unique) in the vicinity of MR, i.e., the reflection pointcorresponding to the source-receiver pair (S,G). Therefore, a one-to-one relationship between thevector parameter ξ and the coordinates xM of the paraxial-ray reflection point MR can be foundcorresponding to equations (2.2.13) for xG and xS . We can therefore find a constant 2 × 2 matrixΓ˜M that describes the relationship between xM and ξ as given by equation (4.5.6a). In other words,

the point MR can be said to be projected along the paraxial reflection ray SMRG into the seismicsection. From equations (4.3.3) and (4.3.4), we have that

(

xMpM − pM0

)

= ˆT˜1

(

xSpS − pS0

)

(4.5.9)

and (

xGpG − pG0

)

= ˆT˜

2

(

xMpM − pM0

)

. (4.5.10)

Upon the use of equation (2.2.13), the solution of the first equation of system (4.5.9) for pS − pS0

is

pS − pS0 = B˜

−11 xM −B

˜−11 A

˜1Γ˜Sξ , (4.5.11)

which, inserted into the second equation of system (4.5.9), yields

pM − pM0 =(

C˜

1 −D˜

1B˜

−11 A

˜1

)

Γ˜Sξ +D

˜1B˜

−11 xM = B

˜−T1 Γ

˜Sξ +D

˜1B˜

−11 xM . (4.5.12)

Here, symplecticity equation (4.2.17) has been used. Substitution of this result in the first equationof system (4.5.10) leads to

xG = Γ˜Gξ =

(

A˜

2 +B˜

2D˜

2B˜

−11

)

xM +B˜

2B˜

−T1 Γ

˜Sξ , (4.5.13)

which can now be solved for xM to yield

xM = H˜

−1F Λ

˜ξ , (4.5.14)

where H˜F is again the Fresnel zone matrix given by equation (4.5.3) and

Λ˜

=(

B˜

−12 Γ

˜G + B

˜−T1 Γ

˜S

)

. (4.5.15)

130 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

In other words,

Γ˜M = H

˜−1F Λ

˜(4.5.16)

Inserting this result into equation (4.5.8) we find

H˜P = Λ

˜TH

˜−1F Λ

˜. (4.5.17)

Equation (4.5.17) tells us how the projected Fresnel zone matrix is derived from the actual one,once the ray SMRG and a macrovelocity model are given. Note that for the computation of Λ

˜,

dynamic ray tracing in a given macrovelocity model is necessary. However, the reflector need notbe known as neither B

˜1 and B

˜2 nor Γ

˜S and Γ

˜G depend on it.

Geometrical spreading

Let us also comment on the consequences of equation (4.5.17) for matrix B˜

that relates to thegeometrical-spreading factor of the central ray SMRG (see formula (3.13.6c)). By substitution ofexpression (4.5.17) for the projected Fresnel zone matrix into the decomposition formula (4.5.3)and solving for B

˜, we find

B˜

= B˜

2Λ˜H˜

−1P Λ

˜TB

˜1 . (4.5.18)

As B˜

can be determined by a direct inversion of measured traveltimes (Tygel et al., 1992), equation(4.5.18) can be used to compute H

˜P without knowledge of the reflector. Note that, this also works

vice versa, i.e., the geometrical spreading along the total ray SMRG can be computed withoutknowledge of the reflector once the projected Fresnel zone has been estimated from the data. Aneasy way to do this will be explained later on.

Zero-offset

It is instructive to note that the above formulas (4.5.16) for the projection matrix and (4.5.18) forthe geometrical-spreading matrix B

˜reduce in the case of a monotypic zero-offset reflection (i.e.,

S = G) to the ones directly derived in Hubral et al. (1993b). For a zero-offset configuration, we haveΓ˜S = Γ

˜G = I

˜, where I

˜is the 2×2 unit matrix. Also, as the upgoing ray branch of the total reflected

central ray is now the reverse ray to the downgoing one, the involved propagator submatrices reduceto B

˜1 = B

˜T2 = B

˜0, and D

˜1 = A

˜T2 = D

˜0, where B

˜0 and D

˜0 are submatrices of the propagator

matrix ˆT˜0 for the downgoing branch of the normal ray. Therefore, we have Λ

˜= 2B

˜−T0 , and together

with H˜F = 2D

˜0B˜

−10 , we finally obtain from equation (4.5.16) using the symplecticity of ˆT

˜0

Γ˜M = D

˜−T0 (4.5.19)

as given in equation (34) of Hubral et al. (1993b) and equation (4.5.18) reduces to

B˜

= 4H˜

−1P (4.5.20)

that parallels equation (36) of Hubral et al. (1993b).

4.6. OTHER APPLICATIONS 131

4.5.4 Time-domain projected Fresnel zone

Correspondingly to equation (4.5.5), we can also define a projected Fresnel zone in the time domainby replacing T/2 by Tε. We have then

1

2|ξ ·H

˜P ξ| < Tε , (4.5.21)

where H˜P is given by equation (4.5.17).

4.5.5 Determination

We still need to show how to obtain the projected Fresnel zone from the data without knowing theactual Fresnel zone on the reflector, i.e., without information about the reflector. For this aim, thepoint of departure for the derivation of a way to determine the projected Fresnel zone is once morethe paraxial traveltime equation (4.2.25) of Bortfeld (1989) for a ray paraxial to the central ray.This ray starts at point S on the anterior surface and ends at a position G on the posterior surface.Recall that the dislocation of S with respect to S is described by the tangent plane coordinatevector x and the corresponding dislocation of G from G by the tangent plane coordinate vectorx′. The values x = 0 and x′ = 0 specify the initial and end points, S and G, of the central ray.We use this traveltime equation again to establish expressions for the reflection and the diffractiontraveltime surface. The reflection traveltime surface T (S,G) is given by equation (4.2.25) whensetting x = xS , x′ = xG, and the matrices A

˜, B

˜, C

˜, and D

˜are the propagator submatrices for

the central ray SMRG. The diffraction traveltime surface for pair S,G from point MR is a sum oftwo traveltimes T (S,MR) + T (MR, G) of the form (4.2.25). The first one T (S,MR) is obtained bysetting x= xS = 0 in equation (4.3.17a). The second one T (MR, G) uses x′ = xG = 0 in equation(4.3.17b).

Substituting equation (4.5.6a) in formula (4.5.2), we see that we can determine the elementsof the projected Fresnel zone matrix from the second derivatives of traveltime, namely

HP ij =∂2[T (S,MR) + T (MR, G) − T (S,G)]

∂ξi ∂ξj(4.5.22)

for i, j = 1, 2. We observe, however, that we cannot obtain the projected Fresnel zone from travel-time measurements, only. Additionally to the picked reflection traveltime surface, a macrovelocitymodel is needed to compute the diffraction traveltime surface of MR that cannot be estimated fromthe data. Note that once H

˜P is known from equation (4.5.22), the actual size can be calculated

with equation (4.5.21) if the length Tε of the seismic wavelet is known.

4.6 Other applications of the surface-to-surface propagator matrix

In this section, we address two more examples for the usefulness of the ˆT˜-propagator-matrix for-

malism in seismic reflection imaging. The first example is of value in true-amplitude migration aswill be discussed in Chapter 7 and also in the general imaging theory derived in Chapter 9. It willhelp to show what influence a diffraction-stack migration principally has on migration amplitudes.Both problems are closely related to the ray-segment decomposition of the submatrix B

˜of ˆT

˜.

132 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

A second application extends the NIP-wave theorem (Hubral, 1983), which plays a key rolein the formulation of moveout formulas for common-mid-point gathers and for solving Dix-type ve-locity inversion problems (Hubral and Krey, 1980; Goldin, 1986). Although the problems addressed

below could, in principle, also be solved with the ˆΠ˜

propagator matrix, the more compact solutions

achieved with the ˆT˜

matrix are, in our opinion, more direct and natural.

4.6.1 Geometrical-spreading decomposition

Later in this book, it will become necessary to express the geometrical-spreading factor L in equation(3.6.15) of an elementary seismic wave propagating along the central ray SMG (Figure 4.7) in termsof the propagator submatrices pertaining to the ray segments SM and MG. To achieve this werewrite equation (4.5.3) as

B˜

= B˜

2H˜F B

˜1 . (4.6.1)

Equation (4.6.1) states that the propagator submatrix B˜

of the total ray SMG can be decomposedinto the two matrices B

˜1 and B

˜2 of the ray segments SM and MG, provided the Fresnel zone

matrix H˜F is known at M . In other words, the contributions to the matrix B

˜due to the two

individual ray segments SM (i.e., B˜

1) and MG (i.e., B˜

2) and due to the Fresnel zone (H˜F ) at

point M can be separated. Therefore, equation (4.6.1) is referred to as the B˜

-matrix decomposition.For further details about its use in true-amplitude migration, see Schleicher et al. (1993a), whereit is used without derivation.

Substituting the decomposition (4.6.1) for B˜

in equation (3.13.6c) leads to the followingdecomposition for the modulus of the geometrical-spreading factor

|L| =

√

cosϑS cosϑGvSvG

|det(B˜

2H˜F B

˜1)|

1

2 (4.6.2a)

=

√√√√

v+M

cosϑ+M

v−Mcosϑ−M

|detH˜F |

1

2 |L1| |L2| , (4.6.2b)

where L1 and L2 are the geometrical-spreading factors along the ray segments SM and MG fora point source at S and M , respectively. Equation (4.6.2b) states that the spreading factor of thetotal ray SMG can be computed from the two spreading factors along the ray segments providedthe Fresnel zone matrix H

˜F and the incidence and reflection/transmission angles ϑ−

M and ϑ+M at

the intermediate point M are known.

We now introduce the “Fresnel geometrical-spreading factor”—to which we will later referfor the sake of brevity as the “Fresnel factor”—defined by

LF = OF |detH˜F |−

1

2 exp

iπ

2κF

. (4.6.3)

Here, the Fresnel obliquity factor OF is given by

OF =

√√√√

cosϑ+M

v+M

cosϑ−Mv−M

=cosϑMvM

, (4.6.4)

where the right-hand side holds for monotypic reflections. The phase of the Fresnel geometrical-spreading factor is given by π/2 times

κF = 1 − Sgn(H˜F )/2 , (4.6.5)

4.6. OTHER APPLICATIONS 133

where Sgn(H˜F ) is the signature of H

˜F . The signature of a matrix is defined as the number of

positive eigenvalues minus the number of negative ones, i.e.,

Sgn(H˜F ) = sgn(λ1) + sgn(λ2) , (4.6.6)

with λ1, λ2 being the real nonzero eigenvalues of the (real symmetric) 2 × 2-matrix H˜F . Also,

sgn(λj) = ±1 according to whether λj > 0 or λj < 0.

Using definition (4.6.3), we may write equation (4.6.2b) in an intuitively even better under-standable form as

|L| =|LS | |LG||LF |

. (4.6.7)

The phase of LF in the above definition (4.6.3) can only be understood with a deeper analysisthat will be presented in Chapter 6. However, let us already at this place mention the result,namely that the decomposition (4.6.7) is valid not only for the moduli but for the full complexgeometrical-spreading factors which include the moduli and arguments (phases).

It can be shown that the reciprocity relation for the geometrical-spreading factor is [compareequations (3.13.6c) and (4.2.24)]

L(S,G) = L(G,S) , (4.6.8)

where L(S,G) is the point-source geometrical-spreading factor for the ray SG and L(G,S) is thecorresponding factor for the reverse ray GS. Equations similar to (4.6.8) hold, of course, also forL1 and L2.

Zero-offset

We now consider again the monotypic zero-offset primary reflection (see Figure 4.9a), where thetwo ray segments SM and MG with S = G and M =NIP (normal incidence point) coincide. We

can write, then, ˆT˜1 = ˆT

˜0 and ˆT˜2 = ˆT

˜∗0, where ˆT

˜0 denotes the segment propagator matrix of the

downgoing normal ray and ˆT˜∗0 the propagator matrix of the reverse, i.e., upgoing, normal ray. It

follows that

|L| =2 cosϑ0

v0|det(D

˜T0B

˜0)|

1

2 . (4.6.9)

Here, ϑ0 is the angle the ray makes with the surface normal at S = G and v0 is the wave velocityat that point. Equation (4.6.9) was also found by Bortfeld and Kiehn (1992). The phase of L inequation (4.6.9) was established in Hubral et al. (1993a).

4.6.2 Extended NIP-wave theorem

In this section, we extend the NIP-wave theorem (where NIP = normal incidence point) to largeroffsets. At first, let us recall the NIP-wave theorem (Hubral, 1983).

It is well known that for a inhomogeneous overburden and a possibly dipping reflector, thereflection points of waves registered in a CMP configuration, i.e., with sources and receivers dis-tributed symmetrically around a fixed common midpoint, generally do not coincide with the re-flection point of the normal ray at the CMP, the latter reflection point being referred to as normalincidence point or briefly NIP. This has in fact led to the replacement of the older denomination

134 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

PIP

S=GS G

(a)

S GS

CMP

CMP

G

PIPCIP

(b)

NIP

Fig. 4.9. Paraxial reflected rays: (a) Central normal ray from S to G = S reflected at NIP andprimary reflected paraxial ray from S to G via the paraxial incidence point PIP (solid line). Thepoint S = G halves the distance between S and G. Also indicated is a hypothetical ray joiningS with G via NIP (dashed line). (b) Arbitrary primary reflected central ray from S to G via thecentral incidence point CIP and paraxial ray from S to G via PIP with a common midpoint (solidline). Also indicated is a hypothetical ray joining S with G via CIP (dashed line), which acts likea diffractor point.

4.6. OTHER APPLICATIONS 135

of that configuration as CDP (common depth point) with CMP. The situation is depicted in Fig-ure 4.9a. The NIP-wave theorem states that nontheless, the CMP reflection traveltime along raySPIPG is, up to the second order in half-offset, equal to the traveltime along a non-Snell ray fromS to NIP to G. This means that each paraxial ray of a CMP ray family can be viewed as “passing”through the common point NIP instead of the offset-dependent real reflection point PIP.

Of course, the half-offset is measured from the CMP, i.e., the source-receiver location of theassociated normal ray. Thus, the equivalence of traveltimes guaranteed by the NIP-wave theoremis only valid in a certain region centered at the CMP. In fact, it is exactly the region of validity ofthe paraxial approximation around the normal ray.

In this section, we investigate the situation outside that region. Therefore, we have to workwith a general central ray from S to G separated by some arbitrary offset (see Figure 4.9b). We maysay that we are looking at the outer traces of a CMP section where the second-order approximationof the NIP-wave fails. To distinguish this situation for the moment from a CMP with rays insidethe paraxial vicinity around the normal ray, we introduce the acronym CMPO (common midpointoffset). Note that the CMPO experiment is merely a conceptual experiment. We introduce it herewith the purpose to study the limit CMPO −→ CMP. In practical situations, even in inhomogeneousmedia very large offsets between S and G are needed to actually realize a CMPO experiment (seealso Vermeer, 1995). In a horizontally layered medium, the paraxial vicinity of the normal rayextends to infinity and thus, no CMPO experiment exists at all. Even if offsets outside that vicinityare available, the CMPO experiment is nothing but a part of the standard CMP experiment.

As shown below, the NIP-wave theorem no longer holds for a general primary reflectedcentral ray of arbitrary offset between S and G (see Figure 4.9b), i.e., the difference in traveltimeTCMPO−TCIP between the rays S-PIP-G and S-CIP-G (CIP = central incidence point, see Figure4.9b) is generally no longer zero up to the second order. What we call the CIP-wave theorem (orextended NIP-wave theorem) is the expression of this traveltime difference which is given in termsof the propagator submatrices B

˜1, B

˜2, and B

˜used in formula (4.6.1). We start by considering

the traveltime TCIP (x), which is the sum of the traveltimes of the two independent (not obeyingSnell’s law at CIP) ray segments that compose the dashed ray in Figure 4.9b. As can be seen there,segment 1 is the one which joins S at x to point CIP and segment 2 links G at −x to point CIP.

Computation of TCMPO

The traveltime TCMPO(x) of the CMPO experiment, is obtained by inserting the condition x′ = −xinto equation (4.2.25). This gives the traveltime for the reflected ray that starts at S(x), hits thereflector at PIP and returns to G(−x). We find

TCMPO(x) = T (x,−x)= T0 − p0 · x − p′0 · x + x ·B

˜−1x

+1

2x ·B

˜−1A

˜x +

1

2x ·D

˜B˜

−1x . (4.6.10)

Computation of TCIP

Since point CIP is described by the condition xM = 0, the traveltime along a ray from any pointat the anterior surface (x) to point CIP (dashed rays in Figure 4.9b) is found by inserting this

136 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

condition into equation (4.3.16a). From this we find for the traveltime T1(x) along ray segment 1

T1(x) = T01 − p0 · x +1

2x ·B

˜−11 A

˜1x . (4.6.11)

Use of equation (4.3.16b) with xM = 0 yields for the traveltime T2(x′) along ray segment 2

T2(x′) = T02 + p′0 · x′ +

1

2x′ ·D

˜2B˜

−12 x′ . (4.6.12)

Equation (4.6.12) has to be taken at x′ = −x, which is required by a CMPO (common-midpointoffset) configuration. The traveltime TCIP (x) along the “non-Snell” ray S-CIP-G is then simplygiven by

TCIP (x) = T1(x) + T2(x′ = −x)

= T01 + T02 − p0 · x − p′0 · x

+1

2x · (B

˜−11 A

˜1 +D

˜2B˜

−12 )x . (4.6.13)

Comparing TCIP with TCMPO

Comparison of equations (4.6.10) and (4.6.13) immediately shows that

TCIP (x= 0) = TCMPO(x= 0) = T01 = T02 = T0 (4.6.14)

and

∇x TCIP |x=0 = ∇x TCMPO|x=0 = − (p0 + p′0) , (4.6.15)

which means that the traveltime surfaces TCIP (x) and TCMPO(x) are tangent at x= 0.

However, there is a difference in the second-order terms of these two functions, i.e.,

TCMPO(x) = TCIP (x) − 1

2x · D

˜x , (4.6.16)

where

D˜

= D˜

2B˜

−12 + B

˜−11 A

˜1 − B

˜−1 − B

˜−T − D

˜B˜

−1 − B˜

−1A˜. (4.6.17)

Upon the use of equations (4.3.7), this may be rewritten as

D˜

= B˜

−1B˜

2B˜

−T1 + B

˜−T2 B

˜1B˜

−1 − B˜

−1 − B˜

−T . (4.6.18)

We now use formula (4.6.1) to eliminate B˜

−1 and B˜

−T from equation (4.6.18). This results in

D˜

= B˜

−11 H

˜−1F B˜

−T1 + B

˜−T2 H

˜−1F B˜

−12 − B

˜−11 H

˜−1F B˜

−12 + B

˜−T2 H

˜−TF B

˜−T1

= (B˜

−11 − B

˜−T2 )H

˜−1F (B

˜−12 − B

˜−T1 ) , (4.6.19)

where, on the last line, we have used the symmetry of the Fresnel matrix H˜F . The above equation

is obtained using the symplectic property and chain rule of the propagator matrices ˆT˜, ˆT˜1, and ˆT

˜2.We refer to equations (4.6.16) and (4.6.19) as the CIP-wave theorem. One can deduce from it that,in a CMPO experiment, neighboring rays do not, up to second order, generally “pass” througha common depth point CIP. In fact, this only happens, if D

˜= 0, i.e., if B

˜2 = B

˜T1 . This latter

4.7. SUMMARY 137

condition can easily be checked by dynamic ray tracing. Note once more that again the B˜

-matricesplay the main role in the expression for the traveltime difference.

Formula (4.6.16) provides a measure of the validity of the CMP technique. This aims at (butgenerally does not succeed in) having all rays of a CMP ray family “pass” through a common depthpoint (CDP) or common reflection point (CRP). To find a CRP traveltime one cannot a priori setx′ = −x in equation (4.2.25). To find shots and receivers on the measurement surface, such that allrays pass through the CRP is also the aim of the migration-to zero-offset (MZO) and dip-moveout(DMO) technology (see, e.g., Black et al., 1993; Hubral et al., 1996b; Tygel et al., 1998).

NIP-wave theorem

To show that the CIP-wave theorem is indeed a generalization of the NIP-wave theorem (Hubral,

1983), we consider the zero-offset situation, for which, as indicated before, ˆT˜

1 = ˆT˜

0 and ˆT˜

2 = ˆT˜∗0.

In that case,

D˜

= (I˜

− B˜

−10 B

˜0

︸ ︷︷ ︸

I˜

) B˜

−1(I˜

− B˜

−T0 B

˜T0

︸ ︷︷ ︸

I˜

) = O˜. (4.6.20)

In other words, since CIP equals NIP for a normal ray,

TCMP (x) = TNIP (x) . (4.6.21)

Equation (4.6.21) is the statement of the NIP-wave theorem that plays a fundamental role in thegeneralization of the Dix (1955) formula to laterally inhomogeneous media (Hubral and Krey, 1980).

4.7 Summary

We have established in this chapter some basic properties of paraxial rays in the vicinity of a3-D central ray. Most of the results obtained in this chapter will be of great value for the true-amplitude imaging approach to be derived below. First of all, we applied Hamilton’s formalism toderive expression (4.2.25) for the traveltime along a paraxial ray in the vicinity of a known central

ray. We have shown that the 4× 4 matrix ˆT˜

is a useful and compact propagator matrix for certainparaxial-ray parameters. It was introduced by Bortfeld (1989) for the case where the ray endpoints

lie on anterior and posterior surfaces. Thus, we refer to ˆT˜

as the surface-to-surface propagatormatrix. Its practicability lies mainly in the fact that most of its properties can be determined fromtraveltimes for paraxial source-receiver pairs even when the medium is not known, as is the case ininversion problems. Certain useful properties of the ˆT

˜propagator, like the symplecticity (4.2.21)

or the chain rule (4.3.5) have been derived. The key formulas of this chapter, however, are thetraveltime equations (4.2.25), (4.2.30), and (4.3.16) to (4.3.19). From these equations, we will learnlater on how the weights for the true-amplitude imaging stack integrals (see Chapters 7 and 9) canbe computed by dynamic ray tracing.

In order to demonstrate already in this chapter that compact results can be obtained withthe ˆT

˜propagator, we addressed three practical problems of the seismic reflection method. The

ˆT˜

propagator is particularly useful for a paraxial Fresnel-zone construction as we have seen fromequation (4.5.2) together with (4.5.3). When ray theory is applied to seismics, it is accepted that

138 CHAPTER 4. SURFACE-TO-SURFACE PARAXIAL RAY THEORY

a region around the ray (the Fresnel volume or “physical ray”) contributes to the observed (mono-frequency) wavefield. Of particular interest is the intersection of the Fresnel volume with a targetreflector. This region (commonly called Fresnel zone) has been given broad attention in stratigraphicresolution studies where the effects of laterally changing reflector properties on the reflected eventsare investigated. We have derived a 3-D inversion method to compute the projected Fresnel zone of aprimary reflection for any arbitrary seismic measurement configuration. The necessary informationis obtained from two traveltime surfaces. The first one is the reflection traveltime surface that ispicked from the data. The second one is the diffraction traveltime surface that is computed bymeans of a macrovelocity model. However, no knowledge about the reflector is needed. ProjectedFresnel zones can consequently be computed with almost no extra effort, when a diffraction stackmigration is to be performed.

Further applications are related to seismic true-amplitude migration [equation (4.6.1)], andgeneralized Dix-type traveltime inversion [equation (4.6.16) with (4.6.19)]. In all of these applica-

tions the decomposition formula (4.6.1) obtained for the submatrix B˜

of the propagator ˆT˜

plays

the most important role. We hope to have already somewhat convinced the reader that the ˆT˜

prop-agator is a very attractive tool to solve certain seismic modeling, imaging, and inversion problems.This will become more evident in the later chapters of this book, where again the fundamentalproperties of the ˆT

˜matrix will be used to derive simple analytic solutions for various model-based

seismic reflection imaging problems. The full potential of the ˆT˜

propagator as a black-box formalismwon’t even be exploited in this book. It is especially useful for model-independent seismic reflectionimaging methods as discussed in Hubral (1999).

Chapter 5

Duality

We have already mentioned in the Introduction that all the map and image transformation pro-cedures to be discussed in this book rely on two basic geometrical concepts, namely the Huygenssurface (also called diffraction-time surface or maximum convexity surface) and the isochron (alsocalled aplanat, aplanatic or equal-traveltime surface). In this chapter, we further elaborate thecommon properties and the mutual relationship of these two fundamental surfaces as well as theirrelationship to the 3-D reflection-time surface and the 3-D target reflector. Note that we alwaysassume a fixed measurement configuration as discussed in Section 2.2. thus, all traveltime surfacesconsidered here are functions of a 2-D vector parameter ξ rather than the complete set of sourceand receiver coordinates xS and xG.

5.1 Basic concepts

Fundamental geometrical relationships between the 3-D reflection-time surface on the one hand andthe 3-D subsurface reflector on the other hand can be expressed in form of certain duality theorems.To explain them, we refer to Figure 5.1. For instance, for each point MΣ on the reflector ΣR, thereis a diffraction-time surface ΓM that is tangent to the reflection-time surface ΓR, and for each pointNΓ on the reflection-time surface there is an isochron ΣN that is tangent to the reflector. Besidesproviding a simple proof of these facts, the first duality theorem also states that the changes of theisochron with varying time and of the diffraction-time surface with varying depth are reciprocalto each other. The second duality theorem expresses a relationship between the curvatures of thereflection and diffraction time surfaces and of the reflector and isochron. This allows to representthe Fresnel-zone matrix and the Fresnel geometrical-spreading factor (see Section 4.5) as a functionof the difference in spatial second derivatives between (a) the diffraction and reflection time surfacesat the tangency point NR or (b) the isochron and reflector surfaces at the reflection point MR. Theduality theorems are of fundamental importance in migration and demigration as well as in seismicmodeling, reflection imaging and traveltime inversion in general.

The 3-D seismic reflection method uses data collected from an organized areal or lineardistribution of source-receiver pairs (the seismic experiment). The general aim is to invert thereflection data for properties pertaining to the subsurface region, which is illuminated by the seismicexperiment. One of the main objectives is to map or image (depth migrate) subsurface interfaces(i.e., discontinuities of the medium properties such as the density and/or the velocity) that reflect

139

140 CHAPTER 5. DUALITY

reflection-signal strip

t

R

( ;M )

( )( )

D

RR

z

rr

MR

M

NRR

depth-migrated strip

R

R

R

R

R

( )r

( ; N )r

R

RR

I

N

S( )G( )

ξ R

ξ

R

ΓM

ξ

ξ

ξ

ϑβ

Σ

ξ Σξ

ΣR

( )r

ΓR

Fig. 5.1. (a) Schematic 2-D sketch of a 3-D seismic record section in one of the mentioned configu-rations. From all seismic traces that define the reflection-signal strip, only the one at the stationarypoint ξR is depicted. At NR, the Huygens surface ΓM computed for the depth point MR [see part(b)] is tangent to the reflection time surface ΓR. (b) Schematic 2-D sketch of a general 3-D seismicmodel. The depth-migrated strip attached to the reflector ΣR results from a depth migration of thereflection-signal strip [see part (a)]. At point MR, the isochron ΣN computed for the time point NR

[see part (a)] is tangent to the reflector ΣR. The points MR and NR are thus dual to each other asexplained in the text. Also shown is the ray SMRG that uniquely defines the dual pair (MR, NR).The angles ϑR and βR denote the reflection angle and the local dip angle at MR.

5.2. DUALITY OF REFLECTOR AND REFLECTION-TIME SURFACE 141

the seismic waves back to the measurement surface. This chapter provides new interesting resultson the relationship between reflections and subsurface reflectors.

The positioning in depth of the seismic reflectors with no regards to the wave amplitudes in-volved is called a kinematic depth migration or map migration. When amplitudes (e.g., geometrical-spreading factors, reflection/transmission coefficients, attenuation) are also taken into considera-tion, that is, when changes in amplitudes are quantitatively controlled during the migration, wetalk about amplitude-preserving depth migration. In particular, when the depth-migrated seismicsignals are freed from the geometrical-spreading effects (and other amplitude factors are not af-fected), we call the process a true-amplitude depth migration. It is well known that an amplitude-preserving or even a true-amplitude migration can only be performed as a prestack migration. Itcannot be performed as a post-stack migration because the common-midpoint (CMP) stack, orNMO/DMO/stack, although improving the signal-to-noise ratio, is not an amplitude-preservingprocess.

Given a dense distribution of source-receiver pairs on a measurement surface and a smoothsubsurface target reflector ΣR below an inhomogeneous velocity overburden (Figure 5.1), it is wellknown that the Huygens surface ΓM pertaining to a reflection point MR on the target reflector ΣR

and its primary-reflection traveltime surface ΓR are tangent surfaces in the time-domain. This istrue irrespective of the measurement configuration (see Section 2.2) that is used. It was alreadymentioned by Hagedoorn (1954) that both surfaces are closely related. In this chapter, we investigatethis relationship, which we call duality. This duality between the reflection time surface ΓR andHuygens surface ΓM on the one hand and isochron ΣN and subsurface reflector ΣR on the otherhand can be expressed in form of two duality theorems. We will observe that both the first andsecond derivatives of these surfaces are related to each other. Their curvatures are also closelyrelated to the Fresnel zone at the reflection point.

5.2 Duality of reflector and reflection-time surface

In this section, we will explain geometrically why we call the reflector ΣR and the reflection-timesurface ΓR dual surfaces of each other. Once this duality is established, some properties of bothsurfaces can be mathematically studied, which will be very much of use for the unified imagingtheory as developed in Chapter 9.

5.2.1 Basic assumptions

To make the migration problem mathematically tractable, a number of simplifying assumptions hasto be made. The first one is, of course, that the wave propagation can be described by zero-orderray theory as treated in the two preceding chapters.

Considering the general 3-D map migration problem, we suppose an areal distribution of(reproducible) source-receiver pairs densely distributed on the measurement surface, ΣM (Figure5.1b). The location of the source-receiver pairs is defined by the measurement configuration (seeSection 2.2) and is described by a 2-D vector (i.e., the configuration parameter) ξ varying on a planarset A, called the aperture of the seismic experiment. Because of the assumed dense distribution ofsource-receiver pairs on ΣM , we take A to be a (full) planar domain. More precisely, we assume

142 CHAPTER 5. DUALITY

that all source-receiver pairs (S,G) are uniquely described by functions S = S(ξ) and G = G(ξ),with ξ defined in A (for further details, see Section 2.2).

One of the interfaces of the 3-D layered subsurface model is to be taken as the target reflectorand denoted by ΣR. We suppose ΣR to be parametrized as z = ZR(r), with r varying on a planardomain E. The coordinates (r, z) refer to a global 3-D Cartesian system (Figure 5.1b), in which ris a 2-D horizontal vector and the z-axis points in downward direction. Points on ΣR are denotedby MΣ, i.e., a point MΣ has the coordinates (r,ZR(r)).

The primary-reflection traveltime surface of ΣR, resulting from the chosen configuration, willbe denoted by ΓR (Figure 5.1a) and parametrized as t = TR(ξ), with ξ in A. Also, (ξ, t) denote theglobal 3-D Cartesian coordinates of the record section, in which ξ is a 2-D spatial vector and t is thetime coordinate. Points on ΓR are denoted by NΓ, i.e., a point NΓ has the coordinates (ξ, TR(ξ)).

5.2.2 One-to-one correspondence

One last, but important, assumption concerns the uniqueness of the one-to-one relationship betweenpoints MR on the target reflector ΣR and points NR on its traveltime surface ΓR. This is stated asfollows: Each point NR on ΓR specifies one source-receiver pair (S,G) in the chosen measurementconfiguration, which in turn determines one and only one reflection point MR on ΣR. In otherwords, there exists a unique primary-reflection ray SMRG only. Reciprocally, each point MR onΣR determines one and only one source-receiver pair (S,G) (and thus one point NR on ΓR) in thechosen measurement configuration, for which SMRG is a primary-reflection ray. In other words,each point MR on ΣR is the reflection point for one and only one source-receiver pair in the chosenmeasurement configuration.

It is useful for our purposes to recast the above one-to-one relationship in terms of theparameterizations of ΣR and ΓR. We have

(a) For each ξ inA, there exists one and only one rR = rR(ξ) in E, for which S(ξ)MR(rR)G(ξ)is a primary-reflection ray. Here, MR(rR) denotes the point on ΣR specified by its horizontalcoordinate vector, rR.

(b) For each r in E, there exists one and only one ξR = ξR(r) in A, for whichS(ξR)MΣ(r)G(ξR) is a primary-reflection ray. Here, MΣ(r) signifies each point on ΣR specifiedby its horizontal coordinate vector, r.

5.2.3 Duality

The above condition defines a one-to-one correspondence (function) between points MΣ on thetarget reflector and points NΓ on its primary-reflection traveltime surface ΓR. We call this corre-spondence duality. Any two corresponding points MΣ on ΣR and NΓ on ΓR are called dual pointsof each other. Note that, given a fixed point MR = MΣ(rR), its dual point NR = NΓ(ξR) is thepoint on ΓR that is defined by the configuration parameter ξR(rR). The time coordinate of NR isthe traveltime t = TR(ξR) of the primary-reflection ray that has MR(rR) as its reflection point onΣR. Reciprocally, given a fixed point NR(ξR), its dual point MR is the specular reflection pointon ΣR with the coordinates (rR(ξR),ZR(rR)) that is defined by the primary-reflection ray joiningthe source-receiver pair specified by NR on ΓR. Since this dual property holds for each pair of

5.3. BASIC DEFINITIONS 143

points MΣ(r) and NΓ(ξ), we say that the reflector ΣR in (r, z)-space and ΓR in (ξ, t)-space are dualsurfaces of each other.

5.3 Basic definitions

Before going further into the mathematical details of the duality between the considered surfaces,we must introduce and define some useful quantities.

5.3.1 Diffraction and isochron surfaces

For each subsurface point M = M(r, z) with fixed coordinates (r, z), we introduce the diffraction-traveltime or Huygens surface ΓM : t = TD(ξ;M) with ξ in A. The diffraction traveltime TD(ξ;M)is defined by

t = TD(ξ;M) = T (S(ξ),M) + T (M,G(ξ)), (5.3.1)

in which T (S(ξ),M) and T (M,G(ξ)) denote the traveltimes along the ray that join the sourcepoint S(ξ) to the subsurface point M and point M to the receiver point G(ξ), respectively. Thesetraveltimes are given in paraxial approximation by equations (4.3.19). The domain of definitionof the Huygens surface ΓM , i.e., the set A of configuration parameters ξ for which the functiont = TD(ξ;M) is defined, depends on the point M and on the macro-velocity model.

In the same way, for any point N = N(ξ, t) in the record section with fixed coordinates (ξ, t),we introduce the isochron surface ΣN : z = ZI(r;N), implicitly defined by the set of points MI

with coordinates (r, z = ZI(r;N)) in the (r, z)-space which satisfies the condition

TD(ξ;MI) = T (S(ξ),MI) + T (MI , G(ξ)) = t. (5.3.2)

In the same way as above, the domain of definition of the isochron ΣN , i.e., the set E of horizontalspatial vectors, r, for which the function z = ZI(r;N) is defined, depends on the point N and themacro-velocity model. Both sets A and E can, in principle, even be void sets.

Note that both the Huygens and isochron surfaces (5.3.1) and (5.3.2) are defined by the verysame traveltime function TD. To obtain the Huygens surface (5.3.1), one has to keep the subsurfacepoint M (i.e., the coordinates r and z) fixed. On the other hand, to obtain the isochron (5.3.2),one has to keep point N (i.e., the coordinates ξ and t) fixed. We can thus conceptually introducea function of six variables F(r, z, ξ, t) as

F(r, z, ξ, t) = TD(ξ;M) − t = 0 , (5.3.3)

that generates both surfaces ΓM and ΣN , depending on which set of three variables, (r, z) or (ξ, t),is kept fixed. It will thus not be very surprising that there exists a fundamental duality not onlybetween surfaces ΣR and ΓR but also between ΓM and ΣN .

Let us therefore now focus our attention to all Huygens surfaces ΓM defined by points MR onΣR and to all isochron surfaces ΣN defined by points NR on ΓR. It is assumed that all these surfacesare defined in (nonvoid) subdomains of A and E, respectively. For these surfaces, the duality cannow be readily extended as follows: The Huygens surface ΓM for point MR and the isochron ΣN forpoint NR are dual surfaces of each other when the points MR and NR are, themselves, dual points.

144 CHAPTER 5. DUALITY

It is important to recognize that the dual surfaces, ΣR (i.e., the target reflector) and ΓR (itsconfiguration-dependent primary-reflection traveltime surface) have, of course, a distinct physicalmeaning as the model data and as the observation data. On the other hand, the isochron andHuygens surfaces are only to be seen as auxiliary surfaces in the (r, z) and the (ξ, t) domains,respectively. As well known, for each point MR on ΣR, the corresponding Huygens surface ΓM istangent to the reflection traveltime surface ΓR (see first duality theorem below) at the dual pointNR of MR. This important geometrical property is the basis for a (Kirchhoff-type) diffraction-stackmigration. Reciprocally, for any point NR on ΓR, the corresponding isochron ΣN is tangent to thereflector ΣR at the dual point MR of NR. This provides the basis for an isochron-stack demigration(Tygel et al., 1996).

The above geometric properties of the dual points and surfaces are fundamental properties ofreflection waves. They involve first-order derivatives (i.e., slopes) of these surfaces in their respec-tive domains of definition. However, as shown below, also the amplitudes of primary reflections arerelated to the Huygens and isochron surfaces. This involves second-order derivatives (i.e., curva-tures) of these surfaces in their domains of definition. In fact, the geometrical-spreading factor ofa primary reflected elementary wave can be directly inferred from (a) the second-order derivativematrix, with respect to the configuration parameter ξ, of the difference between the diffraction andreflection traveltime surfaces at the tangency point NR, or from (b) the second-order derivativematrix, with respect to the horizontal spatial vector parameter r, of the difference between theisochron and reflector surfaces at the tangency point MR. The precise mathematical formulationof the above statements will be given in the form of two duality theorems. However, before we areready to present these, we find it convenient to introduce a set of useful definitions.

5.3.2 Useful definitions

It is convenient to introduce some additional quantities that will make the derivations later moreeasy.

Traveltime functions

The fundamental function to start with is the general six-dimensional function F(r, z, ξ, t) definedfor each ξ in A and for each M(r, z) in equation (5.3.3). Let us consider now only points MΣ onthe reflector ΣR, i.e., with coordinates (r,ZR(r)). Then, equation (5.3.3) defines the “auxiliaryHuygens traveltime function” TΣ(ξ, r) that is given for all ξ in A and all r in E by

t = TΣ(ξ, r) = TD(ξ;MΣ) = TD(ξ, r,ZR(r)) . (5.3.4)

This four-dimensional hyper-surface is the ensemble of all Huygens surfaces for all points MΣ onthe reflector ΣR.

From the assumptions above, we know that each source-receiver pair (S,G) determines ex-actly one reflection point MR. This means that the horizontal coordinate rR that specifies thespecular reflection point MR(rR,ZR(rR)) on ΣR for a given source-receiver pair (S(ξ), G(ξ)) is afunction of the vector parameter ξ that specifies that pair, i.e., rR = rR(ξ). It follows that thereflection-time surface TR(ξ) is given by

TR(ξ) = TΣ(ξ, rR(ξ)) = TD(ξ, rR(ξ),ZR(rR(ξ))) , (5.3.5)

5.3. BASIC DEFINITIONS 145

which describes the traveltime surface ΓR for all ξ in A. This traveltime is given in paraxial ap-proximation by equation (4.2.30).

Correspondingly to equation (5.3.4), it is also useful to consider in equation (5.3.3) onlypoints MI on the isochron, i.e., with coordinates (r,ZI(r;N)). This defines the traveltime functionTI(ξ, t, r) as

TI(ξ, t, r) = TD(ξ;MI) = TD(ξ, r,ZI(r;N)) ≡ t (5.3.6)

for a fixed value of t. This four-dimensional hyper-surface is the ensemble of all Huygens surfacesfor all points MI on the isochron ΣN .

The traveltime function TI(ξ, t, r) reflects the fact that the diffraction time for each point MI

on the isochron ΣN is identical to t. We observe that TI(ξ, t, r) is actually a function independentof r. Furthermore, we conclude from equation (5.3.6) that, for a given point NR, the first derivativeof TI(ξ, t, r) with respect to t equals unity. All higher derivatives with respect to t as well as allderivatives with respect to r vanish.

Depth function

The fundamental equation (5.3.3) implicitly defines the isochron (5.3.2) too. Thus, correspondingto equation (5.3.4), we introduce the “auxiliary isochron function” ZΓ as the restriction of theisochron function to points NΓ on the traveltime surface ΓR of the reflector ΣR, i.e., with coordinates(ξ, TR(ξ)). The function ZΓ is then defined as

z = ZΓ(r, ξ) = ZI(r;NR) = ZI(r, ξ, TR(ξ)) . (5.3.7)

This four-dimensional hyper-surface is the ensemble of all isochron surfaces for all points NΓ onthe reflection-traveltime surface ΓR.

Hessian matrices

We next consider the Hessian matrices obtained from the above defined traveltime surfaces. Atfirst, we define the Hessian matrices H

˜R and H

˜D of t = TR(ξ) and t = TD(ξ;MR), respectively,

with respect to ξ, viz.,

H˜R =

(

∂2TR(ξ)

∂ξi∂ξj

)

(5.3.8a)

and

H˜D =

(

∂2TD(ξ;MR)

∂ξi∂ξj

)

, (5.3.8b)

evaluated at ξ = ξR. Moreover, let H˜

Σ denote the second-order Hessian matrix of TΣ(ξ, r) withrespect to r, viz.,

H˜

Σ =

(

∂2TΣ(ξ, r)

∂ri∂rj

)

, (5.3.9)

evaluated at r= rR and let Λ˜

(r) denote the negative mixed-derivative matrix of TΣ(ξ, r), viz.,

Λ˜

(r) = −(

∂2TΣ(ξ, r)

∂ri∂ξj

)

, (5.3.10)

146 CHAPTER 5. DUALITY

evaluated at ξ = ξR and r = rR. Furthermore, from the discussion in connection with equation(5.3.6), we observe that the Hessian matrix H

˜I of TI(ξ, t, r) with respect to r vanishes, viz.,

H˜I =

(

∂2TI(ξ, t, r)∂ri∂rj

)

= O˜, (5.3.11)

for all r, where we have used that TI(ξ, t, r) ≡ t is constant for all r.

Correspondingly, we also consider the Hessian matrices Z˜I and Z

˜R of the isochron ΓM : z =

ZI(r;NR) and the reflector z = ZR(r), respectively, with respect to r, viz.,

Z˜I =

(

∂2ZI(r;NR))

∂ri∂rj

)

(5.3.12a)

and

Z˜R =

(

∂2ZR(r)

∂ri∂rj

)

, (5.3.12b)

evaluated at r= rR. Finally, we define the 2 × 2 Hessian matrix of ZΓ(r, ξ) with respect to ξ as

Z˜

Γ(r, ξ) =

(

∂2ZΓ(r, ξ)

∂ξi ∂ξj

)

, (5.3.13)

evaluated at ξ = ξR.

Stretch factors

In addition to the above matrices, let us introduce the following notations for the first verticalderivatives of the Huygens and isochron surfaces,

mD =∂TD(ξ, r, z)

∂zz=ZR(r)

, (5.3.14a)

nI =∂ZI(r, ξ, t)

∂tt=TR(ξ)

, (5.3.14b)

also evaluated at ξ = ξR and r = rR. Note that mD and nI are the vertical stretch factors ofmigration and demigration as we will see later.

It is to be remarked that the quantities, Λ˜

(r), mD, and nI do not depend on the curvaturesof either the reflector or the reflection-time surface at the dual points MR and NR, but onlyon the macro-velocity model. We will see in Chapter 7, how Λ

˜(r) can be expressed in terms of

second-derivative matrices of the traveltimes along rays SMR and MRG. In Hubral et al. (1992),these quantities in turn are related to ray-segment propagator matrices that can be computed fromdynamic ray tracing. Moreover, it was shown in Tygel et al. (1994b) that for a monotypic reflection,mD can simply be expressed as

mD =2 cos ϑR cos βR

vR, (5.3.15)

where ϑR is the reflection angle, βR is the local dip angle, and vR is the local velocity just abovethe reflector, at MR (Figure 5.1).

5.3. BASIC DEFINITIONS 147

5.3.3 Expressions in terms of paraxial-ray quantities

Many of the above defined quantities are fundamental to the derivations presented below. Theywill appear in a number of equations in the framework of the unified theory of seismic imagingpresented in this book. In particular, the weight functions of the various imaging techniques tobe derived below will depend on these quantities. Therefore, it is very useful to investigate theirrelationships to the paraxial-ray quantities defined in Chapter 4. Only in this way, we will learnhow the weight functions can be computed by dynamic ray tracing.

The first quantities to be considered are the Hessian matrices H˜R and H

˜D defined in equa-

tions (5.3.8). By taking second derivatives of equation (4.2.30), we find

H˜R = Γ

˜TSB

˜−1A

˜Γ˜S + Γ

˜TGD

˜B˜

−1 Γ˜G − Γ

˜TSB

˜−1Γ

˜G − Γ

˜TGB

˜−TΓ

˜S . (5.3.16)

Inserting equations (4.3.19) for the ray-segment traveltimes into the definition (5.3.1) of the Huygenssurface, we obtain for a fixed point M with coordinates (rM , zM )

TD(ξ;M) = T01 + T02 − (

p0 · Γ˜S + p′0 · Γ

˜G

)

ξ

− rM ·(

G˜

(r)−TB˜

−12 Γ

˜G +G

˜(r)−TB

˜−T1 Γ

˜S

)

ξ

+1

2ξ ·(

Γ˜TSB

˜−11 A

˜1Γ˜S + Γ

˜TGD

˜2B

˜−12 Γ

˜G

)

ξ

+1

2rM ·

(

G˜

(r)−TD˜

1B˜

−11 G

˜(r)−1 +G

˜(r)−TB

˜−12 A

˜2G˜

(r)−1)

rM , (5.3.17)

so that the second derivatives with respect to ξ, for a fixed rM , yield

H˜D = Γ

˜TSB

˜−11 A

˜1 Γ˜S + Γ

˜TGD

˜2B

˜−12 Γ

˜G . (5.3.18)

The next quantities to be expressed in terms of paraxial ray propagator submatrices are theHessian matrices H

˜Σ and Λ

˜(r). We just have to replace the fixed rM in equation (5.3.17) by a

variable r, because for points MΣ on the reflector, TD(ξ;MΣ) = TΣ(ξ, r = rM ). Taking the requiredsecond derivatives with respect to r, we arrive at

H˜

Σ = G˜

(r)−TD˜

1B˜

−11 G

˜(r)−1 + G

˜(r)−TB

˜−12 A

˜2G˜

(r)−1 (5.3.19a)

= G˜

(r)−TH˜FG

˜(r)−1 , (5.3.19b)

with H˜F given by equation (4.5.3). Correspondingly, from the mixed derivatives with respect to

first r and then ξ, we obtain

Λ˜

(r) = G˜

(r)−TB˜

−T1 Γ

˜S + G

˜(r)−TB

˜−12 Γ

˜G (5.3.20a)

= G˜

(r)−TΛ˜, (5.3.20b)

where Λ˜

is the matrix defined in equation (4.5.15). From equation (5.3.20b) together with (5.3.10)and (3.11.11), we observe that

Λ˜

= −(

∂2TΣ

∂xi∂ξj

)

. (5.3.20c)

How the Hessian matrices Z˜R and Z

˜I of the reflector z = ZR(r) and the isochron z =

ZI(r;N) are expressed in terms of the propagator submatrices is in fact also described by theabove equations once the duality theorems are established.

148 CHAPTER 5. DUALITY

5.4 Duality theorems

Having introduced all the necessary mathematical terminology, we are now ready to state thefundamental geometrical results. These concern the reflection-time and Huygens surfaces on theone hand and the reflector and isochron on the other hand, all expressed at the dual tangencypoints MR and NR. The results will be given in the form of two duality theorems that concern thetwo vertical stretch factors mD and nI as well as the curvatures of all the involved surfaces at MR

and NR.

5.4.1 First duality theorem

Given a fixed pair of dual points MR and NR, the first duality theorem consists of the followingthree statements:

Ia) The Huygens surface t = TD(ξ;MR) for the point MR is tangent to the reflection travel-time surface t = TR(ξ) at NR.

Ib) The isochron z = ZI(r;NR) determined by NR and the reflector z = ZR(r) are tangentat MR.

Ic) The vertical stretch factor mD =∂TD(ξ;M)

∂zof the Huygens surface at M = MR and the

vertical stretch factor nI =∂ZI(r;N)

∂tof the isochron at N = NR are reciprocal quantities, i.e.,

the product of both is equal to one.

5.4.2 Second duality theorem

There exists also a duality relationship between the second-derivative matrices of the reflector ΣR,isochron ΣN , reflection-time surface ΓR, and Huygens surface ΓM at the dual points NR on ΓR andMR on ΣR. This relationship is quantified by the second duality theorem, which states that

mD (H˜D − H

˜R) = − Λ

˜(r)T (Z

˜I − Z

˜R)−1 Λ

˜(r) , (5.4.1)

provided all matrices involved are well defined and nonsingular at the dual points. Since the Hessianmatrix of a given surface is closely related to its curvature matrix, theorem (5.4.1) essentiallyshows how the difference in matrix curvature between the reflection-traveltime surface and itstangent Huygens surface at the point NR is related to the difference in matrix curvature betweenthe reflector and its tangent isochron at the dual point MR. For instance, one of the conclusionsthat may be immediately drawn from the above second duality theorem (5.4.1) and that will beelaborated below is the following. Since the Huygens and isochron surfaces can be computed forany dual pair (NR,MR) if the macro-velocity model is specified, one can directly construct thereflector curvature matrix at MR once the curvature matrix of the reflection-time surface at NR

has been determined from the reflection event.

The second duality theorem (5.4.1) can be alternatively formulated in the form of two inde-pendent claims, involving the matrix H

˜Σ. These are stated as

5.5. PROOFS OF THE DUALITY THEOREMS 149

IIa) The difference between the Hessian matrices H˜D andH

˜R of the Huygens and reflection-

time surfaces is given by

H˜D − H

˜R = Λ

˜(r)TH

˜−1Σ Λ

˜(r) . (5.4.2a)

IIb) The difference between the Hessian matrices Z˜I and Z

˜R of the isochron and reflector is

given by

Z˜I − Z

˜R = − 1

mDH˜

Σ . (5.4.2b)

Once the results (5.4.2a) and (5.4.2b) are established, the second duality theorem (5.4.1) followsimmediately from a simple substitution of H

˜Σ given by equation (5.4.2b) into formula (5.4.2a).

Moreover, there is an additional statement involving the auxiliary matrix Z˜

Γ:

IIc) The matrix Z˜

Γ relates to the Hessian matrices of the reflection-time and Huygens surfacesby

H˜D − H

˜R = − mDZ

˜Γ . (5.4.3a)

Using the above formulas, equation (5.4.3a) can be translated into a relationship between Z˜

Γ andthe Hessian matrices of reflector and isochron, namely

Z˜I − Z

˜R =

1

m2D

Λ˜

(r)Z˜−1Γ Λ

˜(r)T . (5.4.3b)

5.5 Proofs of the duality theorems

In this section, we prove the duality theorems stated above. For that purpose, consider a fixed pairof dual points MR on ΣR and NR on ΓR (Figure 5.1). Let these points be characterized by theparameters rR in E and ξR in A, respectively. As indicated above, both points define the samespecular reflection ray SMRG that connects the source S(ξR) via MR on ΣR to the receiver G(ξR).

5.5.1 First duality theorem

Proof of statement Ia

Because of the above observation, referring to the vector parameter ξ = ξR, the diffraction timeTD(ξR;MR) equals the reflection time TR(ξR). To prove the tangency of both surfaces at NR

(statement Ia), we compute the gradients of t = TR(ξ) and t = TΣ(ξ, r) with respect to ξ. We startwith ∇ξTR(ξ), the jth component of which is given by

∂TR(ξ)

∂ξj=

∂

∂ξjTΣ(ξ, rR(ξ))

=∂TΣ(ξ, r)

∂ξjr

R

+∂TΣ(ξ, r)

∂rkr

R

∂rRk∂ξj

, (5.5.1)

in which the summation convention has been used. We now invoke Fermat’s principle, which statesthat a reflection ray is stationary among all rays that join a fixed source-receiver pair to all reflector

150 CHAPTER 5. DUALITY

points in the vicinity of the specular reflection point. In our case, this simply means that

∂TΣ(ξ, r)

∂rkr

R

= 0 , (5.5.2)

Note that the stationarity condition (5.5.2) actually defines the coordinate vector rR = rR(ξ) of thereflection point MR as a function of the given vector parameter ξ. In our case, this given parameteris ξR so that rR = rR(ξR). Of course, equation (5.5.2) remains valid for any other given value of ξ.Substituting equation (5.5.2) into equation (5.5.1) yields

∂TR(ξ)

∂ξj=

∂TΣ(ξ, r)

∂ξjr

R

=∂TD(ξ, r, z)

∂ξjr

R,ZR(r

R)

, (5.5.3)

which holds at ξ = ξR, since we have supposed that rR = rR(ξR). Corresponding equations holdfor any other value of ξ as long as rR is calculated correspondingly, i.e., rR = rR(ξ) as definedby condition (5.5.2). The right equality in equation (5.5.3) follows from the definition of TΣ(ξ, r)in equation (5.3.4). Now, equation (5.5.3) is our desired result as it proves the tangency of thereflection and diffraction traveltime surfaces t = TR(ξ) and TD(ξ;MR) at NR. This concludes theproof of statement Ia.

Proof of statement Ib

Consider the isochron z = ZI(r;NR) for the given point NR on the reflection-time surface ΓR thatcorresponds to the target reflector ΣR. By definition, this isochron contains all depth points MI

with coordinates (r;NR), for which the diffraction traveltime TD(ξ;MI) equals the traveltime valuet = TR(ξR). Hence, the point MR, dual to NR, belongs to the isochron defined by NR as well asto the reflector ΣR i.e., ZI(rR;NR) = ZR(rR). To prove the tangency of both surfaces at MR, weconsider the gradients of the traveltime functions TΣ(ξ, r) and TI(ξ, t, r) with respect to r, the jthcomponents of which are given by

∂TΣ(ξ, r)

∂rj=

∂TD(ξ;M)

∂rj+

∂TD(ξR, rR, z)

∂zz = ZR(rR)

∂ZR(r)

∂rj(5.5.4a)

and∂TI(ξ, t, r)

∂rj=

∂TD(ξ;M)

∂rj+

∂TD(ξR, rR, z)

∂zz = ZI(rR;NR)

∂ZI(r;N)

∂rj, (5.5.4b)

respectively. The left side of equation (5.5.4b) vanishes identically for all ξ in A because of theisochron definition [see equation (5.3.6)]. Moreover, at ξ = ξR and r = rR, also the left side ofequation (5.5.4a) equals zero because of the stationarity condition (5.5.2). It follows from the aboveconsiderations that

∂ZI(r;NR)

∂rj=

∂ZR(r)

∂rj(5.5.5)

at r = rR. This expresses the fact that isochron and reflector are tangent at MR and provesstatement Ib.

5.5. PROOFS OF THE DUALITY THEOREMS 151

Proof of statement Ic

In order to find the relationship between the vertical stretch factors (∂TD/∂z)(ξR;MR) and(∂ZI/∂t)(rR;NR), we differentiate the equation (5.3.6) for the traveltime function TI(ξ, t, r) withrespect to t. We readily find

∂TI(ξ, t, r)∂t

=∂TD(ξ;M)

∂z

∂ZI(r;N)

∂t= 1 (5.5.6)

at r= rR, from which we conclude that

∂TD∂z

(ξR;MR) =

[∂ZI∂t

(rR;NR)

]−1

. (5.5.7)

In other words, the vertical stretch factors of the Huygens surface for MR at NR and of the isochronfor NR at MR are reciprocal. This concludes the proof of statement Ic.

5.5.2 Second duality theorem

Proof of statement IIa

In order to prove equation (5.4.2a), we differentiate the left and right sides of equation (5.5.1)with respect to ξi . Because of the general validity of equation (5.5.2) for all ξ, its total derivativevanishes, i.e.,

∂

∂ξi

∂TΣ(ξ, r)

∂rkr

R

= 0 , (5.5.8)

where the functional dependency rR = rR(ξ), defined by the stationarity condition (5.5.2), hasto be taken into account. It is for this reason that the mixed derivative in equation (5.5.8) isdifferent from matrix Λ

˜(r), defined in equation (5.3.10), where the two differentiations are carried

out independently. Because of equation (5.5.8), a differentiation of equation (5.5.1) is equivalent toa differentiation of equation (5.5.3). Again taking into account that rR = rR(ξ) and applying thechain rule, we can write the derivative of equation (5.5.3) with respect to ξi at ξ = ξR and r= rR,in matrix notation, as

(

∂2TR(ξ)

∂ξi∂ξj

)

≡ H˜R = H

˜D − Λ

˜(r)TY

˜R , (5.5.9)

where we have used equations (5.3.8) and (5.3.10). Moreover, we have introduced the matrix

Y˜R =

(

∂rRi(ξ)

∂ξj

)

, (5.5.10)

evaluated at ξ = ξR. To get rid of the auxiliary matrix Y˜R in equation (5.5.9), we carry out the

derivative with respect to ξi in equation (5.5.8), which identically vanishes for all ξ in A. We have

0 =∂2TΣ(ξ, r)

∂ξi∂rj+

∂2TΣ(ξ, r)

∂rk∂rj

∂rRk∂ξi

, (5.5.11)

152 CHAPTER 5. DUALITY

which, after evaluation at ξ = ξR and r = rR and after use of equations (5.3.10), (5.3.9), and(5.5.10), can be recast in matrix form as

O˜

= −Λ˜

(r) + H˜

ΣY˜R . (5.5.12)

As long as H˜

Σ is nonsingular, equation (5.5.12) can be solved for Y˜R, yielding

Y˜R = H

˜−1Σ Λ

˜(r) . (5.5.13)

Note that the auxiliary matrix Y˜R is nothing else but the projection matrix Γ

˜M expressed in

equation (4.5.16). It was defined in equation (4.5.6a) as the matrix that relates the coordinatesxM of a point on a transmitting or reflecting interface to the configuration parameter ξ. Insertionof equation (5.5.13) into expression (5.5.9) yields immediately equation (5.4.2a) and completes theproof of statement IIa.

Proof of statement IIb

We now only have to compute the difference between the Hessian matrices H˜

Σ of TΣ(ξ, r) [equation(5.3.4)] and H

˜I of TI(ξ, t, r) [equation (5.3.6)] keeping in mind that the latter is identical to zero

for all ξ in A [equation (5.3.11)]. Thus, subtracting zero from H˜

Σ, we can write

H˜

Σ = H˜

Σ − H˜I . (5.5.14)

The desired Hessian matrices on the right-hand side can be computed by taking the derivatives ofequations (5.5.4) with respect to ri. Appropriate use of the chain rule yields

∂2TΣ(ξ, r)

∂ri∂rj=

∂2TΣ(ξ, r)

∂ri∂rj− ∂2TI(ξ, t, r)

∂ri∂rj

=∂mD(ξ, r)

∂rj

[∂ZR(r)

∂ri− ∂ZI(r;N)

∂ri

]

+∂mD(ξ, r)

∂ri

[

∂ZR(r)

∂rj− ∂ZI(r;N)

∂rj

]

+∂2TD(ξ;M)

∂z2

[

∂ZR(r)

∂ri

∂ZR(r)

∂rj− ∂ZI(r;N)

∂ri

∂ZI(r;N)

∂rj

]

+ mD(ξ, r)

(

∂2ZR(r)

∂ri∂rj− ∂2ZI(r;N)

∂ri∂rj

)

. (5.5.15)

We now evaluate the above equation at ξ = ξR and r= rR. Using the tangency of the isochron andthe reflector expressed by equation (5.5.5), we observe that only the last line of equation (5.5.15)yields a nonvanishing result. By expressing the result in matrix notation by means of equations(5.3.12) and (5.3.9), we directly arrive at equation (5.4.2b). This completes the proof of statementIIb.

Proof of statement IIc

In this section, we prove statement IIc of the second duality theorem. Let MI be a point on theisochron defined by point NR on the reflection-traveltime surface ΓR, i.e., MI has the coordinates

5.6. FRESNEL GEOMETRICAL-SPREADING FACTOR 153

(r,ZI(r;NR)). Thus, at MI , we can start from the identity

TD(ξ,MI(r, ξ)) = TR(ξ) . (5.5.16)

We differentiate both sides with respect to ξj (j = 1, 2) using the chain rule to obtain

∂TD∂ξj

+∂TD∂z

z = ZI(r;NR)

∂ZΓ

∂ξj=∂TR∂ξj

. (5.5.17)

At the stationary point NR, we have

∂ZΓ/∂ξj = 0 , (5.5.18)

which leads to the well-known tangency property between the reflection-traveltime and diffractionsurfaces, equation (5.5.3). We next differentiate both sides of equation (5.5.17) with respect to ξ iusing again the chain rule. At the stationary point NR, we find upon the use of the condition(5.5.18)

∂2TD∂ξi ∂ξj

+∂TD∂z

z = ZR(r)

∂2ZΓ

∂ξi ∂ξj=

∂2TR∂ξi ∂ξj

. (5.5.19)

Recognizing mD = ∂TD

∂z (ξ,MR), we obtain, at ξR,

−mD∂2ZΓ

∂ξi ∂ξj=

∂2TD∂ξi ∂ξj

− ∂2TR∂ξi ∂ξj

, (5.5.20)

which, in the previous notation, is exactly formula (5.4.3a). Equation (5.4.3b) follows directly fromequation (5.4.1). The proof is, thus, complete.

5.6 Fresnel geometrical-spreading factor

The Fresnel geometrical-spreading factor (or briefly Fresnel factor) at the reflection point MR onΣR was introduced by Tygel et al. (1994a) as the factor that describes the influence of the Fresnelzone at the reflection point onto the overall geometrical spreading of a reflection ray. A thoroughinvestigation of the role of Fresnel zones in Kirchhoff-type integrals is given in Klimes (1994). TheFresnel factor assumes a key role in a number of forward and inverse seismic modeling problems(Hubral et al., 1995) and particularly in true-amplitude migration (Schleicher et al., 1993a) as wellas in seismic imaging (Hubral et al., 1996a; Tygel et al., 1996).

In this section, we will see how the Fresnel factor is related to the reflection-time and Huygenssurfaces as well as to the isochron and reflector. We will see that besides the curvature matricesof either pair of these surfaces, there also enters the configuration-dependent Beylkin determinant(Beylkin, 1985a; Bleistein, 1987) into one possible representation of the Fresnel factor.

The Fresnel factor was already defined in equation (4.6.3) as

LF (MR) = OF1

√

|det(H˜F )| exp

iπ

2κF

, (5.6.1)

where H˜F is the so-called Fresnel zone matrix defined in equation (4.5.3) (see also Cerveny and

Soares, 1992; Hubral et al., 1992b) that defines the size of the Fresnel zone at MR for any given

154 CHAPTER 5. DUALITY

frequency. Moreover, OF is the Fresnel obliquity factor defined in equation (4.6.4) and κF is theadditional number of caustics that the wave encounters along the total ray due to the existence ofthe interface defined in equation (4.6.5). The symbol Sgn(H

˜F ) denotes the signature of the matrix

H˜F , this being the difference between the number of its positive eigenvalues and the number of

its negative ones, as defined in equation (4.6.6). Note that we are implicitly assuming that the realsymmetric 2 × 2 matrix H

˜F is non-singular, i.e., it has two real and nonvanishing eigenvalues.

We notice that the Fresnel-zone matrix H˜F is closely related to the Hessian matrix H

˜Σ.

Let us now investigate this relationship in more detail. By definition, H˜

Σ is the second derivativematrix of the traveltime function TΣ(ξ, r) with respect to the first two components r = (r1, r2) ofthe global 3-D Cartesian coordinate system r = (r1, r2, r3), that is, the matrix of the derivativesof the slowness vector components. Correspondingly, H

˜F is the matrix of the derivatives of the

projections of these slowness vectors, projected into the tangent plane to the reflector at pointMR (see Hubral et al., 1992a), with respect to the first two components x = (x1, x2) of the local3-D Cartesian coordinate system x = (x1, x2, x3) defined at point MR such that the (x1, x2) planecoincides with the tangent plane to the reflection surface at MR (see Tygel et al., 1995).

To derive the relationship between the two matrices H˜

Σ and H˜F , let us at first consider

the rotation from to r to x. Let us choose

i1, i2, i3

as the canonical base vectors for the global

3-D Cartesian coordinate system. Correspondingly,

j1, j2, j3

form a set of orthonormal base

vectors for the local 3-D Cartesian coordinate system defined above. To relate the planes spanned

by

i1, i2

and

j1, j2

we only need an orthogonal transformation that maps the unit vector

i3 into the unit vector j3. One way to do this is by rotating the plane spanned by

i1, i2

with

respect to the vertical axis through an angle ϕr, until the new vector i1 lies in the same verticalplane as j1. Then we rotate that vertical plane by the angle βR (which is just the local, in-planereflector dip at MR) with respect to an axis along the new i2 direction, until the vector i3 coincideswith the vector j3. Finally, we rotate the new horizontal plane with respect to the j3-axis by anangle ϕx until the coordinate systems coincide (see also Section 3.11.4). The two full 3-D Cartesiancoordinate systems x = (x1, x2, x3) and r = (r1, r2, r3) are related by the equation (3.11.10) or, forsmall dislocations, approximately

r = G˜

(r)x . (5.6.2)

Here, G˜

(r) is the 3-D transformation matrix from global to local Cartesian coordinates given inequation (3.11.12). The final results will not change if we further assume that the unit vector i2 isalready parallel to j2, that is if ϕx = ϕr = 0 1. In other words,

r1 = x1 cos βR − x3 sinβR

r2 = x2 (5.6.3)

r3 = x1 sinβR + x3 cos βR.

Given that the Fresnel matrix H˜F is the derivatives matrix of the slowness vector projections

into the tangent plane, we have to compute it by taking the second derivatives of TΣ(ξ, r) withrespect to the local Cartesian coordinates (x1, x2) within that plane, under the additional projectioncondition x3 = 0. With this condition, r3 can be eliminated from system (5.6.3). It becomes

x1 =r1

cos βR,

x2 = r2 ,(5.6.4)

1This is the case of data collected along the up–dip direction with respect to the tangent plane.

5.6. FRESNEL GEOMETRICAL-SPREADING FACTOR 155

or, in vectorial notation,x = G

˜(r)−1 r . (5.6.5)

This relationship is valid even for nonzero values of the rotation angles ϕx and ϕr. As before, G˜

(r)

is the upper left 2 × 2 submatrix of G˜

(r). In other words, G˜

(r) is the matrix that describes thetransformation from the 2-D horizontal coordinates r of the global Cartesian system to the 2-Dlocal Cartesian coordinate system xR at MR (see Section 3.5.2). The determinant of G

˜(r) is simply

given by cos βR [see equation (3.11.15)].

Now, by applying the chain rule twice, and using the summation convention notation (sumover repeated indexes) we find that

∂2TΣ(ξ, r)

∂ri∂rj=∂xk∂ri

∂2TΣ(ξ, r)

∂xk∂xl

∂xl∂rj

+∂TΣ(ξ, r)

∂xk

∂2xk∂ri∂rj

. (5.6.6)

It is obvious from equation (5.5.2) that at MR, the first factor of the second term in this expressionis zero. Thus, in matrix form, we find that

H˜F = G

˜(r)TH

˜ΣG

˜(r) , (5.6.7)

which has been independently confirmed in equations (5.3.19). Inserting the two equalities (5.4.2a)and (5.4.2b) of the second duality theorem into formula (5.6.7) and using equation (5.3.20b), wecan also write H

˜F as

H˜F = G

˜(r)TΛ

˜(r)(H

˜D −H

˜R)−1Λ

˜(r)TG

˜(r)

= Λ˜

(H˜D −H

˜R)−1Λ

˜T (5.6.8a)

andH˜F = − mD G

˜(r)T (Z

˜I −Z

˜R)G

˜(r) , (5.6.8b)

respectively. It is shown in Appendix C, how Hessian matrices Z˜

in arbitrary Cartesian coordinatesrelate to curvature matrices K

˜. Due to equation (C-9), expression (5.6.8b) can be rewritten as

H˜F = − OD (K

˜I −K

˜R) , (5.6.8c)

where we have introduced the curvature matrices K˜I and K

˜R of the isochron and reflector, respec-

tively, at MR. Moreover, the depth obliquity factor OD is defined as

OD =mD

cos βR. (5.6.9a)

Using the definition of factor mD in equation (5.3.14a), it becomes clear that OD is the derivativeof TD in the normal direction to the reflector, i.e.,

OD =∂TD∂n

=cosϑ−Rv−R

+cosϑ+

R

v+R

= |∇TD| , (5.6.9b)

where ϑ±R are the incidence and reflection angles between the ray segments and the interface normalat MR.

For convenience, we introduce the notations,

H˜

∆ = H˜D −H

˜R , (5.6.10a)

Z˜

∆ = Z˜I −Z

˜R , (5.6.10b)

K˜

∆ = K˜I −K

˜R , (5.6.10c)

156 CHAPTER 5. DUALITY

and

κH = [2 − Sgn(H˜

∆)]/2 , (5.6.11a)

κZ = [2 + Sgn(Z˜

∆)]/2 , (5.6.11b)

κK = [2 + Sgn(K˜

∆)]/2 . (5.6.11c)

We explicitly stress that equations (5.6.8) imply the identity

κH = κZ = κK = κF . (5.6.12)

With these notations, equations (5.6.8) lead to the following alternative representations forthe Fresnel geometrical-spreading factor defined in equation (5.6.1).(A) Representation in terms of traveltime derivatives:

LF (MR) =OF

cos βR

√

|det(H˜

∆)||det(Λ

˜(r))| exp

iπ

2κH

= OF

√

|det(H˜

∆)||det(Λ

˜)| exp

iπ

2κH

. (5.6.13a)

(B) Representation in terms of depth-surface derivatives:

LF (MR) =OF

OD

1

cos2 βR

1√

|det(Z˜

∆)| exp

iπ

2κZ

. (5.6.13b)

(C) Representation in terms of depth-surface curvatures:

LF (MR) =OF

OD

1√

|det(K˜

∆)| exp

iπ

2κK

. (5.6.13c)

For monotypic reflections, the latter two expressions further simplify as then OD = 2OF .

The expressions for H˜F and LF (MR) in terms of the difference in curvature matrices K

˜∆ =

K˜I − K

˜R of the isochron and the reflector provide, in our opinion, a much better geometrical

understanding of these quantities. However, although representation (5.6.13c) may seem the mostintriguing one, also the other two are very useful. The matrix H

˜∆ is the second-derivative matrix of

the traveltime difference T∆(ξ;M) = TD(ξ;M)−TR(ξ) that appears in true-amplitude diffraction-stack migration (see Chapter 7). In full correspondence, the matrix Z

˜∆ is the second-derivative

matrix of the difference Z∆(r;N) = ZI(r;N) − ZR(r) that appears in a true-amplitude isochron-stack demigration (see Chapter 9). Thus, the above representations (5.6.13) turn out to be themost fundamental formulas upon which the computation of true-amplitude weights for both thetrue-amplitude diffraction-stack migration and the isochron-stack demigration is based.

5.6.1 Curvature duality

Introducing the isochron and reflector curvatures K˜I and K

˜R as above into the second duality

theorem (5.4.1), it can be rewritten in the form of a “curvature duality theorem,” i.e.,

H˜D −H

˜R = − 1

ODΛ˜

(r)T G˜

(r) (K˜I −K

˜R)−1

G˜

(r)T Λ˜

(r) ,

= − 1

ODΛ˜T (K

˜I −K

˜R)−1 Λ

˜. (5.6.14)

In this form, it directly shows the relationship between the curvatures of the reflection-time andHuygens surfaces on the one side and of the isochron and reflector on the other side.

5.6. FRESNEL GEOMETRICAL-SPREADING FACTOR 157

5.6.2 Beylkin determinant

The Fresnel geometrical spreading factor given by representation (5.6.13a) contains an up to nownot sufficiently addressed factor, the determinant of the second mixed-derivative matrix Λ

˜or Λ

˜(r).

As will be shown in this section, this factor is closely related to the familiar Beylkin determinanthB (Beylkin, 1985a; Bleistein, 1987), thus revealing the close relationship of the representation(5.6.13a) to true-amplitude migration or migration/inversion. In this way, it also becomes clearhow detΛ

˜can be computed by dynamic ray tracing, because the problem of how to determine hB

is already solved (Cerveny and de Castro, 1993).

For a dual pair of points NR on ΓR and MR on ΣR, the Beylkin determinant hB is defined as

hB = det

∇TD∂∂ξ1

∇TD∂∂ξ2

∇TD

NR,MR

= det

∂TD

∂r1∂TD

∂r2∂TD

∂z∂2TD

∂ξ1 ∂r1∂2TD

∂ξ1 ∂r2∂2TD

∂ξ1 ∂z∂2TD

∂ξ2 ∂r1∂2TD

∂ξ2 ∂r2∂2TD

∂ξ2 ∂z

NR,MR

. (5.6.15)

As shown in Appendix D, equation (D-8), the above determinant hB of a 3 × 3 matrix is simplyrelated to the determinant of the 2 × 2 matrices Λ

˜(r) and Λ

˜by

hB(ξ;M) = mD det(Λ˜

(r)) (5.6.16a)

= OD det(Λ˜

) , (5.6.16b)

where the second identity is a consequence of equation (5.3.20b). Substituting equation (5.6.16a)back into the Fresnel factor representation (5.6.13a) yields

LF (MR) =ODOF

hB|det(H

˜∆)|1/2 exp

iπ

2κH

(5.6.17)

As readily seen from its definition in equation (5.6.15), the Beylkin determinant hB only depends onthe macro-velocity model in the reflector overburden as well as on the measurement configurationand not on the reflector itself. In other words the Fresnel geometrical-spreading factor is the quantitythat carries all the information concerning the reflector curvature influencing the geometrical-spreading factor L of the overall reflection ray SMRG (see Figure 5.1).

Taking determinants on both sides of equation (5.6.14) and also making use of equation(5.6.16a), we may recast the second duality theorem in terms of the Beylkin determinant as

det(H˜D −H

˜R) =

h2B

O4D

1

det (K˜I −K

˜R)

. (5.6.18a)

Sgn(H˜D −H

˜R) = − Sgn (K

˜I −K

˜R) . (5.6.18b)

It is now instructive to divide both sides of equation (5.6.18a) by h2B . We then recognize that the

right-hand side of the resulting equality is independent of the chosen measurement configuration.Consideration of different configurations containing the same central ray (e.g., common shot andcommon offset), give rise to useful relations between the corresponding reflection-time and Huygenssurfaces. In this way, the formulas of Shah (1973) can be generalized. Note that this will obviouslynot work when hB = 0.

158 CHAPTER 5. DUALITY

5.7 Summary

In this chapter, we have recognized and mathematically quantified the duality between the funda-mental surfaces involved in all seismic migration, demigration and imaging methods. These surfacesare the subsurface reflector together with its configuration-dependent reflection-time surface, as wellas the Huygens surface and the isochron. We have seen that the latter two surfaces are defined bythe same set of traveltime functions pertaining to the given measurement configuration. If we fix thecoordinates of the depth point, we obtain its Huygens surface; if we fix the time and source-receivercoordinates, we obtain the isochron surface. Due to this fundamental correspondence, certain du-alities of these surfaces can be observed. Besides the well-known facts that (a) the Huygens surfaceconstructed for an actual reflection point on a given reflector is tangent to the correspondingreflection-time surface and (b) the isochron constructed for a given point on the reflection-time sur-face is tangent to the reflector, further fundamental relationships exist between the first and secondderivatives of these surfaces. We called these relationships the first and second duality theorems.The first theorem involves the variations of the Huygens surface and isochron in time and depth. Itcan be related to the stretch that is observed in seismic depth migration. The second duality theo-rem involves the so-called Fresnel matrix that defines the size of the Fresnel zone at the reflectionpoint. From this second duality theorem, a new expression for the Fresnel matrix was derived thatincorporates not only the reflector but also the isochron curvature. These (first and second) dualitytheorems help to better understand the kinematics and dynamics of a variety of seismic reflectionimaging problems. Indeed, for a given reflector and its corresponding reflection-time surface, wefind that the contribution to the overall geometrical spreading of a primary reflection due to thereflector curvature can be expressed as (a) a second-derivative difference of the reflection-time andHuygens surfaces in the time section or as (b) a second-derivative difference of the reflector andthe isochron in the depth section.

We have seen that the (high-frequency) Kirchhoff-migration or diffraction-stack result placedinto the reflection point is proportional to (and to a great extent controlled by) the Fresnelgeometrical-spreading factor (Tygel et al., 1994a). This (generally complex) factor accounts forthe contribution to the overall reflection-ray geometrical-spreading factor which is due to the tar-get reflector only. Except for a multiplicative quantity (which is closely related to the Beylkindeterminant (Beylkin, 1985a; Bleistein, 1987)) that depends on the model and measurement con-figuration, the Fresnel factor can be expressed as the difference in second derivatives between thediffraction and reflection traveltime surfaces at the point of tangency (or stationary point). Thesederivatives are taken with respect to the 2-D configuration parameter that specifies the locationof source-receiver pairs on the measurement surface, evaluated at the tangency point. In this way,the Fresnel factor can be found from a traveltime analysis without knowing anything about thereflector at the reflection point. As we will see in Chapter 6, this property is inherently made useof in true-amplitude migration, where the geometrical-spreading factor of a reflected ray needs tobe eliminated from the migration amplitudes without knowing the reflector. On the other hand,the Fresnel factor can also be expressed in terms of quantities pertaining to the subsurface, a re-sult that is required, e.g., for a true-amplitude demigration or imaging in general. Both results areconsequences of the mentioned duality between the reflection-time and Huygens surfaces on theone hand and the subsurface reflector and isochron on the other hand. This duality also allowsto represent the Fresnel geometrical-spreading factor as the difference in spatial second derivativesbetween the isochron and reflector surfaces at the reflection point. This is particularly revealing forthe understanding not only of migration and demigration processes, but also in seismic reflectionimaging, modeling and traveltime inversion in general.

Chapter 6

Kirchhoff-Helmholtz theory

In this chapter, we take a closer look at the high-frequency approximation of the traditional Kirch-hoff integral. It provides an integral representation of the seismic reflection response at a receiver,given the locations of a source-receiver pair, a laterally inhomogeneous velocity model, and a reflec-tor. On the use of the Kirchhoff-Helmholtz approximation for the elementary wave after reflectionat the reflector, an approximate forward modeling integral results that we call more appropriatelythe Kirchhoff-Helmholtz integral.

The understanding of the Kirchhoff-Helmholtz integral provided in this chapter will helpto interpret another integral associated with the name of Kirchhoff. This is the diffraction-stackintegral, also known as the Kirchhoff-migration integral. With it, the observed seismic elementaryreflection response of an unknown reflector (recorded by an arbitrary source-receiver configuration)can be transformed into an image of the reflector. This imaging is performed with the help of alaterally inhomogeneous macrovelocity model. As we will see in the next chapter, both operations(i.e., the Kirchhoff-Helmholtz and the diffraction-stack integrals) can, under certain circumstances,be understood, both qualitatively and quantitatively, as being “physically inverse” operations toeach other. In the same way as the Kirchhoff-Helmholtz integral can be conceived as a superpositionof Huygens secondary sources distributed along a specified reflector, the diffraction-stack integralcan be interpreted as a process that recovers the location and amplitude of the very same Huygenssources upon this reflector from their individual contributions to the overall seismic response, thusimaging the reflector together with its reflection coefficients. Note, however, that the Kirchhoffintegral and the Kirchhoff migration integral are not inverses to each other in a strict mathematicalsense. In fact, they are adjoint operations (Tarantola, 1984). The asymptotic inverse to the Kirchhoffmigration integral is the Kirchhoff demigration integral that will be discussed in Chapter 9. Anasymptotic inverse to the Kirchhoff-Helmholtz integral has been set up by Tygel et al. (2000).

Wave phenomena are generally described by different wave equations. The acoustic waveequation plays an important role in this respect. Acoustic waves as well as the components ofelectro-magnetic waves are described by this equation. Even a description of elastodynamic wavesby an acoustic wave equation will be fairly accurate, whenever the compressional and shear-wavecomponents of the wavefield are sufficiently well decoupled (as in the case of ray theory, see Chapter3). Different volume and surface integral solutions to the acoustic wave equation have been dis-cussed by Wapenaar and Berkhout (1993). In this chapter, we deal with one of the surface integralsolutions, namely the well-known Kirchhoff integral (Sommerfeld, 1964; Haddon and Buchen, 1981).It computes the propagation of waves away from the actual sources using the wavefield and its nor-

159

160 CHAPTER 6. KIRCHHOFF-HELMHOLTZ THEORY

mal derivative on a closed surface encompassing the observation point. It can, as is well-known,be extended to the case of reflected waves in an inhomogeneous medium, which are propagatedaway from a reflecting or transmitting interface into the direction towards a receiver that can belocated close to the (primary) sources. Following Huygens’ principle, the reflected waves can beinterpreted as being generated by “secondary sources” distributed along the specified reflector.Although mathematically not consistent and, therefore, obviously strictly not correct, it is oftenuseful to insert the so-called “Kirchhoff-Helmholtz boundary conditions” (Sommerfeld, 1964) intothe Kirchhoff integral. These conditions, which represent a generalization of the so-called physical-optics approximation for a perfectly soft or rigid reflector, replace the (unknown) “total field” onthe illuminated portion of the reflector by the “specularly reflected field.” This can be approximatedby the (known) incident field multiplied by an appropriate plane-wave reflection coefficient. In thesame way, the normal derivative of the specularly reflected field is approximated by the normalderivative of the incident field multiplied by the same reflection coefficient. At each point of thereflector, this reflection coefficient is computed under the assumption that the incident wavefieldimpinges upon the reflector locally as a plane wave, whereby the reflector is also replaced by itstangent plane at the incident point. Corresponding considerations are valid for the transmittedfield and the respective transmission coefficient.

Inserting Kirchhoff-Helmholtz boundary conditions into the Kirchhoff integral provides infact a high-frequency approximation to the reflected wave. This leads to what is referred to asthe “Kirchhoff-Helmholtz approximation” (see also, e.g., Frazer and Sen, 1985). Therefore wecall the resulting integral (which is often considered in forward seismic modeling problems) brieflythe “Kirchhoff-Helmholtz integral (KHI).” For modeling purposes, this celebrated integral is tra-ditionally used to obtain the reflection response of a smooth reflector below a layered, smoothlyvarying overburden (in which ray theory applies). Though mostly formulated for a common-shotconfiguration, we present this integral here for arbitrary seismic measurement configurations.

In contrast to their usefulness in forward problems, neither the original Kirchhoff integral northe KHI is suited for solving the inversion problem that aims at imaging the reflector and/or find-ing the interface-reflection coefficients. One way to solve the inversion problem for a common-shotrecord is to backward propagate the reflected wavefield (Schneider, 1978; Berkhout, 1985; Wape-naar, 1993). This can be done by a trick, namely by replacing in the Kirchhoff integral the retardedGreen’s functions by advanced ones, which leads to the Porter-Bojarski integral (Schneider, 1978;Langenberg, 1986). In other words, the recorded reflected wave is reversed by being representedwith advanced Huygens waves at the measurement surface. In this way the reflected wave propa-gates back into the medium towards the secondary sources, i.e., towards the searched-for reflector.If considered in conjunction with the forward propagated elementary wave from the common sourceand a suitable imaging condition (Claerbout, 1971), the reflector can be imaged. In this procedure,migration is the adjoint operator to forward modeling.

However, this approach does not work for seismic measurement configurations other thancommon-shot or common-receiver configurations (see Docherty, 1991). Another, more recent ap-proach to image reflected waves, valid for arbitrary measurement configurations, is based on thegeometrically motivated diffraction stack (Hagedoorn, 1954; Rockwell, 1971). Here, for each pointin the given macrovelocity model, the amplitude values of the seismic traces are summed up alongthe corresponding diffraction traveltime surface (also called Huygens surface) and the obtainedstack result is assigned to the chosen depth point. The mathematical formulation of this latterprocedure in the high-frequency approximation (see, e.g., Newman, 1975; Bleistein, 1987; Goldin,1991; Schleicher et al., 1993a) leads to the weighted “diffraction-stack integral (DSI).” The result

6.1. KIRCHHOFF-HELMHOLTZ INTEGRAL 161

of the DSI is the true-amplitude image of the subsurface reflector giving a measure of the reflectioncoefficient at any reflector location.

In this and the following chapter, we try to give a physical meaning to the heuristic ansatzchosen for the DSI in Schleicher et al. (1993a) and Tygel et al. (1994a) by revealing its relationshipto the KHI. We extend the acoustic case presented in the cited papers to an elementary elasticwave. As we will see in the next chapter, both the Kirchhoff-Helmholtz and the Kirchhoff-migrationintegrals give rise to closely related imaging operations. Although both integral representations arenot exactly inverses to each other in an asymptotic sense, the DSI can be said to recover theinformation that is the input to the KHI. The proof of this fact will also be given in the nextchapter.

Our analysis will lead us to the following physical interpretation of both integrals, whichmight be intuitively obvious but which will be mathematically quantified below. The KHI (hereconsidered for a smooth reflector below a smooth laterally inhomogeneous overburden) is usuallyunderstood as the superposition of Huygens elementary waves located along the reflector andexploding (in response to the incident wave) with secondary-source strengths proportional to thelocal plane-wave reflection coefficients. Each Huygens source would, if exploding on its own, generateseismic signals distributed along the “diffraction-traveltime surface” (therefore also called “Huygenssurface”) in the seismic record that results from the selected measurement configuration. Theenvelope of these Huygens surfaces is the reflection-time surface. In other words, the two reflectorattributes “location” and “reflection coefficient” are mapped by way of the Huygens sources intothe recorded elementary reflection within the seismic record section.

On the other hand, stacking the seismic trace amplitudes in the very same seismic recordsection along the diffraction-time surface that pertains to a Huygens secondary-source point involvesthen summing up all contributions that come from this particular Huygens wave center. Thisstacking operation, which is done by the DSI with certain weights, recovers then again from therecorded reflection both the reflector location and the reflection coefficient, i.e., the two attributesthat characterize the Huygens source. In this way, the DSI can be interpreted as being a “physicalinverse” to the KHI, although it is mathematically only its adjoint operation.

The main emphasis in this chapter is put on (a) reviewing the classical KHI, (b) formulatingit for arbitrary measurement configurations, and (c) providing its asymptotic evaluation in such away that it can be used in the next chapter in the analogous treatment of the DSI. Together, bothchapters will help us to understand the close relationship between forward Kirchhoff modeling andKirchhoff migration.

6.1 Kirchhoff-Helmholtz integral

The classical Kirchhoff integral (see, e.g., Sommerfeld, 1964) represents a time-harmonic acousticwavefield U(G,ω) at the observation point G (that is, the receiver location) in terms of that veryfield, U(P, ω), and its normal derivative, known at all points P on a given smooth surface Σ thatencloses point G (Figure 6.1), provided the sources from which the field U(P, ω) originated arelocated outside Σ. It also uses the Green’s function G(P,G, ω), as well as its normal derivative,computed at all points P on Σ for a hypothetical fixed point source at G. The symbol ω in theabove expressions stands for a positive circular frequency. The time-harmonic dependency exp(iωt)is omitted in all expressions below.

162 CHAPTER 6. KIRCHHOFF-HELMHOLTZ THEORY

G

P

nΣ 0(P )

sources

Fig. 6.1. The Kirchhoff integral computes the wavefield at an observation point (receiver) G pro-vided the field and its normal derivative are given at an arbitrary surface Σ surrounding G. Thesources of the field must be outside Σ. For simplicity, the 3-D situation is featured by a 2-D sketch.

With this understanding, the standard acoustic Kirchhoff integral (see, e.g., Langenberg,1986) may be written as

U(G,ω) =1

4π

∫

Σ

∫

dΣ1

%

[

G(G,P, ω)∂U

∂n(P, ω) − U(P, ω)

∂G∂n

(G,P, ω)

]

, (6.1.1)

where the vector n, normal to Σ, points outwards, i.e., out of the enclosing surface Σ into theregion where the sources of the wavefield U(G,ω) are found. Also, ∂/∂n = n · ∇ denotes thenormal derivative in that direction. Finally, % is the density of the medium at point P .

In correspondence to the above acoustic Kirchhoff integral, we can set up a scalar Kirchhoffintegral for elastic elementary waves. How this can be done is shown in Appendix E based on thescalar elastic wave equation (E-1). Note, however, that this is a “quick and dirty” derivation thatis justified by nothing else but the fact that the resulting expression for the Kirchhoff-Helmholtzintegral is identical to the one obtained in Appendix G upon strict approximations applied to thecorrect elastic expressions. In Appendix E, we show that the above acoustic Kirchhoff integral turnsinto a scalar elastic one by the substitution of the factor fm = 1/% by fm = %v2, where v is thelocal wave velocity at P . Therefore, it reads

U(G,ω) =1

4π

∫

Σ

∫

dΣ %v2

[

U(P, ω)∂G∂n

(G,P, ω) − G(G,P, ω)∂U

∂n(P, ω)

]

, (6.1.2)

where U denotes the principal component of the considered elastic elementary wave. Note, however,that for the actual computation of elastic waves in a layered medium by means of the scalar Kirchhoffintegral, one cannot use in integral (6.1.2) the wavefield as calculated by the scalar wave equation(E-1) but has to return to the ray-theoretical expressions.

6.1. KIRCHHOFF-HELMHOLTZ INTEGRAL 163

GP

MΣ

n

ΣR

Σ0(P )

incidentwavefield

reflectedwavefield

primarysources

secondary sources

Fig. 6.2. For the computation of the reflected field at G, surface Σ is extended to match the reflectorΣR at one side and to approach infinity everywhere else. For simplicity, the 3-D situation is featuredby a 2-D sketch.

By letting a part of the enclosing surface Σ coincide with the illuminated portion of a reflectinginterface (see Figure 6.2) and by extending the rest of Σ towards infinity and applying Sommerfeld’sradiation condition (Sommerfeld, 1964), the Kirchhoff integral (6.1.2) can be used to describe thefield scattered from the reflecting interface (see, e.g., Bleistein, 1984). Note that the primarysources are now assumed to lie inside Σ so that the incident (or direct) wavefield is not propagated byintegral (6.1.2), whereas the secondary sources (scatterers) are assumed to be outside Σ (Figure 6.2).In the following, the illuminated part of the reflecting interface is denoted by ΣR. As shown inAppendix E, a corresponding procedure can also be conceived for the transmission case. Thus, allfollowing derivations, although discussed with respect to a reflected wavefield, can also be appliedto a transmitted wavefield.

Let us now assume that the wavefield to be described results from an omnidirectional pointsource at a point S. This source produces a signal described by a function F [t]. Denoting the Green’sfunction of the scattered field originating from a point source at S by Gs, we have U = F [ω]Gs. Thus,the Kirchhoff integral for a laterally inhomogeneous overburden can be written as an integrationover all points P = MΣ of the surface ΣR, viz.,

U(G,ω) =−F [ω]

4π

∫

ΣR

∫

dΣR %Mv+M

2[∂Gs(S,MΣ, ω)

∂nG(G,MΣ, ω) −

−Gs(S,MΣ, ω)∂G(G,MΣ, ω)

∂n

]

. (6.1.3)

Here, v+M is the velocity encountered by the wavefield after reflection at MΣ, and %M is the density

above the reflector at MΣ. The different sign of integral (6.1.3) in comparison to equation (6.1.2)accounts for the inverted direction of n. In fact it is common in the literature to invert the directionof the surface normal during this process so as to have it pointing towards the observation point G.

164 CHAPTER 6. KIRCHHOFF-HELMHOLTZ THEORY

6.1.1 Kirchhoff-Helmholtz approximation

Equation (6.1.3) represents the wavefield at G in terms of an integral over the Green’s functionGs(S,MΣ, ω) of the scattered field and its derivative at the surface of the scatterer. In most practicalsituations, however, these quantities are unknown.

The Kirchhoff-Helmholtz approximation replaces the (unknown) Green’s functionGs(S,MΣ, ω) at each point P = MΣ on the reflecting interface ΣR by the following (known)single-scattering approximation of the specularly reflected field (Figure 6.2). It can be understoodas a local plane-wave approximation due to its analogies to the reflection of a plane wave at aplanar interface as explained in Appendix F. To obtain the approximate reflected wavefield, let thereflector at MΣ be locally replaced by its tangent plane. Also, suppose that the incident field isreplaced by a plane wave with the same frequency, amplitude, and incidence angle as the actuallyincident wave at MΣ. Immediately after reflection, the approximate reflected field is equal to theincident field multiplied by the plane-wave reflection coefficient Rc(MΣ). The propagation directionof the reflected wavefield is determined by Snell’s law. In other words, both the wave vector andthe reflection coefficient of the reflected wave are determined by the incident field and the normaldirection of reflection surface ΣR. In the same way, the normal derivative of the reflected field isreplaced by the normal derivative of the local plane-wave reflection at MΣ. Due to the differentpropagation directions before and after specular reflection, the specular reflected field at MΣ has thesame sign as the incident field whereas its normal derivative has opposite sign. The indicated pro-cedure is a natural generalization of the physical optics approximation (see, e.g., Bleistein, 1984; orLangenberg, 1986), where the above substitutions are made for perfectly rigid (Rc = 1) or perfectlysoft (Rc = −1) scattering interfaces ΣR.

In symbols, we have for the scattered field Gs(S,MΣ, ω) in Kirchhoff-Helmholtz approximation(see also Appendix F)

Gs(S,MΣ, ω) = Rc(MΣ) G(S,MΣ, ω) , (6.1.4a)

∂Gs(S,MΣ, ω)

∂n= iω

cosϑ+M

v+M

Rc(MΣ) G(S,MΣ, ω) , (6.1.4b)

where G(S,MΣ, ω) is the Green’s function of the incident wavefield at MΣ and Rc(MΣ) denotes theplane-wave reflection coefficient at point MΣ on the interface ΣR, given the ray incident from thesource at S. Moreover, v+

M is the medium velocity at point MΣ that governs the wave propagationafter reflection. Finally, ϑ+

M in equation (6.1.4b) denotes the acute angle which the specular reflectedray makes with the normal n to ΣR atMΣ immediately after reflection atMΣ on ΣR (see Figure 6.3).In other words, it is connected to the incidence angle ϑ−

M at MΣ via Snell’s law. Note that the aboverepresentation for Gs(S,MΣ, ω) is a high-frequency approximation for the field scattered from ΣR

which is equivalent to zero-order ray theory (see also Appendix F).

The two Green’s functions G(S,MΣ, ω) and G(G,MΣ, ω) (i.e., the time-harmonic responses atMΣ for point-sources at S and G, respectively) are, in general, very difficult to obtain analyticallyin inhomogeneous media. They are, thus, in analogy to the above high-frequency approximationfor Gs(S,MΣ, ω), in most computations replaced by their leading terms in powers of 1/ω, i.e., bytheir zero-order ray-theoretical (high-frequency) approximations

G(S,MΣ, ω) ' G0(S,MΣ) exp[−iωτ(S,MΣ)] (6.1.5a)

andG(G,MΣ, ω) ' G0(G,MΣ) exp[−iωτ(G,MΣ)], (6.1.5b)

6.1. KIRCHHOFF-HELMHOLTZ INTEGRAL 165

GS

M Σ

n

ϑ_

M

ϑ+M

ϑGM

ΣR

Fig. 6.3. Geometric situation at the reflector. The angles ϑ−M , ϑ+

M , and ϑGM at MΣ denote theincidence angle of the ray from S, the angle of specular reflection, and the angle of the nonspecularray to G, respectively. For details, see text. In this 2-D sketch featuring the 3-D situation, aconstant-velocity medium and a planar measurement surface are used for simplicity.

where G0(G,MΣ) and τ(G,MΣ) denote the amplitude factor and traveltime along the ray GMΣ,with corresponding meanings for G0(S,MΣ) and τ(S,MΣ). As explained in Appendix F, theKirchhoff-Helmholtz approximation is a high-frequency approximation in the highest order in ω.In the same sense, we thus need to consider the high-frequency approximations of the normalderivative of the Green’s function G(G,MΣ, ω),

∂G(G,MΣ, ω)

∂n=

∂

∂n

[

G0(G,MΣ) exp[−iωτ(G,MΣ)]

]

=

[∂G0(G,MΣ)

∂n+ G0(S,MΣ) (−iω)

∂τ(G,MΣ)

∂n

]

exp[−iωτ(G,MΣ)]

' (−iω)∂τ(G,MΣ)

∂nG0(G,MΣ) exp[−iωτ(G,MΣ)]

' (−iω)cosϑGMv+M

G0(G,MΣ) exp[−iωτ(G,MΣ)] . (6.1.6)

Here, ϑGM denotes the acute angle the ray GMΣ makes with the normal to ΣR atMΣ (see Figure 6.3).The last equality in equation (6.1.6) can be readily derived using the eikonal equation for the rayGMΣ.

Equations (6.1.4) together with (6.1.5) and (6.1.6) constitute the Kirchhoff-Helmholtz approx-imation and correspond to equations (F-9) and (F-10) in Appendix F. The expression resulting frominserting these high-frequency expressions into the Kirchhoff integral of equation (6.1.2) is calledthe “Kirchhoff-Helmholtz integral (KHI).” Using a slightly modified notation, we find for the KHIthe important expression

U(G,ω) ' F [ω]iω

2π

∫

ΣR

∫

dΣRRc(MΣ)KKH(S,MΣ, G) exp[−iωTD(S,MΣ, G)], (6.1.7a)

166 CHAPTER 6. KIRCHHOFF-HELMHOLTZ THEORY

or in the time domain

U(G, t) ' 1

2π

∫

ΣR

∫

dΣRRc(MΣ)KKH(S,MΣ, G) F [t− TD(S,MΣ, G)] , (6.1.7b)

where the dot denotes the derivative with respect to the argument time. In this case, this is equalto the time derivative. The point-diffractor traveltime TD(S,MΣ, G) in formulas (6.1.7) is simplythe sum of the traveltimes along the rays SMΣ and GMΣ, namely,

TD(S,MΣ, G) = τ(S,MΣ) + τ(G,MΣ). (6.1.8)

It can be associated with the traveltime of a “diffraction event” that originates at the source pointS, is diffracted at the hypothetical Huygens secondary-source point MΣ on ΣR and travels fromthere to the observation point G. The integral kernel KKH(S,MΣ, G) is given by

KKH(S,MΣ, G) = %Mv+M

2 G0(S,MΣ) G0(G,MΣ) OKH(S,MΣ, G), (6.1.9)

in which OKH(S,MΣ, G) denotes the so-called obliquity factor

OKH(S,MΣ, G) =cosϑ+

M + cosϑGM2v+M

. (6.1.10)

The obliquity factor OKH(S,MΣ, G) accounts for the difference in directions between the specularreflected ray and the ray segment GMΣ at MΣ.

Let us now assume that the reflecting interface ΣR is given in a parametrized form with a 2-Dparameter vector. We assume that in the vicinity of any given point MΣ, this 2-D parameter vectorcan be reasonably well approximated by the local Cartesian coordinate vector xM . Moreover, weassume that the considered source and observation points lie on a given measurement surface ΣM

and that to each source point S there corresponds one receiver point G. In that case, also S andG can be expressed in parametrized form with the 2-D parameter vector, ξ as discussed in Section2.2. In other words, we write S = S(ξ) and G = G(ξ). Using these parameterizations, we can recastthe KHI representations (6.1.7a) (in the frequency domain) and (6.1.7b) (in the time domain) into

U(ξ, ω) ' iω

2πF [ω]

∫

ΣR

∫

dΣR(xM ) KKH(ξ,xM ) Rc(xM ) exp[−iωTD(ξ;xM )], (6.1.11a)

and

U(ξ, t) ' 1

2π

∫

ΣR

∫

dΣR(xM ) KKH(ξ,xM ) Rc(xM ) F [t− TD(ξ,xM )] , (6.1.11b)

respectively. All quantities within the integrands of these two integrals are now functions of ξ andxM and not any more of S, MΣ and G. It is worthwhile to keep in mind that in the KHI, thevector ξ remains fixed and only xM varies. Note that in Appendix G, integral (6.1.11a) is obtainedindependently from applying the elastic generalization of the Kirchhoff-Helmholtz approximationto the full elastic, anisotropic representation integral, and reducing the resulting expression approx-imately to the principal component of an elementary elastic wave in an isotropic, elastic, layeredmedium.

Introducing the notation

u(xM , t) = Rc(xM ) F [t] , (6.1.12)

6.2. ASYMPTOTIC EVALUATION OF THE KHI 167

which we will identify as a quite natural description of a secondary Huygens source at MΣ withRc(xM ) being the “Huygens source strength” and with the “Huygens source wavelet” given by F [t],the above expression (6.1.11b) assumes the convenient compact form

U(ξ, t) ' 1

2π

∫

ΣR

∫

dΣR(xM ) KKH(ξ,xM )∂u

∂t(xM , t− TD(ξ,xM )) (6.1.13)

that is suitable to be used in forward modeling and which is appropriate for the later treatment.Usually, the Kirchhoff integral (6.1.2) or its time-domain equivalent is interpreted as follows. Thewavefield recorded at receiver G is constructed by a superposition of the contributions of Huygenssecondary sources in the form of monopoles, G(G,P, ω), and dipoles, ∂G(G,P, ω)/∂n, respectively.These Huygens secondary sources originate at dΣ(P ) on the surface Σ being excited by the in-cident field U(P, ω). The Kirchhoff-Helmholtz integral (6.1.13), on the other hand, gives a morecompact representation. Here, the time-domain contributions of the monopoles and dipoles arecombined. They are moreover separated into effects due to the overburden and the reflector. Alloverburden effects are accounted for by the integral kernel (we may also call it a weight function)KKH(ξ,xM ) and the time-function TD(ξ,xM ). The reflector attribute “location” is included inthe integration over ΣR and the quantity u(xM , t) accounts for the reflector attribute “reflectioncoefficient” Rc(xM ). The question why u(xM , t) also includes the analytic source pulse F [t] will beanswered in the next paragraph.

Surely in formula (6.1.13) all quantities on the right-hand side are known and the separationof the integrand into overburden and reflector effects appears artificial. This is, however, no longerthe case when the inverse problem, i.e., the reconstruction of the reflector location and the reflec-tion coefficients along it from the measured seismic wavefield, is to be addressed. Like in forwardmodeling, also in inversion the overburden of the target reflector will be known on account of themacrovelocity model and the measurement configuration. The integral kernel KKH(ξ,xM ) and thediffraction traveltime TD(ξ,xM ) can therefore be looked upon as the known parameters governingboth modeling and inversion. On the other hand, the attributes of the reflector together with thesource wavelet, being the “input” for modeling using the KHI, are the desirable unknown “output”of an inversion. In the same way, the seismic reflections, that are the “output” of modeling, are the“input” to an inversion. In other words, the Huygens sources map the reflector attributes and thesource wavelet—which can therefore be viewed as attributes that characterize the Huygens sourcesas indicated in connection with equation (6.1.12)—into the seismic reflection. The inversion aimsat recovering these attributes.

6.2 Asymptotic evaluation of the KHI

We return now to the KHI in the frequency domain (6.1.11a) and apply the stationary-phasemethod (Bleistein, 1984) to obtain its high-frequency asymptotic evaluation

U(ξ, ω) ' ΥKH(ξ) Rc(xR) F [ω] exp[−iωTD(ξ,xR)] , (6.2.1a)

where the amplitude factor is

ΥKH(ξ) =KKH(ξ,xR)

|det(H˜F (ξ))|1/2 exp

iπ

4[2 − Sgn(H

˜F (ξ))]

(6.2.1b)

168 CHAPTER 6. KIRCHHOFF-HELMHOLTZ THEORY

with Sgn(H˜F ) denoting the signature (number of positive eigenvalues minus number of negative

eigenvalues) of the Hessian matrix

H˜F (ξ) =

(∂2TD

∂xMi∂xMj

)

xM

=xR

. (6.2.2)

This matrix is supposed to be nonsingular (i.e., to have a nonvanishing determinant) throughoutthis work. In other words, receivers at caustic points are excluded from the present analysis. Asdiscussed in Chapter 4, matrix H

˜F (ξ) accounts for the influence of the Fresnel zone at the reflector

on the reflected wavefield.

Transforming result (6.2.1a) back into the time domain, we find the following asymptoticresult of the Kirchhoff-Helmholtz integral (6.1.13)

U(ξ, t) ' ΥKH(ξ) Rc(xR) F[

t− TD(ξ,xR)]

. (6.2.3)

In the above equations, xM = xR is the stationary-phase point of integral (6.1.11a), i.e., the one,where

∂TD(ξ,xM )

∂xMix

R

= 0 for i = 1, 2 . (6.2.4)

It determines the reflection point MR = MΣ(xR) on ΣR that pertains to the source-receiver pair(S(ξ), G(ξ)), so that SMRG constitutes the reflection ray. As stated earlier, we assume that thisray is uniquely determined for the domains of definition of the parameter vectors ξ and xM thatdescribe the measurement configuration on ΣM and the reflector ΣR, respectively. In other words,equation (6.2.4) defines a one-to-one relationship between ξ and

xR = xR(ξ) . (6.2.5)

The important formula (6.2.3) possesses the form of the zero-order ray-theory solution of thereflection event that pertains to the source-receiver pair (S(ξ), G(ξ)). Let us now write representa-tion (6.2.3) in the familiar ray-theoretical form

U(ξ, t) = U0(ξ) F [t− TR(ξ)] , (6.2.6a)

where the amplitude factor is given by equation (3.13.11), viz.,

U0(ξ) = Rc(xR)A(ξ)

L(ξ). (6.2.6b)

In the above expression, TR(ξ) is the traveltime along the reflection ray SMΣG. Moreover, thefactor A accounts for the amplitude loss due to transmission at the overburden interfaces alongthe total ray path and Rc(xR) is the reciprocal (energy-normalized) reflection coefficient at theinterface ΣR.

As before, the (real or imaginary) quantity L(ξ) denotes the normalized reciprocalgeometrical-spreading factor of the total reflection ray SMΣG. Factors A and L are given by ex-pressions (3.13.7) and (3.6.15), respectively.

Equation (6.2.6a) is the final asymptotic result of the KHI. It states (in the high-frequencyapproximation) that the superposition of the contributions of all Huygens secondary sources origi-nating along the reflector, each of which would distribute its energy along a diffraction traveltimesurface t = TD(ξ,xM ), constructively interferes and results in the total elementary wavefield re-flected from ΣR. The seismic reflection event resulting from this constructive interference in the(ξ, t)-domain aligns itself along the reflection traveltime surface t = TR(ξ) (Figure 6.4).

6.2. ASYMPTOTIC EVALUATION OF THE KHI 169

Huygens secondary sourcesreflector

diffraction traveltime surfaces

reflection traveltime surface

dept

hz

ttim

e

ξx

inhomogeneous reflector overburden

Fig. 6.4. Physical interpretation of the Kirchhoff-Helmholtz integral (2-D sketch): Each Huygenssecondary source, when exploding after excitation, produces energy in a seismic section alongcertain diffraction-traveltime surfaces. Superposing at the receiver the responses of all Huygenssources distributed along the reflector results in the usual elementary seismic reflection responsewith energy distributed along the reflection traveltime surface.

170 CHAPTER 6. KIRCHHOFF-HELMHOLTZ THEORY

6.2.1 Geometrical-spreading decomposition

By the uniqueness of the ray solution, we may equate the right-hand sides of expressions (6.2.3)and (6.2.6a) to obtain the relationships (Goldin, 1991; Schleicher et al., 1993a)

TR(ξ) = TD(ξ,xR) (6.2.7a)

for the traveltime and

A(ξ)

L(ξ)=

(

v−M cosϑ−Mv+M cosϑ+

M

) 1

2

ΥKH(ξ) (6.2.7b)

for the amplitude of the wavefield at G. The factor in front of ΥKH(ξ) accounts for the differencebetween Rc and Rc.

Equation (6.2.7a) tells us the obvious fact that the total traveltime TR(ξ) along the reflectionray SMRG equals the sum of the traveltimes TD(ξ,xR) along the two ray segments, one from S(ξ)to MR and the other from G(ξ) to this reflection point. Equation (6.2.7b) on the other hand statesthat the amplitude of the elementary wave can be written (in the high-frequency approximation)as the ratio between the transmission loss A(ξ) and the geometrical-spreading factor L(ξ), a factthat does not seem to be very exciting.

Whereas equation (6.2.7a) contains nothing new indeed, equation (6.2.7b), however, can beused to derive an interesting relationship between the geometrical-spreading factors of the two raysegments SMR and GMR and the total ray SMRG. For that purpose, it is convenient to recallthe definition of the Fresnel geometrical-spreading factor (Tygel et al., 1994a) as introduced inChapter 4. We may write it as

LF (ξ) =OF

|det(H˜F (ξ)|1/2 exp

iπ

2κF

, (6.2.8)

where OF and κF are defined in equation (4.6.4) and (4.6.5), respectively. For simplicity, we alsointroduce the notations

G0(S,MΣ) = GS0 (ξ,xM ) =AS(ξ,xM )

LS(ξ,xM )(6.2.9a)

and

G0(G,MΣ) = GG0 (ξ,xM ) =AG(ξ,xM )

LG(ξ,xM ), (6.2.9b)

in which GS0 (ξ,xM ) represents the amplitude factor of the Green’s function for the ray segmentconnecting S(ξ) to MΣ. Analogously, AS(ξ,xM ) and LS(ξ,xM ) denote the amplitude loss (due toall transmissions across overburden interfaces) and the point-source geometrical-spreading factor forthis ray segment SMΣ, respectively. The quantities GG0 (ξ,xM ), AG(ξ,xM ), and LG(ξ,xM ) pertainto the ray segment GMΣ. For these quantities, corresponding equations to formulas (3.13.7) and(3.6.15) can be given.

Collecting equations (6.1.9), (6.2.1b), and (6.2.8), we can replace the factor ΥKH(ξ) in equa-tion (6.2.7b) by

ΥKH(ξ) = %Rv+R

2 OKH

OFGS0 (ξ,xR) GG0 (ξ,xR) LF (ξ) . (6.2.10)

We now return to equation (6.2.7b). We first note that

A(ξ) = %Rv−Rv

+R AS(ξ,xR) AG(ξ,xR) . (6.2.11)

6.2. ASYMPTOTIC EVALUATION OF THE KHI 171

This physically means that the total transmission loss in amplitude (due to crossing all overburdeninterfaces along the whole ray path) is the product of the transmission losses along the two raysegments. Formula (6.2.11) can be readily induced from the equations for AS and AG correspondingto expressions (3.13.7).

From Snell’s law, the obliquity factor OKH at the specular reflection point MR = MΣ(xR) isgiven by

OKH(ξ,xR) =cosϑ+

R

v+R

, (6.2.12)

where ϑ+R = ϑ+

M (xR) denotes the angle the reflected ray SMRG makes with the normal to thereflection surface ΣR at the reflection point MR and v+

R = v+M (xR) is the medium velocity at this

point just above the reflector encountered after reflection. Using equations (4.6.4) and (6.2.12), theremaining angle and velocity factors in equation (6.2.10) combine to

v+M

v−M

OKH

OF=

(

v−M cosϑ−Mv+M cosϑ+

M

) 1

2

. (6.2.13)

Together with equations (6.2.9), we thus obtain from the identity (6.2.7b) the importantgeometrical-spreading decomposition formula

L(ξ) =LS(ξ,xR) LG(ξ,xR)

LF (ξ). (6.2.14)

Note that this is the generalization of formula (4.6.7), which addresses only the moduli of thesefactors.

The decomposition formula (6.2.14) has been previously derived in different ways for acousticwaves (Goldin, 1991; Schleicher et al., 1993a; Tygel et al., 1994a). For elementary elastic waves, ithas recently been deduced by (Ursin and Tygel, 1997) for isotropic media and by Schleicher et al.(2001) for general anisotropic media. Note that an equation similar to formula (6.2.14), however,not involving factor LF but the area of the Fresnel zone at the reflection point, has been derived bySun (1995). Formula (6.2.14) explains why LF is called the “Fresnel geometrical-spreading factor.”It accounts for the influence of the Fresnel zone at MR, described by the Fresnel matrix H

˜F , on

the total geometrical spreading along the complete ray SMRG. Although it has not been explicitlystated there, formula (6.2.14) is fundamental for the theory of true-amplitude Kirchhoff prestackmigration as presented in Schleicher et al. (1993a), see also Chapter 7. How the above formulaenters into migration theory can be found in the next chapter.

Let us stress that all factors used in formula (6.2.14) have a modulus and a phase. Thus,formula (6.2.14) provides a decomposition not only for the modulus but also for the phase ofthe geometrical-spreading factor, which is determined by the number of caustics. In other words,knowing the number of caustics for the ray segment SMR (assuming a point source at S) and alsothat for the ray segment GMR (assuming a point source at G), the number of caustics can bedetermined for the total reflected ray SMRG including the influence of the reflector (assuming apoint source at either S or G). The derivation given here for a reflected wave can be done in acompletely parallel way for a wavefield transmitted through an interface (with the points S andG located on opposite sides of the interface). Therefore, equations (6.2.11) and (6.2.14) are alsovalid for the case of transmission. This fact has been used in Hubral et al. (1995) to derive a multi-segment decomposition for the geometrical-spreading factor and the number of caustics along anarbitrary ray into point-source contributions along each segment plus additional contributions fromthe Fresnel zones at the intersection points.

172 CHAPTER 6. KIRCHHOFF-HELMHOLTZ THEORY

6.3 Phase shift due to caustics

In accordance with the definition of L in equation (3.6.15), for the two spreading factors LS andLG we have the expressions

LS =

√√√√

cosϑS cosϑ−RvSv

−R |detN

˜SR|

exp[−iπ2κS ] (6.3.1a)

and

LG =

√√√√

cosϑ+R cosϑG

v+RvG |detN

˜RG|

exp[−iπ2κG] , (6.3.1b)

where κS and κG are the KMAH indices (number of caustics) along the two ray branches. TheN˜

-matrices are again mixed-derivative matrices of the traveltime with respect to the end points ofthe respective ray branch corresponding to equation (4.2.26c) for the total ray.

Inserting equations (6.3.1) into (6.2.14) we obtain for the phase of U(S,G, ω) the expression

argL(S,G, ω) = −π2[κS + κG + κF ] . (6.3.2)

This result is the phase of the primary reflection at G in the high-frequency approximation.As is generally known, the evaluation of the Kirchhoff integral (6.1.13) by the method of station-ary phase yields the ray-theoretical expression for the wavefield. Therefore, this phase has to beidentical to the phase of the analytic primary reflection as given by equation (3.13.15) the phase ofwhich can be inferred from equation (3.13.6c). By comparison, we immediately obtain the followingdecomposition theorem for the KMAH index

κ = κS + κG + κF (6.3.3)

with κF as defined in equation (4.6.5). Formula (6.3.3) was also derived by Goldin (1991) using themethod of discontinuities.

From this phase decomposition formula (6.3.3), we conclude that κF can be interpreted as thecontribution to the total number of ray caustics that is in particular due to the curved interface.Of course, as mentioned above for the ˆT

˜-matrix decomposition (4.3.5), also the above segment

quantities Li and κi (i = S,G) can be further decomposed for any number of elementary raysegments. How this can be used for an efficient computation of the geometrical-spreading factor ofthe total ray was investigated in Hubral et al. (1995).

6.4 Summary and conclusions

In this chapter, we have addressed a classical forward modeling problem for a reflector below alayered, laterally smoothly varying, inhomogeneous overburden. Using the ray-principal component,a scalar description of elementary elastic waves was introduced that allows to formulate a generalizedscalar Kirchhoff representation integral. This representation was shown to be approximately validfor elementary acoustic and elastic waves. The validity conditions for this approximation are thesame as for classical ray theory. Using the Kirchhoff-Helmholtz approximation in this integral,

6.4. SUMMARY AND CONCLUSIONS 173

a representation of the resulting Kirchhoff-Helmholtz integral (KHI) for arbitrary measurementconfigurations has been obtained.

In the present representation, the effects of the overburden and the reflector on the reflectedwave were separated. This separation may look artificial in the forward problem, but it will becomesignificant when studying the inversion of the KHI. By comparison of the high-frequency evaluationof the KHI to the zero-order ray-theoretical high-frequency representation of the reflected wavefield,a decomposition formula for the geometrical-spreading factor (including the number of ray caustics)was derived.

The results obtained in this chapter will obtain their full significance only in the next chapter,where we will address the inverse problem, i.e., the recovery of the reflector image and the reflectioncoefficients from the recorded scattered field. Though this latter problem is, of course, in principleas well-solved as the forward problem [it is in fact often based on the Generalized Radon Transform(Gubernatis et al., 1977a; Gubernatis et al., 1977b; Beylkin, 1985a), and can be either mathemati-cally (Bleistein, 1987; Miller et al., 1987) or geometrically motivated (Schleicher et al., 1993a)], therelationship between both forward and inverse scattering problems in a laterally inhomogeneousenvironment and for arbitrary measurement configuration (i.e., not only for shot records) has, inour opinion, not been as sufficiently elaborated in wave-theoretical terms as is done here. Thisapplies particularly to the situation when considering measurement configurations other than thatof a shot record. We think that this relationship is particularly well exposed once the connectionbetween the diffraction-stack integral (DSI) (that solves the migration/inversion problem) and theKHI (that solves the forward scattering problem) is established. In the next chapter, we will providethis connection. There we will briefly review the theory of the DSI along the lines applied in thischapter to the KHI. Thereafter, we will be ready to formulate both integrals as “physically inverse”to each other.

174 CHAPTER 6. KIRCHHOFF-HELMHOLTZ THEORY

Chapter 7

True-amplitude Kirchhoff migration

In this chapter, we address the Inverse Problem, i.e., the recovery of the reflector image and thedetermination of the reflection coefficients from the recorded reflected field. This will lead us to thetrue-amplitude diffraction-stack migration operation which is the first of the two building blocksfor the theory of the Unified Approach to Seismic Imaging that will be presented in Chapter 9.

Like the Forward Problem, also the Inverse Problem, generally called the migration/inversionproblem, has been extensively discussed in the literature (see, e.g., Schneider, 1978; Bleistein, 1987;Miller et al., 1987; Schleicher et al., 1993a). As seen below, its solution can be represented in theform of the Kirchhoff-migration integral, also known as the diffraction-stack integral (DSI). In thischapter, our main goal is to derive and discuss the DSI, especially with respect to true migrationamplitudes. As part of our discussion, we will also elaborate on the relationship between both theforward and inverse problems. This relationship is particularly well exposed once the connectionof the DSI to the Kirchhoff-Helmholtz integral (KHI), as discussed in Chapter 6, is established.By elaborating on this connection, we provide a physical meaning to the heuristic ansatz chosenfor the DSI in Schleicher et al. (1993a) and Tygel et al. (1994b). Our objective here is, in the firstplace, to show that the DSI and the KHI can be conceived of as physically “inverse” operationsto each other. Thereafter, we will understand the DSI as the first of two Kirchhoff-type operationsthat form the basis for a Unified Approach to the solution of a variety of seismic reflection imagingproblems (Hubral et al., 1996a; Tygel et al., 1996). This Unified Approach will be elaborated indetail in Chapter 9.

As stated above, a true-amplitude migration consists, besides the localization of the seismicreflectors at depth, of the removal of the geometrical-spreading factor from seismic primary reflec-tions. The procedure, like all migration methods, makes use of a macrovelocity model, but (andthis is common to all diffraction stacks) there is no need to identify the primary reflections withinthe unmigrated seismic data volume. Also, the methods needs no a-priori information about thereflectors to be migrated. The problem of recovering the geometrical-spreading factor of primaryreflections from identified traveltimes only (i.e., without the need for a macrovelocity model) wasstudied by Tygel et al. (1992). Here, we show how to design a time or depth migration algorithmand construct true-amplitude migrated reflections.

The present general imaging approach, which allows for arbitrary measurement configura-tions, is based on a weighted DSI (Schleicher et al., 1993a), applied to the recorded seismic traces ofthat configuration along diffraction-time surfaces (Huygens surfaces). These are constructed with

175

176 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

the help of a laterally inhomogeneous macrovelocity model. Based on asymptotic high-frequencyevaluations, one can show that the DSI can provide not only the reflector location (more precisely:the reflector image), but also undistorted source signals aligned along the reflector the amplitudes ofwhich are free of geometrical-spreading losses. Appropriate weight functions need only be specified.They depend on the macrovelocity model and on the measurement configuration.

Let us briefly outline the strategy of our approach with the help of Figure 7.1. We will assumethat the measurement surface ΣM (the (r1, r2)-plane in Figure 7.1) is densely covered with source-receiver pairs (S,G) according to a given measurement configuration. As described in Section 2.2,the location of the source-receiver pairs is specified by one common coordinate vector ξ = (ξ1, ξ2)and certain constant configuration matrices. Of the many interfaces in the subsurface, we considerone as the target reflector that is to be imaged by our migration procedure (hatched surface inFigure 7.1). In our analysis below, we treat this target reflector as if it was the only interfacein depth that caused reflection events in the data. In the general case of an arbitrary number ofreflectors, these are accounted for by a simple superposition of the corresponding migration results.

For a given measurement configuration of sources S(ξ) and receivers G(ξ), let us now considerthe primary reflection that starts at S(ξ), reflects at the point MR on the reflector and returnsto the measurement surface to be recorded at the receiver at G(ξ). Note that MR is completelydetermined by the locations of S(ξ) and G(ξ) by means of Fermat’s principle. In the following, weconsider the ray (Figure 7.1) from S to MR to G as a central ray and denote it as ray SMRG. Itpertains to a primary reflected, elementary elastic wave that crosses a certain number of interfaceson its way down to the reflector and, after being reflected, traverses another number of interfaceson its way back up to the earth’s surface. In the vicinity of this central ray SMRG, we considerparaxial rays SMRG of the same wave mode that pass through the same layers and interfaces. Allrays are assumed to be well-described by the zero-order ray theory as outlined in Chapters 3 and 4.

For source and receiver locations S(ξ) and G(ξ) varying on the measurement surface asspecified by the configuration parameter ξ within the data aperture A, the traveltime of the primaryreflection is a function of ξ and, thus, constitutes a traveltime surface ΓR: t = TR(ξ) within the datavolume. Let us stress once more that to carry out the Kirchhoff migration procedure, there is no needto identify the reflection-traveltime surface ΓR: t = TR(ξ) in the data volume. Instead, the procedurewill make use of auxiliary traveltime surfaces that will be constructed in the macrovelocity modelassumed to be a priori known. These auxiliary surfaces are the diffraction-traveltime or Huygenssurfaces ΓM : t = TD(ξ;M). Both traveltime surfaces ΓR and ΓM depend on the source-receiver pair(S,G) and are, therefore, functions of ξ as described in Chapter 4. General properties that relatethe diffraction and reflection traveltime surfaces have been derived and discussed in Chapter 5.These properties will be used to determine meaningful expressions for the true-amplitude weightfunction of the DSI.

As detailed in Chapter 2, the geometrically most appealing way to describe a Kirchhoffmigration is to think of it as an operation that verifies whether a certain point M in depth actedas a reflection point under the performed seismic experiment, i.e., whether there is an event inthe recorded seismic data that corresponds to a primary reflection at M . For this purpose, oneconsiders the subsurface region to be imaged as being represented by an ensemble of points on agrid, which is generally, but not necessarily, a regular one. For each depth point M where a depthimage is to be obtained, one has to construct the corresponding Huygens surface ΓM : t = TD(ξ;M)as described in equation (5.3.1). Then, the trace values that are found within the data volumealong ΓM are summed (or stacked). In a depth migration, the resulting stack value is assigned to

177

t

z

r

R

S

S

MR

2

Σ

r1

trace

GG

ξ1

ξ2

MR

Fig. 7.1. Earth model and rays for migration.

178 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

the chosen depth point M , in a time migration, it is assigned to the apex of the Huygens surface.

As we have seen in Chapter 5, the traveltime surfaces ΓM and ΓR are tangent if and onlyif M is an actual reflection point MR on ΣR (see Figure 7.2). Consequently one can expect thata diffraction stack performed with arbitrary weights along the Huygens surface will, due to con-structive interference of its contributions, provide a significant result when M = MR. Otherwisethe stack result will, due to destructive interference, be negligible. This well-known fact is provenhere once more by expressing the diffraction stack as an integral and evaluating it by means ofthe stationary-phase method. This finally leads to an appropriate weight for each point M as afunction of ξ, which removes the geometrical-spreading factor of the ray SMRG.

The explicit form of the weight function is found with the help of two decomposition theoremsderived in Chapters 4 and 6. By these, the geometrical-spreading factor for the ray SMRG isdecomposed into a contribution for the down- and upgoing ray segments SMR and MRG, as wellas a factor that accounts for the influence of the reflector at the reflection point. As a result of thestationary-phase analysis applied to the DSI, the latter factor is seen to be eliminated from themigrated amplitudes automatically, even by an unweighted Kirchhoff migration. Thus, the weightfunction needs to account only for the ray segment geometrical spreading, thus being independentof the reflector at MR. Consequently, the weight function can be computed for each source-receiverpair (S,G) and each subsurface point M , irrespective of whether or not M lies on a reflector. In thisway, all subsurface points M at which migration outputs are computed, are treated as “candidatereflection points” in the given macrovelocity model. The determination of the true-amplitude weightfunction is carefully examined in Section 7.2. As we will see there, the weight function for a givenimage point M uses only the traveltimes and dynamic quantities of the ray segments that join pointM to the set of source-receiver pairs (S,G). Thus, as mentioned before, all necessary quantities canbe computed using the given macrovelocity model without any knowledge of the target reflector.

7.1 True-amplitude migration theory

Before entering into the details of the general theory of true-amplitude Kirchhoff depth migration,let us summarize the basic assumptions upon which this theory will be based.

7.1.1 Underlying assumptions

The data to be migrated are assumed to pertain to a gather corresponding to a selected seismicconfiguration (Common-Shot, Common-Offset, etc.). We assume the configuration parameter vec-tor, ξ, of all source-receiver pairs involved in the gather to fall into a region A called the aperture ofthe seismic experiment. Within A, each value of ξ defines the position of the seismic trace recordedat G(ξ) with a source at S(ξ). The 3-D seismic (ξ, t)-data volume collected over A illuminates acertain subsurface region which is to be imaged by the migration process.

For our imaging purposes, each seismic trace in the selected gather is considered as a superpo-sition of primary reflections. All other coherent and incoherent signals are considered as noise andare, as such, no longer taken into account in this treatment. They may, of course, cause nonnegligi-ble contributions in practical migrated sections. In such cases, some kind of pre- or post-processingmight be necessary to eliminate these effects.

7.1. TRUE-AMPLITUDE MIGRATION THEORY 179

t

z

reflector

reflector

diffraction traveltime surface

diffraction traveltime surface

reflection traveltime surface

reflection traveltime surface

S( ) G( )

M

t

z

S( ) G( )

M

(a)

(b)

r

r

D

D

R

R

R

R

*

R

*

Σ

Σ

τ

τ

τ

τ

ξ ξ ξ ξ

ξ ξξ ξ

Fig. 7.2. Stationary situation in the migration integral (7.1.4) for a common-offset experiment. (a)The diffraction traveltime is tangent to the reflection traveltime at ξ∗. (b) Both traveltime surfaceshave the same inclination at ξ∗.

180 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

For simplicity and definiteness, we will assume that the seismic traces in the gather representthe principal components of the particle displacement of the primary reflection signals as recordedat a nonfree surface. How the effect of a free surface can be taken into account is discussed inSection 7.5 and described in detail in Appendix B.

The sources of the seismic experiment under consideration are assumed to be reproduciblecompressional point sources, i.e., the source wavelet f [t] they generate is the same for all source-receiver pairs. Since seismic data, particularly those recorded in a land survey, frequently do notshow this property, some kind of preprocessing might be necessary to equalize the source wavelets.Furthermore, we assume the wavelet f [t] to be a causal pulse of length Tε, i.e., f [t] vanishes outsidean interval 0 ≤ t ≤ Tε.

For one primary reflection event in the trace at ξ, let u(ξ, t) denote its principal componentas described in Chapter 3. It corresponds to a ray SMRG in Figure 7.1 defined by the compressionalpoint source at S(ξ) and the geophone at G(ξ). The principal-component reflection u(ξ, t) at G(ξ)describes the particle displacement in the direction of the emerging ray at G(ξ). Correspondingly,we denote the analytic particle displacement, i.e., the sum of the (real) reflection u(ξ, t) and its(imaginary) Hilbert transform (see Section 3.2.4) by U(ξ, t). It can be expressed in the zero-orderray approximation (Cerveny, 2001) as described in Chapter 3 [see equation (3.13.15)] as

U(ξ, t) = RcAL F [t− TR] . (7.1.1)

Here, F [t] represents the analytic point-source wavelet, i.e., it consists of the real source waveletf [t] and its Hilbert transform as described Section 3.2.4. The wavelet F [t] is not a function of ξbecause of our assumption of reproducible sources.

The function TR = TR(S(ξ), G(ξ)) in equation (7.1.1) provides the traveltime along theprimary reflection ray SMRG (Figure 7.1). As it will turn out, TR needs not be known to apply thetrue-amplitude migration. The other right-hand side factors of formula (7.1.1) (all of them functionsof ξ) can be identified as follows. Quantity L is the normalized geometrical-spreading factor1, Rc isthe reciprocal (or energy-flux normalized) plane-wave reflection coefficient at the reflection point2,and A is the total loss in amplitude due to transmissions across all interfaces along the ray3. All ofthese quantities have been discussed in detail in Chapter 3.

Since in this book our interest lies with the removal of geometrical-spreading effects frommigrated amplitudes, we refrain from studying the influence of the total transmission loss A on theAVO behavior. In this respect, it is to be noted that, for many realistic earth models, the factorA in equation (7.1.1) is a slowly varying quantity. In such cases, the only quantities determiningthe amplitude variations with offset (AVO) behavior of the primary reflection (7.1.1) are the angle-dependent reflection coefficient Rc and the geometrical-spreading factor L. Therefore, for simplicity,

1The normalized geometrical-spreading factor is defined in equation (3.13.6b). It is the same as the relativegeometrical-spreading factor of Cerveny (2001), except for a normalization with respect to velocity. Like the latter,the normalized geometrical-spreading factor is a reciprocal quantity, i.e., it does not alter is value when the positionsof source and receiver are interchanged. Its advantage over Cerveny’s relative geometrical-spreading factor lies inthe fact that the normalized geometrical-spreading factor reduces in a homogeneous medium simply to the distancebetween source and receiver and in a horizontally layered medium to Newman’s (1973) factor.

2For monotypic reflections, Rc equals the amplitude-normalized reflection coefficient Rc. For converted reflections,Rc can be determined from Rc by a simple correction [see equation (3.13.9)]. For expressions for Rc or Rc, the readeris referred to Aki and Richards (1980) or Cerveny (2001). Linearized expressions are collected in Appendix A.

3The transmission loss is defined in equation (3.13.7). Other influences on the amplitudes of seismic traces wouldalso enter into A. Factors that may affect seismic amplitudes are discussed in Sheriff (1975).

7.1. TRUE-AMPLITUDE MIGRATION THEORY 181

we will assume that A can either be neglected or corrected for4. Because of this assumption, weconsider the primary reflection (7.1.1) for the purpose of migration to be well approximated by

U(ξ, t) ≈ U0(ξ) F [t− TR] , (7.1.2a)

where the ray amplitude U = ARc/L is approximated by

U0(ξ) =RcL . (7.1.2b)

The objective of a true-amplitude depth migration is to move the primary reflection event(7.1.2a) to its reflection point MR on the reflector ΣR: z = ZR(r) while simultaneously removingthe geometrical-spreading factor L from its amplitude, equation (7.1.2b). The reflection coefficientRc, however, is to be conserved under this process. To formulate this objective mathematically, wedefine the analytic true-amplitude event in the depth domain as

ΦTA(r, z) = RcFmig[z −ZR(r)] . (7.1.3)

Here, Fmig[z] is the migrated pulse in depth, which ideally looks the same as the the seismic sourcepulse F [t] in time5. As we can see from equation (7.1.3), the desired migrated true-amplitude eventΦTA(r, z) does not differ in form from the source pulse, if the angle-dependent reflection coefficientRc at MR is real. If Rc is complex, ΦTA(r, z) carries its phase.

7.1.2 Diffraction stack

Here, we assume that a weighted modified diffraction stack is the appropriate method to performa true-amplitude migration. In the following, this assumption is proved by setting up a certaindiffraction-stack integral and deriving a weight function from it such that the stack output becomesthe true-amplitude event as defined in equation (7.1.3).

As detailed above, a diffraction stack is then a weighted summation along the Huygens surfaceTD with respect to each pointM . This summation can be mathematically expressed by the followingintegral (Schleicher et al., 1993a)

Φ(M) =−1

2π

∫

A

∫

d2ξ KDS(ξ;M)∂U(ξ, t)

∂tt = TD(ξ;M)

. (7.1.4)

The name diffraction stack is employed to stress the role played by the diffraction traveltime surfacesTD along which the summation is performed. Here, KDS(ξ;M) denotes the weight function orintegral kernel that is yet to be found. The time derivative of the seismic data, ∂U/∂t, is needed in

4If one had, additionally to a reasonably accurate macrovelocity model, also a reliable density model (which inpractice can hardly ever be obtained), it would in principle be possible to determine the factor A using equation(3.13.7). In this case, the effect of the factor A on the primary reflection (7.1.1) could be removed. Note that even ina homogeneous medium A is not just equal to one but a constant factor that depends on the medium velocity anddensity.

5It is to be observed that this placement of the signal in depth rather than time will give rise to a distortionof the original wavelet F [t]. This distortion will be quantitatively discussed in Section 8.2. As we will see there,Fmig[z − ZR(r)] = F

[mD(z −ZR(r))

], where mD is the factor that has already been discussed in part (c) of the

first duality theorem in Chapter 5.

182 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

order to correctly recover the source pulse as will become evident from the result obtained. Integral(7.1.4) can thus be understood as a “time-differentiated, space-weighted Kirchhoff migration.”

The region of integration A should, in the absence of noise, ideally be the total (ξ1, ξ2)-plane.This is, of course, impossible due to the limitation of the aperture of the seismic experiment. In thepresence of noise or taking aliasing into account, one should confine A to an even smaller integrationregion (migration aperture) as is well known. The ideal size of the aperture A will be discussed inmore detail in Section 8.1.

Note that integral (7.1.4) is justified by nothing else than the fact that it describes thediffraction stack and it will solve our problem. There is no need to look for a deeper physicalmeaning of the integral. However, as we will see below, it can be interpreted as a physical inverseto the Kirchhoff-Helmholtz forward modeling integral.

For the following considerations, we introduce an artificial time variable t that may vary. Inother words, we consider the time-dependent stack

Φb(M, t) =−1

2π

∫

A

∫

d2ξ KDS(ξ;M)∂U

∂t(ξ, t+ TD) (7.1.5)

for arbitrary values of t.

Geometrically, the introduction of the artificial time variable in integral (7.1.5) amounts tonothing more than to consider a continuous set of stacks that are carried out along stacking surfacesthat are parallel to the Huygens surface ΓM of M , shifted by an amount t. In other words, for eachpoint M we consider a time band within which parallel stacks are performed. In this way, we cantransform the above integral into the frequency domain for a subsequent stationary-phase analysis.Of course, the actual migration result Φ(M) is obtained from the stack along ΓM with no shift, i.e.,the value Φb(M, t = 0) is the diffraction-stack migration output Φ(M) for the chosen depth pointM . In other words, we may say that the imaging condition for this time-dependent migration ist = 0. The other values of t are only introduced for the mathematical treatment. The correspondingstacks along the shifted Huygens surfaces need not be carried out in practice. The only stack to beactually carried out is the one for t = 0, described by integral (7.1.4).

The target reflector ΣR (unknown in its position) is supposed to be fixed throughout the wholechapter. We suppose that the coordinates of points MΣ on that part of ΣR that is illuminated bythe considered elementary primary wave can be parametrized in the image space in the simple form(r, z = ZR(r)), where r varies on a certain aperture in the horizontal r-plane. This parameterizationdoes not allow for surfaces ΣR that doubly project onto the horizontal r-plane. This difficulty maybe circumvented by a more flexible parameterization of the reflector, which will not be done herefor simplicity.

Integral (7.1.5) yields substantially different values according to whether or not point M isan actual reflection point. Provided a suitable weight function KDS(ξ;M) is determined, the stackin equation (7.1.5) removes the factor L from primary reflections. The real part of the resulting

value is taken at t = 0 and Re

Φb(M, 0)

is then placed at point M . However, since we consider

the migration result an input to further imaging operation like demigration or remigration, whichare to be applied to it later on, we will omit the real-part operation and continue to work with thefull complex result Φ(M) = Φb(M, 0).

7.1. TRUE-AMPLITUDE MIGRATION THEORY 183

To justify the above claims, we substitute the expression for U(ξ, t) given by equation (7.1.2)into integral (7.1.5) and find

Φb(M, t) =−1

2π

∫

A

∫

d2ξ KDS(ξ;M) Rc1

L F [t+ T∆(ξ;M)] , (7.1.6)

where the dot denotes the time derivative, i.e., F = dF/dt. Moreover, we have used the notation

T∆(ξ;M) = TD(ξ;M) − TR(ξ) . (7.1.7)

Next we transform expression (7.1.6) into the frequency domain

Φb(M,ω) = F [ω]−iω2π

∫

A

∫

dξ1 dξ2 KDS(ξ;M) Rc1

L exp[iω T∆(ξ;M)], (7.1.8)

where F [ω] and Φb(M,ω) denote the Fourier transforms of F [t] and Φb(M, t), respectively.

Integral (7.1.8) cannot be solved analytically. We can, however, evaluate it approximately forhigh frequencies using the method of stationary phase. The restriction to high frequencies is in factalready implicitly done, because we are describing wave propagation by the ray method.

7.1.3 Evaluation at a stationary point

We now have to distinguish between two cases, according to whether or not a point ξ∗ = (ξ∗1 , ξ∗2)

exists within the aperture A where the gradient of T∆ vanishes, i.e., where the slopes of TD and TRcoincide. Let us at first consider the case where such a stationary point exists within A. Note thatthis is the case in both situations featured in Figure 7.2. In other words, T∆ satisfies the followingstationarity condition

∇ξT∆(ξ;M)

ξ = ξ∗= 0 . (7.1.9)

This equation defines ξ∗ as a function of the coordinates r of M , i.e., ξ∗ = ξ∗(r). Applying thestationary phase method to integral (7.1.8) means expanding the phase function T∆ of that integralinto a Taylor series up to second order with respect to the stationary point ξ∗, which yields

T∆(ξ;M) = T∆(ξ∗;M) +1

2(ξ − ξ∗) ·H

˜∆ (ξ − ξ∗) . (7.1.10)

Here, H˜

∆ is the Hessian matrix, i.e., the 2× 2-matrix of the second derivatives of T∆ evaluated atξ = ξ∗. In other words, H

˜∆ = H

˜D −H

˜R as defined in equation (5.6.10a).

Assuming that H˜

∆ is nonsingular [i.e., det(H˜

∆) 6= 0], we find in the high-frequency approx-imation (ω 1) and using the 2-D method of stationary phase (see, e.g., Bleistein, 1984)

Φb(M,ω) ≈ F [ω] KDS(ξ∗;M)Rc

L√

|detH˜

∆| ×

× exp

[

iωT∆(ξ∗;M) − iπ

2(1 − Sgn(H

˜∆)/2)

]

. (7.1.11)

Here, the ‘Sgn’-function is the so-called signature as defined in equation (4.6.6). In this work, weshall not consider stationary points where the Hessian matrix H

˜∆ vanishes.

184 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

7.1.4 Evaluation elsewhere

Equation (7.1.11) represents the (high-frequency) evaluation of integral (7.1.8), if a stationary pointξ∗ (where the gradient of the phase function vanishes) falls into the migration aperture A. If this isnot the case, the main contributions to integral (7.1.6) come from the boundaries of the aperture.These contributions, however, are of order ω−1. They are generally suppressed by tapering U(ξ, t) inthe vicinity of the migration aperture border. In other words, if no stationary point exists within A,the diffraction stack will produce a negligible value. If the stationary point falls within one Fresnelzone from the boundary, the amplitude will decrease against equation (7.1.11). At the boundaryitself, the result will be exactly one half of the complete value. The value will decrease to zero asthe stationary point moves outside the Fresnel zone at the boundary. This produces the well-knownboundary effects of Kirchhoff migration (Sun, 1999).

7.1.5 Evaluation result

We see from equation (7.1.11) that the stack (7.1.5) yields the amplitude of the primary wavereflected at M = MR (Figure 7.1), multiplied by a factor that includes the (up to now arbitrary)weight function at the stationary point. If we select KDS(ξ;M) in formula (7.1.5) such that at thestationary point ξ∗

KDS(ξ∗;M) = L√

|detH˜

∆| exp

[iπ

2(1 − Sgn (H

˜∆)/2)

]

, (7.1.12)

then the approximation (7.1.11) reduces to

Φb(M,ω) ≈

Rc F [ω] exp [iωT∆(ξ∗;M)] for ξ∗ ∈ A ,0 else .

(7.1.13)

Let us further study this result in the situation where a stationary point ξ∗ is present within A. Thecorresponding part of equation (7.1.13) represents the spectrum of the source wavelet multipliedwith the reflection coefficient and a phase shift factor that accounts for the difference betweenthe reflection and diffraction traveltime surfaces at the stationary point. Going back to the timedomain, we find

Φb(M, t) = Rc F [t+ T∆(ξ∗;M)] , (7.1.14a)

and thus, using the imaging condition, t = 0, the final migration result at M is

Φ(M) = Φb(M, 0) = Rc F [T∆(ξ∗;M)] , (7.1.14b)

if a stationary point ξ∗ where condition (7.1.9) is satisfied is present in the migration aperture A.

To appreciate the significance of result (7.1.14b), we make use of the fact that the sourcesignal is a function of finite duration, i.e., F [t] vanishes outside an interval 0 ≤ t ≤ Tε. Now let Mbe an actual reflection point, i.e., M = MR. In this case, the multitude of rays from all S(ξ) toMR to all G(ξ) defining TD(ξ;MR) contains the stationary ray from S(ξ∗) to MR to G(ξ∗). Thismeans that T∆(ξ∗;M) vanishes, i.e., both the reflection time and Huygens surface are tangent atξ∗. Therefore, the true-amplitude diffraction stack (7.1.5) provides the value RcF [0]. Now let M bea point dislocated in the vertical direction from the reflection point MR. At points M close to thereflector, this corresponds to a small traveltime difference t = T∆(ξ∗;M) inside interval 0 ≤ t ≤ Tε.

7.2. TRUE-AMPLITUDE WEIGHT FUNCTION 185

The value RcF [t] is then provided by the true-amplitude migration. Finally as point M movesfurther away from the reflector, the result of the true-amplitude stack practically vanishes.

By recalling the definition of the true-amplitude event given by equation (7.1.3) we observethat the described situation can be expressed in the following way:

Φ(M) ≈

ΦTA(r, z) at points M = (r, z) for which 0 ≤ T∆(ξ∗;M) ≤ Tε .0 else .

(7.1.15)

Here, we have used that, according to equation (7.1.14b), the wavelet after migration will be givenby Fmig[z −ZR(r)] = F [T∆(ξ∗;M)].

We finally emphasize once more that a true-amplitude migration performed in this way notonly provides a measure for the angle-dependent reflection coefficients Rc, but also correctly recoversthe source pulse F [t]. Note, however, that the depth-migrated signal is not displayed as a functionof time t but of depth z. This leads to a pulse distortion that will be discussed in Section 8.2.

7.2 True-amplitude weight function

Equation (7.1.12) describes the weight function of a point M , independent of whether or not itis an actual reflection point MR. It looks quite complicated and seems to imply that L must becalculated along the ray SMG and that TR must be known and TD must be calculated, too, inorder to determine H

˜∆. As we will show below, this is, however, not true.

To see this, let us further investigate the quantities L and H˜

∆ in equation (7.1.12). For thatpurpose, let us, for the time being, consider M to be an actual reflection point, i.e., M = MR.Using ray theory as presented in Chapters 3 and 4, we can then find alternative expressions for theunknown quantities appearing in the weight function (7.1.12) in order to derive a more suitableexpression.

7.2.1 Traveltime functions

As a first step, we need suitable expressions for the traveltime functions involved in equation (7.1.7).For that purpose, we consider rays paraxial to (i.e., in the vicinity of) ray SMRG (Figure 7.1).As we have shown in Chapter 4, we can write a second-order Taylor polynomial (4.2.27) for thetraveltime from any point S in the vicinity of S to another point G in the vicinity of G.

We insert the configuration equations (2.2.13) into equation (4.2.27) to obtain the traveltimeTR = TR(S,G) of the primary reflection following the path SMRG in Figure 7.1. It is given byequation (4.2.32), where T0 = TR(S,G) represents the traveltime of the central ray SMRG.

The Huygens traveltime is defined in equation (5.3.1). For the specular reflection point MR,it is obtained by adding the ray-segment traveltimes (4.3.18a) and (4.3.18b) for xM = 0, i.e.,

T (S,MR) = T (S,MR) − p0 · Γ˜Sξ +

1

2ξ · Γ

˜TSN

˜RSΓ˜Sξ (7.2.1a)

and

T (MR, G) = T (MR, G) + p′0 · Γ˜Gξ +

1

2ξ · Γ

˜TGN

˜RGΓ˜Gξ . (7.2.1b)

186 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

Note that, from Fermat’s principle, the slowness vector projections p0 and p′0 in equations (7.2.1)are identical to pS and pG in equation (4.2.32). This is because the rays SMR and MRG are raysegments of the complete ray SMRG. For the same reason, we also have

TD(S,G,MR) = T (S,MR) + T (MR, G) = TR(S,G) . (7.2.2)

The Huygens traveltime TD(S,G,MR) is thus obtained by adding equations (7.2.1a) and (7.2.1b).We find

TD(S,G,MR) = TD(S,G,MR) − pS · Γ˜Sξ + pG · Γ

˜Gξ

+1

2ξ ·[

Γ˜TSN

˜RSΓ˜S + Γ

˜TGN

˜RGΓ˜G

]

ξ. (7.2.3)

The second traveltime in equation (7.1.7) is the reflection traveltime TR that was already expressedin terms of these quantities in equation (4.2.32).

7.2.2 Traveltime difference and Hessian matrix

The function T∆(ξ;M) = T∆(S,G,MR) is the difference between the Huygens and reflection trav-eltime functions [equation (7.1.7)]. From equations (4.2.32) and (7.2.3) we have

T∆(S,G,MR) = TD(S,G,MR) − TR(S,G)

=1

2ξ ·[

2Γ˜TSN

˜SGΓ

˜G − Γ

˜TSN

˜GSΓ˜S − Γ

˜TGN

˜SGΓ˜G

+Γ˜TSN

˜RSΓ˜S + Γ

˜TGN

˜RGΓ˜G

]

ξ . (7.2.4)

We are interested in H˜

∆, the Hessian matrix of T∆(S,G,MR). Recalling that the Hessian matrixof a quadratic form TM (ξ) = 1

2ξ ·M˜ξ is simply H

˜M = 1

2(M˜

+M˜T ) and that Γ

˜TSN

˜SGΓ

˜G is the

only nonsymmetric matrix in equation (7.2.4), we find for H˜

∆ the formula

H˜

∆ = Γ˜TSN

˜SGΓ

˜G + Γ

˜TGN

˜TSGΓ

˜S − Γ

˜TSN

˜GSΓ˜S − Γ

˜TGN

˜SGΓ˜G

+ Γ˜TSN

˜RSΓ˜S + Γ

˜TGN

˜RGΓ˜G. (7.2.5)

To simplify the above complicated expression for H˜

∆ we will use some properties of the N˜

-matrices: The matrices N

˜SG, N

˜GS , N

˜SR, N

˜RS , N

˜GR, and N

˜RG are symmetrical, whereas the matrices

N˜TSG = N

˜GS, N

˜TSR = N

˜RS , and N

˜TRG = N

˜GR are not. Moreover we have three relationships be-

tween these matrices which are proven in Chapter 4. In terms of the N˜

-matrices, equations (4.3.7a)and (4.3.7b) read

N˜RS − N

˜GS = N

˜SGN

˜−1RGN

˜RS , N

˜RG − N

˜SG = N

˜GSN

˜−1RSN

˜RG , (7.2.6)

and the B˜

-matrix decomposition (4.6.1) translates to

N˜SG = N

˜SR

(

N˜SR +N

˜GR

)−1N˜RG . (7.2.7)

Inserting equations (7.2.6) and (7.2.7) into (7.2.5) we obtain, after some tedious but elementarymatrix algebra, the following decomposition formula

H˜

∆ =(

Γ˜TSN

˜SR + Γ

˜TGN

˜GR

) (

N˜SR +N

˜GR

)−1 (

Γ˜TSN

˜SR + Γ

˜TGN

˜GR

)T, (7.2.8)

7.2. TRUE-AMPLITUDE WEIGHT FUNCTION 187

which is just the first statement (5.4.2a) of the second duality theorem proven in Chapter 5, nowstated in local coordinates and using the N

˜-matrices. From equation (7.2.8) result the following

two crucial expressions containing H˜

∆

√

|detH˜

∆| =

∣∣∣det(Γ

˜TSN

˜SR + Γ

˜TGN

˜GR)

∣∣∣

√∣∣det(N

˜SR +N

˜GR)∣∣

(7.2.9a)

and

Sgn(H˜

∆) = Sgn(N˜SR +N

˜GR) . (7.2.9b)

Relying on the relationships (4.2.26), we can identify the sums of matrices in equations (7.2.9)with previously defined quantities. In the numerator of equation (7.2.9a), we recognize the matrixΛ˜

as defined in equation (4.5.15), and in the denominator we recognize the Fresnel matrix H˜F as

defined in equation (4.5.3). Therefore, we may also write

√

|detH˜

∆| =|detΛ

˜|

√

|detH˜F |

(7.2.10a)

and

Sgn(H˜

∆) = Sgn(H˜F ). (7.2.10b)

This relates the Hessian matrixH˜

∆ of the traveltime difference to the Fresnel matrixH˜F . Moreover,

a comparison of equation (7.2.8) to expression (4.5.17) reveals that H˜

∆ = H˜P , i.e., the Hessian

matrix of the traveltime difference T∆ is equal to the projected Fresnel zone matrix.

7.2.3 Geometrical-spreading factor

Now let us pay attention to the (normalized) geometrical-spreading factor L in formula (7.1.12).We will express it in terms of the above matrices. For that purpose, the mixed-derivative matrixN˜SG of the traveltime is very important, as it provides the absolute value of L at G due to a point

source at S by means of a formula originally derived by Goldin [1986, equation (14.25); 1987, p.118] for a 2-D constant-velocity layered model, viz.

|L| =

√

cosϑS cosϑGvSvG

1√

|detN˜SG|

. (7.2.11)

Here, ϑS and ϑG denote the (acute) angles the central ray makes with the normals to the measuringsurface z = 0 at S and G, respectively. Formula (7.2.11) is proven in Hubral et al. (1992a) to bevalid also for 3-D inhomogeneous layered media (see also Chapter 3).

The complete expression for the geometrical-spreading factor L has to consider the numberof caustics traversed by the primary wave along the central ray connecting S to G. This number isknown as the KMAH index and is denoted by κ (Chapman and Drummond, 1982). Then for thecomplete geometrical-spreading factor we have [see also equation (3.13.6c)]

L =

√

cosϑS cosϑGvSvG

1√

|detN˜SG|

exp[−iπ2κ] . (7.2.12)

188 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

Together with the above decomposition for N˜SG [equation (7.2.7)] and the following formula for κ

derived in Chapter 6, i.e., [equation (6.3.3)],

κ = κS + κG + (1 − Sgn(H˜F )/2) , (7.2.13)

we have finally

L =

√

cosϑS cosϑGvSvG

√

|detH˜F |

√

|detN˜SR|√

|detN˜GR|

×

× exp[−iπ2(κS + κG + (1 − Sgn(H

˜F )/2))] . (7.2.14)

Here, κS and κG are the KMAH indices of the two ray segments SMR (MRS) and GMR (MRG),assuming a point source at S (MR) orG (MR), respectively. In terms of the ray segment geometrical-spreading factors LS and LG as well as the Fresnel geometrical-spreading factor introduced inChapter 4, we may write

L =LSLGLF

, (7.2.15)

which has been independently inferred from the backward perspective in Chapter 6.

7.2.4 Final weight function

By inserting equations (7.2.9a) and (7.2.9b) together with formula (7.2.14) into equation (7.1.12) wefind the following expression for the weight function of a true-amplitude diffraction-stack migration

KDS(ξ∗;MR) =

√

cosϑS cosϑGvSvG

∣∣∣det(Γ

˜TSN

˜SR + Γ

˜TGN

˜GR)

∣∣∣

√

|detN˜SR|√

|detN˜GR|

exp[−iπ2(κS + κG)]. (7.2.16)

Before clarifying all quantities in equation (7.2.16) let us briefly review the assumptions that havebeen explicitly or implicitly made during the derivation of equation (7.2.16):

(a) The sources are reproducible, i.e., they produce the same source pulse and radiation patternat all shot locations.

(b) All receivers possess the same characteristics and transfer functions.

(c) The sources and receivers fall on a non-free surface. All sources, receivers and reflection pointslie on smoothly curved surfaces.

(d) The wave propagation is described by zero-order ray theory.

(e) The ray amplitude varies slowly with offset.

(f) The weight function does not vanish at the stationary point.

(g) All amplitude effects other than geometrical spreading have been corrected for independently.

If one of these conditions is not fulfilled, the above expression for the weight function is not strictlyvalid. However, in practice, violations of these conditions frequently occur only at isolated pointsthat do not influence the overall performance of the method.

Now we are ready to explain all quantities that appear in equation (7.2.16). They all pertainto the reflection ray SMRG [i.e., to the vector ξ∗ = (ξ∗1 , ξ

∗2)T ]. We have (see Figure 7.3a):

7.2. TRUE-AMPLITUDE WEIGHT FUNCTION 189

MR

S G

Ω

n

ϑ ϑS

G

GSx xS G

ϑ ϑxM

M

M

z(b)

x

S G

RΣreflector

RΩR

n

ϑ ϑS

G

z

r

RM

GS

R

x xS G

ϑ ϑR R+ -

(a)

r

^

^M

M M+ -

Fig. 7.3. (a) The normal vector nR to plane ΩR (which is tangent to the reflector ΣR at MR)points into the direction of the slowness vector sum of the two ray segments SMR and GMR. (b)Same situation as in (a) for an arbitrary point M in depth (not necessarily on a reflector). PlaneΩM is constructed perpendicular to nM that points into the direction of the slowness vector sumof the two ray segments SM and GM .

(a) Angles ϑS and ϑG are the starting and emergence angle of the central ray SMRG.

(b) Parameters vS and vG are the P-wave velocities at S and G, respectively.

(c) Γ˜S and Γ

˜G denote certain constant 2× 2 matrices (see Section 2.2) describing the source and

geophone locations with respect to the position vector ξ. They depend on the measurementconfiguration.

(d) N˜SR andN

˜GR are second-order mixed-derivative Hessian matrices of traveltimes constructed

as follows. We consider the tangent planes to the measurement surface at S and G. For ahorizontal measurement surface, both these tangent planes coincide with the plane z = 0(see also Figure 7.3a). Within each of these tangent planes, we consider an arbitrary 2-DCartesian coordinate system. The first one is centered at S. A point S is then described inthis system by the coordinate vector xS = (xS1, xS2)

T . The second system is centered at G. Apoint G is then described by the coordinate vector xG = (xG1, xG2)

T . Finally, we consider athird 2-D Cartesian system, this one on the plane ΩR tangent to the reflector at the reflectionpoint MR and centered at this point. A point MR on this plane has the coordinate vector

190 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

xR = (xR1, xR2)T . The Hessian matrices N

˜SR and N

˜GR are then given by

N˜SR =

(∂2T (S,MR)

∂xSj∂xRk

)∣∣∣∣x

S=x

R=0

(7.2.17a)

and

N˜GR =

(∂2T (MR, G)

∂xGj∂xRk

)∣∣∣∣x

G=x

R=0

. (7.2.17b)

Thus, N˜SR and N

˜GR are the second-order traveltime mixed-derivative matrices that express

the cross variations of T (S,MR) with respect to (xS1, xS2) and (xR1, xR2) and of T (MR, G)with respect to (xG1, xG2) and (xR1, xR2), respectively. All these derivatives are evaluated atthe origins of the respective coordinate systems. One could say that detN

˜SR and detN

˜GR

are the ray Jacobians of the ray segments SMR and GMR, transformed into the coordinateplanes z = 0 and ΩM . We note that the plane ΩR at MR is completely defined by the two raysegments SMR and GMR at MR. Indeed, the direction of the normal nR to plane ΩR at MR

is given by the sum of the slowness vectors of these two ray segments at MR. For a monotypicreflection, nR bisects the angle between SMR and GMR. We have seen in Chapter 3 how thematrices N

˜SR and N

˜GR relate to the dynamic-ray-tracing matrices.

(e) Quantity κS is the number of caustics (KMAH index) along ray segment SMR assuming apoint source at S or also along the reverse ray segment MRS with a point source at MR.Also, κG is the number of caustics along the ray segment GMR (resp. MRG) assuming apoint source at G (resp. MR). The caustics can also be determined by dynamic ray tracing(Cerveny, 1985, 2001; Cerveny and Castro, 1993). We remark that the actual values of κicannot be determined from traveltime derivatives.

It is important to note that the quantities appearing in equation (7.2.16) only depend on thetwo individual ray segments SMR and MRG and not on any reflector properties. In other words,they can be computed for any arbitrary composite ray SMRG irrespective of whether or not thereis a reflector at M . This has an important consequence. It means that formula (7.2.16) allows fora generalization to all points S and G specified by any coordinate pair (ξ1, ξ2) and any point M inthe macrovelocity model. We simply have to replace MR by M in formula (7.2.16). We find

KDS(ξ;M) =

√

cosϑS cosϑGvSvG

∣∣∣det(Γ

˜TSN

˜SM + Γ

˜TGN

˜GM )

∣∣∣

√

|detN˜SM |

√

|detN˜GM | exp[−iπ

2(κS + κG)] . (7.2.18)

This final weight is now employed in the stack (7.1.5). The matrices N˜SM and N

˜GM are computed

correspondingly to N˜SR and N

˜GR as defined in equations (7.2.17). The 2-D coordinate vector xR

is to be replaced by xM defined at M in the plane ΩM , the normal of which points into the directionof the sum of the slowness vectors of the ray segments SM and GM . This more general situationis depicted in Figure 7.3b.

The plane ΩM is constructed as follows (see Figure 7.3b). Assuming the source-receiver pair(S(ξ), G(ξ)) fixed, we trace from M the rays MS and MG and consider the vector

nM = ∇M [T (S,M) + T (G,M)] = ∇MTD, (7.2.19)

where ∇M denotes the 3-D vector gradient with respect to dislocation of M . In case of a monotypicreflection (e.g., a pure P-wave reflection) at M , this vector bisects the angle between the slownessvectors of the two rays SM and GM at M , because then

|∇MT (S,M)| = |∇MT (G,M)| = 1/v(M) . (7.2.20)

7.2. TRUE-AMPLITUDE WEIGHT FUNCTION 191

In other words, the vector nM is the interface normal to a (real or hypothetical) interface at whicha specular reflection may take place that follows the ray SMRG.

We now construct the plane ΩM through point M normal to nM (see Figure 7.3). On thatplane, we define an arbitrary 2-D-Cartesian coordinate system xM = (xM 1, xM 2). Using this plane,we can than calculate all quantities in the weight function KDS(ξ;M) as given by equation (7.2.18).

Let us elaborate in slightly more detail on the actual computation of the weight factor (7.2.18).It is important to note that the Hessian matrices N

˜SM and N

˜MG that define the modulus of the

weight function may or may not be computed by dynamic ray tracing. On the one hand, they relateto the dynamic-ray-tracing matrices P

˜andQ

˜, as has been discussed in Chapter 4. The computation

of P˜

and Q˜

can be done by performing dynamic ray tracing (Cerveny, 1985; Cerveny and Castro,

1993; Cerveny, 2001) with respect to the two independent rays SM and MG. On the other hand,being second derivatives of traveltimes, the matrices N

˜SM and N

˜MG can be alternatively obtained

using the (known) traveltimes of the rays in the paraxial vicinity of the central ray segments SMand MG. The necessary traveltimes are obtained by perturbations of the ray segment end points Sand M , and M and G along the tangent planes of the respective surfaces. These perturbations ofthe end points are easily done due to the fact that, when actually performing the diffraction stack,traveltimes (not necessarily computed by rays) have to be calculated from all points on a 2-D gridin the (r1,r2)-plane (earth’s surface) to all points on a 3-D grid in the (r1,r2,z)-domain (subsurface)for which the migrated image is to be constructed. Therefore, the modulus of the weight can beobtained without any need to consider dynamic ray tracing. A fast method to calculate the weightfunction from traveltimes on a coarse grid has been recently presented by Vanelle and Gajewski(2002).

The phase of the weight function in equation (7.2.18) depends on the number of causticsalong the ray segments. These cannot be determined from traveltime derivatives but must be foundby ray tracing. For this reason, we conclude that Kirchhoff migration needs dynamic ray tracing tocorrectly recover the phase of the migrated pulse.

7.2.5 Alternative expressions for the weight function

There are several other useful representations of the true-amplitude weight function KDS(ξ;M).Recalling the definitions of matrix Λ

˜in equations (4.5.15) and (5.3.20c), i.e.,

det(Λ˜

) = det(

Γ˜TSN

˜SM + Γ

˜TGN

˜GM

)

= det

(

∂2TD∂ξj ∂xMk

)

, (7.2.21)

its relationship to Λ˜

(r) from equation (5.3.20b), as well as the matrix definitions (4.3.20), we canimmediately rewrite formula (7.2.18) as

KDS(ξ;M) =

√

cosϑS cosϑGvSvG

∣∣∣det(Γ

˜TSN

˜

(r)SM + Γ

˜TGN

˜

(r)GM )

∣∣∣

√∣∣∣detN

˜

(r)SM

∣∣∣

√∣∣∣detN

˜

(r)GM

∣∣∣

exp[−iπ2(κS + κG)] . (7.2.22)

From equation (7.2.22), we recognize that the above described construction of plane ΩM won’t

be needed in practice since the derivative matrices N˜

(r)SM and N

˜

(r)GM with respect to the global

coordinates of M can be used instead of the corresponding derivative matrices N˜SM and N

˜GM

with respect to local coordinates.

192 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

By further recalling the definitions of the Green’s function amplitudes GS0 and GG0 that aregiven by equations (6.2.9), we may also specify the kernel KDS(ξ;M) as

KDS(ξ;M) =|det(Λ

˜)|ODS(ξ,M)

GS0 (ξ,M)GG0 (ξ,M). (7.2.23)

Here, we have introduced the “diffraction-stack obliquity factor” ODS , given by

ODS(ξ,M) =

√√√√

v−Mcosϑ−M

v+M

cosϑ+M

=vM

cosϑM, (7.2.24)

where the rightmost expression holds for a monotypic reflection. As before, v−M and v+M are the

velocities of the medium at point M encountered by the ray segments SM and GM , respectively.In the same way, ϑ−M and ϑ+

M equal the “hypothetical reflection angles,” i.e., the angles betweenthe slowness vectors of the ray segments SM and GM and the normal vector of a “hypotheticalreflector” at M . For a monotypic (P-P or S-S) reflection, v−M = v+

M = vM and ϑ−M = ϑ+M = ϑM .

Another useful expression for the true-amplitude weight function in terms of the geometrical-spreading factors of the ray segments from S to M and from M to G reads

KDS(ξ;M) = ODS LSLG |det(Λ˜

)| , (7.2.25)

which, in view of the relationship (5.6.16b) between Λ˜

and the Beylkin determinant hB , can berewritten as

KDS(ξ;M) =ODS

ODhB LSLG . (7.2.26)

Here, OD is the depth obliquity factor given by equations (5.6.9). As before, LS and LG representthe point-source geometrical-spreading factors of the ray segments SM and MG, respectively.Finally, Λ

˜is the second-order mixed-derivative matrix of the total traveltime TΣ(ξ, r) along the

composite ray SMRG with respect to the coordinates ξ of S and G and xM of M for points Mconfined to the plane ΩM [see equation (5.3.20c)]. This plane is tangent to the reflector in case Mcoincides with the actual reflection point MR.

7.3 True-amplitude migration result

In the time domain and upon the use of the imaging condition t = 0, the asymptotic evaluation(7.1.11) of the DSI, in the form given by equation (7.1.8), evaluated at a point M with horizontalcoordinate r close to the reflector ΣR, reads

Φ(M) ≈ ΥDS(ξ∗) U0(ξ∗) F [T∆(ξ∗;M)] , (7.3.1a)

where ξ∗ = ξ∗(r) and where the amplitude factor reads

ΥDS(ξ∗) =KDS(ξ∗;M) exp

−iπ2 [1 − Sgn(H˜

∆)/2]

|det(H˜

∆)|1/2 . (7.3.1b)

As before, H˜

∆ denotes the Hessian matrix of the traveltime difference T∆(ξ;M) given in equation(7.1.10).

7.3. TRUE-AMPLITUDE MIGRATION RESULT 193

Using equations (6.2.8), (6.2.14), and (7.2.10), we obtain

ΥDS(ξ∗) =KDS(ξ∗;M)OF (ξ∗)

LF (ξ∗) |det(Λ˜

)| .

= KDS(ξ∗;M)OF (ξ∗)OD(ξ∗)

LF (ξ∗)hB. (7.3.2)

Further recognizing from comparison of equations (4.6.4) and (7.2.24) that

ODS(ξ∗) = 1/OF (r∗) , (7.3.3)

and substituting KDS(ξ;M) from equation (7.2.26) in formula (7.3.2), we end up with the result

ΥDS(ξ∗) = L(ξ∗(r)) . (7.3.4)

Therefore, by inserting equations (7.1.2b) and (7.3.4) into expression (7.3.1a), we finally arrive at

Φ(M) = Rc(xM )F [t] , (7.3.5)

where t = T∆(ξ∗;M) is the distance of the Huygens surface ΓM of point M to the reflection-timesurface ΓR of the target reflector ΣR. To recover the amplitude-normalized Zoeppritz reflection coef-ficient Rc instead of the reciprocal (or energy-normalized) one, Rc, the weight function KDS(ξ;M)simply needs to be corrected by the corresponding factor [see equation (3.13.9)]. Then, the migrationresult can be represented as

Φ(M) = u(xM , t) , (7.3.6)

where u(xM , t) is given by equation (6.1.12).

Written in the above form, result (7.3.6) states that the “output” of the DSI at a reflectionpoint is exactly the “input” for the KHI. This fact can be physically interpreted as follows. In thesame way as the KHI superposes the contributions of all Huygens sources (originating along thereflector ΣR) to compute the reflection response at the receiver, the DSI decomposes the reflectionresponse in order to reconstruct the source strength of a Huygens source at ΣR. Consequently, wemay say that a diffraction stack sums up all contributions in the record section that pertain to oneparticular Huygens secondary source on ΣR (Figure 6.4).

In other words, one can call the DSI the approximate “physical” inverse to the KHI in thefollowing sense. The KHI maps the reflector attributes onto the seismic reflection distributed alongthe reflection-time surface (in the time-trace domain), and the DSI transforms this reflection backinto the depth domain and reconstructs in this way the amplitude values (reflection coefficients)on the reflector ΣR. Both image transformations are not only kinematically but also dynamicallycorrect. Note that the DSI is not an inverse to the KHI in a stricter mathematical asymptoticsense. The result of the DSI has to be interpreted in order to provide the input to the KHI. In fact,there exist two more related integrals being the respective inverses to the KHI and the DSI. Theasymptotic inverse to the DSI is the isochron stack integral (ISI) that makes a fundamental partof the theory that is the topic of this book. The ISI will be discussed in detail in Chapter 9. Theasymptotic inverse to the KHI is beyond the scope of this book. The interested reader is referredto Tygel et al. (2000) and (Schleicher et al., 2001a), where this inverse KHI and its main featuresare discussed in detail.

The asymptotic results obtained above could also be derived by application of the four-dimensional stationary-phase method to evaluate the combined integrals. This is obtained by in-serting the KHI into the DSI or vice versa, which results in four-dimensional integrals (see Bleistein,1987).

194 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

Finally, let us repeat that the DSI not only provides correct (true) amplitudes along thesearched-for reflector but also reconstructs the source wavelet’s shape (Tygel et al., 1994b) providedthe macrovelocity model is sufficiently accurate. If the phase property of the source wavelet is known,the correct reflector position can be determined. In this sense, the DSI can be seen as a full inversionintegral of the KHI, as all attributes of the Huygens sources that are input to the KHI [reflectorlocation, reflection coefficient, wavelet shape, see equation (6.1.13)] are recovered by the DSI.

7.4 Comparison with Bleistein’s weight function

Assuming an acoustic model, Bleistein (1987), based on the determinant of Beylkin (1985a), pro-vided a similar weight function KB(ξ;M). Interpreting the amplitude factors appearing in Bleis-tein’s weight by means of elastic ray theory, it can be shown that KB relates to our weight functionKDS(ξ;M) given in equation (7.2.18) in the following simple way

KDS(ξ;M) = KB(ξ;M) e−iπ

2(κS+κG) . (7.4.1)

The derivation of formula (7.4.1) is included in Schleicher (1993). The phase-shift factor is notfound in the Bleistein weight because Beylkin (1985a) did not allow for any caustics along rays.According to formula (7.4.1), both weights can be computed following Cerveny and Castro (1993),who show how the Beylkin determinant can be evaluated by dynamic ray tracing.

We have elaborated here on how our weight function can be computed by dynamic ray tracing.This must be done if both the modulus and the phase of the weight are to be accurate. However,we have shown that, if only the modulus of the migration result is desired, dynamic ray tracingcan even be omitted. The true-amplitude reflections obtained will be correct then with respect totheir phase in the absence of ray-segment caustics. Finally we want to indicate that based on anidea of Bleistein (1987), we have discussed in Tygel et al. (1993) a so-called vector diffraction stackmigration. The idea is to use three simple weights simultaneously in the diffraction stack with theaim of economizing on dynamic ray tracing while still obtaining correct results. The method willbe explained in Section 8.4.

In summary, the above 3-D true-amplitude migration scheme requires nothing more thanthe implementation of the two simple formulas (7.1.5) and (7.2.18). To shed more light uponboth, we will discuss certain standard measurement configurations and then briefly summarize theoperational procedure involved in performing the true-amplitude migration. However, prior to this,let us briefly address the question how to proceed if the data are recorded at a free surface.

7.5 Free surface, vertical displacement

We remark once more that expression (7.1.2) for the principal component particle displacement onlyholds, if the measurement surface is not a free surface. Therefore, the true-amplitude migration asdescribed above can only be applied if such data are available. To apply it to displacement datarecorded at a free surface, one can proceed in two different ways.

(a) If three-component data are available, one can remove the effect of the free surface on thesedata before migrating them. How this can be done, using the conversion coefficients as given,

7.6. PARTICULAR CONFIGURATIONS 195

e.g., by Cerveny et al. (1977), is discussed in Appendix B. The resulting principal componentdata obtained from the three-component seismograms can be migrated using the diffractionstack (7.1.5).

(b) If only the vertical component of the free-surface displacement vector is available, one has tomodify the weight function (7.2.18) for the true-amplitude migration. The vertical displace-ment component recorded at the free surface differs from the principal component (7.1.2) bythe factor c3 cosϑG, where c3 is the vertical component of the conversion coefficient vector c(Cerveny et al., 1977; see also Appendix B). Since c3 consists only of factors that contain thecompressional and shear wave velocities and the density at G together with ϑG, the weightfunction (7.2.18) can be divided by c3 cosϑG for each receiver point G. With such a modifiedweight function the vertical displacement free-surface data can also be migrated directly.

Finally, let us mention that pressure data can also be migrated in true-amplitude. Either onecomputes the principal displacement component from taking the gradient of the pressure or oneuses the corresponding ray-theoretical equivalent to formula (7.1.1) for the pressure. This formulalooks very similar to equation (7.1.1), except that the factor A has a slightly different expression(cf., e.g., Beydoun and Keho, 1987). After a correction for A, equation (7.1.2) looks the same foracoustic reflections. In the latter case, the true-amplitude migration can, of course, only determinethe angle-dependent reflection coefficients for pressure and not those for the particle displacement.

7.6 Particular configurations

In this section, we take a quick look at the seismic configurations most commonly used. In Section 2.2we have already shown that the matrices Γ

˜S and Γ

˜G then take on a very simple form. Accordingly,

the corresponding true-amplitude weight functions also reduce from equation (7.2.18) to simplerformulas.

7.6.1 Zero-offset (ZO) configuration

In this case, source and receiver positions are coincident (i.e., S = G) and the configuration matricesare Γ

˜S = Γ

˜G = I

˜, with I

˜being the 2× 2 unit matrix. Also using vS = vG = v0, ϑS = ϑG = ϑ0, and

κS = κG = κ0, we find from equation (7.2.18) that the true-amplitude weight function reduces to

KZODS = 4

cosϑ0

v0exp[iπκ0] = 4

cosϑ0

v0(−1)κ0 . (7.6.1)

If only the vertical component instead of the total principal-component displacement is considered,the cosine factor does not appear since it is included in the vertical component.

The above formula (7.6.1) has been previously derived by various authors (e.g., Cohen etal., 1986; Miller et al., 1987; Goldin, 1987; 1989; Kiehn, 1990; Hubral et al., 1991), except for theexponential factor that can be ±1, depending on the even or odd number of caustics along the raySM = GM .

196 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

7.6.2 Common-offset (CO) configuration

For this configuration source S and receiver G are dislocated by the same amount in the samedirection (Figure 2.2c). We therefore have to substitute Γ

˜S = Γ

˜G = I

˜into equation (7.2.18) and

obtain the common-offset true-amplitude weight function

KCODS =

√

cosϑS cosϑGvSvG

|det(N˜SM +N

˜GM )|

√

|det(N˜SM )|

√

|det(N˜GM )| exp

[

−iπ2(κS + κG)

]

. (7.6.2)

This resulting formula should be of particular help to migrate a 3-D data set of a conventionalmarine seismic survey, where the 2-D measurement plane z = 0 is equidistantly and densely coveredby the midpoints of source-receiver pairs of fixed offset and fixed azimuth.

7.6.3 Common-midpoint offset (CMPO) configuration

The true-amplitude weight function for this case is very similar to the preceding one. Source S andreceiver G are again dislocated by the same amount, but now in opposite directions (Figure 2.2d).This is described by setting Γ

˜S = I

˜and Γ

˜G = −I

˜into equation (7.2.18) which yields

KCMPODS =

√

cosϑS cosϑGvSvG

|det(N˜SM −N

˜GM )|

√

|det(N˜SM)|

√

|det(N˜GM )| exp

[

−iπ2(κS + κG)

]

. (7.6.3)

The true-amplitude migration procedure works for the CMPO experiment only if N˜SM 6=

N˜GM , which expresses, in fact, the requirement that detH

˜∆ 6= 0. Otherwise, the method of

stationary phase is invalid for evaluating the stacking integral (7.1.5). This condition is not fulfilledfor the ordinary common-midpoint (CMP) experiment, where rS0 = rG0, i.e., the central ray is thenormal ray. Note that this is exactly the situation described by the extended NIP wave theorem inSection 4.6.2. In the limit CMPO → CMP, the traveltimes TR (TCMPO) and TD (TCIP ) have thesame curvature. Thus, their difference matrix D

˜expressed in equation (4.6.19) vanishes as observed

in equation (4.6.20). Therefore, to practically realize a true-amplitude migration of data resultingfrom this experiment with a nonvanishing weight, one will usually need unrealistically large offsets(Vermeer, 1995). In other words, there is no true-amplitude weight for Kirchhoff migration of CMPdata with conventional offsets.

7.6.4 Common-shot (CS) configuration

In this configuration, only the receiver is dislocated while the source remains fixed (Figure 2.2a). Thetrue-amplitude weight function for the CS configuration is also a special case of formula (7.2.18).If we set Γ

˜S = O

˜and Γ

˜G = I

˜, we obtain

KCSDS =

√

cosϑS0 cosϑGvS0vG

√

|det(N˜GM )|

√

|det(N˜SM )| exp

[

−iπ2(κS + κG)

]

, (7.6.4)

where vS0 and ϑS0 denote the velocity and the emergence angle at the fixed source point rS0.

Ignoring the significance of caustics (i.e., neglecting the above exponential factor), Goldin(1987; 1989) presented an equivalent formula for the 2-D case. His result is expressed by means of

7.6. PARTICULAR CONFIGURATIONS 197

the spreading factors LS and LG of the two ray segments SM and GM . These factors are relatedto the matrices N

˜SM and N

˜GM by [see equations (6.3.1a) and (6.3.1b)]

√

|det(N˜SM )| =

√cosϑS0 cosϑM√vMvS0|LS |

and√

|det(N˜GM )| =

√cosϑG cosϑM√vMvG|LG|

. (7.6.5)

Substituting these expressions in equation (7.6.4) and ignoring the exponential factor yieldsGoldin’s result

|KCSDS | =

cosϑGvG

|LS ||LG|

. (7.6.6)

This result has also been derived by Beydoun and Keho (1987).

7.6.5 Common-receiver (CR) configuration

The result for the common-receiver (CR) configuration (Figure 2.2b) is, of course, quite similarto the one of the preceding CS experiment. The configuration matrices are Γ

˜S = I

˜and Γ

˜G = O

˜.

Therefore, it is only necessary to interchange the roles played by S and G in equation (7.6.4). Notethat in formula (7.2.18) the determinant in the numerator can only be expressed in the CS and CRcases in terms of geometrical spreading factors. Ideas on how to simplify the computation of thisweight have been discussed by Sun and Gajewski (1997, 1998).

7.6.6 Cross-profile (XP) configuration

The configuration matrices for the cross-profile experiment are given by Γ˜S = I

˜and Γ

˜G = ±R

˜,

according to the dislocation of sources and receivers into directions perpendicular to each other(see Figure 2.2e). The weight function for this configuration is

KXPDS =

√

cosϑS cosϑGvSvG

|det(N˜SM ∓R

˜N˜GM )|

√

|det(N˜SM)|

√

|det(N˜GM )| exp

−iπ2(κS + κG)

, (7.6.7)

where R˜

is the 2 × 2-rotation matrix for 90given in equation (2.2.11).

Formula (7.6.7) also holds for any other configuration where the receiver lines are rotated by aconstant angle with respect to the source lines. In this case, matrix R

˜represents the corresponding

rotation matrix.

7.6.7 Cross-spread (XS) configuration

This is another geometry with perpendicular dislocation of sources and receivers (Figure 2.2f). Theconfiguration matrices are given in equation (2.2.12). Introducing the matrix (Vermeer, 1995)

N˜XS =

(∂2TD

∂xS1∂xM 1

∂2TD

∂xS1∂xM 2

∂2TD

∂xG2∂xM 1

∂2TD

∂xG2∂xM 2

)

, (7.6.8)

we can write the weight function for this configuration as

KXSDS =

√

cosϑS cosϑGvSvG

|det(N˜XS)|

√

|det(N˜SM )|

√

|det(N˜GM )| exp

−iπ2(κS + κG)

. (7.6.9)

198 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

7.7 True-amplitude migration procedure

Without entering into any theoretical details we now summarize the most important steps of thetrue-amplitude (time or depth) migration.

1. The principal components of all seismic 3-component traces are computed.

2. Each principal-component trace is then time-differentiated and transformed into a complextrace (analytic signal).

3. The aperture A, which consists of a 2-D grid of points in the ξ-plane, is determined (seeSection 8.1). It results from a grid of source-receiver pairs (S,G) in the measurement surfacespecified by the measurement configuration.

4. In the migration volume (i.e., the part within the macrovelocity model for which the migrationshall be performed) subsurface points M are distributed forming a 3-D grid. For a depthmigration, this grid can be rectangular.

5. The traveltime from all surface points S and G to all subsurface points M must be computedusing an efficient algorithm. Alternatively, if the migration is to recover the source pulse,dynamic ray tracing must be performed to compute the number of caustics. It may also beused for the computation of the weights.

6. The following steps are repeated for every subsurface point M of interest within the migrationvolume V . The result is a depth migrated 3-D section.

(a) For one point M , the Huygens traveltime surface TD is computed for all ξ ∈ A as thesum of traveltimes along both ray segments SM and GM where S and G are specifiedby ξ.

(b) For one point M , the weight function KDS(ξ;M) is computed for all ξ ∈ A. If onlytraveltimes are available, second derivatives of the traveltimes must be computed todetermine the modulus of the weights. Dynamic ray tracing on the other hand determinesboth modulus and phase of the weight function. If for one particular ray a caustic appearsat a surface point (shot S or receiver G), the weight of this ray is undefined and thereforecannot be used.

(c) For one point M , the time derivatives of the analytic principal-component seismogramtraces at time TD are multiplied by the weight KDS(ξ;M) and summed for all ξ ∈ A.

(d) For a depth migration, the resulting stack signal is simply displayed at M . If the point Mlies on a reflector, the stack provides a value proportional to the complex angle-dependentreflection coefficient.

If not taken into account independently, the transmission-loss factor A is still containedin the resulting true-amplitude migrated reflections. If a dynamic ray tracing is performed and amacrodensity model is also available, its value can be estimated and its influence removed. However,this will be difficult in practice. Note that a correction for A is needed even in the case of a

homogeneous reflector overburden, where A = 1/√

%Sv2S%Gv

2G. How the effects of a finely layered

reflector overburden can be corrected for has been shown by Widmaier et al. (1996).

7.8. SUMMARY 199

The true-amplitude migration determines angle-dependent reflection coefficients. For an am-plitude variation with offset (AVO) analysis, the incidence angle of the stationary ray, i.e., thereflection angle, is therefore of great importance. Based on an idea of Bleistein (1987) the problemof finding that angle is solved in Tygel et al. (1993) by using the concept of a vector diffractionstack. There, three weight functions are used in the stack to determine the specular ray. The methodis discussed in detail in Section 8.4.

The algorithm of the present true-amplitude depth migration scheme was implemented byHanitzsch (1992) for the 2-D case. He migrated synthetic data computed for some simple modelsusing different measurement configurations (see also Hanitzsch et al., 1994). These initial exper-iments confirmed the recovery of true-amplitude reflections, even in the presence of caustics, aspredicted by the theory. Further 2.5-D and 3-D developments, implementations and numerical ex-periments were carried out by Hanitzsch (1995) and Martins et al. (1997). Additional discussionson more implementational forms of the weight function and differences to weights of other authorscan be found in Hanitzsch (1997) and Sun and Gajewski (1997, 1998). An independent comparisonof true-amplitude migration methods is carried out in Gray (1997).

7.8 Summary

In this chapter, we have shown how the Kirchhoff-migration integral or diffraction-stack integral(DSI) is related to the Kirchhoff-Helmholtz integral (KHI) that is widely used in forward seismicmodeling to calculate the shot-record wavefield response from a given reflector. Both integral trans-forms are specified in this work for arbitrary measurement configurations and a laterally inhomoge-neous overburden above a smooth reflector. The KHI superposes the contributions of all Huygenssecondary sources located along the reflector ΣR, to provide the reflected wavefield recorded atthe receivers G(ξ). The DSI on the other hand extracts from the recorded wavefield at all pointsG(ξ) the Huygens source contributions to the scattered wave field and allocates their amplitude topoints on the reflector image strip.

It is important to note that the true-amplitude weight function of the DSI depends only onquantities relative to the two individual ray segments from the source S to the reflection point MR

and from there to the receiver G and not on any reflector properties. This implies that it can becomputed for any arbitrary composite ray SMG irrespective of whether or not there is a reflectorat M . The modulus of the weight function may be computed by dynamic ray tracing (Cerveny,2001) with respect to the two independent rays SM and MG, or, alternatively, from second ordertraveltime derivatives (see, e.g., Vanelle and Gajewski, 2002). The phase of the weight function inequation (7.2.18) depends on the number of caustics along the ray segments and can, thus, only bedetermined by ray tracing.

We have seen that the DSI represents a natural (physical) inverse to the KHI. Both integralsprovide a proper theoretical justification for the diffraction-stack migration operation. This newunderstanding helps to physically interpret this migration procedure (often only based on eitherpurely geometrical considerations or on mathematical ones like the Generalized Radon Transform)in terms of dynamically correct Huygens wavefield contributions. Further investigations should becarried out to examine the relationship between both transform integrals also for low frequencies.

Although the DSI can be interpreted as a “physical inverse” to the KHI, it should be keptin mind that these two integrals do not constitute a transform pair in a mathematical sense.

200 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

The true mathematical asymptotic inverse of the KHI is described in Tygel et al. (2000). Froma mathematical point of view, the DSI can be interpreted as the adjoint operation to the KHI(Tarantola, 1984). From a more physical point of view, this can be understood as follows. Withthe DSI, one can image an unknown reflection event from the time domain into its reflector imagealigned along a previously unknown reflector ΣR in the depth domain. The KHI, however, cannottransform the unknown depth image of the reflector back into the reflection event, because thereflector ΣR is required in the KHI in order to perform the integration along it. Consequently, aninterpretation of the migration result obtained from the DSI would be necessary before applyingthe KHI again.

Interestingly enough, a “true” asymptotic imaging transform pair not requiring the knowledgeof the reflecting interface and feasible for automatic application with no need for any interpretationin order to find ΣR can be indeed established. In other words, a true mathematical asymptoticinverse to the DSI—the so-called isochron-stack integral—can be derived by wave-theoretical con-siderations along the lines above. It will be presented in the next chapter. It will be in perfectagreement with the forward and backward (mathematical) generalized Radon transform pair, butwill provide more physical and geometrical insight.

As indicated, the diffraction stack describes the most important seismic imaging operation,as the most desirable task in general is to construct an (r, z)-domain depth-migrated image froma (ξ, t)-domain seismic record. In this book, it will gain its overall significance as the first of twoimaging integrals that form the basis of a general theory of seismic reflection imaging. As we willsee in the next chapter, the weighted stack over the migrated (r, z)- domain image along isochronsurfaces, shortly referred to as “isochron stack,” can be looked upon as the inverse operation tothe diffraction stack provided the same macrovelocity model, the same measurement configurationand the same ray code (elementary wave) is used in both stacks. However, the isochron stack mayserve a much more important purpose. It can be used to demigrate a depth-migrated image Φ(M)for a macrovelocity model, a measurement configuration and/or ray code different from those usedin the diffraction stack. This capability is very desirable and is the heart of the Unified Approachto Seismic Reflection Imaging (Hubral et al., 1996a; Tygel et al., 1996) as discussed in Chapter 9.There a diffraction stack and an isochron stack are analytically chained or cascaded to transform,for instance, a certain seismic record or image by a single stack exclusively within either the (ξ, t)-domain or the (r, z)-domain.

To our knowledge, the first successful attempt to employ ray theory for seismic full wavefieldmigration for the purpose of estimating zero-offset reflection coefficients was made by Newman(1975) when he proposed the so-called modified diffraction stack for a vertically inhomogeneousmacrovelocity model and zero-offset 3-D marine data. It was originally Newman’s migrated resultsfrom short streamers that have encouraged us to formulate the true-amplitude migration methodpresented in this chapter. Though the problem of migration/inversion has over the years beenaddressed by many other authors, we are unaware of a solution to the problem explicitly stated interms of basic ray theoretical considerations. By having used ray theory from the beginning to theend we hope to have provided a clear geometrical picture of all basic steps and limitations involvedin the true-amplitude migration process. We pointed out the significance of caustics when the aimis to correctly recover seismic source pulses that may suffer a multiple of π

2 -phase distortions dueto caustics in the subsurface.

The fact that ray theory is highly developed (Cerveny, 2001) for many complicated media(e.g., anisotropic, weakly absorbing, etc.) points to the potential for the approach to be extended

7.8. SUMMARY 201

to more general situations than discussed above. Even though the theoretical derivation of ourweight function requires a good comprehension of ray theory, it remains, nevertheless, a surprisingfact that it is conceptually quite simple to compute [by formula (7.2.18)] and to implement [byformula (7.1.4)]. It also simply relates to the weight function of Bleistein (1987), who has obtainedit using far less direct arguments and not allowing for caustics along the ray paths.

Also note that ray theory, which for a long time has been a popular tool in explorationseismics, particularly in 3-D seismic forward modeling, traveltime or map migration, traveltimeinversion (i.e., computation of interval velocities) and traveltime tomography, is presently the onlytool that can be used to perform a 3-D full-wavefield prestack time or depth migration. It should,however, be kept in mind that the present method is not valid at a receiver that lies in the vicinityof a caustic as ray theory becomes an inadequate description of wave propagation in that region.Note, however, that this is also the case for the migration methods of Bleistein (1987) and Miller etal. (1987) as they are based on the WKBJ approximation which relies on the same high-frequencyassumptions as ray theory.

202 CHAPTER 7. TRUE-AMPLITUDE KIRCHHOFF MIGRATION

Chapter 8

Further aspects of Kirchhoff migration

In Chapter 7, we have introduced the true-amplitude Kirchhoff migration integral, which constitutesthe first of the two building blocks of the Unified Approach to seismic imaging. The second one isthe true-amplitude Kirchhoff demigration integral, which will be introduced in Chapter 9. In thisChapter, we further elaborate on the properties of Kirchhoff migration. However, all its aspectsthat are needed to understand the general imaging theory have been discussed in Chapter 7. Thereader whose main interest lies with the latter may thus directly proceed to Chapter 9.

After the quantitative evaluation of the Kirchhoff migration integral in Chapter 7, which ledus to the derivation of an appropriate form of the true-amplitude weight function, we discuss inthis chapter a number of other important aspects of Kirchhoff migration. All these effects have astrong influence on the quality and appearance of the migration result. The quantitative propertiesof Kirchhoff migration studied in this chapter include the relationship between migration aperturesand the Fresnel zones that leads to the choice of an optimal aperture for the migration operator(Schleicher et al., 1997b), the quantification of the pulse stretch that is observed in the migrationoutputs (Tygel et al., 1994b), and the vertical and horizontal resolution of the migration result. Itshould be kept in mind that these analyses, which are done in this chapter for Kirchhoff migration,can be carried out in a completely parallel way for any other Kirchhoff-type stacking method asdiscussed in Chapter 9.

Moreover, this chapter also treats the possibility of using two or more simultaneous Kirchhoff-type stacks along the same stacking surfaces, but with different weights. It is shown that, in thissituation, the obtained results can be combined to yield useful seismic attributes. The method wasoriginally suggested for diffraction-stack migration (Bleistein, 1987; Tygel et al., 1993). Its applica-tions include the determination of incidence angles of primary reflections leading to more reliableAVO/AVA analysis after migration (Bleistein, 1987; Hanitzsch, 1995), as well as the derivation ofsimpler and less expensive computation of true-amplitude weights (Tygel et al., 1993; Hanitzsch,1995). The extension of the technique to any other Kirchhoff-type imaging method is natural andstraightforward. Examples have been presented for migration to zero offset (Tygel et al., 1999;Bleistein et al., 1999) and common-shot migration to zero offset (Schleicher and Bagaini, 2004)

203

204 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

8.1 Migration aperture

An important aspect of Kirchhoff-type diffraction-stack migration is its aperture, i.e., the rangeof data over which the stack is performed. In the Kirchhoff migration integral, equation (7.1.4),the aperture is represented by the region of integration, A. Ideally, in the absence of noise, themigration aperture should be limitless, i.e., the integration in equation (7.1.4) should cover thetotal ξ-plane so that no contributions due to the abrupt truncation of the sum occur. This is, ofcourse, impossible. In practice the aperture is always limited since the region, over which seismicdata have been acquired, is finite. Therefore, some usual migration aperture-boundary effects mustbe accepted (Stolt and Benson, 1986). They can be reduced by tapering, but one must keep inmind that tapering can also destroy true amplitudes. A quantitative analysis of the main aperture-boundary effects is given in Sun (1998, 1999).

A confinement of the integration region A to a restricted migration aperture, even excludingranges of source and receiver positions where data actually have been acquired, can be of interest inpractical migration implementations. There are three major reasons why such a procedure can beadvantageous. The first reason is that less traces to sum leads to a speedup of the whole migrationprocess. Secondly, a smaller operator excludes steeper dips, which helps to avoid operator aliasing(see, e.g., Abma et al., 1999), It is well-known that in Kirchhoff migration, the spatial sampling rateof the traces restricts the range of dips that may be correctly migrated (see, e.g., Bleistein et al.,1985). Finally, less summation of data away from the signal reduces the stacking of unwanted noise.In this way, the migration becomes more robust and the resulting amplitudes are more reliable.However, such an aperture restriction should never go beyond a certain “minimum aperture” ifcorrect migration amplitudes are desired. For the best possible reduction of aliasing and noise aswell as the best computational efficiency, one would like to use a model-based aperture restriction.

In this section, we discuss the optimal choice for the migration aperture, i.e., the minimumaperture that is needed to still get a dynamically correct migration result of a key reflector. Thegeometry of the problem is explained with the help of Figure 8.1. Shown is a primary reflection raySMRG together with its Fresnel zone (indicated for a certain frequency) on the reflector at MR andone paraxial ray SMRG. To correctly image point MR from primary seismic reflections with thehelp of a diffraction-stack migration, its diffraction traveltime (or Huygens) surface is needed. ThisHuygens surface is tangent to the reflection traveltime surface at the a priori unknown emergencepoint of ray SMRG in the seismic section. The summation of all traces within an a priori specifiedaperture A along the Huygens surface yields a prestack depth-migration when putting the resultingvalue to point MR. Putting it to the emergence point of the image ray in the time section withthe two-way traveltime of that ray as abscissa results in a time migration. However, the size of theaperture A has an influence on the resulting migration amplitudes that must not be underestimated.In this section, we derive the minimum aperture that is necessary to guarantee correct migrationamplitudes and then we show how this region is related to the projected Fresnel zone and how itcan be computed.

8.1.1 Minimum aperture

In connection with the discussion of the diffraction-stack migration result, equation (7.1.15), wehave already discussed the importance of the finite length of the seismic source wavelet F [t]. Ifthe diffraction stack (7.1.4) is carried out for a reflection point, M = MR, its Huygens surface

8.1. MIGRATION APERTURE 205

reflector

G

t

z

S

Huygens surface

U

r

Ar

SG

1

1

R

D

22

R

Fresnel zone

reflection traveltime surface

central ray

paraxial ray

projectedFresnel zone

ξ

Γ

τ

ξ

Σ

MRMR

Fig. 8.1. In the 3-D elastic earth model, a known reflector (hatched surface) is buried within alaterally inhomogeneous velocity field, so that the primary P-wave reflection U at G for a pointsource at S is described by the ray SMRG. The ray SMRG is a paraxial ray for a slightly dislocatedsource-receiver pair (S,G). The depth migration places the true-amplitude signal ΦTA into pointMR. Note that in this figure, the halfspace above the plane z = 0 is the time-trace domain, i.e., theξ, t-space.

206 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

t

z

r

Huygens surface

*

RR R R

min

R

M M M

AS S G S G

reflector

reflectiontraveltime surface

wavelet length

ξ

Σ

ξ

τεD

Γ

τ

Fig. 8.2. 2-D cut through Figure 8.1 in a plane perpendicular to the r2-axis. The tangency regionis that part of the ξ-plane where the difference between the traveltime surfaces is less than thelength of the wavelet.

TD(ξ;M) is tangent to the reflection-time surface TR(ξ) at the stationary point ξ = ξ∗ (Figure 8.2).Correspondingly, for all points M sufficiently close to such a point MR, there exists a stationarypoint ξ = ξ∗ within the aperture A where both surfaces have an identical slope within the length ofthe wavelet. Then, there will be a constructive interference of the stacked energy and the migrationresult will be represented by expression (7.1.15). On the other hand, for pointsM that do not belongto a reflector and also not to its close vicinity, the distance between both surfaces at the stationarypoint, i.e., where they have an identical slope, is larger than the wavelet length. Therefore, no energycan be summed up with constructive interference. Mathematically, this observation is expressed bythe fact that, in the high-frequency limit, the term of zeroth order in ω of the evaluation of integral(7.1.4) vanishes. The first remaining term, which is of order ω−1, describes the aperture-boundaryeffects. Analytical expressions and a geometrical discussion of these effects can be found in Sun(1999) and Hertweck et al. (2003). Therefore, the stack (7.1.4) provides an image of all reflectors(irrespective of the weight function) once it is performed over all subsurface points M to which thevalue Φ(M) is assigned.

8.1. MIGRATION APERTURE 207

There is, however, another important consequence of the finite wavelet length. Let us consideran actual reflection point MR for which the Huygens and reflection-time surfaces are tangent atξ∗ (see Figure 8.2). Because of their different curvatures, their distance will increase when leavingthe stationary point. Consequently, at some distance of ξ∗, this distance will become larger than awavelet length.

We observe from the above reasoning (see again Figure 8.2) that all constructive contributionsto the diffraction stack for one particular reflection point MR stem from a certain tangency region.No seismic energy pertaining to the same seismic reflection is located along the diffraction timesurface outside that region. To catch all energy in the data that is necessary to obtain the migrationresult (7.1.15), the diffraction stack needs thus only be performed over this tangency region. In otherwords, this tangency region is the minimum aperture for the diffraction stack.

Using the traveltime equation (4.2.25) given in Chapter 4, we derive an estimate for the aboveexplained tangency region or minimum aperture. The boundary of the tangency region is implicitlygiven by equation (7.1.15) as TD(ξ;M) − TR(ξ) = Tε, where Tε is the length of the source pulse.More explicitly, all source-receiver pairs (S,G) for which the diffraction time along ray SMRG(dashed rays in Figure 8.2) and the reflection time along ray SMRG (solid rays in Figure 8.2) donot differ by more than Tε lie within the tangency region, i.e., mathematically, all points ξ, forwhich

|t(S(ξ), G(ξ)) − t(S(ξ),MR) − t(MR, G(ξ))| ≤ Tε , (8.1.1)

pertain to that region. Inserting the traveltime expressions (4.2.30) and (4.3.17) with M = MR,i.e., xM = 0 also using equations (2.2.13) and (4.5.6a), we obtain after a little algebra that involvesequations (4.3.7) and (4.6.1)

1

2|ξ ·H

˜P ξ| ≤ Tε , (8.1.2)

where H˜P is the projected Fresnel zone matrix given by equation (4.5.17). In other words, the

tangency region, which is equal to the minimum aperture, is directly given once the projectedFresnel zone matrix H

˜P is known and vice versa. In fact, the minimum aperture is the time-

domain projected Fresnel zone of equation (4.5.21), where the maximum time difference betweenthe rays is no longer defined by the frequency of a monofrequency wave (T/2 = π/ω) but by thelength Tε of the transient signal. The fact that the minimum aperture is governed by the projectedFresnel zone matrix is a direct consequence of the fact that H

˜P = H

˜∆, which we have observed in

connection with equation (7.2.8).

8.1.2 Application

The interesting relationship (8.1.2) provides a method of controlling the aperture of diffraction-stackmigration. One can check whether the size of the measurement aperture is sufficient to cover theminimum aperture. Once the reflection traveltime surface is picked and the diffraction traveltimesurface is computed, we can use equation (8.1.2) together with (4.5.22) to estimate the tangencyregion. If the measurement aperture was larger than the tangency region, the migration aperturecan be restricted to this region. As is well-known, the random noise level of a stacked trace increasesproportional to the number of traces stacked (Krey, 1987). A migration aperture restricted to theminimum aperture is, therefore, expected to reduce the noise effects in the migrated section and,therefore, enhance the signal-to-noise ratio. Synthetic examples that demonstrate this effect havebeen shown in Schleicher et al. (1997). Also it economizes the computation of the diffraction stacks

208 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

as only a limited number of traces must be stacked. It should be kept in mind that, to apply thediffraction stack really with the minimum aperture, one must have some a priori knowledge of thedip of the reflector to be migrated.

Our estimate of the minimum aperture is an alternative method to the heuristic one of Katzand Henyey (1992) who suggested to stack only those traces where the signal-to-noise ratio exceedsa certain threshold value. However, as is well-known, a restriction of the migration aperture meansa restriction to a certain range of dips that are correctly migrated. Although the computation ofthe minimum aperture can be done for each depth point for which the diffraction stack is to beperformed, we suggest to make only a few estimates over the whole reflection surface and use anaverage size for the migration aperture in order to keep the procedure economically reasonable.This should be sufficient when the lateral variations of the medium are small.

Also, the result represented by formula (8.1.2) can be used in the opposite direction. Supposethat the tangency region can be estimated directly from good quality data. In that case, one has adirect measure of the projected Fresnel zone of the considered reflection available.

8.2 Pulse distortion

When migrating seismic primary reflections obtained from arbitrary source-receiver configurations(e.g., common shot or constant offset) into depth, there occurs a pulse distortion along the imagedreflector. This exists even if the migration was performed using the correct macrovelocity model.Regardless of the migration algorithm, this distortion is a consequence of a varying reflection angle,reflector dip, and/or migration velocity. The relationship between the length of the original timepulse and that of the depth pulse after migration can be explained and quantified by means of theprestack Kirchhoff-type diffraction-stack migration theory. Note that this pulse distortion is closelyrelated to the well-known NMO stretch (see also Barnes, 1995).

Seismic primary reflections obtained from arbitrary source-receiver configurations (e.g., com-mon shot or constant offset) are recorded in form of seismic wavelets that have a certain duration.Figure 8.3 shows a sketch of a smooth subsurface reflector below an inhomogeneous velocity over-burden. Like all other figures in this chapter, Figure 8.3 shows a 2-D sketch of a 3-D situation. Thisis done for simplicity. Note, however, that all formulas below are generally valid in 2-D and 3-Dsituations. Suppose that seismic data acquisition was performed with different source and receiverpositions S(ξ) and G(ξ) as described by the 2-D parameter vector ξ (see Section 2.2). Accordingto ray theory and under the assumption of reproducible sources and receivers, as well as subcriticalincidence, a reflection event is described (apart from slowly changing amplitude factors) by identi-cal reflected causal pulses of equal length Tε in time (i.e., scaled copies of the causal source pulse)at all receivers. For a discussion on how the length of a wavelet can be defined and how it can bedetermined from the seismic data see, e.g., Berkhout (1984). All reflections thus fall into a strip(i.e., the reflector image) of constant width in time with the reflection traveltime surface t = TR(ξ)as the lower boundary. For noncausal pulses, the true reflection time surface lies somewhere withinthe strip.

When migrated to depth using any standard migration scheme, the reflector image assumesa certain “thickness” in form of the depth-migrated strip. For causal wavelets, the reflector is (forsubcritical angles of incidence) the upper boundary of the depth-migrated strip. For noncausalpulses, it is located somewhere within the strip. It is important to note that the thickness of this

8.2. PULSE DISTORTION 209

reflection-signal strip

t

R

( ;M )

( )( )

D

RR

z

rr

MR

M

NRR

depth-migrated strip

R

R

R

R

R

( )r

( ; N )r

R

RR

I

N

S( )G( )

ξ R

ξ

R

ΓM

ξ

ξ

ξ

ϑβ

Σ

ξ Σξ

ΣR

( )r

ΓR

Fig. 8.3. Migration of a strip in time leads to a distorted strip in depth.

210 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

strip will, in general, vary along the reflector. For that reason, the interpreter may fail to correctlylocate the reflector, even using the correct velocity model, when relying on the wavelet’s maximum,only, because its distance to the true reflector location varies with the length of the migratedwavelet. In what follows, we call a reflector any smooth subsurface interface that would resultfrom a map migration of the reflection traveltime surface TR(ξ) irrespective of whether the velocitymodel is correct or not. In other words, the considerations and formulas presented here are validfor correct and incorrect macrovelocity models as well.

Why a laterally varying depth-migrated strip is obtained can be easily explained by simplegeometrical considerations. Consider the points along the reflection traveltime surface TR(ξ) (Fig-ure 8.3, bold traveltime curve). For each of these points, there exists one isochron surface (oftenalso called aplanat) in the subsurface that is, for a given macrovelocity model, entirely defined bythe source and receiver positions and the observed reflection time. It is the locus of all subsurfacepoints that have the following property. The sum of the traveltimes along the two ray segments thatconnect the selected subsurface point to both the shot and receiver equals the given reflection time.The envelope of all isochrons thus specifies the reflector (bold curve). Now consider the points alongthe parallel traveltime surface TR(ξ)+Tε (dashed traveltime curve in Figure 8.3). These points alsodefine a set of isochrons, the envelope of which is the lower boundary of the depth-migrated stripof the reflection (dashed curve). The thickness of the latter strip naturally depends on Tε, the localvelocity, the depth of the reflector, and on the seismic measurement configuration.

In this section, we provide an approximate expression for the 3-D depth-migrated image ata point M vertically below, but still in the near vicinity of, a point MR located on the reflector(Figure 8.4). In other words, we derive a formula that quantitatively describes the above indicatedpulse distortion. Note that any deviation of the ray-theoretical assumption of a constant waveletlength Tε along the reflection-time surface in the seismic section results in an additional distortionthat is not described by the present approach.

Let us now comment on why we define the wavelet distortion in vertical direction. Of course,if a certain part of a seismic record trace contributes to the migrated image of the desired reflectorin the sense of a specular reflection, then the immediate time neighborhood of that part from thesame trace will also be migrated to a neighborhood of the previous image. The new image location isgenerally not vertically below the first image. The direction in which the image location is displaceddepends on a number of factors including the time dip, the velocity distribution, and the particulartype of gather of traces which is migrated. This is, however, not the direction in which the migratedseismic pulse appears. Because the migrated seismic traces are typically displayed in the directionof the vertical axis, the distortion is most naturally observed along this direction (Brown, 1994).

8.2.1 Geometrical approach

The stretch factor, mD, for a wavelet f [t], when migrated from time to depth, can be derived bysimple geometrical arguments in a heuristic way. Although the result can be proven only by themore rigorous mathematical examination that is given in Section 8.2.2, it is useful to attach to thestretch factor a geometrical meaning so as to make it more plausible. For that purpose, we considerthe fixed depth point MR with global Cartesian coordinates (rR, zR) in Figure 8.4 together with itsdiffraction time surface TD(ξ;MR). The point MR is located at a reflector that is fixed throughoutthe analysis. However, this reflector is assumed to be not specified in the macrovelocity model, i.e,the velocity is taken to be continuous at MR. If this is not the case, different values for the stretch

8.2. PULSE DISTORTION 211

M

( ;M)

( ;M )

S( )

t

z

reflector

G( )x

R

M

R( )

D R

D

T

T

T

TD

T

ξξ

ξ

ξ

ξ

ξ

ε

Fig. 8.4. 2-D sketch of a Kirchhoff-type diffraction stack depth migration. Consider a point M atcoordinates (rR, z), vertically displaced from an actual reflection point MR(rR, zR) at depth zR.The resulting diffraction time surface TD(ξ;M) is shifted in time by a certain amount ∆TD fromTD(ξ;MR).

factor are obtained on either side of the interface.

When the depth point MR is vertically displaced to a point M with coordinates (rR, z), itsdiffraction time surface TD(ξ;M) is shifted in time by an amount ∆TD = TD(ξ;M) − TD(ξ;MR).The vertical distance ∆z = z−zR between MR and M (Figure 8.5) is considered to be small. Thus,the two rays from S to M (ray SM) and from S to MR (ray SMR) can be considered parallel,i.e., the double-circled angles can be considered identical. The traveltime difference between theserays is then obtained by the difference between the length of these rays, ∆z cos Θ−

R, divided by thelocal velocity v−R at MR. Correspondingly, the difference in travel distance between the two raysconnecting MR to G (ray MRG) and M to G (ray MG) is ∆z cos Θ+

R. Therefore, we have alongeach of the two ray segments associated with M the additional traveltime ∆z cos Θ±

R/v±R . For the

change of the traveltime, when displacing the depth point from MR to M , we therefore find

∆TD =

[

cos Θ−R

v−R+

cos Θ+R

v+R

]

∆z . (8.2.1)

212 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

ΘM- ΘM

+

z∆

Θ M+

cos

ΘM

-

z cos∆

R

raySM

raySM

ray M

GR

ray M

G

M

β

β

Θ Θ

RMΣR

reflectorϑ-

RR

ϑ+R

R

+-

R

R

z∆

Fig. 8.5. Detailed view of Figure 8.4. Shown is a geometrical construction with details on the raysand angles near point MR.

Why does equation (8.2.1) determine the stretch factor? Well, under the assumption that the shapeof the wavelet is correctly recovered in the depth-migrated section, it is exactly the local ratiobetween a small interval ∆TD, measured in the seismic time section (i.e., the length of the reflectedpulse), and a small interval ∆z in the seismic depth section (i.e., the length of the depth-migratedpulse) that defines the stretch factor. Therefore, the ratio ∆TD/∆z equals mD.

The above considerations provide a geometrical derivation for the stretch factor mD. Wehave implicitly assumed that the value resulting from the diffraction stack at M , i.e., the migrationresult, recovers a scaled and stretched version X (MR)f [mD(z − zR)] of the source wavelet f [t] andnot of another pulse. To verify that this is actually the case, we must now investigate in more detail,how the strip in the time record is mapped into the depth domain by the diffraction-stack integral.We are going to show in Section 8.2.2 that a Kirchhoff-type diffraction-stack migration (witharbitrary weights applied to the seismic data along the diffraction traveltime surface TD(ξ;M))indeed reconstructs a scaled version of the source pulse, distorted by the above heuristically derivedstretch factor mD. Unlike the simple kinematic treatment performed above, the following proof isalso valid for the more general and realistic cases of overcritical reflections and in the presenceof caustics in the wavefield. These cause the reflector image to not only include scaled copies ofthe source pulse, but also pulses with a certain phase shift. The reason is that in the subsequentanalysis, as before, the analytic pulse F [t], consisting of the original (real) source wavelet f [t] asthe real part and its Hilbert transform as imaginary part, is used instead of the real source wavelet

8.2. PULSE DISTORTION 213

f [t] itself.

8.2.2 Mathematical derivation

In this section, we asymptotically evaluate the Kirchhoff-type diffraction stack at point M with co-ordinates (rR, z) in the near vicinity of the specular reflection point MR, with coordinates (rR, zR),that is located on the reflector. This result will not only provide the desired expression for thestretch factor mD, but it will also prove that the migration result at M is indeed the scaled anddistorted analytic source wavelet XF [mD(z − zR)]. An expression for the amplitude factor X hasbeen derived in the previous chapter.

As the starting point, we consider the diffraction-stack integral (7.1.6), where we let the timet vary so as to use the Fourier transform. In other words, we consider the time-dependent stack

Φb(M, t) =−1

2π

∫

A

∫

d2ξ B(ξ;M) F [t+ T∆(ξ;M)], (8.2.2)

whereB(ξ;M) = KDS(ξ;M) Rc(ξ)/L(ξ) . (8.2.3)

In other words, the searched-for diffraction-stack migration result is given by

Φ(M) = Φb(M, 0) . (8.2.4)

Applying the Fourier transform with respect to t to equation (8.2.2), we obtain according tofamiliar rules [compare to equation (7.1.8)]

Φb(M,ω) = F [ω]−iω2π

∫

A

∫

d2ξ B(ξ;M) eiωT∆(ξ;M) . (8.2.5)

Approximation of Φb(M,ω) in the vicinity of MR.—Let us now make use of the fact that Mhas the coordinates (rR, z), i.e., only the z-coordinate varies from MR to M (Figure 8.4). Applyinga Taylor expansion of Φb(M,ω) in z in the vicinity of zR, we find for the first-order approximation,

Φb(M,ω) ≈ Φb(MR, ω) +∂Φb

∂z(MR;ω) (z − zR) . (8.2.6)

The first term in equation (8.2.6) represents the diffraction-stack integral (8.2.5) at MR on thereflector. As shown in the previous sections, this integral can be asymptotically evaluated upon theuse of the Method of Stationary Phase (Bleistein, 1984). Its result is approximated by

Φb(MR, ω) ≈ F [ω] X (ξ∗;MR) , (8.2.7)

where ξ∗ denotes the stationary or critical point, i.e., the point that satisfies equation (7.1.9). Asdiscussed before, we assume that one and only one critical point ξ∗ exists in the aperture range Awhich satisfies equation (7.1.9). If no critical point ξ∗ exists in A, the diffraction-stack output willbe asymptotically small. On the other hand, if more than one critical point exists in A, the stackresult will be a sum of the contributions from each single one. These contributions will show, ingeneral, different amplitudes and different distortions. Therefore, the migrated pulse is no longer

214 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

under control. However, for most of the usual seismic measurement configurations (e.g., commonshot or constant offset), the latter situation is extremely unlikely, as this means that a second rayconnecting the same source-receiver pair would reflect at the same depth point.

As we have seen in equations (7.3.1), the amplitude factor X (ξ∗;MR) in equation (8.2.7) isgiven by

X (ξ∗;MR) =B(ξ∗;MR)

|detH˜

∆|1/2 exp

[

−iπ2[1 − Sgn(H

˜∆)/2]

]

= ΥDS(ξ∗)U0(ξ∗) . (8.2.8)

where H˜

∆ is the second-derivative (Hessian) matrix of T∆(ξ), taken at ξ∗. This matrix is assumedto be nonsingular, i.e., det(H

˜∆) 6= 0. Its signature Sgn(H

˜F ) is the number of positive eigenvalues

minus the number of negative ones. Factor ΥDS(ξ∗) is given by equation (7.3.2). Note that X = Rc,if the weight in equation (8.2.3) is chosen to be the true-amplitude weight (7.2.18).

To derive an approximate expression for the second term in equation (8.2.6), we take thederivative of equation (8.2.5) with respect to z. We observe that

∂Φb

∂z(MR, ω) = F [ω]

−iω2π

∫

A

∫

d2ξ∂

∂z

[

B(ξ;M)eiωT∆(ξ;M)]

M=MR

= F [ω]−iω2π

∫

A

∫

d2ξ

[∂B(ξ;M)

∂zeiωT∆(ξ;M) +

+ B(ξ;M) (iω)∂T∆(ξ;M)

∂zeiωT∆(ξ;M)

]

M=MR

. (8.2.9)

For high frequencies, the term of order ω2 dominates as long as ∂T∆/∂z 6= 0, so that we can write

∂Φb

∂z(MR, ω) ≈ F [ω]

−(iω)2

2π

∫

A

∫

d2ξ mD(ξ;MR) B(ξ;MR)eiωT∆(ξ;MR) , (8.2.10)

where we have used the notation

mD(ξ;MR) =∂T∆(ξ;M)

∂zM=MR

. (8.2.11)

Since the reflection time surface TR(ξ) does not depend on M and therefore not on z, we may alsowrite

mD(ξ;MR) =∂TD∂z

M=MR

. (8.2.12)

This observation justifies the choice of the symbol mD in equation (8.2.11), although it had alreadybeen used in the sense of equation (8.2.12) in Chapter 5.

Except for a factor mD(ξ;MR) inside the integral and a factor of iω in front of it, theintegral in equation (8.2.10) is identical to that in equation (8.2.5). Therefore, the stationary-phaseevaluation of the former is readily performed in the same way as detailed above for the latter.Under consideration of the additional factor iω in front of the integral, the asymptotic result ofequation (8.2.10) at a point MR on the reflector thus reads

∂Φb

∂z(MR, ω) ≈ iωmD(ξ∗;MR) F [ω] X (ξ∗;MR) , (8.2.13)

8.2. PULSE DISTORTION 215

where X (ξ∗;MR)is again given by equation (8.2.8). The symbol mD(ξ∗;MR) denotes the valueof mD at the stationary point, i.e., mD(ξ∗;MR) = ∂TD/∂z|ξ∗

,MR. It remains to prove that the

distorted migration output at M is indeed proportional to the wavelet F [mD(z− zR)] and that thestretch factor mD, as given by equation (8.2.12), is in agreement with the expression for ∆T /∆zas given by formula (8.2.1).

For that purpose, we insert equations (8.2.7) and (8.2.13) into equation (8.2.6) to obtain

Φb(M,ω) ≈ [1 + iωmD(z − zR)]F [ω] X (ξ∗;MR) . (8.2.14)

Back in the time domain, we have consequently

Φb(M, t) ≈ [F [t] +mD(z − zR)F [t]] X (ξ∗;MR) . (8.2.15)

Following equation (8.2.4), we now set t = 0 to obtain

Φ(M) = Φb(M, t = 0)

≈ [F [0] +mD(z − zR)F [0]] X (ξ∗;MR) . (8.2.16)

We finally note that, for small |mD(z− zR)|, we have in accordance with a first-order Taylor seriesexpansion of F [t] in the vicinity of t = 0

F [t = 0] +mD(z − zR)F [t = 0] ≈ F [t = mD(z − zR)] . (8.2.17)

Hence, we find the desired expression

Φ(M) ≈ X (ξ∗;MR) F [mD(z − zR)] . (8.2.18)

This result proves that at a point, M , vertically below a specular reflection point, MR, the outputof a diffraction-stack migration is the distorted source wavelet with the same amplitude factorX (ξ∗;MR) as at the point MR. The stretch factor mD is given by equation (8.2.12). We are nowgoing to analyze that equation in order to prove that it represents indeed the same formula for thestretch factor as the one previously derived in a purely heuristic manner.

8.2.3 Geometrical interpretation

Using a point M that differs from MR only in the z-coordinate (Figure 8.4) and considering asource pulse F [t], we showed that the diffraction-stack output at M is proportional to the “distortedpulse” F [mD(z− zR)], where the stretch factor is given by equation (8.2.12). This factor turns outto have the simple geometrical meaning concealed in formula (8.2.1). To prove this, we carry outthe differentiation of the diffraction traveltime function TD with respect to z, viz.,

mD = i3 · ∇MTD(ξ∗;MR)

= i3 ·(

∇MT (S,M) + ∇MT (M,G))

R

, (8.2.19)

where T (S,M) (T (M,G)) is the traveltime along the ray segment from S to M (from M to G) andi3 is the unit vector in the vertical direction. By the eikonal equation (3.4.2), the gradient of aneikonal function T at a certain point equals the slowness vector of the ray at that point (Cerveny,1987). Therefore, the latter expression is exactly the sum of the vertical components of the slowness

216 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

vectors of the two ray segments at MR. Since the modulus of the slowness vector at the point MR

is 1/vR, we arrive at

mD =

[

cosΘ−R

v−R+

cos Θ+R

v+R

]

, (8.2.20a)

where Θ±R is the acute angle that the incident/reflected ray segment makes with the vertical axis

at MR (Figure 8.5). Equation (8.2.20a) proves that our heuristic argumentation led us indeed tothe correct expression for the stretch factor mD.

From Figure 8.5, we observe that Θ±R = ϑ±R ∓ βR, where ϑ±R are the incidence and reflection

angles and βR is the local reflector dip in the plane of reflection, i.e., in the plane ΩR that is definedby the two slowness vectors of the ray segments at MR. Since angles ϑ±R are related to each otherby Snell’s law, equation (8.2.20a) can be recast into the form

mD =sin(ϑ−R + ϑ+

R)

v+R sinϑ−R

cos βR . (8.2.20b)

We conclude from equation (8.2.20b) that, once the factor mD is known, it provides a relationshipbetween the following quantities: v±R (local velocities at MR), ϑ−R (incidence angle at MR), and βR(reflector dip at MR). Hence, mD can be used, for example, to determine the incidence angle ϑ−

R

once the propagation velocities and the reflector dip at MR are known. For a monotypic reflection,equation (8.2.20b) reduces to

mD =2

vRcosϑR cosβR , (8.2.20c)

This is the expression derived by Tygel et al. (1994) and Brown (1994).

Due to equation (8.2.12), the stretch factor mD can be estimated once MR and ξ∗ are known,because in a Kirchhoff-type diffraction stack the traveltime surface TD(ξ;M) is computed for allsubsurface points M and for all vector parameters ξ. Both quantities MR and ξ∗ may then, forinstance, be computed with a vector diffraction stack (Tygel et al., 1993) or modifications of it (seealso Section 8.4). There is no need to identify reflections in the (ξ, t)-domain and to construct theenvelopes of the isochrons. Alternatively, the parameter mD could be estimated from the data bycomparing the length of the seismic source wavelets in the time and depth domains, i.e., the widthTε of the reflection time strip with the varying width of the depth-migrated strip.

We remark that for the considerations in this chapter, the macrovelocity model is assumedto be represented by a continuous velocity function across the true location of the reflector. If thisis not the case, the considerations of this chapter remain completely unchanged for points M inthe portion of the depth-migrated strip that lies above or below the reflector. All formulas arevalid with the understanding that they are evaluated separately for points M above or below theinterface.

8.2.4 Synthetic example

To examine whether the derived formula for the wavelet’s distortion in the Kirchhoff-type depthmigration is valid, we performed a simple acoustic 2.5-D synthetic example (i.e., simulating 3-Dwave propagation in a 2-D medium) using a symmetrical Gabor wavelet (Gabor, 1946; Morletet al., 1982) with a dominant frequency of 40 Hz. The earth model (see Figure 8.6) consists of twohomogeneous layers separated by a horizontal interface at a depth of 0.6 km. The fairly shallow

8.2. PULSE DISTORTION 217

0 500 1000 1500 2000 2500 3000

0

200

400

600

Distance (m)

Dep

th (

m)

Fig. 8.6. Earth model for the synthetic shot record data example. The fairly shallow reflector waschosen to cover a large range of reflection angles.

0 500 1000 1500 2000 2500 3000

0

100

200

300

400

500

600

700

800

900

1000

Receiver Coordinate (m)

Tim

e (m

s)

Fig. 8.7. Synthetic shot record data example. The seismic reflections are computed by ray theoryfor the model indicated in Figure 8.6. Every second trace is shown.

reflector was chosen to cover a large range of reflection angles. The wave velocity is 4 km/s in theupper layer, 4.1 km/s in the lower one. A common-shot situation was simulated with the sourceposition at 0 km and 120 receivers distributed equidistantly between 50 m and 3000 m offset. Inthis geometry, the reflection angle varies from 0 to about 68. Figure 8.7 depicts the synthetic shotrecord where each trace has been normalized to its maximum. Since the data were computed by ray-theoretical forward modeling, the wavelet length is identical for all traces. These data were migratedusing the 2.5-D Kirchhoff-type diffraction-stack migration as described by Hanitzsch et al. (1994).

218 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

0 500 1000 1500

500

550

600

650

700

Distance (m)

Dep

th (

m)

Fig. 8.8. Migrated reflections. The migration result is shown within the target area, that is, thedepth range from 0.45 km to 0.75 km and offset range from 0km to 1.5 km.

The normalized depth-migrated data are shown in Figure 8.8. The target zone of the migration wasreduced to the illuminated part of the reflector. The pulse distortion is clearly visible. Note thatonly the target area, that is, the depth range from 0.45 km to 0.75 km is shown. Therefore, the pulsedistortion effect looks much larger than it would show up in a conventional seismic depth-migratedimage. Figure 8.9 compares the wavelet length along the imaged reflector, as obtained from themigrated image in Figure 8.8, with the theoretical value as predicted by formulas (8.2.20). Bothcurves coincide quite well. The steps in the picked curve are due to sampling. We observe, e.g., thatat a distance of 1.5 km, i.e., for a reflection angle of about 68, the pulse is about 2.7 times longerthan the zero-offset reflection pulse. For reflection angles less than 25, the pulse distortion is lessthan 10 percent and may be neglected. The situation does not change for a dipping reflector as canbe seen from equation (8.2.20b). The effect decreases for synclinal structures, but it increases foranticlinal ones. Note that the pulse distortion decreases with increasing reflector depth as the rangeof reflection angles decreases. For a numerical example with a depth-dependent velocity, see Barnes(1995). Additional numerical examples and discussions have been given by Hanitzsch (1995).

8.3 Resolution

Seismic resolution after depth migration has been theoretically discussed by various authors(Berkhout, 1984; Beylkin, 1985a; Cohen et al., 1986; Bleistein, 1987). A recent comprehensivestudy on the subject was carried out by Vermeer (1998, 1999), where additional references on the

8.3. RESOLUTION 219

0 500 1000 15000

0.5

1

1.5

2

2.5

3

Distance (m)

Rel

ativ

e W

avel

et L

engt

h

Fig. 8.9. Comparison between the wavelet length determined from the migrated reflector image ofFigure 8.8 (filled circles) and the result predicted by the theory (solid line).

subject can be found.

The above discussion of the pulse distortion is directly related to the question of verticalresolution after seismic migration. It is not difficult to see that two sharp reflectors whose im-ages overlap cannot be completely resolved. In this section, we discuss horizontal resolution in acompletely analogous manner.

It is widely accepted among geophysicists that “depth migration reduces the Fresnel zone.”Although this is a very sloppy expression, because the Fresnel zone is a fixed-size frequency-dependent quantity associated with the reflected ray, we will see in this section that there is a lotof truth in it. Firstly, horizontal resolution can indeed be quantified using a Fresnel zone concept.Secondly, for usual seismic reflection angles, seismic migration improves the horizontal resolution. Itis, however, interesting to observe that for higher reflection angles, migration may actually worsenthe horizontal resolution.

Note that we implicitly define resolution in a slightly different way from what is usuallydone in the literature. Conventionally, resolution is quantified by the minimal distance of twoobjects such that their images can still be recognized as two distinct ones. In this way, resolution isclearly a frequency-domain concept. For a more practical, time-domain concept, we need a differentdefinition. Guided by the above section on pulse distortion, we quantify horizontal resolution bymeans of the region around the migrated reflection point MR that is influenced by the migratedelementary wave at MR.

220 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

To obtain an estimate for the mentioned zone of “horizontal influence” after migration, weinvestigate the migration output at the chosen depth point M = MR in the vicinity of the specularreflection point MR (see Figure 8.10), i.e. when the output point is moved along the reflector ΣR.

migrated reflector image

M x

S G r

RΣz

R

RM

ΩRR

Fig. 8.10. Horizontal resolution: influence of the migrated event at the specular reflection pointMR on the migration result at the neighboring point MR on the reflector.

8.3.1 Mathematical derivation

We start again from expression (8.2.5). Analogously to the above analysis of the pulse stretch, weset up a Taylor-series expansion of Φb(MR, ω), this time however in xR in the plane ΩR tangentto the reflector at MR. Due to Fermat’s principle represented by equation (5.5.2), we will need asecond-order series

Φb(MR, ω) = Φb(MR, ω) + ∇RΦb(MR, ω) · xR +1

2xR ·H

˜Φ(MR, ω)xR , (8.3.1)

where ∇RΦb is the gradient and H˜

Φ is the second-order derivative (Hessian) matrix of Φb(MR, ω)with respect to xR1 and xR2 taken at MR.

As before, Φb(MR, ω) is given, after asymptotic evaluation of integral (7.1.4), by equation(8.2.7). The derivatives of Φb(MR, ω) with respect to the components of xR are given by

∂Φb

∂xRj(MR, ω) = F [ω]

−iω2π

∫

A

∫∂

∂xRj

[

B(ξ;MR)eiωT∆(ξ;M)]

. (8.3.2)

By application of the product rule, the above derivative operation yields

∂Φb

∂xRj(MR, ω) = F [ω]

−iω2π

∫

A

∫(

iω∂T∆

∂xRjB +

∂B

∂xRj

)

eiωT∆(ξ;M) , (8.3.3)

where the derivatives have to be taken for MR varying along the reflector. Since TR is not a functionof xR, we observe that

∂T∆

∂xRj=

∂TD∂xRj

=∂TΣ

∂xRj. (8.3.4)

8.3. RESOLUTION 221

The last identity in equation (8.3.4) is in accordance with the definition of TΣ in equation (5.3.4),as a consequence of the confinement of MR to the reflector.

Differentiating equation (8.3.3) a second time yields

∂2Φb

∂xRj∂xRk(MR, ω) = F [ω]

−iω2π

∫

A

∫[

iω∂2TΣ

∂xRj∂xRkB + iω

∂TΣ

∂xRj

∂B

∂xRk+ (iω)2

∂TΣ

∂xRj

∂TΣ

∂xRkB +

+∂2B

∂xRj∂xRk+ iω

∂TΣ

∂xRk

∂B

∂xRj

]

eiωT∆(ξ;M) , (8.3.5)

where we have again used equation (8.3.4).

In high-frequency approximation, equations (8.3.3) and (8.3.5) are dominated by the highest-order non-vanishing terms in ω. At MR, the first derivative of TΣ vanishes due to Fermat’s principle,equation (5.5.2). Thus, we find, to the second order in ω,

∂Φb

∂xRj(MR, ω) = 0 (8.3.6)

and∂2Φb

∂xRj∂xRk(MR, ω) = F [ω]

−(iω)2

2π

∫

A

∫∂2TΣ

∂xRj∂xRkB(ξ;MR)eiωT∆(ξ;MR) . (8.3.7)

The asymptotic evaluation of equation (8.3.7) is completely parallel to that of integrals (8.2.5) and(8.2.10) and yields

∂2Φb

∂xRj∂xRk= iωF [ω]X (ξ;MR)

∂2TΣ

∂xRj∂xRk. (8.3.8)

Here, we recognize the elements of the Hessian matrix H˜F as defined by equation (4.5.2). Intro-

ducing the corresponding Hessian matrix H˜

Φ of Φb, we can thus write in matrix form,

H˜

Φ = iωF [ω]X (ξ;MR)H˜F . (8.3.9)

We now substitute equations (8.2.7), (8.3.6) and (8.3.8) in the Taylor series (8.3.1) to obtain

Φb(MR, ω) =

[

1 + iω1

2xR ·H

˜FxR

]

F [ω]X (ξ;MR) . (8.3.10)

Back in the time domain, this reads

Φb(MR, t) =

[

F [t] +1

2xR ·H

˜FxRF [t]

]

X (ξ;MR) , (8.3.11)

or, at t = 0,

Φ(MR) = Φb(MR, 0) =

[

F [0] +1

2xR ·H

˜FxRF [0]

]

X (ξ;MR) . (8.3.12)

This result can again be interpreted as a first-order Taylor expansion in t of

Φ(MR) = F

[

t =1

2xR ·H

˜FxR

]

X (ξ;MR) . (8.3.13)

The physical interpretation of this result is straightforward. Since F [t] is zero outside the interval0 ≤ t ≤ Tε, the influence of the migrated wavefield at MR ends at that particular point MR, where

1

2xR ·H

˜FxR = Tε . (8.3.14)

222 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

Distance (m)

Dep

th (

m)

−5000 0 5000

600

700

800

900

1000

1100

1200

Fig. 8.11. Migration result of common offset data with one trace multiplied by 3. The correspondingmigration smile has the same shape as the boundary effects. The isochrons of the perturbed andboundary points are indicated by the dashed lines.

This is exactly the definition of the time-domain Fresnel zone as defined in equation (4.5.4). Thus,the area affecting the reflected field in the vicinity of MR is the area of the paraxial Fresnel zoneat MR.

As we have seen in Section 8.1, the information pertaining to each “diffraction point” M isdistributed in the seismic data over one projected Fresnel zone, which is therefore the minimumaperture for seismic Kirchhoff prestack depth migration. The present result tells us that the in-formation pertaining to each depth point is smeared in the migrated section over a time-domainFresnel zone.

8.3.2 Synthetic example

To demonstrate the validity of the above mathematical considerations, we have devised the follow-ing simple numerical experiment. Consider a horizontal interface at a depth ZR = 1 km below ahomogeneous halfspace with an acoustic wave velocity of 6 km/s. The velocity below the interfaceis 5 km/s. The velocity inversion was chosen to study large reflection angles while avoiding over-critical reflections. Common-offset experiments with offsets ranging from 0 m to 7000 m have beensimulated placing source-receiver pairs with midpoints at every 10 m between −5.5 km and 5.5 km.The reflection angle for the largest offset is about 74. The synthetic data were computed by raymodeling. The source pulse is a symmetrical Ricker wavelet with a peak frequency of about 30 Hz,i.e., an effective wavelet length of Tε ≈ 20 ms.

To simulate the presence of an amplitude irregularity, we have perturbed the amplitudeof the central trace of the common-offset data increasing it by a factor three. In this way, wehave created an abrupt amplitude discontinuity. Figure 8.11 shows the migrated image after 2.5-DKirchhoff migration of the common-offset section with a source-receiver offset of 1000 m. Both

8.3. RESOLUTION 223

−6000 −4000 −2000 0 2000 4000 60000.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Distance (m)

Nor

mal

ized

mig

ratio

n am

plitu

de

−500 0 5000.96

1

1.04

1.08

Fig. 8.12. Migration amplitudes. The inlay shows the amplitude perturbation in the center of themigrated image together with the estimated size of the perturbation zone (dashed lines) and itstheoretical prediction (solid lines).

survey endpoints and the amplitude perturbation generate the well-known migration smiles. Wecan clearly see that all three smiles, although due to different effects, exhibit identical geometries.They follow the isochrons (dashed lines) of the respective data points.

The sizes of the zones on the reflector image that are influenced by the migration smilesare hard to estimate in Figure 8.11. To make these zones more evident, we have picked the peakamplitudes along the seismic event. This amplitude is shown in Figure 8.12 as a function of lateralposition. Also indicated in Figure 8.12 is a zoom of the center region with the perturbed amplitude.From this data, we can estimate the size of the zone with wrong amplitude. Since we need a numer-ical criterion for the endpoint of the perturbation zone, we take that point where the perturbationfalls below ten percent of its maximum value. The so estimated size is indicated by a dashed linein the inlay in Figure 8.12 and compared to the theoretical size according to equation (8.3.14)(solid lines). For a common-offset experiment over a model with a horizontal reflector below anoverburden with a constant velocity v, the Fresnel zone is an ellipse with semi-axes

b =√

vTεZR , a =b

cos3/2 ϑR, (8.3.15)

where Tε is the length of the source wavelet, ZR is the reflector depth and ϑR is the reflection angle.Indicated in Figure 8.12 is the size of the greater semi-axis a that quantifies the extension of theFresnel zone in the direction of the seismic line. We observe quite a good coincidence between theestimated and theoretical values.

To put this investigation on a broader basis and make its results more conclusive, we haverepeated this numerical comparison for a range of source-receiver offsets between 0 m and 7000 m.

224 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

0 1000 2000 3000 4000 5000 6000 70000

200

400

600

800

1000

1200

1400

1600

1800

2000

Offset (m)

Fre

snel

zon

e ha

lf−ax

is (

m)

Fig. 8.13. Size of the perturbation zone as a function of offset (dotted line), as compared to thetheoretical prediction (solid line) and the size of the boundary zone (dashed line).

Figure 8.13 shows the size of the perturbation zone after migration as a function of offset (dottedline) as compared to the theoretical prediction (solid line). Also indicated is the size of the boundaryzone (dashed line). The latter can be estimated from the knowledge of the stationary-phase analysisat the boundary. We know that the obtained amplitude at the boundary is half the true amplitude.We thus estimate the radius of the boundary zone as the distance between the point with an errorof -50% (because we have used normalized amplitudes, this is the point where the amplitude has avalue of 0.5) and the point where the error falls to ten percent of the maximum amplitude error.

Note the almost perfect coincidence between all three curves over the whole range of offsets.In other words, a single point in the seismic data, be it the integral boundary, be it a trace withwrong amplitude, affects a whole Fresnel zone around its dual point in the migrated section. Thisconfirms our theoretical result (8.3.14).

8.4 Multiple weights in Kirchhoff imaging

Three-dimensional prestack Kirchhoff-type imaging methods can, in addition to estimating thelocation of arbitrarily curved reflectors or their images and the true amplitudes along them, alsobe used to provide useful kinematic and dynamic information about the specular reflection raythat connects the source and receiver via the unknown reflecting interface. This is achieved byperforming a Kirchhoff-type stack more than once upon the same seismic data set using identicalstacking surfaces but different weight functions (Bleistein, 1987). Some of these weight functionscan be applied simultaneously, i.e., as a weight vector (Tygel et al., 1993). The approach offers thepossibility of determining various useful quantities that help to compute and interpret the resulting

8.4. MULTIPLE WEIGHTS IN KIRCHHOFF IMAGING 225

seismic images.

In this section, we discuss the multiple-weight method for Kirchhoff migration. Note, however,that conceptually equivalent methods can be conceived of for any arbitrary Kirchhoff-type imagingmethod. How this can be done, e.g., for migration to zero offset has been discussed by Bleistein etal. (1999) and Tygel et al. (1999).

8.4.1 Multiple diffraction-stack migration

The basic idea of the multiply-weighted migration technique is to perform the diffraction stack onthe same data set and the same diffraction surfaces more than once with different weights. Themigration results are then divided in order to gain certain useful information on the primary wavereflection ray. Bleistein (1987) was the first to suggest this procedure. He indicated how Kirchhoffmigration can be used to obtain the angle of incidence of the specular ray at the reflector in additionto the reflection coefficients. Parsons (1986) applied Bleistein’s idea to determine the midpointcoordinate in this way. From this he then computed the incidence angle. Later Geoltrain andChovet (1991) used the method to obtain the specular traveltime and trace abscissa to reconstructunmigrated data from migrated data. Tygel et al. (1993) showed how multiply-weighted diffractionstacks may be used to compute other quantities, such as the reflector dip as well as the incidence(at the source) and emergence (at the receiver) angles of the primary reflection ray. They furthershowed how three diffraction stacks can be used to determine the source and receiver coordinatesfor the specular ray, leading to a method for efficient amplitude-preserving migration. Numericalexamples and implementational details for the method can be found in Hanitzsch (1995). Mostrecently, the technique has been applied by Chen (2004) to restrict the aperture to the Fresnel zonearound the specular ray. It is worthwhile to note that the multiple-weights approach can be appliedto any Kirchhoff stacking method. For Kirchhoff MZO, this has been theoretically discussed byBleistein et al. (1999) and demonstrated for a numerical example by Tygel et al. (1999).

Theoretical description.—The idea of the multiple diffraction stack is simply to generalize theKirchhoff migration integral of equation (7.1.5) to

Φj(M, t) =−1

2π

∫

A

∫

dξ1 dξ2 wj(ξ;M) U(ξ, t+ TD(ξ;M)) . (8.4.1)

Here, index j indicates that we will get for any numbered weight wj a different diffraction-stackresult Φj(M, t).

Let us now perform the diffraction stack (8.4.1) twice with two different weights w1(ξ;M)and w2(ξ;M). According to equation (8.4.1), this will result in the two migration outputs Φ1(M, t)and Φ2(M, t), respectively. It is seen from equation (7.1.11) that at a reflection point M = MR,the migration output is proportional to the weight factor, viz.,

Φj(MR, t) ∝ wj(ξ∗;MR) , (8.4.2)

where the stationary point ξ∗ determines the ray SMRG. Therefore, the ratio of two results for Φj

provides the ratio of two values of wj (Bleistein, 1987)

Φ2(MR, t)

Φ1(MR, t)=

w2(ξ∗;MR)

w1(ξ∗;MR)

with w1(ξ∗;MR) 6= 0 , (8.4.3)

226 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

if the diffraction stack (8.4.1) with weight w1 yields a nonzero migration output Φ1 (Geoltrain andChovet, 1991). If not, or if the weight function w1 vanishes at the stationary point, the ratio inequation (8.4.3) is obviously not defined as the denominator equals zero.

A diffraction-stack migration with two different weights offers, thus, the possibility of deter-mining any seismic quantity

c(ξ∗;MR) = w2(ξ∗;MR)/w1(ξ

∗;MR) (8.4.4)

that can be defined as a function of “seismic parameters” that pertain to the actual (sought-for)reflection ray SMRG (Figure 7.1). All one has to do is to specify the weight functions w1(ξ;M)and w2(ξ;M) accordingly, i.e., as

w2(ξ;M) = c(ξ;M) w1(ξ;M) . (8.4.5)

The only condition to be observed is that it must be possible to completely express both weightfunctions in terms of seismic parameters encountered along each “diffraction ray” SMG, irrespectiveof the fact whether M is a reflection point or not. In this context, one should also keep in mindthat for the application of the stationary phase method to integral (8.4.1) the weights wj(ξ;M)should be slowly varying functions of (ξ1, ξ2). A depth section of values of the seismic parameterc(M) for each depth point M is then obtained when performing the following procedure:

c(ξ;M) =

Φ2(MR, t = 0)/Φ1(MR, t = 0) if M = MR

0 if M 6= MR ,(8.4.6)

i.e., the value of the above ratio is allocated to M , if M equals an actual reflection point. Otherwise,the value at M is set to zero. Obviously, it is no simple task to distinguish between points M = MR

and M 6= MR in practice. We will discuss this problem in more detail in the next section.

Division by zero.—There is one main difficulty arising when applying formula (8.4.6) that has tobe overcome. This is the problem of how to decide whether M equals an actual reflection point ornot. In general, it is not sufficient to simply avoid divisions by zero, because in that case the ratioof two very small values will remain significant for many depth points M far away from a reflector.The result of a multiply-weighted diffraction stack will then be a very unstable depth image.

Neither Bleistein (1987) nor Parsons (1986) addressed this important question. Geoltrainand Chovet (1991) used the following method to avoid the problem. They evaluated the ratio ofequation (8.4.6) for all depth points M under consideration. Thereafter, they masked the resultingdepth section with the migration result of the unweighted diffraction stack. Obviously, this is atwo-step approach. However, there is a one-step approach to distinguish between points M = MR

and M 6= MR by way of defining an a priori threshold value for Φ1. Wherever the stack result Φ1

exceeds this threshold, the division is performed. Otherwise, the value zero is assigned to point M .The threshold value can be specified either as a certain percentage of the maximum value obtainedin the Φ1 section (relative threshold) or as an absolute number (absolute threshold). The valueof a relative threshold can be estimated during the diffraction stack. For the absolute threshold,however, a good estimate of the expected values is needed. If an absolute threshold can be found,the two diffraction stacks can be performed simultaneously and the results may be instantaneouslydivided where appropriate, i.e., in one step. Note that if a too-large value is chosen as an absolutethreshold, only the quantities belonging to very strong reflectors will be imaged. However, if thethreshold is too low, the image will be unstable. Other possibilities of avoiding a division by zerohave been discussed by Hanitzsch (1995).

8.4. MULTIPLE WEIGHTS IN KIRCHHOFF IMAGING 227

Detectable quantities.—The most desirable quantity to be determined by equation (8.4.6), assuggested by Bleistein (1987), is the reflection angle ϑR of a monotypic reflection at M = MR

(Figure 7.1). This angle is needed, e.g., for an AVO analysis. It can be obtained by choosing

w2(ξ;M) = cosϑM w1(ξ;M) (8.4.7)

for each fixed depth point M , where ϑM is the half-angle at M between the descending ray segmentfrom S to M and the ascending one from M to G (Figure 7.3b). Possible choices for w1 and w2

include, for example, the pairs

w1(ξ;M) = 1 and w2(ξ;M) = cosϑM (8.4.8a)

or

w1(ξ;M) = KDS(ξ;M) and w2(ξ;M) = cosϑM KDS(ξ;M) , (8.4.8b)

where KDS(ξ;M) is the true-amplitude weight function of equation (7.2.26). The latter pair ofweight functions is the one originally suggested by Bleistein (1987). Upon such a choice for w1

and w2, the result of equation (8.4.6) will be cosϑR at all points M = MR on the reflector. Thus,this procedure allows us to obtain the angle of incidence of the primary wave reflection ray atMR and to use it as proposed by Bleistein (1987). He proposes to use the reflection angle ϑR forrecovering the velocity below a reflector in a constant-density model from the angle-dependentreflection coefficient Rc(ϑR). In this case, we assume that the medium above the reflector is known.Of course, the described process can be readily generalized to determine the incidence and reflectionangles ϑ±R of a converted wave.

Parsons (1986) suggested to apply the method not to directly recover the reflection angle ϑRbut to determine the midpoint coordinate between S andG by two diffraction stacks and to computethe angle ϑR from this coordinate. He states that more stable results are obtained in this way. Inthe same spirit, Geoltrain and Chovet (1991) have computed depth sections of specular reflectiontraveltimes by specifying w1 = 1 and w2 = TR = τ(S,MR) + τ(G,MR). They also suggested thecorresponding determination of the trace abscissas in order to reconstruct unmigrated zero-offsetsections from migrated ones.

There are several other useful quantities that might also be determined with the help offormula (8.4.6). In fact, this formula enables the determination of all quantities that pertain to thecentral reflection ray SMRG. Examples include its starting angle ϑS at S or the emergence angle ϑGat G, as well as the reflector dip angle βR at MR. The latter angle describes the inclination of thetangent plane to the reflector, i.e., the plane perpendicular to the specular direction defined by theray segments SMR and MRG (see Figure 7.3). Obviously, the azimuth of the reflector dip direction,i.e., its strike, can also be determined in this way. Further detectable quantities include the first andsecond-order traveltime derivatives of the central ray at S or G, that is, the ray slowness vectors andwavefront curvatures or radii. It is to be noted that even the wavefront curvatures or radii of theray segments MS or MG that originate from a thought point source at M and travel to S or G areexpressible in terms of medium parameters encountered along the central ray SMRG. Therefore,their stationary values for M = MR may be computed using formula (8.4.6). These curvaturematrices are closely related to the (paraxial) Fresnel zone (Gelchinsky, 1985; Cerveny and Soares,1992; Hubral et al., 1993b) in the plane tangent to the reflector at MR. This Fresnel zone cantherefore be obtained in the same way. If a density model was available, even the transmission lossA specified in equation (3.13.7) could be computed for each ray segment SM and MG, so that itsstationary value for the unknown reflection ray SMRG could be determined. With this factor, the

228 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

seismic trace amplitudes of equation (7.1.1) could be corrected for the transmission loss such thatsignal (7.1.2) is obtained, which is assumed to be the input to true-amplitude migration.

We observe that for the determination of all the above quantities various weight functionshave to be computed for the two ray segments SM and GM and many weighted diffraction stacksmust be performed. Of course, such a procedure is very uneconomic. In the next section, we showthat all information pertaining to the unknown reflection ray SMRG can be obtained with muchless effort. The corresponding procedure will be described in Section 8.4.2. The basic observationis that all the above mentioned quantities characterizing the reflection ray are easily determinedonce two key parameters defining the reflection ray are known.

8.4.2 Three fundamental weights

We now show that the diffraction stack (8.4.1) needs to be performed with only three differentweights to compute the values at the stationary point of many desired quantities that characterize aprimary reflection. The only condition is the same as for the individual determination of the desiredquantities, namely that the considered quantity is expressible in terms of the seismic parametersencountered along the ray segments SM and MG, independently of whether or not M is an actualreflection point. The three weights provide the two components of the parameter vector ξ∗ of thestationary point of integral (8.4.1), which in turn completely specifies the reflection ray SMRG.This method is similar to the ones of Parsons (1986) or Geoltrain and Chovet (1991). To obtainthe parameter vector ξ∗, the diffraction stack has to be carried out with the weights w1(ξ;M) = 1,w2(ξ;M) = ξ1 , and w3(ξ;M) = ξ2 . In other words, we have to perform nothing more than a vectordiffraction stack

Φ(M, t) =−1

2π

∫

A

∫

dξ1 dξ2 w(ξ;M) U(ξ, t+ TD(ξ;M)) , (8.4.9)

where Φ is the vector of migration outputs Φ = (Φ1,Φ2,Φ3), and where w = (w1, w2, w3) is thesimple vector of weights w = (1, ξ1 , ξ2 ). The two big advantages of these particular weights are thatthey are (a) simultaneously applicable and (b) directly available, i.e., no additional computationaleffort is required except for the multiplications of the weight factors with each trace followed bythe summation procedures. Therefore, we refer to the stack (8.4.9) as a vector-weighted diffractionstack or simply as a vector diffraction stack. It needs only little more computer time than requiredfor the ordinary diffraction stack performed without (i.e., with a unit) weight. The fact that onlytraveltimes are needed for the vector-weighted diffraction stack allows for the use of any of the manyfast traveltime computation techniques (e.g., Vidale, 1988; van Trier and Symes, 1991; Podvin andLecomte, 1991; Klimes and Kvasnicka, 1994; Vinje et al., 1993; Leidenfrost et al., 1999, see alsoreferences there). Numerical tests have shown good results (Hanitzsch, 1995).

After the vector-weighted summation (8.4.9) is performed for all points M of interest, wedivide Φ2 and Φ3 by Φ1, according to rule (8.4.6), at all those points M where Φ1 exceeds a certainthreshold value and obtain ξ∗. Once this vector is known, the actual reflection ray can be tracedthrough the macro-velocity model from S(ξ∗1 , ξ

∗2) at xS = Γ

˜Sξ

∗ to the chosen depth point M = MR

and from there to G(ξ∗1 , ξ∗2) at xG = Γ

˜Gξ

∗ However, in practice the reflection ray needs only tobe identified from the multitude of diffraction rays SMG if they are already traced in order todefine the Huygens surface TD for point M . From the knowledge of the actual reflection ray, we canin a second step readily determine all desirable quantities for which one otherwise would have to

8.4. MULTIPLE WEIGHTS IN KIRCHHOFF IMAGING 229

perform various diffraction stacks with multiple weights, e.g., the reflection angle or the reflectordip and strike. It is to be observed that this procedure promises not only to be more economic butalso to present less dangers of instability, because the trace coordinates are weights as smooth asany other weight might be.

True-amplitude migration.—The vector diffraction stack has a very attractive implication fortrue-amplitude migrations that estimate reflection coefficients along unknown reflectors as describedin Chapter 7. Upon the use of the above result, it is no longer necessary to perform the dynamic raytracing for all diffraction rays. This is required in the migration/inversion methods for the compu-tation of the appropriate weight in order to compensate the reflection events for their geometrical-spreading loss and phase distortions due to caustics. In fact, using the vector diffraction stack(8.4.9), a purely kinematic ray tracing is sufficient for migration. After the coordinates ξ∗ areavailable from the vector diffraction stack, the dynamic ray tracing needs to be applied only alongthe actual specular reflection rays, which are now known. The computation of the true-amplitudeweight factor KDS(ξ∗;MR) as given in equation (7.2.26) can then be restricted to these specu-lar rays. As the final step, the migration output Φ1(MR, t) resulting from the simple stack withunit weight function w1(ξ;M) = 1 must be multiplied by this factor. The result is the same mi-grated section with geometrical-spreading-free (true) amplitudes as described in equation (7.1.3).Of course, a true-amplitude migration can be realized in this way too, if the specular ray can bedetermined by any other means.

The same simplification applies to the determination of other dynamic properties of the raySMRG like, e.g., the Fresnel zone at the reflection point M = MR. Any of these quantities has tobe calculated for each diffraction ray when to be determined by means of equation (8.4.6). Onceξ∗ is known, it is sufficient to calculate them for each reflection ray.

It is to be stressed that, fortunately, the problems related to the possible instabilities causedby division by small stack results or zero, as discussed in Section 8.4.1, do not affect the mainmigration result, i.e., the depth image of the angle-dependent reflection coefficients. The true-amplitude signal ΦTA is constructed by multiplying the weight that corresponds to the specularray [i.e., the true-amplitude migration weight KDS(ξ∗;MR)] with the result of the unweighteddiffraction stack Φ1. This second factor has very small values for depth points M far from reflectorpoints, and therefore, the amplitude-preserving migration realized by means of the vector-weightedstack constructs a stable image of the reflection coefficients in the target zone under consideration.In symbols,

ΦTA(M) = KDS(ξ∗;M)Φ1(M, t = 0) with ξi∗ =

Φi(M, t = 0)

Φ1(M, t = 0)(8.4.10)

In this way, the unstable depth section obtained from the division is masked by the stable depthsection obtained from the unweighted migration.

8.4.3 Synthetic example in 2-D

The diffraction stack with two weights was implemented and tested by Geoltrain and Chovet (1991)on Amoco’s synthetic data set. Their results show how the method operates on quite complex modeldata. As a first indication of how the vector-weighted diffraction stack works in connection witha quick approach to amplitude-preserving migration, we present in this section a simple synthetic

230 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

1

2

3

Distance (km)

Dep

th (

km)

Fig. 8.14. Earth model and reflected rays for the 2-D synthetic example. One planar horizontalinterface lies at a depth of 2.5 km between two homogeneous layers with wave velocities of 5 km/sand 6 km/s, respectively. The region within the box is the target zone to be migrated.

example in two dimensions. More synthetic results can be found in Hanitzsch (1995). Note thatthe derivative operation in 3-D (which is given in the frequency domain by multiplication withthe factor iω) is to be replaced in 2-D by the half-derivative (given in the frequency domain bymultiplication with (iω)1/2). Also the Huygens surfaces in 3-D become Huygens curves in 2-D.With this understanding, we will freely use the expressions ‘derivative’ and ‘Huygens surface’ inthe description below.

Figure 8.14 shows the simple acoustic earth model used for the numerical example. It consistsof two homogeneous half-spaces separated by a planar interface. The wave velocity within the upperhalf-space is 5 km/s and in the lower one 6 km/s. The density is chosen to be unit in both layers.At the earth’s surface, 2.5 km above the interface, a common-shot experiment is simulated with 150equidistant receivers separated by 25 m intervals, the first being located at a distance of 1 km fromthe source. Selected reflection rays are indicated in Figure 8.14 to show the part of the interfaceilluminated by the experiment. The region in the box is the target region to be seismically migratedby the diffraction-stack method as described above.

The corresponding synthetic seismograms are depicted in Figure 8.15. These data have beenmodeled by the ray method for the seismic measurement configuration of Figure 8.14. The Huygenssurface of one particular depth point situated on the (sought-for) interface is also indicated in Figure8.15 (by a continuous line). The macro-velocity model used for the computation of the Huygenssurface is simply a constant-velocity model with a migration velocity of 5 km/s, which is the correctvelocity for the reflector overburden. We observe that the Huygens surface is indeed tangent to thereflection time surface as expected (see also Figure 7.2a). This results in a constructive interferencewhen the seismic data are summed up along the Huygens surface.

To determine the Huygens surface, a multitude of rays are traced. Since our example involvesa common-shot acquisition geometry, the (downgoing) ray from the source to the chosen depthpoint and all (upgoing) rays emitted from there to each receiver point are needed. By adding thetime along the downgoing ray to the time of each of the upgoing rays we obtain the Huygenssurface. Along this time surface the summation is performed twice, once with a unit weight andonce weighted with the trace number which represents the ξ-coordinate.

8.4. MULTIPLE WEIGHTS IN KIRCHHOFF IMAGING 231

1 1.5 2 2.5 3 3.5 4 4.5 5

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

Receiver Position (km)

Tim

e (s

)

Fig. 8.15. Synthetic seismogram of a common-shot experiment computed for the model in Figure8.14 (every third trace is shown). Also indicated is the Huygens surface of a selected depth pointbelonging to the interface.

The two resulting depth sections are divided according to equation (8.4.6). As this divisionis only defined where actual reflections are observed, we used the criterion of a relative thresholdvalue to distinguish points M = MR from points M 6= MR. The division was done wheneverthe amplitude of the unweighted migrated section exceeded a threshold value of 50 percent of themaximum amplitude in that section. All other values were set to zero. The result of this procedureis depicted in Figure 8.16. It is clearly visible in this figure that the reflector is migrated to thecorrect depth position of 2.5 km.

The amplitudes along the reflector image in Figure 8.16 increase linearly from the left to theright. This is the expected behavior, since the amplitude should represent the coordinate of thespecular ray. Obviously, in the case of this simple model, this coordinate increases linearly fromthe left to the right, as responses from reflection points further to the right emerge at more andmore distant geophones. This is expressed by the observed increasing amplitudes. Therefore, eachamplitude at a depth point upon the imaged interface tells us indeed at which surface position thereflection event, corresponding to the respective depth point, was recorded.

Figure 8.17 compares the picked amplitudes along the reflector image in Figure 8.16 withthe theoretically predicted values of the coordinate ξ∗ of the specular ray. The straight line fromthe bottom left to the top right corner of the figure is the ideal curve. The actual result matchesthe ideal curve quite well, except for the boundary region, where some aperture effects cannot beavoided. However, these are not specific to the vector diffraction stack, but are common to allmigration methods. The range of these aperture effects is frequency-dependent. We observe thatfor the diffraction stack with multiple weights, the boundary effects are even less severe than for asingle migration. Obviously, the procedure of dividing two migration results heals these effects to

232 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

0.5 1 1.5 2 2.5

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

Distance (km)

Dep

th (

km)

Fig. 8.16. Result of the vector diffraction stack. The division of equation (8.4.3) is performed when-ever the amplitude of the unweighted migrated depth section exceeds 50 percent of the maximumamplitude in that section. All other values are set to zero.

some extent.

To check the quality of the vector diffraction stack, we have computed the true-amplitudeweight function by the equivalent of equation (7.2.26) in 2-D from the amplitudes resulting from thevector diffraction stack and applied it to the unweighted diffraction-stack migrated depth section.The resulting true-amplitude depth-migrated section is compared in Figure 8.18 to the correspond-ing one obtained by direct application of true-amplitude migration employing the full weights. Atfirst glance, no difference is visible between the two migrated reflector images in Figure 8.18. Weobserve that both migrations have perfectly restored the source pulse along the migrated reflectorimage. The amplitude variations along the reflector seem to be identical as far as can be inferredfrom Figure 8.18. Note that the application of the threshold value criterion was not necessary toobtain the migrated sections of Figure 8.18. The migrated section shows a stable depth image be-cause the unweighted migration itself yields a proper mask for the true-amplitude section. Onlydivisions by zero had to be avoided.

There is only one slight difference between the two migrated sections in Figure 8.18. Theboundary effect (i.e., the migration smile above the reflector image) in Figure 8.18a is a littlestronger than the one in part b. The reason is that no care has been taken to multiply the boundaryeffect with the correct weight. In practice, this will not cause any troubles as boundary effects canbe strongly reduced by tapering (Sun, 1999; Hertweck et al., 2003).

For better comparison, Figure 8.19 shows the picked true amplitudes (i.e., the recovered

8.4. MULTIPLE WEIGHTS IN KIRCHHOFF IMAGING 233

0.5 1 1.5 2 2.50.5

1

1.5

2

2.5

Distance (km)

Tra

ce P

aram

eter

(km

)

Fig. 8.17. Comparison of the vector diffraction stack amplitudes with the expected theoreticalresult. The agreement is obvious. The resulting migration boundary effects are much weaker thanin a conventional migrated image.

reflection coefficients) along the reflector. We compare the reflection coefficients obtained with thehelp of the stationary point coordinates (vector diffraction stack) with (a) the ones obtained directlyby the true-amplitude (TA) migration described in Chapter 7 and with (b) the exact values of thereflection coefficients of the known model. The continuous line represents the theoretically exactreflection coefficients. The true-amplitude reflection coefficients obtained via the vector diffractionstack (filled circles) and the TA migration results obtained directly with more computational effort(open circles) are almost indistinguishable. Both migration amplitudes match the theoretical curvequite well, with exception of those at the left and right margins where the amplitudes are too smalland too large due to the well-known boundary effects. Note that the combined aperture effects ofthe vector diffraction stack, including those of the ξ∗ determination and those of the unweightedmigration, are no more severe than those of the direct true-amplitude migration.

The TA migration using the vector diffraction stack provided good results for this simplecase, as it should. Rather than to evaluate the performance of the method, our aim in using thissimple example was to illustrate the procedure. Thorough investigations and testing are, of course,necessary to correctly assess the full practical potential of the multiply-weighted diffraction stackfor use on real data, in particular on its stability against noise and a wrong macro-model. Acouple of such tests have been carried out by Hanitzsch (1995) and Tygel et al. (1999) with quitesome success, indicating that the vector-weighted diffraction stack can indeed be a feasible cheaperalternative to a fully weighted true-amplitude migration.

234 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

0.5 1 1.5 2 2.5

2

2.2

2.4

2.6

2.8

3

0.5 1 1.5 2 2.5

2

2.2

2.4

2.6

2.8

3

(a)

Distance (km)

Dep

th (

km)

(b)

Distance (km)

Dep

th (

km)

Fig. 8.18. True-amplitude depth-migrated sections. The source pulse is correctly restored in bothsections. (a) The unweighted diffraction-stack migration result multiplied with the weight (7.2.26)computed for the stationary point. (b) The true-amplitude migrated section using the full weight(7.2.26) during the migration process.

8.5 Summary

In this chapter, we have discussed a number of important aspects of Kirchhoff depth migration.We remind the reader that corresponding analyses can be carried out in a parallel way for all otherKirchhoff-type imaging procedures that are discussed in Chapter 9.

The first aspect that we have discussed in detail is the optimal aperture for Kirchhoff migra-tion. As a result, we have observed that the minimum diffraction-stack migration aperture that isneeded to guarantee true amplitudes, is closely related to the projected Fresnel zone as discussed inChapter 4. Thus, the migration aperture can be effectively controlled by computing the projectedFresnel zone. Projected Fresnel zones can consequently be computed with almost no extra effort,when a diffraction stack migration is to be performed.

If the measurement aperture is larger than necessary, it is possible to even restrict the mi-gration aperture to the actual region of tangency, where the diffraction and reflection traveltimesurface “strips” touch. It is from this region that all information is actually gathered according toa diffraction stack migration. The restriction to that region not only economizes computer timeby improving the efficiency of the summation procedure and helps to avoid operator aliasing byavoiding to migrate unnecessarily large dips, but it also enhances the S/N ratio when summing upnoisy traces. In the case of migration/inversion or true-amplitude migration, where suitable weights

8.5. SUMMARY 235

0.5 1 1.5 2 2.5

0.05

0.1

0.15

0.2

Distance (km)

Am

plitu

de

Fig. 8.19. Reflection coefficients. The amplitudes of Figure 8.18 were picked along the reflector.The true-amplitude results obtained via the vector diffraction stack (filled circles) and directly bythe true-amplitude migration (open circles) are compared with the theoretical values (continuousline).

are applied prior to summation, such a procedure may result in a more reliable reconstruction ofthe interface reflection coefficients in dependence to the reflection angle. The accurate control overthe migration aperture becomes a must when stacking noisy traces.

It is a well-known fact that a depth migration using a Kirchhoff-type diffraction-stack resultsin distorted wavelets in depth irrespective of whether the macrovelocity model is right or wrong. Inthis chapter, we have also investigated this distortion both geometrically and mathematically. Inboth approaches, the same expression for the stretch factor was obtained. Since this factor dependsonly on the local velocity at the reflection point, the reflection angle, and the reflector dip, oneof these quantities can be estimated from it, provided the other two are already known. We havealso indicated how the distortion factor can be obtained directly from the seismic data. Either itcan be determined without identifying reflections in the data, e.g., by a vector diffraction stack(see Section 8.4), or by a comparison of the estimated lengths of the migrated and unmigratedreflection wavelets. The length of a wavelet can, of course, be determined from events observedin either the time or depth domains. The ratio of the so-determined wavelet lengths would alsoprovide the distortion factor.

Having demonstrated how the length of a depth-migrated pulse varies with different raypaths,we have implicitly addressed the important question of vertical resolution. In the case of two closelyspaced reflectors, we envision that situations may exist where the depth migration can resolve

236 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

reflectors for short shot-receiver distances but not for long ones. Along the same lines, we have alsodiscussed horizontal resolution. We have seen that the region around the reflection point affectedby the reflected wavefield after migration is the time-domain Fresnel zone. The region affected bythe boundary effects at the survey end is of the same size. This theoretical result was confirmed bya simple synthetic example. Consequently, the size of the boundary zone can be taken as a measurefor the horizontal resolution of a migrated seismic section.

Moreover, we have included a numerical study on horizontal resolution close to an amplitudeanomaly. In this situation, the behavior of resolution is different from the above result. For theusual seismic range of offsets and reflector depths, we qualitatively observe the expected behaviorof a decreasing horizontal resolving power with increasing offset. Generally, migration will signif-icantly improve the horizontal resolution. However, for very large offsets and shallow reflectors,the resolution after migration may actually even be worse than before. The observed behaviorof seismic resolution after Kirchhoff depth migration must be considered when carrying out anAVO/AVA analysis close to strong amplitude variations. Since amplitudes of images for differentoffsets may be differently affected by the presence of the amplitude variations, the AVO behaviormay be incorrect.

As the last topic of this chapter, we have discussed how Kirchhoff-type migration methodscan be used to provide useful kinematic and dynamic information about the specular reflectionray. This is achieved by performing a Kirchhoff-type stack more than once upon the same seismicdata set using identical stacking surfaces but different weights (Bleistein, 1987). Some of theseweights can be applied simultaneously, i.e., as a weight vector (Tygel et al., 1993). The approachoffers the possibility of determining various useful quantities that help to compute and interpretthe resulting images like, e.g., the reflection angle or the Fresnel zone. The method, which has beenpresented here for Kirchhoff migration, can be readily applied in the very same way in any otherKirchhoff-type imaging method described in Chapter 9.

The advantages of performing several Kirchhoff-type stacks on the same data using the samediffraction surface but different weights have already been pointed out by several authors (Bleistein,1987; Parsons, 1986; Geoltrain and Chovet, 1991). A simple division of stack results leads to anydesired ray quantity of the specular ray. We have indicated some additional useful parameters thatcharacterize a reflection ray which can be determined by this method. However, for each quantityto be determined one must perform at least one additional diffraction stack.

Using a simplified approach to the weighted diffraction stack, we have shown that the numberof diffraction stacks and weight functions actually needed is decreased by a significant amount. Ingeneral, three fundamental weights are sufficient to determine all relevant attributes of a specularreflection ray. These are a unit weight and two parameters that describe the source and receiverlocations of each seismic trace as well as the corresponding specular ray. These weights are verysimple, smooth, and do not depend on ray tracing. Moreover, they are simultaneously applicable,i.e., as a vector of weights. A division of the results of these vector-weighted diffraction stacks yieldsthe two parameters of the specular ray. With this information it is possible to determine everywanted quantity related to the two ray segments of the specular ray, e.g., the reflection angle, thereflector dip/azimuth the incidence and emergence angles, the Fresnel zone at the reflector, etc.

In particular, also an amplitude-preserving migration which uses a weighted diffraction stack(Kirchhoff migration) to determine reflection coefficients can be performed more economically. Aftera vector-weighted diffraction stack has been performed, only one weight factor per (known) specularray needs to be computed. Thus, the number of rays for which dynamic ray tracing is to be carried

8.5. SUMMARY 237

out is drastically reduced. The result of the unweighted diffraction stack is then multiplied withthe weight factor computed for the specular ray. This procedure yields the reflection coefficient atthe corresponding reflection point. Thus, the final result, a true-amplitude migrated section, canbe obtained in a more economic way.

There is a fundamental problem that arises when multiple weights are used in diffraction-stackmigration methods. The division of two stack results, which is required to obtain the searched-forquantity for the specular ray, is not defined if the denominator vanishes. To simply avoid divisionsby zero is, in general, not sufficient. Because of noise, the stack result in the denominator willoften assume a small value and this will cause an unstable depth image of the desired quantity.To overcome this problem, we have suggested to take as reflection points only those points, wherethe unweighted diffraction stack result exceeds a certain threshold value. This provides the op-portunity to perform the necessary diffraction stacks simultaneously, which further increases theefficiency of the method. However, this problem does not arise if the diffraction stack aims at thedetermination of a migrated depth image the amplitudes of which carry some information (e.g.,reflection coefficients in true-amplitude migration). In this case a stable migration depth imageis obtained by multiplying the weight related to the specular ray by the unweighted stack result.This multiplication suppresses the values of all ratios away from reflectors where the denominatormay be close to zero. Obviously, one only has to prevent actual divisions by zero in this case, forexample by stabilizing the denominator.

238 CHAPTER 8. FURTHER ASPECTS OF KIRCHHOFF MIGRATION

Chapter 9

Seismic imaging

In this chapter, we quantitatively describe the “true” asymptotic inverse process to diffraction stackmigration, namely isochron-stack demigration. We show that the diffraction-stack and isochron-stack integrals constitute an asymptotic transform pair, which is well interlinked by the dualitytheorems derived in Chapter 5. This transform pair can be used to solve a multitude of true-amplitude target-oriented seismic imaging (or image-transformation) problems. These include, forexample, the dynamic counterparts of the kinematic map-transformation examples qualitativelydiscussed in Chapter 2. All image-transformation problems can be addressed by applying bothstacking integrals in sequence, whereby the macrovelocity model, the measurement configurationor the ray-code of the considered elementary reflections may change from step to step. Alternatively,both stacking procedures can be mathematically combined into one single process. This leads toweighted (Kirchhoff-type) summations along certain stacking surfaces (or inplanats). In this chap-ter, we provide the general formulas for the stacking surfaces and the true-amplitude weights ofthese new Kirchhoff processes. To demonstrate the value of the proposed imaging theory, which isbased on analytically chaining the two stacking integrals, we have solved the true-amplitude config-uration transform and remigration problems for the case of a 3-D isotropic laterally inhomogeneousmedium.

In this central chapter of our book, we present the theoretical background for the geometri-cally motivated mapping and imaging concepts discussed in Chapter 2. The diffraction-stack theoryas presented in Schleicher et al. (1993a) and discussed in detail in Chapter 7 serves as the point ofdeparture to present the theory of true-amplitude isochron-stack demigration following very similarlines. These Kirchhoff-type migration and demigration operations provide the basis for the unifiedimaging theory (Hubral et al., 1996; Tygel et al., 1996) that is the central subject of this book.In the last section of this chapter, we show how the diffraction-stack and isochron-stack integralscan be chained to solve various seismic image-transformation problems. As in the cited papers, wepresent this unified theory in the time domain. An equivalent frequency-domain theory is discussedin Bleistein and Jaramillo (2000) and Bleistein et al. (2001).

Such as the diffraction-stack integral in Chapter 7, also the isochron-stack integral is for-mulated in a way that allows seismic images of arbitrary 3-D measurement configurations to betransformed. The integral will be asymptotically evaluated in the high-frequency range with thestationary-phase method. This will lead to an analytic expression for the demigrated events thatare given in form of a signal strip attached to the reflection-traveltime surface ΓR in the sameway as the migrated signal strip is attached to the target reflector ΣR as discussed in Chapter 7

239

240 CHAPTER 9. SEISMIC IMAGING

(see Figure 7.1). As in the case of the diffraction-stack migration integral, also the isochron-stackdemigration integral is initially formulated to permit an arbitrary weight function. This weight isthen related to that of the diffraction-stack integral as soon as the two integrals are shown to be(asymptotically) inverse operations to each other when applied to the same macrovelocity model,measurement configuration, and elementary wave. Particular emphasis will be put on deriving theformula for the true-amplitude weight. This weight is not only necessary to perform the demigrationin an true-amplitude way, but also any other image transformations as long as the “best possibleamplitudes” are desired.

The principal aim of this chapter (and of the whole book) is not only to provide the isochron-stack integral as an inverse transform to the diffraction-stack integral, but also to show how thisintegral pair can be used to solve a multitude of seismic target-oriented imaging problems in trueamplitude. Such problems can be addressed by explicitly applying both stacking integrals in se-quence (cascaded solution), whereby the macrovelocity model, the ray code of the elementary wave,or the measurement configuration may change from step to step. However, as we will show in thelast section, one stacking procedure can be spared, because the stacking integrals can be chainedanalytically so as to lead to single-stack solutions. The resulting stacking surfaces turn out to bethe inplanats as introduced and defined in Chapter 2.

There exists a large number of seismic image transformation problems that may be solvedwith the proposed theory. For example, one important image transformation procedure, requir-ing the application of both the diffraction-stack and isochron-stack integrals using an identicalmacrovelocity model and identical elementary waves, but different input and output measurementconfigurations, is the 3-D true-amplitude migration to zero offset (MZO) (corresponding to Prob-lem #1 in Chapter 2) or the closely related dip-moveout (DMO) correction. Another, albeit verysimilar task is the 3-D shot-continuation operation (SCO) that transforms the seismic primaryreflections of one 3-D seismic shot record into those of another one for a displaced source location.A comparison of the resulting SCO shot record with that of an actually acquired field record forthe very same displaced shot location allows for the validation (and even the updating) of themacrovelocity model employed for the SCO (Bagaini and Spagnolini, 1993). Both the MZO andthe SCO can be described by the same general image-transformation approach (that was calledProblem #1 in Chapter 2). This general image transformation is also referred to as the “configura-tion transform.” Other configurations transforms include inverse DMO or MZO, offset continuation(transformation of one common-offset section into another one with a different offset), common-shotDMO (transformation of a common-shot section into a zero-offset section), inverse common-shotDMO, azimuth moveout (AMO, transformation of a common-shot section into another one with adifferent azimuth), redatuming, the respective inverse operations, etc.

A somewhat different problem, where both stacks are applied using identical measurementconfigurations and elementary waves but different macrovelocity models in the input and outputspaces, is the 3-D true-amplitude remigration (Problem #2 in Chapter 2). With this operation,an updated image for an improved macrovelocity model (e.g., one taking lateral velocity variationsinto account) can be constructed from a depth-migrated image that was obtained using a differentinitial macrovelocity model (e.g., one employing simple velocity laws where the migration can bevery efficiently realized). Like a configuration transform, also a remigration requires no more thanone single weighted (Kirchhoff-type) summation along the problem-specific inplanats to achieve thedesired image transformation.

9.1. ISOCHRON STACK 241

Moreover, the proposed theory can also be used for elementary-wave transformations (e.g.,a P-S shot record could be changed into a P-P shot record). In the same way, transformations canbe conceived where the elementary wave and the measurement configuration may change at thesame time (e.g., a P-S shot record could be transformed into a P-P zero-offset record). In addition,it should be mentioned that the proposed theory can be used to correct images from previousinaccurate transformations. For example, a correction for lateral velocity changes could be appliedto time sections after a constant-velocity DMO. By an extension of the theory to anisotropic media,corrections for anisotropy could be introduced into sections that were imaged with an isotropicmacrovelocity model.

9.1 Isochron stack

In this section, we introduce the asymptotic inverse to Kirchhoff-type diffraction-stack migration,to which we refer as isochron-stack demigration. The operation is based on completely analogousconsiderations to those that lead to the diffraction-stack integral (7.1.4). The assumptions about themacrovelocity model, measurement configuration, or ray code of the elementary wave are the sameas for the diffraction-stack integral (DSI) in Section 2.1. We consider again some target reflectorΣR and a primary reflected ray joining a source at point S to a receiver at point G via a specularreflection point MR on ΣR (see again Figure 7.1 on page 177).

Let the function Φ(M) be a complex-valued depth-migrated image obtained by any type ofdepth migration (e.g., by the diffraction stack (7.1.4) described above or by any other migrationmethod). This image is assumed to exist at all points M with coordinates (r, z) in a sufficientlylarge part of the depth domain. In particular, the horizontal coordinate vectors r = (r1, r2) of Mare assumed to be confined to a given reflector aperture E. Moreover, let N denote an arbitrary,fixed point with coordinates (ξ, t) in the time-trace domain, with its horizontal coordinate vectorξ = (ξ1, ξ2) specified by the desired output measurement configuration.

In analogy to the procedure for migration, we assume that a weighted modified isochronstack is the appropriate method to perform a true-amplitude demigration. In the following, thisassumption is proved by setting up a certain isochron-stack integral and deriving a weight functionfrom it such that the stack output becomes the ray-theoretical reflection signal in equation (7.1.2).

An isochron stack is then a weighted summation along the isochron surface ΣN defined bythe function z = ZI(r;N) with respect to each point N . This summation can be mathematicallyexpressed in analogy to the diffraction-stack integral (7.1.4) by the integral (Tygel et al., 1996)

Ψ(N) =1

2π

∫

E

∫

d2rKIS(r;N)∂Φ(r, z)

∂zz = ZI(r;N)

. (9.1.1)

In this formula, KIS(r;N) is a kernel (or weight) function to be specified later. We see that theintegral (9.1.1) represents a stack over the (r, z)-domain for all r in E along the isochron z =ZI(r;N) determined by the given point N . For a fixed source-receiver pair (S,G) specified byξ, the isochron ΣN : z = ZI(r;N) is the set of all points MI with coordinates (r,ZI(r;N)), forwhich the traveltime sum along the ray segments connecting S(ξ) to MI and MI to G(ξ) (i.e., thediffraction or Huygens traveltime TD(ξ;MI) as defined before) is constant and equal to the time tspecified by N . Explicitly, we have

TD(ξ;MI) = T (S(ξ),MI) + T (MI , G(ξ)) = t (9.1.2)

242 CHAPTER 9. SEISMIC IMAGING

for a fixed ξ and all points MI with r in E. Equation (9.1.2) implicitly defines the isochron ZI(r;N)determined by N . Note that, as in the case of the diffraction-stack integral (7.1.4), a measurementconfiguration, a macrovelocity model, and a ray code (of the chosen elementary-wave reflector imageto be demigrated) enter also into the isochron-stack integral (9.1.1).

Now, let the function Φ(r, z), representing the depth-migrated reflector image to be dem-igrated, be given by a depth-migrated reflection strip in the (r, z)-domain of the following form

Φ(r, z) = Φ0(r) Fm[m(r)(z −ZR(r))] , (9.1.3)

where (a) the points MR(r, z = ZR(r)) specify the reflector ΣR, where (b) the vertical stretchfactor m(r) is a known, always positive function of r in E, and where (c) the migrated signalFm[.] denotes an analytic wavelet defined for real arguments with the dimension of time. Let themigrated amplitude Φ0(r) be a slowly varying function of z, so that we may neglect its derivative∂Φ0

∂z when compared to the derivative of the source pulse. Then, the isochron-stack integral (9.1.1),which describes the demigration of the depth-migrated strip (9.1.3), becomes

Ψ(N) =1

2π

∫

E

∫

d2rKIS(r;N) Φ0(r) m(r) Fm[m(r)(ZI(r;N) −ZR(r))] , (9.1.4)

where we have applied the chain rule to the z-derivative. As before, the dot denotes the derivativewith respect to the argument time. The value Re Ψ(N) is the demigration output generallyassigned to point N . Note, however, that for a sequential application of the diffraction-stack andthe isochron-stack integrals – as considered later – the analytic (i.e., complex) quantities Φ(M) andΨ(N) are needed. We therefore do not consider Re Ψ(N) but the full complex quantity Ψ(N) asthe demigration output.

9.1.1 Asymptotic evaluation at the reflection-time surface

We will now perform the asymptotic evaluation of the isochron-stack integral (9.1.4) for the casethat the chosen point N lies on the reflection traveltime surface t = TR(ξ) (Figure 7.1). This point,denoted by NR, has the coordinates (ξR, TR(ξR)). It is assumed to remain fixed throughout thefollowing analysis.

The asymptotic evaluation is carried out in full analogy to that of the diffraction-stack integral(7.1.4). Therefore, we refrain from stating all steps explicitly. As for integral (7.1.4), we take theFourier transform of formula (9.1.4) with respect to an artificially introduced time variable, andapply the stationary-phase method in the frequency domain. The result, transformed back into thetime domain and taken at time zero, is

Ψ(NR) ' ΥIS(r∗) Φ0(r∗) Fm[0] , (9.1.5a)

where the amplitude factor is given by

ΥIS(r∗) =KIS(r∗;NR) m(r∗)

|det(H˜IS)|1/2 exp[i

π

2(1 + Sgn(H

˜IS)/2)] . (9.1.5b)

In formulas (9.1.5), r∗ = r∗(ξR) denotes the stationary point (supposed to be uniquely definedin the demigration aperture E) of the function

δIS(r;NR) = m(r)(ZI(r;NR) −ZR(r)) . (9.1.6)

9.1. ISOCHRON STACK 243

In other words, the point MR with horizontal coordinates r∗ is the only one in E that satisfies thecondition

∇rδIS(r;NR)

r= r∗= 0 . (9.1.7a)

This equation thus defines the dual point MR(r∗) of the chosen point NR (see Chapter 5). Sincethe isochron for a point NR and the reflector surface ΣR are, in fact, tangent at MR, we also havethat

δIS(r∗;NR) = 0 . (9.1.7b)

The symbolH˜IS in equation (9.1.5b) designates the Hessian matrix of the function δIS(r;NR) with

respect to r, evaluated at r= r∗, viz.,

H˜IS =

(

∂2

∂ri∂rj[m(r)(ZI(r;NR) −ZR(r))]

)

r = r∗, (9.1.8)

and Sgn(H˜IS) denotes the signature of this matrix. The matrix H

˜IS is assumed to have a nonva-

nishing determinant. A more suitable expression for H˜IS will be given in Section 9.1.4, where the

true-amplitude kernel KIS(r;N) is derived.

9.1.2 Isochron stack in the vicinity of the reflection-time surface

To derive the vertical thickness of the demigrated signal strip, i.e., the length of the pulse attached tothe reflection traveltime surface t = TR(ξ), we follow closely the corresponding case of the evaluationof the diffraction-stack integral (7.1.4) in the vicinity of a point MR on the reflector [see Section8.2]. In the vicinity of the point NR(ξR, TR(ξR)), we consider a point N(ξR, t) that is displaced inthe direction of the t-coordinate. Again, the actual mathematics is completely analogous to whathas been done in Section 8.2. We thus refrain from restating it in detail. Using the correspondingarguments as for the evaluation of the diffraction stack, we find that

Ψ(N) ' ΥIS(r∗) Φ0(r∗) Fm[nI(r

∗;NR)m(r∗)(t− TR(ξR))] , (9.1.9)

with the same amplitude factor ΥIS(r∗) as before, given by equation (9.1.5b). Moreover, nI is thevertical stretch factor of the isochron stack,

nI(r∗;NR) =

∂ZI(r∗;N)

∂tNR

. (9.1.10)

At this point, we make use of the result of the first duality theorem derived in Chapter 5 [equation(5.5.7)]. From that theorem, we know that the vertical stretch factor, nI of the isochron stack atNR is reciprocal to the vertical stretch factor of the diffraction stack at MR. In symbols,

nI(r∗;NR) =

1

mD(ξR;MR(r∗)), (9.1.11)

where MR is the point of tangency between the (known) isochron of point NR and the (unknown)reflector, i.e., the dual point to NR. By substitution of expression (9.1.11) in equation (9.1.9), thefinal result of the isochron-stack demigration becomes

Ψ(N) ' ΥIS(r∗) Φ0(r∗) Fm[

m(r∗)

mD(ξR;MR(r∗))(t− TR(ξR))] . (9.1.12)

244 CHAPTER 9. SEISMIC IMAGING

Formula (9.1.12) expresses the demigrated signal strip (i.e., the simulated reflection) on and in thevicinity of the reflection-time surface TR(ξ) corresponding to a user-specified output measurementconfiguration.

9.1.3 Isochron stack elsewhere

As in the case of the diffraction stack for points significantly displaced from the reflector ΣR, theisochron stack for points N significantly displaced from the reflection-time surface TR(ξ) yields avanishingly small result, either because there is no contribution to the stack from the stationarypoint due to the limited length of the source signal, or because no stationary point exists in thespatial aperture E. The highest order contributions are again the boundary effects that should bereduced by tapering.

9.1.4 True-amplitude kernel

Up to now, the kernel or weight function in integral (9.1.1) has been left unspecified. For itsdefinition, we employ the condition that demigration must be the inverse process to migration. Inother words, a demigration [by the isochron stack (9.1.1)] must undo what the migration [by thediffraction stack (7.1.4)] has done to the original seismic primary reflections in the (ξ, t)-domain,provided (a) the same macrovelocity model, (b) the same measurement configuration, and (c) thesame elementary wave are used in both operations. The weight functions in both stacks musttherefore relate to each other. Of particular interest is the determination of the true-amplitudeweight of the isochron stack (9.1.1), which is the counterpart to the true-amplitude weight (7.2.26)of the diffraction stack (7.1.4).

The derivation of the TA kernel for the isochron stack (9.1.1) is done in parallel to the onefor the diffraction stack in Chapter 7. We start from the result of the asymptotic evaluation inequation (9.1.12) for a point NR on the reflection traveltime surface.

Our first aim is to simplify expression (9.1.5b) for ΥIS(r∗). For that purpose, we must derivea suitable expression for H

˜IS . We start by considering equations (9.1.7). From these equations

together with the property mD(ξ, r) > 0, one can readily verify that r∗ also defines the stationarypoint of the function

Z∆ = ZI(r;NR) −ZR(r) =1

m(r)δIS(r;NR) , (9.1.13)

i.e., we have

∇rZ∆(r;NR)

r= r∗= 0 (9.1.14a)

and furthermore,

Z∆(r∗;N) = 0 . (9.1.14b)

These equations express the tangency of the isochron z = ZI(r;NR) and the reflector z = ZR(r)at the point MR with coordinates (r∗,ZR(rR)) that is dual to NR. They have already been provenin connection with the first duality theorem in Chapter 5 (see also Tygel et al., 1995).

9.1. ISOCHRON STACK 245

To relate H˜IS to known quantities, we use the Hessian matrix Z

˜∆ of Z∆(r;NR), taken with

respect to horizontal coordinates r and evaluated at r = r∗ (see also Chapter 5). Upon the use ofequations (9.1.7), we can readily verify from equations (9.1.8) and (9.1.14) that

H˜IS = m(r∗) Z

˜∆ . (9.1.15)

The second duality theorem of Chapter 5, equation (5.4.2b), claims that

Z˜

∆ = − 1

mD(ξ;MR)H˜

Σ , (9.1.16)

where expression (5.6.10a) has been used. Combining equations (9.1.15) and (9.1.16), we find

H˜IS = − m(r∗)

mD(ξ;MR)H˜

Σ . (9.1.17)

Using equation (5.6.7), and recalling that mD(ξ;MR) > 0, we are able to relate H˜IS to the Fresnel

matrix H˜F . We obtain for the determinant of H

˜IS

det(H˜IS) = m2(r∗) det(Z

˜∆)

=m2(r∗)

m2D(ξ;MR)

det(H˜

Σ)

=m2(r∗)

m2D(ξ;MR) cos2 βR

det(H˜F ) , (9.1.18a)

as well as for its signature

Sgn(H˜IS) = Sgn(Z

˜∆) = − Sgn(H

˜Σ) = − Sgn(H

˜F ) . (9.1.18b)

These are the desired expressions for H˜IS .

Insertion of the above equations (9.1.18) into expression (9.1.5b) for ΥIS(r∗) yields

ΥIS(r∗) =KIS(r∗;NR)mD(ξ;MR) cos βR

|det(H˜F )|1/2 exp[i

π

2(1 − Sgn(H

˜F )/2)] . (9.1.19)

To derive a suitable expression for the TA kernel KIS(r;N), it is useful to write this last equationin terms of the Fresnel geometrical-spreading factor LF (MR) defined by equation (6.2.8). We find

ΥIS(r∗) =OD

OFKIS(r∗;NR) LF (MR) cos2 βR . (9.1.20)

The result of an isochron-stack demigration can thus be conveniently represented by equation(9.1.12) with ΥIS(r∗) given by formula (9.1.20). Note that for monotypic reflections, the ratioOD/OF = 2.

To obtain the TA kernel KIS(r;N) for the isochron stack (9.1.1), we proceed as follows. As inthe derivation of the TA diffraction-stack weight (7.2.26), we make use of the fact that we know thedesired true-amplitude result. Since a true-amplitude migration removes the geometrical-spreadingfactor L from the data amplitudes, we want a true-amplitude demigration to reintroduce it. Bymodifying equation (9.1.12) accordingly, we can formulate the desired result as

Ψ(N) ' Φ0(r∗)

L(ξR)Fm[

m(r∗)

mD(ξR;MR(r∗))(t− TR(ξR))] . (9.1.21)

246 CHAPTER 9. SEISMIC IMAGING

A comparison of the isochron-stack result (9.1.12), incorporating ΥIS(r∗) as specified by formula(9.1.20), with the desired true-amplitude result (9.1.21) reveals that the true-amplitude isochron-stack kernel must be chosen such that at the stationary point, it satisfies

KIS(r∗;NR) =OF

OD LS(ξR, r∗)LG(ξR, r

∗) cos2 βR. (9.1.22)

Here, the decomposition formula (6.2.14) for the geometrical-spreading factor L has been used.Factors LS(ξ, r∗) and LG(ξ, r∗) denote the point-source geometrical spreading for the ray segmentsfrom S(ξR) to MR and from MR to G(ξR), respectively.

In analogy to the diffraction stack, we call any expression for the isochron-stack kernelKIS(r;N) that, at NR and r∗, satisfies equation (9.1.22) a “true-amplitude kernel.” The mostnatural extension of formula (9.1.22) to arbitrary points N , lying or not on the reflection timesurface t = TR(ξ), is given by the following operations. Let (S,G) be specified by the ξ-coordinateof point N and let MI be an arbitrary point with the horizontal coordinate r on the isochronz = ZI(r;N) specified by N , i.e., MI has the coordinates (r,ZI(r;N)). Then, the true-amplitudeweight function can be expressed as

KIS(r;N) =OF

OD LS(ξ, r) LG(ξ, r) cos2 βM, (9.1.23)

where LS(ξ, r) and LG(ξ, r) denote the point-source geometrical-spreading factors for the ray seg-ments SMI and MIG with a point source at S and MI , respectively. Also, OF and OD are theFresnel and depth obliquity factors at MI as defined in equations (4.6.4) and (5.6.9), respectively.Note that for monotypic reflections, OF /OD = 1/2. In this case, equation (9.1.23) reduces to theexpression of Jaramillo et al. (1998).

The last quantity in equation (9.1.23) to be explained is βM . This is the dip angle of theisochron, i.e., the angle that the specular-normal direction makes with the vertical axis at MI .Because of the tangentiality between the isochron and the reflector in the case that MI coincideswith a reflector point MR, this choice guarantees that the stationary value of βM equals the reflectordip βR at MR. Here, the specular-normal direction is that direction which divides the total anglebetween the incident and reflected ray segments SMI and GMI according to Snell’s law. Note thatthis direction is easily determined since the sum of the slowness vectors of these ray segments,i.e., the gradient of TD(ξ;M), points in this direction. For a monotypic reflection, the specular-normal direction is the half-angle direction between the ray segments SMI and GMI . Thus, forany arbitrary depth point MI , the angle βM can be determined from

cos βM =mD

OD=i3 · ∇TD(ξ;MI)

|∇TD(ξ;MI)|, (9.1.24)

where i3 is the global unit vector in the direction of the vertical axis. We can use equation (9.1.24)to eliminate cos βM from equation (9.1.23). In this way,

KIS(r;N) =OF OD

LS(ξ, r) LG(ξ, r)m2D(ξ;MI)

(9.1.25)

is an alternative representation of the true-amplitude weight for an isochron-stack demigration.

Application of the isochron stack (9.1.1) to the depth-migrated image represented by functionΦ(r, z) that is given in equation (9.1.3) using the true-amplitude kernel (9.1.23) yields then indeed

9.2. DIFFRACTION- AND ISOCHRON-STACK CHAINING 247

the desired result

Ψ(N) ' Φ0(r∗)

L(ξ)Fm[

m(r∗)

mD(ξ;MR(r∗))(t− TR(ξ))] . (9.1.26)

The demigration result (9.1.26) admits the following interpretation in terms of fictitious operations:Let a point N with coordinates (ξ, t) be given in the time-trace domain. Then, (a) find the pointon the reflection traveltime surface TR(ξ) of the target reflector ΣR that has the same coordinateξ as N . Call this point NR. It has the coordinates (ξ, TR(ξ)). Now, (b) find the specular reflectionray determined by the source-receiver pair (S(ξ), G(ξ)), i.e., find the stationary coordinate vectorr∗ = r∗(ξ) so that S(ξ)MR(r∗)G(ξ) constitutes the reflection ray for this source-receiver pair. Now,for this ray, (c) compute the point-source geometrical-spreading factor L(ξ) and the vertical stretchfactor mD(ξ;MR(r∗)) and (d) divide the original amplitude and phase by L(ξ) and mD(ξ;MR(r∗)),respectively. Finally, (e) place the so obtained true-amplitude reflection at N . The true-amplitudeisochron stack (9.1.1) performs the fictitious operations (a) to (e) in one imaging step.

Let us now assume that the depth-migrated signal strip Φ(r, z), which is to be demigrated bythe isochron stack (9.1.1), was the result of an arbitrarily weighted diffraction stack based on thesame macrovelocity model, the same measurement configuration, and the same elementary wave.Let us consider the diffraction-stack and isochron-stack for the dual pointsMR andNR, respectively.We observe that in this situation the wavelet is identical in both stacks, i.e., Fm[·] = F [·], and alsothat m(r∗(ξ)) = mD(ξ;MR(r∗)). Note that this latter equality also justifies our assumption thatm(r) > 0 for all r ∈ E, because we have already observed mD(ξ;MR(r∗)) to fulfill this condition.Therefore, the result (9.1.26) of the true-amplitude isochron stack reduces to the simple expression

Ψ(NR) ' Φ0(r∗)

L F [t− TR(ξ)] . (9.1.27)

Here Φ0(r) may still be an arbitrary function of r. If it is specified as the result of a true-amplitudemigration, i.e., according to equation (7.1.14b) as Φ0(r

∗) = Rc(ξ), the demigration result (9.1.27)clearly reveals that the true-amplitude isochron stack (9.1.1), with the weight function (9.1.23),recovers indeed the original seismic reflection (7.1.2) in the (ξ, t)-domain, when applied to the true-amplitude depth-migrated signal strip. In other words, the isochron-stack demigration representsthe asymptotic inverse operation to diffraction-stack migration.

9.2 Diffraction- and isochron-stack chaining

In Chapter 2, we have already indicated from a geometrical point of view how the diffraction andisochron stacks could possibly be combined in order to solve a number of image-transformationproblems. In this section, we are going to explicitly perform this combination for the problemsexemplified in Chapter 2. As we will see below, the mathematical derivations will heavily rely onthe geometrical picture we have developed in Chapter 2.

We will begin with the treatment of the configuration transform (Problem #1 in Chap-ter 2), i.e., we will analytically chain the two stacking integrals (7.1.4) and (9.1.1) for the directtransformation of a seismic record for a certain measurement configuration to that of a differentconfiguration. We will continue to use the MZO as an example for this transformation. In the sameway, (Kirchhoff-type) stacking integrals can be found for all other types of image-transformationproblems including DMO, MZO, SCO, CS-DMO, OCO, AMO, VSP transformation, redatuming,

248 CHAPTER 9. SEISMIC IMAGING

and elementary-wave transformations, e.g., the transformation of a P-S reflection to a P-P reflec-tion, the inverses of these processes, or combinations thereof, etc. Thereafter, we will chain bothintegrals to achieve the remigration (Problem #2 in Chapter 2), i.e., the direct transformation ofa depth-migrated image obtained for a certain macrovelocity model to the image for a different,e.g., updated macrovelocity model. The respective true-amplitude weights that are needed for thesetrue-amplitude single-stack image transformations can be derived using the true-amplitude weights(7.2.26) and (9.1.23) of the diffraction and isochron stacks. Alternatively, they can be obtained bysetting up the corresponding integrals kinematically and designing the weights such that the outputamplitudes satisfy certain desired true-amplitude conditions.

9.2.1 Chained solutions of Problem #1

In order to find a solution for Problem #1 (i.e., the configuration transform), we consider a fixedmacrovelocity model and a fixed elementary wave, but two different measurement configurations.The source-receiver pairs (S,G) and (S, G) of both measurement configurations are specified by2-D vector parameters ξ and η, respectively, that vary on respective aperture sets A (in the inputspace) and A (in the output space). Throughout this section, we denote quantities pertainingto the output configuration by a tilde to distinguish them from those pertaining to the inputconfiguration. In this way, we write U(ξ, t) for the input data. For a migration to zero offset, thiswould be the common-offset configuration. The desired simulated seismic record in the outputspace, is accordingly represented by U(η, τ). For an MZO, this would be the (simulated) zero-offsetrecord.

The obvious way to achieve a configuration transform is to perform a diffraction-stack mi-gration and an isochron-stack demigration in sequence, i.e., to firstly migrate the data and thendemigrate the resulting migrated image. In this section, we mathematically chain the correspond-ing integral operators. Our aim is to obtain simplified expressions for the total process so thatits realization requires less computational effort than the sequenced application of migration anddemigration.

Cascaded solution

According to equation (9.1.1), the demigrated output U(η, τ) can be obtained by an isochron stackapplied to a given depth-migrated image Φ(r, z). In other words, for each point N with coordinates(η, τ) in the output space, we have

U(η, τ) =1

2π

∫

E

∫

d2r KIS(r; N)∂Φ(r, z)

∂zz = ZI(r; N)

. (9.2.1)

Note that the isochron z = ZI(r; N ) as well as the integral kernel KIS(r; N) must be computedusing the output-space configuration and are, thus, marked with a tilde.

Now, let the depth-migrated image Φ(r, z), to which the isochron stack (9.2.1) is applied,be given by the result of a diffraction stack as represented by integral (7.1.4), performed on theoriginal data U(ξ, t). In symbols,

Φ(r, z) =−1

2π

∫

A

∫

d2ξ KDS(ξ;M)∂U(ξ, t)

∂tt = TD(ξ;M)

. (9.2.2)

9.2. DIFFRACTION- AND ISOCHRON-STACK CHAINING 249

Note that KDS(ξ;M) and TD(ξ;M) refer to the true-amplitude weight function and the Huygenssurface at M(r, z), respectively, corresponding to the input-space configuration.

It is now our aim to substitute equation (9.2.2) into equation (9.2.1) in order to expressU(η, τ) as a cascaded (chained) stack over U(ξ, t). However, as a preliminary step, we have to takecare of the derivative of Φ(r, z) with respect to z, which we move into the integral (9.2.2) to obtain

∂Φ(r, z)

∂z=

−1

2π

∫

A

∫

d2ξ∂

∂z

KDS(ξ;M)∂U(ξ, t)

∂tt = TD(ξ;M)

. (9.2.3)

Applying the product rule to the z-derivative in equation (9.2.3), it can be split into two terms, onecontaining the z-derivative of KDS(ξ;M) and the other one containing the z-derivative of ∂U/∂t.The latter can, by the use of the chain rule, be seen to be proportional to ∂ 2U/∂t2. The proportion-ality factor is just the vertical stretch factor mD defined in equation (5.3.15). Assuming, accordingto ray theory, that the reflection U(ξ, t) is of a high-frequency content, we can approximate thetotal z-derivative by its second term only, namely

∂

∂z

KDS(ξ;M)∂U(ξ, t)

∂tt = TD(ξ;M)

'

' KDS(ξ;M) mD(ξ;M)∂2U(ξ, t)

∂t2t = TD(ξ;M)

. (9.2.4)

We know already that the quantity mD(ξ;M), as given by equation (5.3.15), is always positive.Hence the integral (9.2.3) for the z-derivative of the depth-migrated output can be convenientlyapproximated by

∂Φ(r, z)

∂z' −1

2π

∫

A

∫

d2ξ KDS(ξ;M) mD(ξ;M)∂2U(ξ, t)

∂t2t = TD(ξ;M)

. (9.2.5)

Now we are ready to insert equation (9.2.5) into the isochron-stack integral (9.2.1). We obtain whatwe call the cascaded solution of Problem #1 as

U(η, τ) =−1

4π2

∫

E

∫

d2r

∫

A

∫

d2ξ KCC(ξ, r; N )∂2U(ξ, t)

∂t2t = TCC(ξ, r; N)

, (9.2.6)

where we have introduced the notations

TCC(ξ, r; N) = TD(ξ; MI(r; N)) (9.2.7a)

and

KCC(ξ, r; N ) = KIS(r; N) KDS(ξ; MI(r; N)) mD(ξ; MI(r; N)) (9.2.7b)

for the composite traveltime and weight functions, respectively, of the chained or cascaded con-figuration (CC) transform. Also, MI(r; N) represents, for varying r, all points on the isochronz = ZI(r; N) in the depth domain defined by N with coordinates (η, τ) and calculated with respectto the output measurement configuration. In other words, a point MI(r; N) has the coordinates(r, ZI(r; N)). Consequently, the composite traveltime function TCC(ξ, r; N) of equation (9.2.7a) is

250 CHAPTER 9. SEISMIC IMAGING

−1000 −500 0 500 10000

200

400

600

800

Midpoint coordinates ξ, η [m]T

ime

[ms]

ÑΓ

RZO

ΓRCO

ΓMCO(M

I)~

CO/ZO time−trace domain

−1000 −500 0 500 1000

0

200

400

600

800

Distance [m]

Dep

th [m

] ZO isochron MI

~

ΣR

Depth domain

Fig. 9.1. Visualization of the geometry of the cascaded configuration-transform integral, equations(9.2.6), for the case of a constant-velocity MZO. Top: A point N (diamond) is chosen in the outputzero-offset time-trace domain. For each point MI on its zero-offset isochron, the Huygens curveΓCOM (MI): t = TD(ξ; MI(r; N )) (solid lines) is constructed in the common-offset time-trace domain.The stack (9.2.6) is performed along all these Huygens curves. Bottom: The zero-offset isochronΣN : z = ZI(r; N) (bold line) of point N is constructed in the depth domain. An arbitrary pointon ΣN is denoted by MI (plus signs). Also indicated by a dashed line is the true position of thereflector ΣR.

the set of all Huygens surfaces, calculated with the input configuration, of all points MI(r; N) onthe isochron z = ZI(r; N ) of N , calculated with the input configuration.

Figure 9.1 visualizes the situation for the case of a constant-velocity MZO in the same simple2-D model used in the figures of Chapter 2. For a given point N in the output zero-offset time-tracedomain, the zero-offset isochron ΣN : z = ZI(r; N) is constructed in the depth domain. For eachpoint MI on this zero-offset isochron ΣN , the Huygens curve ΓCOM (MI): t = TD(ξ; MI(r; N)) isconstructed in the common-offset time-trace domain. The stack (9.2.6) is performed along all theseHuygens curves.

As we can see, the cascaded solution (9.2.6) consists of a weighted diffraction-stack depth mi-gration of the modified original input data (inner integral) followed by an unweighted isochron-stackdemigration of the resulting depth-migrated image (outer integral). In other words, all amplitudemanipulations of the cascaded operation have been entirely moved to the migration operation,so that the subsequent demigration can be carried out without a weight. Of course, the kernelKCC(ξ, r; N) may, if no attention is given to amplitudes, be chosen completely arbitrary or evenbe omitted. If, however, a true-amplitude transformation is desired, the necessary weight function

9.2. DIFFRACTION- AND ISOCHRON-STACK CHAINING 251

is given by equation (9.2.7b) with KDS(ξ;M) and KIS(r; N) specified in equations (7.2.26) and(9.1.23), respectively. Then, the weight function (9.2.7b) reads

KCC(ξ, r; N) =OF

OD LS(ξ, r) LG(ξ, r) cos2 βM

ODS

ODhB LSLG mD(ξ; MI(r; N ))

=OF

OF

hB

mD(η; MI(r; N ))

LSLGLSLG

cos βM

cos βM, (9.2.8)

where we have used equations (5.6.9a) and (7.3.3). In the above expressions, OF , OD and ODS

are the Fresnel, depth, and diffraction-stack obliquity factors defined by equations (4.6.4), (5.6.9)and (7.2.24), respectively, and hB is the Beylkin determinant. All these quantities are calculatedat MI(r; N) using the input configuration. Correspondingly, OF and OD are the Fresnel and depthobliquity factors as calculated with the output configuration. Moreover, LS, LG and LS , LG arethe geometrical-spreading factors of the source and receiver ray segments in the input and outputconfigurations, respectively.

It is worthwhile to observe that the ratio of the cosines in equation (9.2.8) needs not be takeninto consideration. All that is needed for a true-amplitude kernel is that it assumes the correctvalue at the stationary point of the stacking integral. At the stationary point of integral (9.2.6),both isochron dip angles βM and βM in the input and output configurations are equal to the actualreflector dip angle, βR. Thus, at the stationary point, cos βM

cos βM

= 1. Therefore, this ratio can be

omitted in weight function (9.2.8), i.e., the use of the modified weight function

KCC(ξ, r; N) =OF

OF

hB

mD(η; MI(r; N ))

LSLGLSLG

(9.2.9)

in the stacking integral (9.2.6) will yield the same high-frequency result.

It remains to be demonstrated that equation (9.2.6) represents indeed a true-amplitude con-figuration transform, if the kernel function of equation (9.2.9) is used. As proven in Section H.1of Appendix H, the cascaded operation (9.2.6) is asymptotically approximated at a point N withcoordinates (η, τ) as

U(η, τ) ' L(ξR)

L(η)U0(ξR) F [

mD(ξR;MR)

mD(η;MR)(τ − TR(η))] , (9.2.10)

where the reflection point MR with coordinates (rR,ZR(rR)) on ΣR is specified by the stationarypoint, rR = rR(η), of the r-integral in equation (9.2.6). Point MR, in turn, is dual to the point NR

in the output configuration, which is the point on the reflection-time surface τ = TD(η;M) thathas the same coordinate η as N . Moreover, ξR = ξR(rR(η)) is the stationary point of the ξ-integralin equation (9.2.6). It specifies the point NR in the input configuration that is dual to this pointMR. The vertical stretch factors in equation (9.2.10) are given by

mD(ξR;MR) =∂TD(ξR;M)

∂zM = MR

, (9.2.11a)

mD(η;MR) =∂TD(η;M)

∂zM = MR

. (9.2.11b)

Also, L and L are the geometrical-spreading factors of the two reflection rays S(ξR)MRG(ξR) andS(η)MRG(η) reflected at MR in the two measurement configurations. The output trace of the cas-caded operation (9.2.6) at η is represented by the same analytic signal as recorded at ξ. However,

252 CHAPTER 9. SEISMIC IMAGING

it is stretched by the ratio of vertical stretch factors that correspond to the different measure-ment configurations and rescaled with the ratio of the corresponding geometrical-spreading factors.Substitution of equation (7.1.2b) yields the final result of the cascaded configuration transformoperation as

U(η, τ) ' Rc(ξR)

L(η)F [mD(ξR;MR)

mD(η;MR)(τ − TR(η))] . (9.2.12)

Note, however, that the reflection coefficient is not transformed, i.e., it remains the one pertaining tothe input configuration. This is a desired effect as it enables an AVO analysis in the output sectionafter any arbitrary configuration transformation. Equation (9.2.12) proves that the cascaded oper-ation (9.2.6) is indeed a true-amplitude image transformation, provided the true-amplitude weightfunctions (7.2.26) and (9.1.23) are used to determine the integral kernel KCC(ξ, r; N ) of equation(9.2.7b). Note again that KCC(ξ, r; N ) is independent of any reflector properties (in particular itscurvature) and can thus be computed from the macrovelocity model without any knowledge of thereflector to be imaged.

In order to better appreciate the important result (9.2.12), we may consider the followingsequence of fictitious operations: Let the point N with coordinates (η, τ) in the output time-tracedomain be given. Referring to the output measurement configuration, (a) find that point on thetraveltime surface ΓR of the target reflector ΣR which has the same coordinate η as N . Call this pointNR. It has the coordinates N(η, TR(η)). Now, (b) find the point MR with coordinates (rR,ZR(rR))on the reflector ΣR so that the ray S(η)MRG(η) is a specular reflection ray, i.e.,MR is the dual pointto NR, and (c) find for the reflection ray S(η)MRG(η) the point-source geometrical-spreading factorL(η) and the vertical stretch factor mD(η;MR). Now use the input measurement configuration to(d) find the source-receiver pair (S(ξR), G(ξR)) for which the composite ray SMRG constitutesa reflection ray and (e) for this ray, compute the point-source geometrical-spreading factor L(ξR)and the vertical stretch factor mD(ξR;MR). This explains all quantities found in equation (9.2.10).Finally, (f) scale and stretch the input space reflection according to equation (9.2.10) and (g) placethe result into point NR in the output space. The cascaded configuration transform (9.2.6) performsthe fictitious operations (a) to (g) in two imaging steps.

As a final check, it remains to see what happens to integral (9.2.6) if the output configurationis the same as the input one. Removing the tilde, which signifies quantities pertaining to the outputconfiguration, we obtain, in parallel to the derivation in the first section of Appendix H,

U(η, τ) = U0(ξ) F [t− TR(ξ)] , (9.2.13)

which is, according to equation (7.1.2a), equal to U(ξ, t). The cascaded operation for identicalconfigurations therefore turns out to be the identity transform as it should be. This also proves againour above statement that the diffraction stack (7.1.4) and isochron stack (9.1.1) are asymptoticallyinverse operations to each other (i.e., they constitute an asymptotic transform pair).

However, equation (9.2.6) is not the final answer to our problem of finding a simple operatorfor the configuration transform. As indicated above, there exists a more attractive solution, inwhich the configuration transform is achieved by one single stacking procedure along inplanats inthe input space. This operation, as we will see in the next section, is completely analogous to theKirchhoff migration and demigration operations (7.1.4) and (9.1.1).

9.2. DIFFRACTION- AND ISOCHRON-STACK CHAINING 253

Single-stack solution

Now we are ready to derive the single-stack configuration transform. For that purpose, we inter-change the order of integrations in equation (9.2.6). Geometrically, this means to reorganize thesumming strategy. Instead of following the individual Huygens surfaces t = TD(ξ; MI(r; N )) oneby one, now all contributions at a single ξ are summed in the t direction and then, the results aresummed along the ξ axis (see also Figure 9.1). Upon the use of formula (7.1.2a), this leads to theexpression

U(η, τ) =−1

4π2

∫

A

∫

d2ξ U0(ξ)

∫

E

∫

d2rKCC(ξ, r; N ) F [TCC(ξ, r; N) − TR(ξ)] . (9.2.14)

Let us denote by Ix(ξ; N) the inner integral in equation (9.2.14), viz.,

Ix(ξ; N ) =1

2π

∫

E

∫

d2rKCC(ξ, r; N ) F [TCC(ξ, r; N) − TR(ξ)] . (9.2.15)

If it is possible to asymptotically evaluate this integral analytically, we will find the desired single-stack solution by inserting the result into equation (9.2.14). However, it is not as easy as beforeto interpret integral (9.2.15) geometrically. Since ξ is fixed in integral (9.2.15), it represents a sumover a single data trace at ξ, taken at different times TCC(ξ, r; N) as a function of r. These timesare defined, for each r, by the Huygens surface TD(ξ; MI(r; N)) of the depth point MI(r; N ) withcoordinates (r, ZI(r; N)). As we will see below, integral (9.2.15) has the same effect as an isochronstack for the output configuration, where the isochron of the input configuration plays the role ofthe reflector.

To evaluate integral (9.2.15) asymptotically, we need to look for the stationary point of thetraveltime difference

δCC(ξ, r; N ) = TCC(ξ, r; N ) − TR(ξ) = TD(ξ; MI(r; N)) − TR(ξ) , (9.2.16)

this time for a fixed ξ in A with respect to varying r. The stationarity condition reads now

∇rδCC(ξ, r; N)

r= rCT

= 0 , (9.2.17a)

or

∇rTCC(ξ, r; N )

r= rCT

= 0 , (9.2.17b)

where the second equation follows immediately from the observation that TR(ξ) is a constant fora fixed ξ. We observe that the coordinate vector rCT of the stationary point of integral (9.2.15)depends on ξ and η. It describes the point MCT with coordinates (rCT ; N), where the two isochronsfor the two configurations have the same inclinations. Note that condition (9.2.17b) is exactly themathematical description of the envelope of the family of Huygens surfaces TD(ξ; MI(r; N)) for allpoints MI on the isochron of N as geometrically described in Chapter 2.

The Hessian matrix H˜CC of TCC(ξ, r; N) with respect to r at the point rCT , which appears

in the stationary-phase evaluation of integral (9.2.15), can be computed in full analogy to H˜IS of

the isochron stack as derived in Section 9.1.4 [see equation (9.1.15)]. As shown in Appendix I, wefind [see equation (I-11)]

H˜CC = mD(ξ;MCT )

(

Z˜I −Z

˜I

)

, (9.2.18)

254 CHAPTER 9. SEISMIC IMAGING

where mD(ξ;MCT ) is the stretch factor taken at the point MCT , and where Z˜I and Z

˜I are the

Hessian matrices of the isochron functions z = ZI(r;N) and z = ZI(r; N) in the input and outputconfiguration, respectively. Taking into account the above expression for H

˜CC , we find for integral

(9.2.15) the asymptotic evaluation

Ix(ξ, N) ' exp−iπ2 [1 − Sgn(Z˜I −Z

˜I)/2]

mD(ξ;MCT ) |det(Z˜I −Z

˜I)|

1

2

×

× KCC(ξ, rCT (ξ); N) F [TCC(ξ, rCT ; N ) − TR(ξ)] . (9.2.19)

Now, by substituting equation (9.2.19) into the chained integral expression (9.2.14), we finallyarrive at the desired single-stack solution of Problem #1, which we have called the “configurationtransform.” It reads

U(η, τ) =−1

2π

∫

A

∫

d2ξ U0(ξ) KCT (ξ; N) F [TCT (ξ; N ) − TR(ξ)] , (9.2.20)

where we have introduced the inplanat TCT (ξ; N) and the true-amplitude kernel KCT (ξ; N) of theconfiguration transform (CT) with respect to a point N with coordinates (η, τ) in the output space.The inplanat is given by

TCT (ξ; N) = TCC(ξ, rCT ; N ) , (9.2.21)

where TCC(ξ, r; N) is defined in equation (9.2.7a), and where rCT is determined by equation(9.2.17).

We observe that the inplanat TCT (ξ; N) is exactly the surface we have geometrically inferredin Chapter 2. It is the envelope of the family of diffraction traveltimes for all points MI on theisochron of the output point N . Equations (9.2.7a), (9.2.17), and (9.2.21) define this envelope inmathematical terms. In other words, if the isochron z = ZI(r; N ), constructed for point N using theoutput measurement configuration, is taken as a reflector, then the configuration-transform inplanatt = TCT (ξ; N) is its traveltime surface with respect to the input measurement configuration. Thisprovides us with a rule on how to determine the inplanat from standard traveltime tables asconventionally used in Kirchhoff migration.

Figure 9.2 visualizes the situation for the case of a constant-velocity MZO in the same simple2-D model used in the figures of Chapter 2. For a given point N in the output zero-offset time-tracedomain, the zero-offset isochron ΣN : z = ZI(r; N ) is depicted in the depth domain. Also depictedare the Huygens curves ΓCOM (MI): t = TD(ξ; MI(r; N)) in the common-offset time-trace domain forsome selected points MI on this zero-offset isochron ΣN . The stack (9.2.6) is performed along theenvelope t = TCT (ξ; N) of all these Huygens curves, i.e., the MZO inplanat. Because in Figure 9.2,the point N was chosen on the (unknown) zero-offset traveltime curve ΓZOR , the corresponding MZOinplanat is tangent to the common-offset traveltime curve ΓCOR .

The weight factor KCT (ξ; N) in equation (9.2.20) is given by

KCT (ξ; N) = KCC(ξ, rCT (ξ); N)exp−iπ2 [1 − Sgn(Z

˜I −Z

˜I)/2]

mD(ξ;MCT ) |det(Z˜I −Z

˜I)|

1

2

, (9.2.22)

provided KCC(ξ, r; N), as specified in equation (9.2.7b) with the true-amplitude kernelsKDS(ξ; MI(r; N)) and KIS(r; N), is used. Note that the weight KCT (ξ; N) in equation (9.2.22)is totally determined without any quantities that depend on the reflector. We observe that this

9.2. DIFFRACTION- AND ISOCHRON-STACK CHAINING 255

−1000 −500 0 500 10000

200

400

600

800

Midpoint coordinates ξ, η [m]T

ime

[ms]

Ñ

MZO inplanat

ΓRZO

ΓRCO

ΓMCO(M

I)~

CO/ZO time−trace domain

−1000 −500 0 500 1000

0

200

400

600

800

Distance [m]

Dep

th [m

] ZO isochron MI

~

ΣR

Depth domain

Fig. 9.2. Visualization of the geometry of the single-stack configuration-transform integral, equation(9.2.23), for the case of a constant-velocity MZO. Top: A point N (diamond) is chosen in the outputzero-offset time-trace domain. Also depicted are the Huygens curves ΓCOM (MI): t = TD(ξ; MI(r; N))in the common-offset time-trace domain (solid lines) for some selected points MI on its zero-offsetisochron. The stack (9.2.23) is performed along the envelope of all these Huygens curves, the MZOinplanat (bold line). Bottom: The zero-offset isochron ΣN : z = ZI(r; N ) (bold line) of point N inthe depth domain. An arbitrary point on ΣN (plus signs) is denoted by MI . Also indicated by adashed line is the true position of the reflector ΣR.

is a common property to all Kirchhoff-type true-amplitude imaging processes. This is a crucialobservation as it is this property that makes it possible to compute the weight function purely fromthe given macrovelocity model without any knowledge about the seismic reflector to be imaged.

Using the zero-order ray solution (7.1.2a) for U(ξ, t) in the input space, the single-stacksolution (9.2.20) for the construction of U(η, τ) in the output space can be recast into the generalform

U(η, τ) =−1

2π

∫

A

∫

d2ξ KCT (ξ; N )∂U(ξ, t)

∂tt = TCT (ξ; N )

. (9.2.23)

We observe that this configuration transform has the same general structure as the diffraction-stack integral (7.1.4) or the isochron-stack integral (9.1.1), with the appropriate replacements ofthe weight function and stacking surface (inplanat), respectively. Note that the correspondinginverse configuration transformation can be achieved by the same formula, where the roles of theinput and output spaces are interchanged.

To demonstrate that equation (9.2.23) represents in fact a true-amplitude configuration trans-form, one has to evaluate it asymptotically by the stationary-phase method as applied previously.

256 CHAPTER 9. SEISMIC IMAGING

The actual calculus is very similar to what has been done in Section H.1 of Appendix H for integral(9.2.6). After some tedious algebraic manipulations involving the duality relations of Chapter 5,this evaluation leads again to the important result (9.2.12), which makes an true-amplitude configu-ration transform well understood in analytical terms. In other words, the single-stack configurationtransform (9.2.23) performs the above explained fictitious operations (a) to (g) in one single imagingstep.

9.2.2 Chained solutions of Problem #2

In this section, we address Problem #2 (i.e., the remigration problem) in terms of the unifiedtrue-amplitude imaging theory. The derivations can be done along very similar lines as above forProblem #1. We therefore refrain from repeating all steps but only state the formulas correspondingto equations (9.2.6) and (9.2.23) for this case. Let (r, z) and (ρ, ζ) denote the global coordinates ofthe input and output image spaces, respectively. We assume that a migrated image Φ(r, z), obtainedby an arbitrary (true-amplitude) migration computed with a certain inaccurate velocity field v(r, z)(the original macrovelocity model in the input space), is already available. We are searching for amore accurate image Φ(ρ, ζ) at a given point M with coordinates (ρ, ζ) using an updated, moreaccurate macrovelocity model v(ρ, ζ) (the improved macrovelocity model in the output space). Incorrespondence to Problem #1, the tilde denotes quantities computed using the output velocitymodel.

Cascaded solution

The cascaded remigration (CR) operation at the point M with coordinates (ρ, ζ) becomes [compareto equation (9.2.6)]

Φ(M ) =−1

4π2

∫

A

∫

d2ξ

∫

E

∫

d2rKCR(ξ, r; M )∂2Φ(r, z)

∂z2

z = ZCR(ξ, r; M )

, (9.2.24)

where the composite stacking surface z = ZCR(ξ, r; M) is given by

ZCR(ξ, r; M ) = ZI(r;ND) . (9.2.25a)

Here, we have introduced the notation ND for an arbitrary point on the diffraction-time surfacet = TD(ξ; M ) in the output model, i.e., ND has the coordinates (ξ, TD(ξ; M )). In other words, forvarying ξ, the points ND describe the diffraction-time surface corresponding to M with respect tothe output model. Consequently, the composite function ZCR(ξ, r; M ) describes the set of isochrons,calculated for the input model, of all points ND on the Huygens surface t = TD(ξ; M ) of M ,calculated with the output model. Also, the true-amplitude kernel KCR(ξ, r; M ) reads

KCR(ξ, r; M) =KDS(ξ; M ) KIS(r;ND)

mD(ξ;MID), (9.2.25b)

where MID is a point with coordinates (r,ZI(r;ND)) on the isochron z = ZI(r;ND) of ND withrespect to the original model. Here, we have again made use of the first duality theorem.

9.2. DIFFRACTION- AND ISOCHRON-STACK CHAINING 257

Using expressions (7.2.26) and (9.1.23) for KIS and KDS , equation (9.2.25b) may be recastinto the form

KCR(ξ, r; M ) =1

mD(ξ;MID)

ODS

OD

hB LSLGOF

OD LS LG cos2 βM

=OF

OF

hB

m2D(ξ;MID) mD(ξR; M)

LSLGLSLG

cos βMcos βM

. (9.2.26)

Note the close resemblance of this expression to the weight (9.2.8) of the cascaded configurationtransform. In equation (9.2.26), OF , OD, and ODS are the Fresnel, depth, and diffraction-stackobliquity factors, respectively, and hB and mD(ξR; M) are the Beylkin determinant and the stretchfactor at M , all calculated for the output configuration. Correspondingly, OF and mD(ξ;MID) arethe Fresnel obliquity factor and the stretch factor at MID, both calculated for the input configura-tion. Moreover, LS , LG and LS, LG are the point-source geometrical-spreading factors or the sourceand receiver ray segments in the input and output models, respectively. Finally, angles βM and βMare the isochron dips in the input and output models. Note that, differently from weight (9.2.8) forthe cascaded configuration transform, the ratio of the cosines cannot be omitted in weight function(9.2.26) since the isochron dips in the two different velocity models are generally different. How-ever, both angles can be determined by equations of the type (9.1.24). Elimination of the cosinesin equation (9.2.26) yields

KCR(ξ, r; M ) =OF

OF

OD

OD

hBm3D(ξ;MID)

LSLGLSLG

. (9.2.27)

as an alternative representation of the true-amplitude weight function for a cascaded remigration.

Let now Φ(r, z) be represented by equation (9.1.3). The asymptotic evaluation of the cascadedoperation (9.2.24) can be obtained along similar lines as its counterpart given by equation (9.2.10).As shown in Section H.2 of Appendix H, the result is

Φ(M ) ' L(MR)

L(MR)Φ0(rR) Fm[

mD(ξR; M)

mD(ξR;MR)m(rR)(ζ − ZR(ρ))] . (9.2.28)

Here, MR is the point on the reflector ΣR with the same horizontal coordinate ρ as M in the outputmodel andMR is its dual point in the input model. Moreover, L and L are the geometrical-spreadingfactors of the rays SMRG and SMRG in the input and output models, respectively.

Equation (9.2.28) describes the remigrated primary reflection event pertaining to the outputvelocity model. The simulated event in equation (9.2.28) is kinematically and dynamically equiva-lent to the true migrated event that was directly migrated from the original data with the outputvelocity model. Remigration correctly performs the relocation of the migrated event from ΣR toΣR. Moreover, since the first true-amplitude migration with the input velocity model realizes amultiplication of the migrated event by the geometrical-spreading factor L as computed in thatmodel, remigration must divide the amplitude by that factor and multiplies with the correct factorL calculated in the output model. One can see from equation (9.2.28) that exactly this operationis carried out by integral (9.2.24). Even the migration stretch is correctly replaced by a Kirchhoffremigration. This can also be seen from equation (9.2.28). For that purpose, let the pre-stretch fac-tor m(rR) of the section to be remigrated be given by the true Kirchhoff stretch factor mD(ξR;MR)of the input model. Then, it immediately becomes clear that Kirchhoff remigration restretches the

258 CHAPTER 9. SEISMIC IMAGING

migrated pulse according to the output model. In symbols, we have for Φ0(rR) = LU0(ξR) andm(rR) = mD(ξR;MR) that

Φ(M ) = LU0(ξR)F[

mD(ξR; M )(ζ − ZR(ρ))]

. (9.2.29)

This is exactly the result that would have been obtained if the original data had been migrateddirectly using the output velocity model.

As previously, we can interpret the above result by means of a sequence of simple operations:Let the point M with coordinates (ρ, ζ) be given in the output space. Using the improved model,(a) find the point on the reflector ΣR that has the same horizontal coordinate ρ as M . Call thisprojection point MR. It has the coordinates (ρ, ZR(ρ)). Now, (b) find its dual point NR withcoordinates (ξR, TR(ξR)) on the traveltime surface ΓR of the reflector ΣR. Now, (c) consider thereflection ray S(ξR)MRG(ξR) and find, for this ray, the point-source geometrical-spreading factorL(MR) and the vertical stretch factor mD(ξR; M). Now, (d) use the original model to find the pointMR with coordinates (rR,ZR(rR)) on ΣR for which the composite ray S(rR)MRG(rR) constitutesa reflection ray, i.e., MR is the dual point to NR. Then, (e) for this ray, compute the point-sourcegeometrical-spreading factor L(MR) and the vertical stretch factor mD(ξR;MR). This explains allquantities found in equation (9.2.28). Finally (f) stretch and scale the depth-migrated strip at Mand (g) place it at M . The cascaded remigration (9.2.24) performs the fictitious operations (a) to(g) in two imaging steps.

Single-stack solution

The single-stack remigration that performs the operations (a) to (g) in one imaging step is obtainedby a fully analogous procedure to the one that led to the single-stack integral (9.2.23) for theconfiguration transform. By interchanging the order of integrations in the cascaded remigration(9.2.24) and evaluating the now inner ξ-integral again by the method of stationary phase, weobtain (see derivation in Section H.3 of Appendix H)

Φ(M) =1

2π

∫

E

∫

d2rKRM (r; M)∂Φ(r, z)

∂zz = ZRM (r; M)

, (9.2.30)

which has the same structure as the isochron-stack integral (9.1.1). The remigration (RM) inplanatZRM (r; M) is given by

ZRM (r; M) = ZCR(ξRM , r; M ) , (9.2.31a)

where ξRM is the stationary point of the ξ-integral in equation (9.2.24) and where z = ZCR(ξ, r; M)is the composite stacking surface of the cascaded remigration defined in equation (9.2.25a). Thecorresponding true-amplitude kernel KRM (r; M) is

KRM (r; M) = mD(ξRM ;MRM ) KCR(ξRM , r; M) ×

× exp−iπ2 [1 − Sgn(H˜D −H

˜D)/2]

|det(H˜D −H

˜D)| 12

, (9.2.31b)

where MRM is the dual point in the input model to NRM defined by ξRM on ΓR. Moreover,H˜D and H

˜D are the Hessian matrices of the diffraction traveltimes TD(ξ;MRM ) and TD(ξ; M),

respectively, evaluated at ξRM . Note that, because of the duality properties, these two diffraction

9.2. DIFFRACTION- AND ISOCHRON-STACK CHAINING 259

times are tangent to each other at NRM . Using KCR(ξ, r; M ) from equation (9.2.27) and observingthat MID(ξRM ) = MRM , this kernel function can be recast into the form

KRM (r; M) =OF

OF

OD

OD

hBm2D(ξRM ;MRM )

LSLGLSLG

exp−iπ2[

1 − Sgn(H˜D −H

˜D)/2

]

|det(H

˜D −H

˜D)|1/2

. (9.2.32)

All quantities in equation (9.2.32) are explained in connection with equation (9.2.26). Here, theyare evaluated at MRM .

In equations (9.2.31), ξRM = ξRM (r) is the stationary point with respect to ξ of the phasefunction of integral (9.2.24),

δCR(r, ξ; M) = m(r)(ZCR(ξ, r; M ) −ZR(r)) . (9.2.33)

For each r in E, ξRM is, in fact, the vector parameter for which the gradient of ZCR(ξ, r; M ) withrespect to ξ vanishes, because the other quantities in expression (9.2.33) do not depend on ξ. Insymbols,

∇ξδCR(r, ξ; M )

ξ=ξRM

= 0 , (9.2.34a)

or

∇ξZCR(ξ, r; M)

ξ=ξRM

= 0 . (9.2.34b)

Note again the complete analogy between these and equations (9.2.17). As a part of the analogy tothat case, relation (9.2.34b) constitutes the mathematical description of the envelope of the familyof isochrons ZI(r;ND) for all points ND on the Huygens surface of M as geometrically describedand termed remigration inplanat in Chapter 2.

In order to acquire a geometrical picture of this stationary condition, consider the pointNRM (ξRM ; M ) with coordinates (ξRM , TD(ξRM ; M)) on the Huygens surface t = TD(ξ; M). PointNRM is the dual point to M in the output model. In the same way, NRM has a dual point in theinput model, which we denote by MRM . From the duality theorems, we can then conclude thatthe Huygens surfaces t = TD(ξ; M) (constructed using the improved model) and t = TD(ξ;MRM )(constructed using the original model) are tangent at NRM . From its property of being the envelopeof the family of isochrons ZI(r;ND) for all points ND on the Huygens surface of M , we alsoobserve that the remigration inplanat z = ZRM (r; M ) is the “migrated surface” in the input spaceof the diffraction-time surface t = TD(ξ; M ) constructed for point M using the improved model.This means that, with respect to the original model, if z = ZRM (r; M ) were a reflector, thent = TD(ξ; M) would be its traveltime surface. Again, this observation provides us with the rules tocompute the remigration inplanats from conventional Kirchhoff traveltime tables.

To demonstrate that equation (9.2.30) represents in fact a true-amplitude remigration, onehas to evaluate it asymptotically by the stationary-phase method as applied previously. The actualcalculus is very similar to what has been done in Section H.2 of Appendix H for integral (9.2.24).After some tedious algebraic manipulations involving the duality relations of Chapter 5, this evalu-ation leads again to the important result (9.2.28), which makes an true-amplitude remigration wellunderstood in analytical terms. In other words, the single-stack remigration (9.2.30) performs theabove explained fictitious operations (a) to (g) in one single imaging step.

Figure 9.3 visualizes the situation for the case of a constant-velocity remigration from a

260 CHAPTER 9. SEISMIC IMAGING

−1000 −500 0 500 10000

200

400

600

800

Midpoint coordinate [m]T

ime

[ms]

Huygens curve

Time−trace domain

ND

−1000 −500 0 500 1000

0

200

400

600

800

Distance [m]

Dep

th [m

]

RM inplanat

ΣR

ΣR

~

Depth domain

M~

Fig. 9.3. Visualization of the geometry of the cascaded and single-stack remigration integrals,equations (9.2.24) and (9.2.30). Bottom: A point M (diamond) is chosen in the output velocityfield v. Also depicted are the isochrons (solid lines) in the input velocity field v for some selectedpoints ND on the Huygens curve of M as constructed for the output velocity field. The stack(9.2.24) is performed along the set of all these isochrons for all points ND. The stack (9.2.30) isperformed along the envelope of all these isochrons curves, i.e., the remigration inplanat (bold line).Top: The Huygens curve (bold line) corresponding to point M . An arbitrary point on this surfaceis denoted by ND.

wrong migration velocity v to the correct one, v, in the same simple 2-D model used in the figuresof Chapter 2. For a given point M in the output depth domain, the Huygens curve as constructedwith the output velocity field v is depicted in the time-trace domain. Also depicted are the isochronsfor the input velocity field v for some selected points ND on this Huygens curve. The stack (9.2.24)is performed along the set of all these isochrons for all points ND. The stack (9.2.30) is performedalong the envelope of all these isochrons curves, i.e., the remigration inplanat z = ZRM (r; M).Because in Figure 9.3, the point M was chosen on the (unknown) updated reflector image ΣR,the corresponding remigration inplanat is tangent to the original reflector image ΣR in the inputvelocity field v.

9.2.3 Some general remarks on image transformations

By way of the cascaded solutions (9.2.6) and (9.2.24), as well as by the single-stack formulas (9.2.23)and (9.2.30), we have presented the solution to two important seismic reflection imaging problems.These can be solved by nothing more than a single weighted stack performed along inplanats

9.2. DIFFRACTION- AND ISOCHRON-STACK CHAINING 261

confined to either the time-trace or depth domain. Each of the two problems could, of course, besolved by applying the stacks described by the integrals (7.1.4) and (9.1.1) explicitly in sequence.However, only after appreciating the potential of chaining both integrals analytically, they obtaintheir full generality as tools of solving a wide spectrum of 3-D seismic imaging problems. For allpossible image transformations, single-stack solutions can be formulated that possess the samestructure as the two basic integrals (7.1.4) and (9.1.1) and as the two transformation integrals(9.2.23) and (9.2.30) derived in this chapter. In this way, all seismic image transformations can beexpressed in one and the same basic form, being a weighted stack along problem-specific inplanats.The use of true-amplitude weights is mandatory if best possible image amplitudes are required. Forpurely kinematic purposes, the same stacks without weights will be sufficient.

The theory presented in this chapter has already been used to provide 2.5-D solutions foroffset continuation (Santos et al., 1997), common-shot DMO (Bagaini and Schleicher, 1997) andMZO and DMO (Tygel et al., 1998). Note that now that we know the basic structure of all Kirchhoff-type image transformations to be an integral of the type of equations (7.1.4), (9.1.1), (9.2.23), or(9.2.30), concrete imaging problems can be solved by setting up an integral of that structure anddetermining the weight function such that is guarantees the desired amplitude behavior ratherthan actually calculating it from the chained expressions (9.2.22) or (9.2.31b). This procedure wasadopted in the cited papers.

Here, we have only described image transformations with different measurement configura-tions or different macrovelocity models. However, we could as well formulate image transformationswhich use different ray codes in the input or output space. Using the theory developed above wecould, for example, formulate an image transformation, whereby constant-offset P-S reflections areimaged into zero-offset P-P reflections or vice versa. Other transformations include redatuming,layer stripping, transformation of a surface seismic section into a VSP section, and various correc-tions applied to any kind of the above transformations, such as, for example a correction to a DMOdue to an improved macrovelocity model.

Let us also address a general problem that applies to migration, demigration and any otherkind of image transformation irrespective of how these problems are solved. Surely, if subsurfacereflectors are not illuminated by a certain input measurement configuration (i.e., not hit by re-flecting rays), one cannot expect image transformations to provide information concerning thesenon-illuminated reflector regions, even if they were illuminated had the data acquisition actuallybeen carried out with the output configuration. For that reason, image transformations of theabove kind may remain incomplete for non-illuminated reflector segments, an effect well-known inpractice.

Both the diffraction and isochron stacks investigated in this work describe weighted sum-mations (with possibly complex weights). They are applied to the Huygens or isochron stackingsurfaces that depend on the macrovelocity model, the measurement configuration and the ray codefor the chosen elementary wave. Both stacks are mathematically expressed by certain integrals,each of which can be understood in a wider sense as a Generalized Radon Transform (GRT), as fre-quently considered for migration/inversion (Beylkin, 1985a,b; Bleistein, 1987; Miller et al., 1987; deHoop and Bleistein, 1997). Even if one may want to relate both the diffraction-stack and isochron-stack integrals to either the mathematical inverse or forward GRT, we believe to have provided amore direct (geometrical as well as wave-theoretical) access to solving seismic image transformationproblems.

262 CHAPTER 9. SEISMIC IMAGING

It should also be mentioned that both stacking integrals presented in this work are closelyrelated to integrals described by Goldin (1988, 1989, 1990), who used the so-called “Method ofDiscontinuities” to investigate them. However, the same argument of conveying only a very math-ematical and not a geometrical nor wave-theoretical picture also applies to Goldin’s investigations.Therefore, combining the geometrical simplicity of both the diffraction and the isochron stackswith zero-order ray theory has helped us to solve in a unified way the important imaging problemsaddressed in this book. Furthermore, the direct connection with ray theory clearly demonstrateshow to compute the true-amplitude weights, as well as the traveltime functions that are necessaryto devise the stacking surfaces, from the information that is available from dynamic ray tracing.

The unified approach to seismic reflection imaging, as originally published in Hubral et al.(1996a) and Tygel et al. (1996) (corrections in Jaramillo et al., 1998), has triggered a lot of activityin this direction. Especially the possibility of chaining the migration and demigration integrals hasgiven rise to methods that Jaramillo and Bleistein (1998, 1999; see also Bleistein and Jaramillo,1998) now call “data mapping”. A recent discussion of the topic can be found in Bleistein et al.(2001).

9.3 Summary

In this chapter, we have introduced Kirchhoff demigration as an operation to transform a depth-migrated seismic image back into the time domain. In the same way as Kirchhoff migration isrealized by a stack along the diffraction-traveltime surface, and thus also referred to as diffraction-stack migration, Kirchhoff demigration is realized by a stack along the dual surface, the isochron,and thus also referred as isochron-stack demigration. We have shown how the two weighted stackingintegrals operate on seismic reflection events that are confined to signal strips in either the time-trace or depth domain. By specifying the desired output of Kirchhoff demigration as those datathat were the input to Kirchhoff migration, we have demonstrated how the weights have to bedesigned to obtain true amplitudes in both spaces. Zero-order ray theory has helped to appreciateall kinematic and dynamic aspects of both stacks. In this way, also those aspects that go beyondthe geometry of wave propagation remain easy to understand. Moreover, we have shown that thediffraction and isochron stacks are asymptotically inverse operations to each other. This is revealedwhen both transform integrals are applied sequentially for an identical measurement configuration,macrovelocity model, and ray code.

On the basis of this pair of operations, we have formulated a complete wave-equation based(high-frequency) true-amplitude theory for seismic reflection imaging as a generalization of classicalkinematic seismic reflection mapping procedures (map migration and map demigration). Its highlygeometrical features turn out to form its main advantage, thus providing a great help for theinterpretation.

The basic observation is that both stacking operations can be applied in sequence using dif-ferent macrovelocity models, measurement configurations or ray codes in the input or output space.Instead of actually carrying out this sequence of operations on the data, the stacking integrals canbe chained analytically. In this way, the chained integrals allow to transform (a) seismic reflectionevents from one seismic time-trace domain into another one or (b) depth-migrated reflector imagesfrom one depth domain into another one. As a result of chaining, one can formulate various im-age transformations (e.g, migration to zero offset, DMO, SCO, redatuming, remigration, etc.) in

9.3. SUMMARY 263

one step by using only one kind of stacking surfaces (i.e., the problem-specific inplanats) in therespective domain. In this way, all image transformations are represented by an operation of thesame structure, thus being part of a unified theory. Like diffraction-stack migration, any other ofthese Kirchhoff-type operations can be realized by a smear-stack along a problem-specific outplanatinstead of a stack along the inplanat. Inplanats and outplanats are dual surfaces to each other inthe same way as diffraction-time surfaces and isochrons.

For each image transformation, there exists also an inverse transformation, where the rolesplayed by the inplanat and outplanat are exchanged, and where the weight functions for the forwardand inverse transformation, of course, also closely relate to each other. They can be specified bythe proposed theory, regardless which weight was used. In particular, weights can be devised thatrevert an unweighted transformation.

It should be kept in mind that the proposed unified theory of seismic reflection imaging can beextended to all other media (e.g., anisotropic or slightly absorbing), for which wave propagation iswell-described by zero-order ray theory. The resulting diffraction-stack and isochron-stack integralsfor such media could then not only be chained with each other but also with the ones proposed herefor isotropic media. In this way, one could solve a new, extended class of target-oriented seismicimaging problems. Just to name one such problem, let us mention a remigration of a depth-migratedreflector image obtained for an (inaccurate) isotropic macrovelocity model to that of a (moreaccurate) anisotropic one. Such a remigration could be achieved by way of chaining the respectivestacking integrals (i.e., the isotropic isochron-stack demigration with the anisotropic diffraction-stack migration), constructing the required inplanats, and performing the weighted single stack.

We have shown that all these image transformations can be carried out in a true-amplitudesense, that is, correctly transforming the geometrical-spreading factor from the input into theoutput domain. The necessary weight functions depend only on the reflector overburden and areindependent of any reflector properties like its dip or curvature. This is a feature of the Kirchhoff-type stacking structure of the integrals and the adequately determined stacking surfaces. It doesnot depend on any assumptions about the medium and the reflector curvature besides the obviouscondition that (a) all quantities in the problem vary sufficiently smoothly for the zero-order raydescription of the waves and (b) the asymptotic evaluations are valid. Therefore, the stackingsurfaces and true-amplitude weight functions can be computed for any arbitrary output point fromquantities that are entirely determinable by dynamic ray tracing in the given macrovelocity modelwithout any a priori information about the reflector to be imaged.

264 CHAPTER 9. SEISMIC IMAGING

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Appendix A

Reflection and transmissioncoefficients

In this appendix, we provide the formulas for the amplitude-normalized plane-wave reflection andtransmission coefficients as derived in Cerveny et al. (1977) or Cerveny (2001) on the basis of theboundary conditions of Zoeppritz (1919). Note that we assume the incidence angle to assume valuesbetween 0and 90or between 90and 180, depending on the choice of the direction of the normalvector of the interface.

We also state several linearized expressions for the reflection coefficients, since generally,contrast at seismic reflectors are rather small. The linearized formulas have proven to be veryuseful when inverting the reflection coefficients for medium parameters.

A.1 Reflection coefficients

A.1.1 P-P reflection

The amplitude-normalized plane-wave reflection coefficient RPP = RPP (θ) for a P-P reflection asa function of the incidence angle, θ, is given by the ratio

RPP = NPP/DR, (A-1)

with the numerator

NPP = q2p2P1P2P3P4 + %1%2(β1α2P1P4 − α1β2P2P3)

−α1β1P3P4Y2 + α2β2P1P2X

2 − α1α2β1β2p2Z2 , (A-2)

and the denominator

DR = q2p2P1P2P3P4 + %1%2(β1α2P1P4 + α1β2P2P3)

+α1β1P3P4Y2 + α2β2P1P2X

2 + α1α2β1β2p2Z2 . (A-3)

277

278 APPENDIX A. REFLECTION AND TRANSMISSION COEFFICIENTS

These expressions involve the following abbreviations

q = 2(%2β22 − %1β

21) , X = %2 − qp2 ,

Y = %1 + qp2 , Z = %2 − %1 − qp2 ,

P1 = (1 − α21p

2)1/2 , P2 = (1 − β21p

2)1/2 ,

P3 = (1 − α22p

2)1/2 , P4 = (1 − β22p

2)1/2 .

(A-4)

Here, α1,2, β1,2, and %1,2 are the velocities and the density of the two media separated by thereflecting interface. Index 1 indicates the quantities pertaining to the incident wave’s side of theinterface. Moreover, the value of the ray parameter, p is defined by the incidence angle, θ, and thepropagation velocity of the incident wave. In the case of the P-P reflection, the latter is α1 andthus, p = sin θ/α1.

Of course, the radicands of the square roots in the above expressions for Pi (i = 1, 2, 3, 4) canbecome negative. Under those circumstances, we have to choose the positive sign of the imaginarysquare root, i.e.,

P1 = i(α21p

2 − 1)1/2 for p > 1/α1 ,

P2 = i(β21p

2 − 1)1/2 for p > 1/β1 ,

P3 = i(α22p

2 − 1)1/2 for p > 1/α2 ,

P4 = i(β22p

2 − 1)1/2 for p > 1/β2 . (A-5)

For small contrasts in density, ∆% = %2 − %1, as well as P-wave velocity ∆α = α2 − α1 andS-wave velocity ∆β = β2 − β1, Lame’s parameters ∆λ = λ2 − λ1 and ∆µ = µ2 − µ1, or P-waveimpedance, ∆Iα = Iα2 − Iα1, and S-wave impedance ∆Iβ = Iβ2 − Iβ1, this rather complicatedexpression can be substituted by any of the following linearized (or first-order) approximations(Aki and Richards, 1980; Shuey, 1985; Wright, 1986; Tjaland, 1993; Hanitzsch, 1995; Novais; 1998)

RLPP (θ) =1

2

(

1 − 4β2

α2sin2 θ

)

∆%

%+

1

2 cos2 θ

∆α

α− 4

β2

α2sin2 θ

∆β

β(A-6a)

=1

2

∆IαIα

+

[

∆α

α− 4

β2

α2

(

2∆β

β+

∆%

%

)]

sin2 θ +1

2

∆α

αtan2 θ sin2 θ (A-6b)

=1

2

(

1 − 1

2 cos2 θ

)∆%

%+

1

4 cos2 θ

∆Mp

Mp− 2

β2

α2sin2 θ

∆µ

µ(A-6c)

=1

2

∆%

%+

1

2 cos2 θ

∆α

α− 2

β2

α2sin2 θ

∆µ

µ(A-6d)

=1

2

∆IαIα

− 2β2

α2sin2 θ

∆µ

µ+

1

2tan2 θ

∆α

α(A-6e)

=1

2

∆IαIα

− 2 sin2 θ∆µ

Mp+

1

2tan2 θ

∆α

α(A-6f)

=1

2 cos2 θ

∆IαIα

− 4β2

α2sin2 θ

∆IβIβ

− 1

2

(

tan2 θ − 4β2

α2sin2 θ

)

∆%

%. (A-6g)

=1

2(1 + cos 2θ)

∆λ

Mp+

cos2 2θ

1 + cos 2θ

∆µ

Mp+

cos 2θ

2(1 + cos 2θ)

∆%

%. (A-6h)

Here, quantities without an index denote the mean value of the respective medium parameter. TheP and S-wave impedances are defined as Iα = %α and Iβ = %β, respectively. Moreover, Mp = λ+2µ

A.1. REFLECTION COEFFICIENTS 279

is the P- or plane wave modulus (see also Table 3.1). An extensive discussion of these first-orderexpressions (except the last two) and their quality can be found in Tjaland (1993). Hanitzsch(1995) has found parameterization (A-6g) to be particularly useful for inversion purposes. The lastapproximation, (A-6h), expresses the reflection coefficient as a function of the full reflection angle,2θ, rather than the incidence angle θ.

A.1.2 SV-SV reflection

The plane-wave reflection coefficient RSS = RSS(ϕ) for SV-SV reflections as a function of theS-wave incidence angle, ϕ, is given by a similar ratio

RSS(ϕ) = NSS/DR, (A-7)

where the numerator is now

NSS = q2p2P1P2P3P4 + %1%2(α1β2P2P3 − β1α2P1P4)

−α1β1P3P4Y2 + α2β2P1P2X

2 − α1α2β1β2p2Z2 , (A-8)

and where the denominator DR is again as defined in equation (A-3). All abbreviations q, X, Y ,Z, P1, P2, P3, and P4 are the same as before, i.e., they are given by equations (A-4). Of course,since the incident wave is now an SV-wave, its propagation velocity is β1 and thus, in this case,p = sinϕ/β1.

The corresponding first-order approximations for small contrasts are

RLSS(ϕ) =1

2

(

1 − 4 sin2 ϕ) ∆%

%+

(1

2 cos2 ϕ− 4 sin2 ϕ

)∆β

β(A-9a)

=1

2

∆IβIβ

−(

7∆β

β+ 4

∆%

%

)

tan2 ϕ+

(

4∆β

β+ 2

∆%

%

)

tan2 ϕ sin2 ϕ (A-9b)

=

(1

2− 2 sin2 ϕ

)∆IβIβ

+

(1

2tan2 ϕ− 2 sin2 ϕ

)∆β

β(A-9c)

=1

2

∆%

%+

1

2 cos2 ϕ

∆β

β− 2 sin2 ϕ

∆µ

µ(A-9d)

=

(1

2 cos2 ϕ− 4 sin2 ϕ

)∆IβIβ

−(

1

2tan2 ϕ− 2 sin2 ϕ

)∆%

%(A-9e)

=cos 2ϕ

2(1 + cos 2ϕ)

∆%

%+

2 cos2 2ϕ− 1

2(1 + cos 2ϕ)

∆µ

µ. (A-9f)

A.1.3 SH-SH reflection

The plane-wave reflection coefficient RSS = RSS(ϕ) for a SH-SH reflection is given by

RSS(ϕ) = (%2β2P2 − %1β1P4)/(%2β2P2 + %1β1P4) , (A-10)

where P1, P2, P3, and P4 are again the square roots given in equations (A-4). Of course, since theincident wave is now an SH-wave, its propagation velocity is β1 and thus, in this case, p = sinϕ/β1.

280 APPENDIX A. REFLECTION AND TRANSMISSION COEFFICIENTS

The corresponding first-order approximation for small contrasts are

RLSS =1

2

∆%

%+

1

2 cos2 ϕ

∆β

β(A-11a)

=1

2

∆IβIβ

+1

2tan2 ϕ

∆β

β(A-11b)

=1

2 cos2 ϕ

∆IβIβ

− 1

2tan2 ϕ

∆%

%(A-11c)

=cos 2ϕ− 1

2(1 + cos 2ϕ)

∆%

%+

1

2(1 + cos 2ϕ)

∆µ

µ. (A-11d)

A.1.4 P-SV reflection

The plane-wave reflection coefficient RPS = RPS(θ) for P-SV reflections is given by

RPS(θ) = NPS/DR, (A-12)

where

NPS = 2α1pP1(qP3P4Y + α2β2XZ), (A-13)

with DR and all other quantities again as defined in equations (A-3) and (A-4). Here, the incidentwave is again a P-wave, and thus, p = sin θ/α1. The reflected wave is an S-wave with the propagationangle, ϕ, defined by Snell’s law, i.e., by p = sinϕ/β1.

The linearized formulas for small contrasts read

RLPS =sin θ

2 cosϕ

(

1 − 2β2

α2sin2 θ + 2

β

αcos θ cosϕ

)

∆%

%

+sin θ

2 cosϕ

(

−4β2

α2sin2 θ + 4

β

αcos θ cosϕ

)

∆β

β(A-14a)

=sin θ

2 cosϕ

[(

1 + 2β

αcos(θ + ϕ)

)∆%

%+ 4

β

αcos(θ + ϕ)

∆β

β

]

(A-14b)

=sin θ

2 cosϕ

∆%

%− sinϕ(tanϕ sin θ − cos θ)

∆µ

µ(A-14c)

=α

βtanϕ

(1

2

∆%

%+β

αcos(θ + ϕ)

∆µ

µ

)

. (A-14d)

A.1.5 SV-P reflection

The plane-wave reflection coefficient RSP = RSP (ϕ) for an SV-P reflection is given by

RSP (ϕ) = NSP/DR, (A-15)

where

NSP = −2β1pP2(qP3P4Y + α2β2XZ), (A-16)

A.2. TRANSMISSION COEFFICIENTS 281

with the same meaning of all quantities as before. Here, the incident wave is an SV-wave, theoutgoing wave a P-wave, and thus, p = sinϕ/β1 = sin θ/α1.

The linearized formulas for small contrasts read

RLSP = − sinϕ

2 cos θ

(

1 − 2β2

α2sin2 θ + 2

β

αcos θ cosϕ

)

∆%

%

− sinϕ

2 cos θ

(

−4β2

α2sin2 θ + 4

β

αcos θ cosϕ

)

∆β

β(A-17a)

= − sinϕ

2 cos θ

[(

1 + 2β

αcos(θ + ϕ)

)∆%

%+ 4

β

αcos(θ + ϕ)

∆β

β

]

(A-17b)

= − sinϕ

2 cos θ

∆%

%+ sinϕ

(

sin2 ϕ

cos θ− sinϕ cosϕ

sin θ

)

∆µ

µ(A-17c)

= −βα

tan θ

(1

2

∆%

%+β

αcos(θ + ϕ)

∆µ

µ

)

. (A-17d)

A.2 Transmission coefficients

A.2.1 P-P transmission

The plane-wave transmission coefficient TPP = TPP (θ) for a P-P transmission as a function of theincidence angle, θ, is given by the ratio

TPP = MPP/DR, (A-18)

with the numerator

MPP = 2α1%1P1(β2P2X + β1P4Y ) . (A-19)

The denominator DR and the abbreviations involved are the same as in the reflection case, definedin equations (A-4) and (A-3). In the case of the P-P transmission, the propagation velocity of theincident wave is α1 and thus, p = sin θ/α1.

A.2.2 SV-SV transmission

The plane-wave transmission coefficient TSS = TSS(ϕ) for SV-SV transmission is given by a similarratio

TSS = MSS/DR, (A-20)

where the numerator is now

MSS = 2β1%1P2(α1P3Y + α2P1X) , (A-21)

and where the denominator DR, as well as the abbreviations q, X, Y , Z, P1, P2, P3, and P4 aregiven by equations (A-3) and (A-4). Of course, since the incident wave is now an SV-wave, itspropagation velocity is β1 and thus, in this case, p = sinϕ/β1.

282 APPENDIX A. REFLECTION AND TRANSMISSION COEFFICIENTS

A.2.3 SH-SH transmission

The plane-wave transmission coefficient TSS = TSS(ϕ) for SH-SH transmission is given by

TSS = 2%1β2P2/(%2β2P2 + %1β1P4) , (A-22)

where P1, P2, P3, and P4 are again the square roots given in equations (A-4). Of course, since theincident wave is now an SH-wave, its propagation velocity is β1 and thus, in this case, p = sinϕ/β1.

A.2.4 P-SV transmission

The plane-wave transmission coefficient TPS = TPS(θ) for P-SV transmission is given by

TPS(θ) = MPS/DR, (A-23)

where

MPS = −2α1%1pP1(qP2P3 − α2β1Z), (A-24)

with DR and all other quantities again as defined in equations (A-3) and (A-4). Here, the incidentand transmitted waves are a P and an S-wave, respectively, and thus, p = sin θ/α1 = sinϕ/β1.

A.2.5 SV-P transmission

The plane-wave transmission coefficient TSP = TSP (ϕ) for SV-P transmission is given by

TSP (ϕ) = MSP /DR, (A-25)

where

MSP = 2β1%1pP2(qP1P4 − α1β2Z), (A-26)

with the same meaning of all quantities as before. Here, the incident wave is again an SV-wave,and thus, p = sinϕ/β1.

Appendix B

Waves at a free surface

In Chapter 3, we have derived formulas that describe how the scalar amplitude of the particledisplacement changes along a ray. However, in a seismic survey, it is not this amplitude that isrecorded, because the geo- or hydrophones are located at a free surface. Therefore, this Appendixis devoted to the question how the described scalar amplitude of the particle displacement canbe computed from land-seismic three-component free-surface recordings. We also address how thepressure is described in a sea-seismic survey. The formulas in this Appendix are based on theconversion coefficients given by Cerveny et al. (1977).

B.1 P-waves at a free surface

As stated in equation (3.3.16), the polarization vector of the particle displacement of a P-wave (inthe absence of a free surface) is parallel to the propagation direction of the wave, i.e., the vectorialamplitude can be represented as

U(P )

= U (P )t . (B-1)

Again, t is the unit tangent vector to the ray as defined in equation (3.4.12).

However, in a three-component seismic survey at a (curved) free surface, the following vectorcomponents are recorded at the receiver G:

Uc

= (Uc1 ,Uc2 ,Uc3)T = U (P ) c , (B-2)

where c = (c1, c2, c3)T is the vector of conversion coefficients. It has the components (Cerveny et

al. 1977)

c1 = 4PGSGpβGD−1G cosϕ , (B-3a)

c2 = 4PGSGpβGD−1G sinϕ , (B-3b)

c3 = − 2PG(

1 − 2β2Gp

2)

D−1G . (B-3c)

Here, the angle ϕ describes the orientation of the local Cartesian coordinate system xG within theplane tangent to the measurement surface at G. In this representation of the conversion coefficients,

283

284 APPENDIX B. WAVES AT A FREE SURFACE

we have used the following notation in accordance with Cerveny et al. (1977):

PG =(

1 − α2Gp

2) 1

2 , (B-4a)

SG =(

1 − β2Gp

2) 1

2 , (B-4b)

DG =(

1 − β2Gp

2)2

+ 4p2PGSGβ3Gα

−1G . (B-4c)

As before, αG and βG are the P- and S-wave velocities directly below the free surface at the receiverposition G. These are assumed to be known. Parameter p is the horizontal slowness of the ray atthe surface (where “horizontal” means “within the tangent plane at G”). It is given by the sine

of the P-wave emergence angle ϑ(P )G divided by the P-wave velocity αG, and, due to Snell’s law,

also by the S-wave emergence angle ϑ(S)G divided by the S-wave velocity βG. The square roots in

equations (B-4) are all real because evanescent waves are not considered here. In other words, onlyvalues of p are assumed that are less than the inverse of the greatest velocity along the ray path.

Thus, if p is known, the amplitude U (P ) of the displacement vector of the P-wave, as if itwas not affected by the free surface, can be directly obtained. For that purpose, only the verticalcomponent U c3 of the particle displacement at G in a land-seismic survey has to be recorded. Todetermine the desired scalar amplitude U (P ), this component must be divided by c3. In other words,amplitude U (P ) is given by

U (P ) = − Uc3DG

2PG(1 − 2β2

Gp2) . (B-5)

If p is unknown, the emergence direction of the wave must also be determined. In that case,the measurement of all three free-surface components of the displacement vector is necessary. Then,the following operations determine U (P ):

1. Rotation of the local coordinate system around the vertical axis (where “vertical” means“perpendicular to the tangent plane at G”) by the angle

ϕ = arctan

(c2c1

)

= arctan

(Uc2Uc1

)

. (B-6)

2. Determination of the rotated horizontal component of the displacement vector

Uc1′ = Uc1 cosϕ+ U c2 sinϕ . (B-7)

3. Determination of p from the vertical and rotated horizontal components. We have

p =1

βGsin

[

−1

2arctan

(Uc1′Uc3

)]

, (B-8)

where the expression inside the square brackets is the S-wave emergence angle ϑ(S)G obtained

from equations (B-3) by substituting pβG = sinϑ(S)G .

With the so determined p, the scalar amplitude U (P ) can be calculated from equation (B-5). Al-ternatively, the full vectorial amplitude U can be obtained by division of all three components ofUc

by the respective components of c and subsequent vectorial addition. Then, the modulus of U

represents the sought-for quantity

U (P ) =

√

(Uc1)2 + (Uc2)2

2(4PGSGpβGD−1G )2

+(Uc3)2

2(2PG(1 − 2β2Gp

2)D−1G )2

. (B-9)

B.2. S-WAVES AT A FREE SURFACE 285

B.2 S-waves at a free surface

For S-waves at a free surface, the situation is much more complicated than for P-waves. Thecomponents of the shear wave with a polarization vector within the plane of propagation (SV-waves)and perpendicular to that plane (SH-waves) must be distinguished. The plane of propagation is theplane defined by the slowness vector of the incident S-wave at G and the surface normal at G.

B.2.1 SV-waves at a free surface

The polarization vector of the shear-wave displacement is tangent to the wavefront, i.e., normal tothe propagation direction. The vectorial amplitude of an SV-wave, which is polarized within theplane of propagation, can thus, in absence of a free surface, be expressed in the form

U(S)

= U (S)(

n cos ΘG − b sinΘG

)

, (B-10)

where n and b are the Frenet normal and binormal vectors (see Section 3.9). Moreover, ΘG denotesthe angle the Frenet normal vector n makes with the polarization direction of the SV-wave at G.

The recorded wavefield at a free surface is again represented by equation (B-2), where thecomponents of the vector c are now given by (Cerveny et al., 1977)

c1 = 2SG(

1 − 2β2Gp

2)

D−1G cosϕ , (B-11a)

c2 = 2SG(

1 − 2β2Gp

2)

D−1G sinϕ , (B-11b)

c3 = 4PGSGpβ2Gα

−1G D−1

G . (B-11c)

The involved quantities have the same meaning as before.

We conclude that also for an SV-wave, the scalar amplitude U (S) can only be obtained aftercertain operations. If p is known, again a simple division of the vertical component of U

cby the

conversion coefficient c3 is sufficient. In symbols, we have

U (S) =Uc3αGDG

4PGSGpβ2G

. (B-12)

Otherwise, more than one operation is necessary to determine p and U (S) from Uc. These are

1. As in case of the P-wave, a rotation of the local coordinate system around the vertical axisby the angle ϕ as determined by equation (B-6).

2. Computation of the rotated horizontal component U cx′ using equation (B-7).

3. Determination of p. As before, it’s computation involves the ratio between the rotated hori-zontal and vertical components. Note that in this case, the calculation does not directly result

in a unique value for p because the tangent of the P-wave emergence angle ϑ(P )G fulfills the

quadratic equation

(Ucx′Uc3

tan 2ϑ(P )G − 1

)2

=

(

α2G

β2G

− 1

)2 (

1 + tan2 2ϑ(P )G

)

. (B-13)

286 APPENDIX B. WAVES AT A FREE SURFACE

Of the two solutions in −π2 < ϑ

(P )G < π

2 , one has to choose the one that satisfies

α2G

β2G

− 1 + cos 2ϑ(P )G

sin 2ϑ(P )G

=Ucx′Uc3

. (B-14)

Once ϑ(P )G is known, p is easily determined from p = sinϑ

(P )G /αG.

With the so determined p, the scalar amplitude U (S) can be calculated from equation (B-12).Alternatively, the full vectorial amplitude U can be obtained by division of all three componentsof U

cby the respective components of c and subsequent vectorial addition. Then, the modulus of

U represents the sought-for quantity

U (S) =

√

(Uc1)2 + (Uc2)2

2(2SG(1 − 2β2Gp

2)D−1G )2

+(Uc3)2

2(4PGSGpβ2Gα

−1G D−1

G )2. (B-15)

B.2.2 SH-Waves at a free surface

Of all possible cases in an elastic medium, this is the most simple one. All three components of theSH-wave are recorded at a free surface with double the amplitude which the components wouldhave in the absence of a free surface. Thus, vectorial amplitude and division by two provides in thiscase directly the scalar amplitude of the particle displacement.

Remark: The steps that lead from the recorded components of the particle displacement to itsscalar amplitude are different for all three types of elastic waves. Thus, one has to decide before thecomputation of the scalar amplitude which type of elastic wave is to be imaged. Events pertainingto other types of elementary waves must be suppressed by preprocessing or they will be consideredas noise. This problem cannot be solved as the component of the displacement vector in or verticalto the propagation direction, i.e., the scalar amplitude, cannot be directly recorded in the field.

B.3 Acoustic waves at a free surface

As for a compressional wave in the elastic medium, the acoustic wave is also polarized in propagationdirection, i.e., the vectorial amplitude of the particle displacement can be written in form of equation(B-1). If the displacement was recorded at a free surface, the resulting data could again be expressedin form of equation (B-2). The components of the conversion coefficient vector read in this case:

c1 = 0 , (B-16a)

c2 = 0 , (B-16b)

c3 = − 2PG . (B-16c)

We observe that a correction of the vertical component by a division by −2PG yields directly themodulus U (P ) of the displacement vector. The propagation direction is not determinable.

We have to note, however, that in usual marine seismic surveys it is not the vertical componentof the particle displacement at the water surface that is recorded but the pressure slightly below

B.3. ACOUSTIC WAVES AT A FREE SURFACE 287

it. The vertical component of the particle displacement at the free surface must be determinedcomputationally. We use the fact that the time derivative of the particle displacement equals theparticle velocity, i.e.,

∂2

∂t2Uc3 =

∂V3

∂t, (B-17)

where V3 is the vertical component of the particle velocity. If the medium density % is constant, theright-hand side of equation (B-17) can be expressed after Tygel and Hubral (1987) as

∂V3

∂t=

1

%

∂p

∂z, (B-18)

One finds that the vertical component of the particle displacement satisfies

∂2

∂t2Uc3 =

1

%

∂p

∂z. (B-19)

If the density is not constant, it cannot be ignored in the differentiation in formula (B-19). In thatcase, the acoustic potential is needed instead of the pressure. However, the assumption of a constantwater density in the vicinity of the sea level is realistic in most cases.

To record the spatial derivative of the pressure field, either a double streamer experiment isneeded, or one simply uses the fact that at a free surface the pressure vanishes. If the depth of thestreamer is under good control, which is usually the case, the recorded pressure p(δz) divided bythe streamer depth δz yields a good estimate for the sought-for spatial derivative of the pressurefield (Matthies et al., 1991)

∂p

∂z' δp

δz=

p(δz) − p(0)

δz, (B-20)

where, of course, the pressure at the free surface vanishes, i.e., p(0) = 0. This approximation is validas long as the streamer depth δz beneath the water surface is less than a dominant wavelength.The final expression for the (second time derivative of the) vertical component of the displacementreads now

∂2

∂t2Uc3 =

1

%

p(δz)

δz. (B-21)

The particle displacement U c3 can be obtained from its second derivative by a division by (iω)2 inthe frequency domain.

If the pressure is recorded at a streamer depth of more than a dominant wavelength, it canbe directly used as an input to true-amplitude imaging upon the use of the modified formulas forthe acoustic case as described in Chapter 3.

288 APPENDIX B. WAVES AT A FREE SURFACE

Appendix C

Curvature matrices

In this Appendix, we derive the relationship between the Hessian matrix of second derivatives of agiven surface Σ in arbitrary Cartesian coordinates and the curvature matrix of that surface Σ. Let

z = Z(r) (C-1)

denote the surface Σ when described in an arbitrary global 3-D Cartesian coordinate system r =(r1, r2, r3 = z)T , with r= (r1, r2). For a given point P on that surface, the Hessian matrix is

Z˜

=

∂2Z∂r12

∂2Z∂r1∂r2

∂2Z∂r2∂r1

∂2Z∂r22

. (C-2)

Moreover, at P , we consider the local 3-D Cartesian coordinate system x = (x1, x2, x3 = z′)T withx = (x1, x2). It is chosen to be oriented such that the x1x2-plane is tangent to the surface Σ atP . The orientation of the x1- and x2-axes may be arbitrary within the tangent plane. Then, thecurvature matrix of surface Σ at P is defined as the Hessian matrix in the local Cartesian coordinatesystem, viz.,

K˜

=

∂2Z ′

∂x12

∂2Z ′

∂x1∂x2∂2Z ′

∂x2∂x1

∂2Z ′

∂x22

. (C-3)

Here,z′ = Z ′(x) (C-4)

is the same surface Σ as described by equation (C-1), but now represented in the local Cartesiancoordinate system. Note that curvature matrices are unique up to a rotation of the x coordinateswithin the plane ΩT that is tangent to the surface Σ at P . The so-called “principal curvatures” ofthe surface Σ are obtained upon rotation of the local Cartesian coordinate system within plane ΩT

such that the off-diagonal elements of the curvature matrix K˜

become zero. Then, the diagonalelements of K

˜are the principal curvatures of the surface Σ.

In order to find a representation ofK˜

in terms of Z˜

, we first have to establish the relationshipbetween Z(r) and Z ′(x). For that purpose, we consider the x coordinate system to be oriented insuch a way that it can be constructed from the r system by a single rotation of the z-axis onto the

289

290 APPENDIX C. CURVATURE MATRICES

surface normal at P . In other words, we assume the origins of both coordinate systems to coincide.The rotation angle is the local in-plane dip angle (i.e., the angle between the z-axis and the surfacenormal). Let this angle be denoted by βP .

Moreover, without loss of generality, we may suppose that the r2-axis of the old systemand the x2-axis of the new system coincide. In general, this is achieved by a further (horizontal)rotation of the new system in the x1x2-plane (tangent to the surface at P ) which does not affectthe definition of K

˜.

Under the above conditions, the transformations between the two coordinate systems can bewritten as

x1 = r1 cos βP − z sinβP , (C-5a)

x2 = r2 , (C-5b)

z′ = r1 sinβP + z cos βP . (C-5c)

We now insert equations (C-1) and (C-4) into the transformation formula (C-5c) to obtainthe relationship between Z and Z ′ as

Z ′(x) = r1 sinβP + Z(r) cos βP . (C-6)

The desired relationship between matrices Z˜

and K˜

can now be derived from the derivatives ofequation (C-6) with respect to the x coordinates. We have

∂Z ′

∂x1= sinβP

∂r1∂x1

+ cos βP∂Z∂rk

∂rk∂x1

(C-7a)

and∂Z ′

∂x2= cosβP

∂Z∂rk

∂rk∂x2

. (C-7b)

Since βP is a constant angle, the second derivatives are

∂2Z ′

∂xi∂xj= cos βP

∂2Z∂rk∂rl

∂rk∂xi

∂rl∂xj

(C-8)

or in matrix notationK˜

= cos βP B˜T Z

˜B˜, (C-9)

where B˜

is the 2 × 2 submatrix

B˜

=

∂r1∂x1

∂r2∂x1

∂r1∂x2

∂r2∂x2

=

(

cosβP 00 1

)

(C-10)

of the transformation matrix defined in equations (3.11.13).

Equation (C-9) is our desired relationship that expresses the curvature matrix in terms ofthe Hessian matrix in arbitrary Cartesian coordinates. Moreover, from equation (C-9), we observethat the relationships between the determinants and signatures of the matrices Z

˜and K

˜, which

are needed in the text, are given by

detK˜

= cos4 βP detZ˜

(C-11a)

291

andSgnK

˜= SgnZ

˜. (C-11b)

The additional in-plane rotations of the r2-axis onto the x2 axis by angles ϕx and ϕr that isin general necessary (see Section 3.11.4) leads to a modified projection matrix

B˜

= G˜

(r) =

(

cosϕx − sinϕxsinϕx cosϕx

)(

cos βP 00 1

) (

cosϕr − sinϕrsinϕr cosϕr

)

(C-12)

in equation (C-9). Equations (C-11) remain unaffected. An additional translation, i.e., a finitedistance between the origins of the two coordinate systems x and r has no effect at all on equations(C-9) and (C-11) as only derivatives with respect to x and r are involved in the above derivation.

292 APPENDIX C. CURVATURE MATRICES

Appendix D

Relationships to Beylkin’sdeterminant

In this Appendix, we derive the relationship (5.6.16a) between the determinant of Λ˜

(r) as definedin equation (5.3.10) and the Beylkin determinant hB as defined in equation (5.6.15).

We start from equation (5.3.4), the derivative of which with respect to ri (j = 1, 2) can berewritten as

∂TΣ(ξ, r)

∂ri=

∂TD(ξ, r, z)

∂ri+

∂TD(ξ, r, z)

∂z ZR(r)

∂ZR(r)

∂ri. (D-1)

From equation (5.5.2) we know that at r= rR this expression vanishes. We conclude that

∂ZR(r)

∂rirR

= − 1

mD

∂TD(ξ, r, z)

∂rirR

, (D-2)

where we have made use of formula (5.3.14a).

To calculate the matrix Λ˜

(r), we need the second, mixed derivative of TΣ(ξ, r) with respectto ri (i = 1, 2) and ξj (j = 1, 2). These can be written accordingly as

∂2TΣ(ξ, r)

∂ri∂ξj=∂2TD(ξ, r, z)

∂ri∂ξj+∂2TD(ξ, r, z)

∂ξj ∂z

∂ZR(r)

∂ri. (D-3)

We take equation (D-3) at rR, insert equation (D-2), and factor out the common factor 1/mD toarrive at

∂2TΣ(ξ, r)

∂ri∂ξjrR

= − 1

mD

(∂TD(ξ, r, z)

∂rirR

∂2TD(ξ, r, z)

∂ξj ∂zrR

−

− ∂TD(ξ, r, z)

∂zrR

∂2TD(ξ, r, z)

∂ri∂ξjrR

)

= − 1

mDdet

∂TD(ξ, r, z)

∂ri

∂TD(ξ, r, z)

∂z∂2TD(ξ, r, z)

∂ri∂ξj

∂2TD(ξ, r, z)

∂ξj ∂z

rR

. (D-4)

293

294 APPENDIX D. RELATIONSHIPS TO BEYLKIN’S DETERMINANT

Inserting this result into the expression for the determinant of Λ˜

(r), viz.,

detΛ˜

(r) =∂2TΣ(ξ, r)

∂r1∂ξ1rR

∂2TΣ(ξ, r)

∂r2∂ξ2rR

− ∂2TΣ(ξ, r)

∂r1∂ξ2rR

∂2TΣ(ξ, r)

∂r2∂ξ1rR

, (D-5)

yields after some tedious by straightforward algebra

detΛ˜

(r) =1

mD

∂TD(ξ, r, z)

∂r1det

∂2TD(ξ, r, z)

∂r2∂ξ1

∂2TD(ξ, r, z)

∂ξ1 ∂z∂2TD(ξ, r, z)

∂r2∂ξ2

∂2TD(ξ, r, z)

∂ξ2 ∂z

−

− ∂TD(ξ, r, z)

∂r2det

∂2TD(ξ, r, z)

∂r1∂ξ1

∂2TD(ξ, r, z)

∂ξ1 ∂z∂2TD(ξ, r, z)

∂r1∂ξ2

∂2TD(ξ, r, z)

∂ξ2 ∂z

+

+∂TD(ξ, r, z)

∂zdet

∂2TD(ξ, r, z)

∂r1∂ξ1

∂2TD(ξ, r, z)

∂r2∂ξ1∂2TD(ξ, r, z)

∂r1∂ξ2

∂2TD(ξ, r, z)

∂r2∂ξ2

rR

. (D-6)

Introducing the line vector

∇TD(ξ, r, z) =

(∂TD(ξ, r, z)

∂r1,∂TD(ξ, r, z)

∂r2,∂TD(ξ, r, z)

∂z

)

, (D-7)

we may write

detΛ˜

(r) =1

mDdet

∇TD(ξ, r, z)∂

∂ξ1∇TD(ξ, r, z)

∂

∂ξ2∇TD(ξ, r, z)

rR

=hBmD

, (D-8)

where we have recognized the well-known Beylkin determinant hB defined by Beylkin (1985a) andgiven in equation (5.6.15).

Appendix E

Derivation of the scalar elasticKirchhoff integral

In this Appendix, we derive the scalar version of the elastic Kirchhoff integral for direct, transmitted,and primary reflected elementary waves in isotropic media. This scalar integral is only useful forthe derivation of the scalar Kirchhoff-Helmholtz integral in Chapter 6. For any other purposes, thegeneral isotropic Kirchhoff integral as studied in Appendix G should be used.

E.1 A scalar wave equation for elastic elementary waves

Due to the similarity of equations (3.3.5), (3.3.14), and (3.3.15), as well as (3.3.7), (3.3.18), and(3.3.24), it is possible to set up a scalar wave equation that describes the propagation of notonly an acoustic wave but also of the principal component of an elementary elastic P- or S-wave.It is justified by nothing else but the similarity of the elastic and acoustic eikonal and transportequations. In the Fourier domain, the corresponding general homogeneous scalar Helmholtz equationfor acoustic and elastic elementary waves can be written in the form

∇ ·(

fm(r)∇U(r, ω))

+ gm(r)ω2U(r, ω) = 0 , (E-1)

where fm and gm are certain generalized model parameters. This equation has a physical meaningonly where the corresponding general eikonal and transport equations

(∇T )2 =gmfm

, (E-2a)

∇ · (fmU2∇T ) = 0 , (E-2b)

provide an acceptable approximation to high-frequency wave propagation. In particular, equation(E-1) is not valid across interfaces with abrupt changes of the medium parameters. In other words,equation (E-1) is only justified by the fact that substitution of

fm =1

%, gm =

1

kfor acoustic waves, (E-3a)

fm = %v2 , gm = % for elastic waves (E-3b)

295

296 APPENDIX E. DERIVATION OF THE SCALAR ELASTIC KIRCHHOFF INTEGRAL

in equation (E-2) yields the corresponding eikonal and transport equations derived in Chapter (3).Note that %v2 = λ + 2µ for isotropic P-waves and %v2 = µ for isotropic S-waves. In the followingderivation, we will continue to use the above expressions in their explicit form for P-waves, i.e.,with fm = %α2 = λ + 2µ and gm = %, however keeping in mind that a substitution by the otherpossible forms of fm and gm allows to describe acoustic and S-wave reflections, too.

The physical meaning of U(r, ω) is, of course, different in the different cases. In the acousticcase it denotes the zero-order amplitude coefficient of pressure, i.e., P, and in the elastic caseit stands for the principal component of the particle displacement for the considered elementarywave, i.e., U (P ) or U (S). Note that the above Helmholtz equation does not describe, of course, thepolarization vector of the considered elastic elementary wave. This direction, however, is knownin zero-order ray approximation for the principal component to be parallel to the propagationdirection for P-waves and perpendicular to that direction for S-waves.

We stress that the above scalar wave equation for elementary elastic waves is, of course, onlyvalid where zero-order ray theory is valid, too. This means in particular that it does not correctlydescribe elastic transmission and reflection coefficients. So, one might wonder about the advantageone would gain from using this generalized equation. The point is that this scalar wave equationprovides us with simpler arguments to set up a scalar Kirchhoff-Helmholtz integral (see Chapter 6)for elementary seismic primary-reflected waves that can also be obtained from a more rigorousanalysis (see Appendix G).

E.2 Direct waves

Let us now set up the Kirchhoff integral for direct waves. Consider the situation depicted in FigureE.1. We assume all sources of the wavefield under consideration to be confined to a region Q. Also,we consider a region V inside which a receiver (or observation point) G is located. In Figure E.1b,Q is a part of V , while in Figure E.1a, it is not. Our aim is to compute the direct scalar wavefieldU0(r, t) that will be measured at G due to the sources in Q. The wave equation that governs thisproblem is the inhomogeneous scalar wave equation, which is, in the frequency domain, representedby the generalized scalar Helmholtz equation (see Section E.1)

∇ ·(

fm(r)∇U0(r, ω))

+ gm(r)ω2U0(r, ω) = −4πq(r, ω) , (E-4)

where q(r, ω) is the source function that vanishes for all r outside Q. As we have seen in SectionE.1, this Helmholtz equation may describe as well the propagation of acoustic as that of elementaryelastic waves. For acoustic waves, we have fm = 1/% and gm = 1/k = 1/%c2, with % being themedium density, k its bulk modulus and c the acoustic wave velocity. For elementary elastic waves,fm = %v2 and gm = %, where v is now the wave velocity of the considered elementary (P or S)wave.

We know that a solution to this wave equation can be found once the Green’s function G isknown that satisfies the corresponding equation

∇ ·(

fm(r)∇G(r, ω; r′))

+ gm(r)ω2G(r, ω; r′) = −4πδ(r − r′) , (E-5)

E.2. DIRECT WAVES 297

(b)

(a)

Q

Q

G

G

n

V

V

Σ 0

n

Σ 0

Fig. E.1. A receiver position G is located in a volume V with surface Σ. The sources are confinedto a region Q that is (a) outside V , (b) inside V .

where the source is represented by a delta pulse at r′. If equation (E-5) has been solved, a solutionto equation (E-4) is readily found to be

U0(r, ω) =

∫

Q

dQ (r′) G(r, ω; r′)q(r′, ω) . (E-6)

This can be easily checked by applying the differential operator ∇ ·(

fm(r)∇)

to equation (E-6)

and inserting the result into equation (E-5).

We now want to find an alternative solution in terms of the wavefield at the boundary Σof V . For that purpose, we consider Gauss’s divergence theorem. It states that for any arbitraryvolume V with surface Σ and for any arbitrary vector field Ψ(r) that is defined for all points r inV and on Σ, there exists the following relationship between the volume and surface integrals

IV ≡∫

V

dV ∇ · Ψ(r) =

∫

Σ

dΣ n · Ψ(r) ≡ IΣ . (E-7)

Here, n is the outward pointing unit normal vector to the surface Σ of V . The physical meaning ofGauss’s theorem is that of a conservation law. Every field that hits G coming from sources outside

298 APPENDIX E. DERIVATION OF THE SCALAR ELASTIC KIRCHHOFF INTEGRAL

V must cross the surface Σ once more on its way in than on its way out (see Figure E.1a). On theother hand, every field that stems from sources inside V and crosses the surface Σ (an odd numberof times) won’t hit G (Figure E.1b).

With the particular choice

Ψ = Gfm∇U0 − U0fm∇G , (E-8)

we may rewrite the volume integral IV on the left-hand side of equation (E-7) as

IV =

∫

V

dV[

G∇ ·(

fm(r)∇U0)

− U0∇ ·

(

fm(r)∇G)]

. (E-9)

The form (E-9) of the divergence theorem is also known as Green’s theorem.

Solving the above scalar Helmholtz equations (E-4) and (E-5) for the terms ∇ ·(

fm(r)∇U0)

and ∇ ·(

fm(r)∇G)

, and inserting the results into equation (E-9), we arrive, after some straight-

forward simplifications, at

IV = −∫

V

dV[

4πG(r, ω; r′)q(r′, ω) − 4πU0(r′, ω)δ(r − r′)]

. (E-10)

Let us now distinguish the two cases indicated in Figures E.1a and E.1b. (a) If Q belongs toV , the first volume integration reduces to region Q because q(r, ω) vanishes elsewhere. The secondintegral in equation (E-10) containing the delta function is readily solved. Thus, the overall resultis

IV = −4π

∫

Q

dQ[

G(r, ω; r′)q(r′, ω)]

+ 4πU0(r, ω) , (E-11)

which vanishes due to equation (E-6). Due to the equality of the surface and volume integrals inequation (E-7), also the surface integral IΣ on the right-hand side of that equation vanishes in thiscase. (b) On the other hand, if Q does not belong to V , the first integral in equation (E-10) vanishes[remember that q(r, ω) is identical to zero outside Q], so that we arrive at

IV = 4πU0(r′, ω) . (E-12)

Together with the right-hand side of the divergence theorem (E-7), we finally find an expression forthe wavefield U0(rG, ω) originating from sources outside the volume V enclosed by Σ and recordedat an observation point G inside Σ. It reads

U0(rG, ω) =1

4π

∫

Σ

dΣ fm(r)

[

G(r, ω; rG)∂U0(r, ω)

∂n− U0(r, ω)

∂G(r, ω; rG)

∂n

]

, (E-13)

where ∂/∂n = n · ∇ is the derivative in the direction of the surface normal. This is the famousKirchhoff integral representation (Sommerfeld, 1964; Born and Wolf, 1987), here generally rederivedfor any type of scalar Helmholtz equation (E-4). We remind that fm = 1/% for acoustic waves andfm = %v2 for elementary elastic waves.

E.3. TRANSMITTED WAVES 299

Q

G

n

V

Σ 0

Σ T

n

Fig. E.2. By extending the surface Σ to infinity where possible without crossing the transmittinginterface ΣT and to ΣT everywhere else, the Kirchhoff integral can be reduced to an integrationalong ΣT . Note that the direction of the normal vector must be inverted.

E.3 Transmitted waves

We have seen that the wavefield at an observation point G can be computed by integral (E-13)from the values of the wavefield and its normal derivatives at a surface Σ surrounding G, providedthe sources of the wavefield are located outside Σ. The particular shape of the volume V or ofthe surface Σ plays no role for this representation. In particular, we may extend the surface Σto a transmitting interface ΣT that is assumed to be located between G and Q (see Figure E.2),and to infinity elsewhere. The integration over infinity does not yield any contribution because thewavefield and its derivatives are required by Sommerfeld’s radiation conditions to vanish at infinitedistance from the source. Inverting the direction of the normal vector to have it pointing towardsG, a representation for the transmitted field U t(rG, ω) at G is thus found to be

U t(rG, ω) =−1

4π

∫

ΣT

dΣT fm(r)

[

G(r, ω; rG)∂U t(r, ω)

∂n− U t(r, ω)

∂G(r, ω; rG)

∂n

]

, (E-14)

where U t(r, ω) inside the integral represents the wavefield at the transmitting interface ΣT directlyafter transmission.

E.4 Reflected waves

Similar considerations can be used to derive a “Kirchhoff integral” for reflected waves. Consider thesituation depicted in Figure E.3. We assume that the direct problem for a medium with parametersfm and gm has already been solved. We are now interested in solving a related problem with differentmedium parameters. To visualize this difference, we denote the (variable) medium parameters nowby a tilde above the symbol, i.e., by fm and gm. However, we assume that there exists a certainregion R to which all differences are confined, i.e., fm 6= fm and gm 6= gm in R, but fm = fm andgm = gm elsewhere. The “scattering region” R is assumed to be entirely outside the volume V andthe source region Q is assumed to be entirely part of V . Our aim is now to compute the additional

300 APPENDIX E. DERIVATION OF THE SCALAR ELASTIC KIRCHHOFF INTEGRAL

Q G

n

V :

~f ≠ fg ≠ g~

Σ 0

R :Σ R

~f = fg = g~

Fig. E.3. The medium parameters, fm and gm, are supposed to differ from the unperturbed ones,fm and gm, in region R only. The scattered wavefield due to this perturbation is to be computedat G.

contribution to the wavefield at G due to the presence of the scattering region R, i.e., the field“scattered” or “reflected” at the medium perturbations in R.

The wave equation for the total wavefield U(r, ω) is given correspondingly to equation (E-4)above by

∇ ·(

fm(r)∇U(r, ω))

+ gm(r)ω2U(r, ω) = −4πq(r, ω) . (E-15)

Subtracting now equation (E-4) from equation (E-15), and introducing the “scattered field” U s(r, ω)as the difference between the total field U(r, ω) and the direct field in absence of the scatterer,U0(r, ω), i.e., U s(r, ω) = U(r, ω) − U0(r, ω), we obtain

∇ ·(

fm(r)∇U s(r, ω))

+ ω2gm(r)U s(r), ω) = −4πqs(r, ω) , (E-16)

where

qs(r, ω) =1

4π

(

∇ · [(fm − fm)∇U ] + ω2(gm − gm)U)

(E-17)

describes the so-called secondary sources in the region R, i.e., the “sources” of the scattered wave-field that is only present because of the differences fm − fm and gm − gm. Physically, we mayinterpret this source term as the Huygens sources excited by the total field U(r, ω). At this point,it is worthwhile to observe that qs(r, ω) = 0 for all r in V . This is because of our assumption thatall points r where fm 6= fm and gm 6= gm are confined to the region R that was supposed to beoutside V . Note that in single-scattering approximation, one would replace in equation (E-17) thetotal field U(r, ω) by the incident (direct) field U0(r, ω). As equation (E-16) is just the originalHelmholtz equation (E-4) with a different source term qs(r, ω), its solution can be represented inform of equation (E-6) with qs instead of q and integrating over region R instead of Q. Togetherwith the mentioned single-scattering approximation for qs, this is the Born approximation for thescattered wavefield.

To derive a Kirchhoff representation, we now return to Gauss’s divergence theorem (E-7)using, however, a slightly different vector function Ψ, namely

Ψ = Gfm∇U − Ufm∇G , (E-18)

E.4. REFLECTED WAVES 301

with G still being a solution of Helmholtz equation (E-4), but U being now a solution of Helmholtzequation (E-15). In parallel to the above, we arrive at

IV = 4π(U0 + U s) +

∫

V

dV[

−G 4πq − G 4πqs]

. (E-19)

The first integration reduces to domain Q inside V , and thus yields −4πU0 due to equation (E-6).The second integration vanishes because qs(r) = 0 for all r in V and, thus, integral (E-19) yieldsIV = 4πU s. In other words, because of equation (E-7), the result for the scattered wavefieldU s(rG, ω) at G is

U s(rG, ω) =1

4π

∫

Σ

dΣ fm(r)

[

G(r, ω; rG)∂U (r, ω)

∂n− U(r, ω)

∂G(r, ω; rG)

∂n

]

. (E-20)

We now replace in the above integral U by U0+U s and separate the result into two surface integrals,depending on U0 and U s, respectively. We recognize that the integration over U0 vanishes, becauseQ is contained in V . This leads to equation (E-11) for the direct field U0. Thus, equation (E-20)can be recast into

U s(rG, ω) =1

4π

∫

Σ

dΣ fm(r)

[

G(r, ω; rG)∂U s(r, ω)

∂n− U s(r, ω)

∂G(r, ω; rG)

∂n

]

. (E-21)

Because of our assumption that all sources are confined to region Q and that all secondary sources(scatterers) are confined to region R, equation (E-21) is valid independently of the particular shapeof Σ. We may thus extend it to infinity wherever possible, however in such a way that R remainsoutside V (see Figure E.4). At the very end of such an extension, we will have essentially a sum oftwo integrations. The first one is carried out in infinite distance from the source, where the fieldis required to vanish due to Sommerfeld’s radiation conditions, and thus yield a vanishing result.The second integration is carried out along the surface ΣR of R, where the surface normal is nowpointing inward, i.e., into region R (see Figure E.4a). Changing the direction of the normal vectorof this surface to pointing outward region R means changing the sign of the resulting integration.We thus finally arrive at the following expression for the reflected field U r(rG, ω) at G

U r(rG, ω) =−1

4π

∫

ΣR

dΣR fm(r)

[

G(r, ω; rG)∂U r(r, ω)

∂n− U r(r, ω)

∂G(r, ω; rG)

∂n

]

. (E-22)

This form of the Kirchhoff integral for reflected waves does not depend on whether the surface ΣR

of R is a closed or an open surface (Figures E.4a and E.4b).

Note that this integral expression describes as well acoustic as elementary elastics wavesas long as the ray-theoretical approximations for G(r, ω; rG) and U r(r, ω) are valid. It is exactlythis situation in which we are interested in Chapter 6. In that chapter, we apply the Kirchhoff-Helmholtz approximation to integral (E-22), i.e., we substitute the ray-theoretical approximationsfor G(r, ω; rG) and U r(r, ω) in that equation. The resulting Kirchhoff-Helmholtz integral describesthe reflected field at G by an integral along the reflector. Although all direct propagation effects toand from the reflector are approximated by zero-order ray theory, the resulting integral describesthe reflected wavefield with better accuracy, even including diffraction events.

302 APPENDIX E. DERIVATION OF THE SCALAR ELASTIC KIRCHHOFF INTEGRAL

R

GQV

GQV

n

n

n

nΣR

ΣR

Σ0

Σ0

Fig. E.4. The surface Σ can be extended to infinity wherever it does not cross the surface ΣR.The surface ΣR may either be (a) a closed surface or (b) a reflecting interface stretching to infinity.Note that in both cases the direction of the normal vector must be inverted.

Appendix F

Kirchhoff-Helmholtz Approximation

In this Appendix, we give an explanation of the ansatz used in the Kirchhoff-Helmholtz approxi-mation. For that purpose, let us first consider the simple case of a transmission (resp. reflection)of a plane wave at a planar interface between two homogeneous half-spaces.

F.1 Plane wave considerations

Without loss of generality, we assume the interface to be horizontal and to coincide with the planez = 0, where the z-axis is pointing into the lower medium (Figure F.1). Leaving the time-harmonicdependency expiωt aside, a monofrequency plane wave incident from above, e.g., when passingthrough point Gi, is generally described by

Ui(ω) = U exp

−iω(

ηsinϑ−

v−+ z

cosϑ−

v−

)

, (F-1)

where U is the (constant) amplitude and v− is the wave velocity for the incident field. In Figure F.1,v− = v1, i.e., the (constant) medium velocity above the interface. The quantity ϑ− is the acuteangle that the propagation direction makes with the z-axis. Also, we have used the horizontalcoordinate η = x cosϕ+ y sinϕ, where ϕ is the azimuth angle within the xy-plane. The wavefieldat an observation point Gt (Gr) after being transmitted (reflected) at the plane z = 0 is then givenby

Ua(ω) = Ca U exp

−iω(

ηsinϑ+

v+± z

cos ϑ+

v+

)

, (F-2)

with index a being r for the reflected wave and t for the transmitted wave. The upper sign inequation (F-2) holds for transmission (a = t), the lower one for reflection (a = r). Moreover,Ct (Cr) is the transmission (reflection) coefficient T (Rc). Quantities that are marked with anupper index − denote parameters before incidence at the interface and those marked with + denoteparameters after transmission (reflection). For instance, in Figure F.1, for transmission v+ = v2,whereas for (monotypic) reflection v+ = v1.

In the Kirchhoff integral (E-13), the normal derivative of the field to be propagated at thesurface Σ is needed. To compute the corresponding derivatives of the above wavefields (F-1) and

303

304 APPENDIX F. KIRCHHOFF-HELMHOLTZ APPROXIMATION

Gi

Gt

Gr

nt

nr

z = 0η

ϑrϑi

ϑt

MR

v1

v2

incidentplane wave

z

Fig. F.1. A plane wave impinges onto a planar interface located at z = 0.

(F-2), let us first take their gradients. We find

∇Ui = − iω

(

cosϕsinϑ−

v−, sinϕ

sinϑ−

v−,cosϑ−

v−

)T

Ui (F-3)

and

∇Ua = − iω

(

cosϕsinϑ+

v+, sinϕ

sinϑ+

v+,±cosϑ+

v+

)T

Ua . (F-4)

To obtain the normal derivatives of the fields, one simply has to multiply the above gradientswith the surface normal n at the transmission (reflection) point. However, there are two possibledefinitions for the surface normal of the interface at z = 0. Which one is correct? In the Kirchhoffintegrals (E-14) and (E-22) for the transmission and reflection case, respectively, the normal vectorsare defined as outward normals on the surfaces ΣT and ΣR, i.e., pointing towards the respectiveobservation point Gt orGr. Thus, we now have to introduce different normal vectors for the reflected(nr) and transmitted (nr) wavefields. We must use (Figure F.1)

nr = (0, 0,−1)T and nt = (0, 0, 1)T . (F-5)

With these normal vectors, we arrive at

na · ∇Ua =∂Ua∂n

= − iωcosϑ+

v+Ua , (F-6)

where we have also taken into account that ϑ− and ϑ+ denote acute angles with the z-axis. Multi-plying also the incident field with these normal vectors, we have

na · ∇Ui =∂Ui∂n

= ∓ iωcosϑ−

v−Ui . (F-7)

Equations (F-6) and (F-7) can be used to investigate the relationship between the incident andtransmitted (reflected) fields at the interface. We first observe in equation (F-6) that the signs of

F.2. LOCAL PLANE-WAVE APPROXIMATION 305

the normal derivatives of the transmitted (a = t) and reflected (a = r) fields are the same. In spiteof this fact, they relate differently to the corresponding signs of the incident field. This is due to thedifferent directions of the normal vectors (see Figure F.1). On the one hand, the normal derivativeof the transmitted field has the same sign as that of the incident field. On the other hand, thesign of the normal derivative of the reflected wave in equation (F-6) is inverted with respect to thecorresponding one of the incident field in equation (F-7). This is due to the “inverted” propagationdirection (“upward” instead of “downward” propagation) of the reflected field.

Let us now consider the situation at point MR on the interface (z = 0). Inserting z = 0 intoequations (F-1) and (F-2), and taking into account Snell’s law, i.e., sinϑ−/v− = sinϑ+/v+, weobserve that the transmitted and reflected fields relate to the incident one as

Uaz=0

= Ca Uiz=0

. (F-8a)

Correspondingly, the respective normal derivatives relate to the incident field as

∂Ua∂n

z=0

= − iωcosϑ+

v+Ca Ui

z=0

. (F-8b)

which results from inserting result (F-8a) into equation (F-6).

F.2 Local plane-wave approximation

Now it is easy to explain what is done when the Kirchhoff-Helmholtz approximation is used in theKirchhoff integrals (E-14) and (E-22) for the transmission and reflection case, respectively. In thoseintegrals, there appear both the expressions Ua and ∂Ua/∂n at an arbitrary interface ΣR and withan arbitrary incident field. The Kirchhoff-Helmholtz approximation now simply assumes that theequations (F-8a) and (F-8b) are also valid in this general case, i.e., at each point MR on ΣR,

UaMR

= Ca UiMR

, (F-9a)

∂Ua∂n

MR

= − iωcosϑ+

R

v+R

Ca UiMR

. (F-9b)

Here, we have denoted the velocity at MR after specular reflection by v+R to indicate that this

approximation may also be used in (slightly) inhomogeneous media. Physically spoken, it is assumedin this approximation that the incident wavefield locally behaves like a plane wave and that thereflector ΣR locally acts like a planar interface at MR. The amplitude variation due to the curvaturesof the interface and of the true wavefront is neglected.

Moreover, correspondingly to equation (F-9b), it is also assumed that the Green’s functionthat describes the wave propagation from MR to G approximately fulfills

∂Ga(r, r′)∂n

MR

= −iω cosϑGRv+R

GaMR

, (F-10)

where ϑGR is the angle that the ray from MR to G makes with n at MR. This is a high-frequencyapproximation that corresponds to zero-order ray-theory assumptions.

306 APPENDIX F. KIRCHHOFF-HELMHOLTZ APPROXIMATION

It is to be remarked that equation (F-9b) simplifies for the particular case of a monotypicreflection (i.e., a P-P, S-S, or acoustic reflection). In this case, which is usually considered in theliterature, v− = v+ and ϑ− = ϑ+. Therefore, upon the use of equation (F-7), equation (F-8b) maybe written as

∂Ur∂n

z=0

= − Rc∂Ui∂n

z=0

. (F-11)

The Kirchhoff-Helmholtz approximation equation (F-9b) can thus be recast into the following well-known form (Bleistein, 1984)

∂Ur∂n

MR

= − Rc∂Ui∂n

MR

. (F-12)

Note that equations (F-9a) and (F-12) for Rc = −1 are known as the “Physical Optics Approx-imation for perfectly soft scatterers” or “Dirichlet boundary conditions” and for Rc = 1 as the“Physical Optics Approximation for perfectly rigid scatterers” or “Neumann boundary conditions”(Sommerfeld, 1964). For arbitrary Rc, they are often referred to as “Kirchhoff approximation”or “Kirchhoff-Helmholtz approximation.” By choosing to represent the second of these approxima-tions in the form of equation (F-12), one implicitly assumes, however, that monotypic reflections areconsidered. The general Kirchhoff-Helmholtz approximation for arbitrary reflected or transmitted(scalar) waves, may they be converted or not, is given by the pair of equations (F-9).

Appendix G

The scalar elasticKirchhoff-Helmholtz integral

In this Appendix, we derive the scalar Kirchhoff-Helmholtz integral (6.1.11a) for isotropic elasticelementary waves from the general anisotropic representation theorem (Aki and Richards, 1980).All concepts and ideas are similar or identical to those discussed in Appendix E. For that reason,we refrain from repeating them here in detail. We use Einstein’s summation convention.

G.1 The anisotropic elastic Kirchhoff integral

We start with a brief investigation of the Kirchhoff integral for elastic, anisotropic media as far as itis needed for the present purposes. Consider the Green’s functions Gin(r, ω; rS) and Gim(r, ω; rG)governing the wave propagation from the source S at rS to a reflector point MΣ at r and from areceiver G at rG to the same reflector point, respectively, in the general anisotropic case. TheseGreen’s functions are described in the frequency domain by the following two Helmholtz equations(Aki and Richards, 1980),

−%ω2Gin(r, ω; rS) −(

cijklGkn,l(r, ω; rS))

,j= 4πδinδ(r − rS) , (G-1a)

−%ω2Gim(r, ω; rG) −(

cijklGkm,l(r, ω; rG))

,j= 4πδimδ(r − rG) , (G-1b)

where cijkl (i, j, k, l = 1, 2, 3) are the components of the general anisotropic elastic tensor and Gij(i, j = 1, 2, 3) are the components of the anisotropic Green’s function which is also a tensor. Anindex j after a comma indicates the derivative with respect to the jth Cartesian coordinate.

Using the divergence theorem that formulates the relationship between a surface integral ISand a volume integral IV , we may write according to Aki and Richards (1980) or Frazer and Sen(1985),

IS ≡∫

Σ

(

cijklGkm,l(r, ω; rG)Gin(r, ω; rS) − cijklGkn,l(r, ω; rS)Gim(r, ω; rG))

nj dr =

=

∫

V

(

cijklGkm,l(r, ω; rG)Gin(r, ω; rS) − cijklGkn,l(r, ω; rS)Gim(r, ω; rG))

,jdr ≡ IV . (G-2)

307

308 APPENDIX G. THE SCALAR ELASTIC KIRCHHOFF-HELMHOLTZ INTEGRAL

The volume V is assumed to contain the source as well as the receiver, but not the scattering points.Applying the chain rule to the rj-derivative, we can recast the volume integral into the form

IV =

∫

V

[(

cijklGkm,l(r, ω; rG))

,jGin(r, ω; rS) + cijklGkm,l(r, ω; rG)Gin,j(r, ω; rS) −

−(

cijklGkn,l(r, ω; rS))

,jGim(r, ω; rG) − cijklGkn,l(r, ω; rS)Gim,j(r, ω; rG)

]

dr .(G-3)

We now make use of the wave equations (G-1) to replace the first and the third term in the aboveintegral. Also, we rename the summation indices in the fourth term. This yields

IV =

∫

V

[

−4πδimδ(r − rG)Gin(r, ω; rS) − %ω2Gim(r, ω; rG)Gin(r, ω; rS)

+ cijklGkm,l(r, ω; rG)Gin,j(r, ω; rS)

+ 4πδinδ(r − rS)Gim(r, ω; rG) + %ω2Gin(r, ω; rS)Gim(r, ω; rG)

− cklijGin,j(r, ω; rS)Gkm,l(r, ω; rG)

]

dr . (G-4)

Here, we recognize that the second and fifth term of the above volume integral cancel each other.Using the symmetry of the elastic tensor, cijkl = cklij, we observe that also the third and sixthterms do.

If Gmn(r, ω; rS) represents a direct wavefield only, and the volume V contains both source andreceiver, the remaining two integrals, which contain Dirac’s δ-functions, yield the Green’s functions−4πGmn(rG, ω; rS) and 4πGnm(rS , ω; rG), respectively. Due to the symmetry relation

Gij(r, ω; r′) = Gji(r′, ω; r) , (G-5)

these are identical and thus the above volume integral vanishes. From this, we conclude that theleft-hand-side surface integral over a direct field only must vanish, too.

The situation is different, if Gmn(r, ω; rS) represents a superposition of a direct and a scatteredwavefield,

Gmn(r, ω; rS) = Gimn(r, ω; rS) + Gsmn(r, ω; rS) . (G-6)

In this situation, the remaining of the above integrals can be separated into integrals that containonly the scattered or the direct wavefield. As in Appendix E, we assume again that the volume Vcontains both the source and the receiver, but that the scattering region is outside V . Therefore, theintegrals over the direct field Gimn(r, ω; rS) vanish as discussed above. Because of equation (G-3),so does the corresponding surface integral. However, because of our assumption that the scattereris outside V , the volume V does not contain the sources of the scattered field, Gsmn(r, ω; rS), whichare, of course, secondary sources in this case. Thus, in this case, the final result of the above volumeintegration is

IV = − 4πGsmn(rG, ω; rS) . (G-7)

Equaling this result to the surface integral in equation (G-3), we have thus found the followingrepresentation for the scattered field

Gsmn(rG, ω; rS) =1

4π

∫

Σ

(

cijklGskn,l(r, ω; rS)Gim(r, ω; rG)

− cijklGkm,l(r, ω; rG)Gsin(r, ω; rS))

nj dr . (G-8)

G.2. ANISOTROPIC KIRCHHOFF-HELMHOLTZ APPROXIMATION 309

This is the Kirchhoff integral for the scattered field in elastic, anisotropic media (see also Aki andRichards, 1980). Note that there is no approximation involved in the derivation of integral (G-8).This is an exact representation of the scattered field. However, since the field and its derivative ata certain surface are generally unknown, the representation integral in this form is not of great usein practice. A possible way to make practical use of formula (G-8) consist of using the generalizedKirchhoff-Helmholtz approximation for the wavefield and its derivative in anisotropic media. Thefull treatment of this approximation can be found in Schleicher et al. (2001). In the next section,we present a brief summary.

G.2 Anisotropic Kirchhoff-Helmholtz approximation

As before (see Appendix E), let the secondary sources (i.e., the scattering points) be confined toa region R outside V that is separated from V by a given surface ΣR. As is usually done whendescribing scattering by means of the Kirchhoff integral (see, e.g., Langenberg, 1986), we now extendthe surface of integration Σ to infinity wherever no scattering points are met, and else to the surfaceΣR. There, we cannot extend the surface Σ further because of the assumption that the sources areoutside the volume V . Again, the integration over the infinity parts of the boundary Σ does notcontribute. The reason is the physical condition that no wavefield can be generated at infinity andall wavefield generated in finite distances have to vanish at infinity. This condition translates tosome anisotropic farfield conditions that are equivalent to Sommerfeld’s radiation conditions foracoustic media. Thus, the integration in equation (G-8) reduces to a surface integral over ΣR. Notethat this gives rise to a change of sign of the integral because the normal vector to the surfacehas now to be chosen in the opposite direction to make it point outward again if the scatteringsurface ΣR is closed. We next replace in analogy to classical Kirchhoff-Helmholtz (high-frequency)approximation (see Appendix F) the scattered field and its derivative at the surface ΣR by thespecularly reflected field after reflection at ΣR, i.e.,

Gsin(r, ω; rS) = Grefin (r, ω; rS) , (G-9a)

Gskn,l(r, ω; rS) = −iωprefl Gref

kn(r, ω; rS) , (G-9b)

as well as the receiver Green’s function derivative by

Gkm,l(r, ω; rG) = iωpGl Gkm(r, ω; rG) , (G-10)

where prefl and pGl are the components of the slowness vectors at the scattering point r of the

incident ray after specular reflection, pref, and of the receiver ray, pG, respectively. Both vectors rand pref are taken to point downwards. Taking into account the usual reversal of the direction ofthe surface normal vector, we arrive at

Gsmn(r, ω; rS) =iω

4π

∫

ΣR

(

cijklGrefkn(r, ω; rS)pref

l Gim(r, ω; rG) +

+ cijklGkm(r, ω; rG)pGl Grefin (r, ω; rS)

)

nj dr

=iω

4π

∫

ΣR

cijklGrefkn(r, ω; rS)Gim(r, ω; rG)

(

nlpGj + njp

refl

)

dr , (G-11)

where we have again made use of the symmetry of the elastic tensor. For further approximateevaluation, we introduce the generalized zero-order ray approximation (Chapman and Coates, 1994)

310 APPENDIX G. THE SCALAR ELASTIC KIRCHHOFF-HELMHOLTZ INTEGRAL

for the Green’s function linking the reflector point to the receiver,

Gij(r, ω; rG) = hGi (r)G0(r, rG) expiωT (r, rG)hj(rG) . (G-12a)

Here, hGi (r) is the ith component of the polarization vector hG(r) of the receiver ray at r. Corre-

spondingly, hj(rG) is the jth component of the polarization vector at rG. Analogously, we describethe Green’s function of the specular reflected field in ray-theoretical approximation as

Grefij (r, ω; rS) = href

i (r)Gref0 (r, rS) expiωT (r, rS)hj(rS) . (G-12b)

In this expression, the change of the polarization direction has been accounted for by replacing the

incoming polarization vector of the source ray, hS(r), by the reflected one, h

ref(r), assumed to be

known. Again in analogy to classical Kirchhoff-Helmholtz approximation (Appendix E), we nowassume that the amplitude Gref

0 (r, rS) of the reflected Green’s function Grefij (r, ω; rS) in equation

(G-12b) is approximately given by the amplitude of the incident wavefield, multiplied by the scalaranisotropic plane-wave reflection coefficient Rc of the elementary reflected wave under consideration(as, for example a P-P reflection). In symbols,

Gref0 (r, rS) = Rc G0(r, rS) . (G-13)

Using these approximations, we can finally write the anisotropic Kirchhoff-Helmholtz integral as

Gsmn(rG, ω; rS) =iω

4π

∫

ΣR

hm(rG)Rc cijklhGi h

refk

(

nlpGj + njp

refl

)

×

× G0(r, rG)G0(r, rS) expiω[T (r, rG) + T (r, rS)]hn(rS)dr . (G-14)

In an isotropic medium, the scalar quantity

NK = cijklhGi h

refk

(

nlpGj + njp

refl

)

. (G-15)

appearing in the kernel of integral (G-14) further simplifies as we will see in the next section.

Note that in an isotropic medium, the polarization vectors at the source and receiver positions,h(rS) and h(rG), respectively, do not depend on the position of the scattering point and can betaken out of the integral. Thus, the Green’s function Gsmn(rG, ω; rS) of the scattered field can becomputed by the multiplication of the polarization vectors h(rS) and h(rG) with a scalar integral.In the next section, we will show that this scalar integral can be approximated by the scalar elasticKirchhoff integral (6.1.11a).

G.3 The Kirchhoff-Helmholtz integral for an isotropic medium

Above, we have derived the general expression for a Kirchhoff-Helmholtz integral in the anisotropic,elastic case. Here, we will see how this integral (G-14) reduces in the case of an isotropic medium. Wewill then be able to compare it with the directly derived scalar elastic Kirchhoff integral (6.1.11a)derived in Chapter 6.

The elastic tensor for an isotropic medium is given by (Aki and Richards, 1980)

cijkl = λδijδkl + µ(δikδjl + δilδjk). (G-16)

G.3. THE KIRCHHOFF-HELMHOLTZ INTEGRAL FOR AN ISOTROPIC MEDIUM 311

Inserting this into equation (G-15), we obtain

NK = [λδijδkl + µ(δikδjl + δilδjk)]hGi h

refk (nlp

Gj + njp

refl )

= [λhGj hrefl + µ(δjlh

Gi h

refi + hGl h

refj )](nlp

Gj + njp

refl ) .

= λhGj hrefl (nlp

Gj + njp

refl ) + µ[hGi h

refi (njp

Gj + njp

refj ) + hGl h

refj )(nlp

Gj + njp

refl )] .(G-17)

To achieve further simplification of the above expression for NK , we have to make a distinctionbetween two possible cases: (a) the receiver ray at r is that of an elementary P-wave, and (b) thereceiver ray at r is that of an elementary S-wave. Case (a) includes all possible wave modes thatinvolve, at the target reflector ΣR, a P-P reflection or an S-P conversion, and case (b) includes allremaining wave modes involving an S-S reflection or a P-S conversion.

In the case of (a), the reflected wave at r has a polarization vector in the direction of the rayand its slowness vector. Thus, we have h

γ= αpγ , and h

γ · pγ = 1/α, where the superscript γ maybe G or ref, and where α is the P-wave velocity. In this case, the factor NK becomes

NK = [λ+ 2µ(hrefi hGi )](pGj nj + pref

j nj) . (G-18a)

In the case of (b) on the other hand, the wave at r is polarized perpendicularly to the ray. Thus, wehave hγi p

γi = 0, where again the superscript γ may be G or ref. In this situation, equation (G-17)

reduces to

NK = µ[hGi hrefi (njp

Gj + njp

refj ) + hGl nlh

refj p

Gj + hGl p

refl h

refj nj] . (G-18b)

Both expressions (G-18a) and (G-18b) for NK are still rather complicated. To further simplify

them, we approximate the polarization vector of the reflected wave by href ' hG. This relationship

holds exactly at the specular reflection point, from where the main contribution of the Kirchhoffintegral stems. Using this approximation, we find the following common expression for the kernelfactor NK for both cases (a) and (b),

NK = %v2(pGj + prefj )nj = 2fm OK . (G-19)

Here, OK is the obliquity factor of the Kirchhoff integral as defined in equation (6.1.10) andfm = %v2 as defined in Section E.1 for elastic waves. Quantity v is the velocity encountered by theoutgoing wavefield after scattering at the medium discontinuity, i.e., v = α =

√

(λ+ 2µ)/% for case(a) and v = β =

√

µ/% for case (b).

As we have observed above, in isotropic media the polarization vectors at the source andreceiver positions, h(rS) and h(rG), respectively, do not depend on the position of the scatteringpoint and can be taken out of the integral. Thus, for isotropic media the Kirchhoff-Helmholtzintegral for the scattered field in equation (G-14) reduces to

Gsmn(rG, ω; rS) = hm(rG)U (rG, ω; rS)hn(rS) , (G-20)

where the scalar quantity

U(rG, ω; rS) =iω

4π

∫

ΣR

RccijklhGi h

refk

(

nlpGj + njp

refl

)

×

× G0(r, rG)G0(r, rS) expiω[T (r, rG) + T (r, rS)]dr (G-21)

312 APPENDIX G. THE SCALAR ELASTIC KIRCHHOFF-HELMHOLTZ INTEGRAL

can be approximated by the integral

U(rG, ω; rS) =iω

2π

∫

ΣR

Rc %v2 OK G0(r, rG)G0(r, rS) expiω[T (r, rG) + T (r, rS)]dr . (G-22)

This is exactly the same expression as the scalar elastic Kirchhoff-Helmholtz integral in equation(6.1.11a) together with its kernel given in equation (6.1.9). In Chapter 6, it was independentlyderived starting from the scalar elastic wave equation (E-1) of Section E.1. The more generalderivation presented in this Appendix justifies the scalar derivation in Chapter 6. Note, however,that any transmission losses to be included in formula (6.1.11a) of (G-22) need to be calculated usingthe elastic formula (3.13.7). The use of amplitudes G0 calculated by the scalar wave equation (E-1)leads to wrong reflection amplitudes wherever transmitting interfaces in the reflector overburdenare involved.

Appendix H

Evaluation of chained integrals

In this Appendix, we use the stationary-phase method to evaluate certain stacking integrals thatappear when chaining the diffraction and isochron-stack integrals, i.e., when inserting them intoeach other, to solve the configuration-transform (CT) and remigration (RM) problems.

H.1 Cascaded configuration transform

In this section, we prove that the cascaded operation in equation (9.2.6) indeed yields the amplitude-preserving configuration transform (CT) result of equation (9.2.10), provided the TA kernelsKDS(ξ;M) and KIS(r;N) specified in equations (7.2.26) and (9.1.23) are used. For that purpose,let us denote by Iξ(r; N) the inner integral in equation (9.2.6), viz.,

Iξ(r; N) =−1

2π

∫

A

∫

d2ξ KCC(ξ, r; N) U0(ξ)F [TCC(ξ, r; N ) − TR(ξ)] , (H-1)

where we have made use of the ray-theoretical expression (7.1.2a). Here, the double dot denotesthe second derivative with respect to the argument. Taking into account the definition (9.2.7a) ofthe chained traveltime function TCC(ξ, r; N) = TD(ξ; MI), a comparison of equation (H-1) withexpression (7.1.4) shows immediately that Iξ(r; N) represents a diffraction-stack integral with re-spect to the point MI with coordinates (r, ZI(r; N )) on the isochron z = ZI(r; N), however with adifferent kernel or weight function KCC(ξ, r; N). The wavelet in this case is the first derivative ofthe original source wavelet F [t]. Because of this observation, we know that integral (H-1) can, inthe same way as we have done this before with integral (7.1.4), be asymptotically evaluated. Theresult is [compare with equation (8.2.18) under consideration of equation (8.2.8)]

Iξ(r; N) ' ΥCC(r; N)U0(ξ∗)F [mD(ξ∗;MΣ)(ZI(r; N ) −ZR(r))] , (H-2)

where ξ∗ = ξ∗(r) denotes the stationary point of the function

δCC(ξ, r; N ) = TCC(ξ, r; N) − TR(ξ)

= TD(ξ; MI) − TR(ξ) , (H-3)

i.e., the point where the gradient with respect to ξ vanishes, viz.,

∇ξδCC(ξ, r; N)

ξ∗= 0 . (H-4)

313

314 APPENDIX H. EVALUATION OF CHAINED INTEGRALS

As for the asymptotic evaluation of integral (7.1.4), it is supposed that to each r in E, there existsa unique point MΣ with coordinates (r,ZR(r)) on the reflector ΣR in the vicinity of which integral(H-1) is evaluated. This point MΣ is dual to the point NΓ with coordinates (ξ∗, TR(ξ∗)) in theinput space, where the Huygens surface ΓM of MΣ is tangent to the reflection-time surface ΓR.The Hessian matrix of δCC(ξ, r; N) with respect to ξ is identical to the already computed Hessianmatrix H

˜∆, because the ξ-dependence of δCC(ξ, r; N) in equation (H-3) is exactly the same as that

of T∆(ξ;M) in equation (7.1.7).

The amplitude factor ΥCC(r; N) is thus given correspondingly to equation (7.3.2) by

ΥCC(r; N) =OF (ξ∗)OD(ξ∗)

hB(ξ∗;MΣ)

KCC(ξ∗(r), r; N)

LF. (H-5)

Inserting the asymptotic evaluation result (H-2) of Iξ(r; N) into the chained integral (9.2.6), weobtain the asymptotic expression

U(N) ' 1

2π

∫

E

∫

d2rΥCC(r; N) U0(ξ∗) F [mD(ξ∗;MΣ)(ZI(r; N) −ZR(r))] . (H-6)

We recognize that the isochron z = ZI(r; N), constructed for the output pair (S(η), G(η)) that isdetermined by point N with coordinates (η, τ), represents the demigration inplanat as describedin Chapter 2. Therefore, the asymptotic evaluation of integral (H-6) can be readily performed infull analogy to that of the isochron-stack integral (9.1.4). In analogy to expression (9.1.12), we findat N with coordinates (η, τ), slightly displaced in τ -direction from the point NR with coordinates(η, TR(η)), that

U(N) ' ΥCT (N ) U0(ξR) F [mD(ξR;MR)

mD(η;MR)(τ − TR(η))] . (H-7)

Equation (H-7) represents the CT output at N . Here, τ = TR(η) represents the reflection traveltimesurface ΓR of the reflector ΣR observed in the output space. Moreover, we have introduced inequation (H-7) the notation ξR = ξ∗(rR), where rR = r∗(η) is the stationary point of the differencefunction

δCT (r; N) = mD(ξ∗;MR)(ZI(r; N) −ZR(r)) , (H-8)

i.e., rR is the horizontal coordinate of the tangency point between the isochron z = ZI(r; N ) andthe target reflector ΣR given by z = ZR(r). The coordinate vector rR = r∗(η) selects from allreflector points MΣ that particular point MR with coordinates (rR,ZR(rR)) that is dual to NR

with coordinates (η, TR(η)), the latter being the point on the traveltime surface ΓR in the outputspace with the same coordinate η as N . Correspondingly, the same point MR on ΣR is also dualto point NR with coordinates (ξR, TR(ξR)) on the traveltime surface ΓR.

The amplitude factor ΥCT (N ) is given by a similar expression to equation (9.1.5b), involvingtwo modifications. The integral kernel KIS(r∗;NR)m(r∗) is replaced by ΥCC(rR; NR) and the Hes-sian matrix H

˜IS of δIS(r;NR) [defined in equation (9.1.6)] is replaced by the Hessian matrix H

˜CT

of δCT (r; N) [defined in equation (H-8)]. All that remains to be done is to compute a convenientexpression for ΥCT (N). Similar to equation (9.1.20), we may express it as

ΥCT (N) =OD(η)

OF (η)

ΥCC(rR; NR)

mD(ξR;MR)LF cos2 βR . (H-9)

H.2. CASCADED REMIGRATION 315

Here, we have expressed H˜CT using equation (9.1.18a) for the determinant of H

˜IS with the pre-

stretch factor m(r∗) replaced by mD(ξR;MR), i.e.,

det(H˜CT ) =

mD(η;MR)

mD(ξR;MR)det(H

˜F ) . (H-10)

Moreover, we have used the definition (6.2.8) of the Fresnel geometrical-spreading factor. Bothequations have been applied to the output configuration, i.e., using ZI(r; N) instead of ZI(r;N).Note that at the stationary point rR, the isochrons are tangent to to each other and to the reflectorat MR, and thus βR = βR.

Upon comparison with equation (9.1.20), there seems to be an additional factor mD in thedenominator of equation (H-9). This factor arises from the expression for H

˜CT corresponding to

equation (9.1.18a) forH˜IS. The corresponding factor m(r∗) appearing in the asymptotic evaluation

of integral (9.1.4) cancels with the same factor present in the integral kernel. Since no factor mD

is present in the kernel of integral (H-6), it remains in equation (H-9).

We now insert equation (H-5) at rR into expression (H-9) to obtain the alternative expression

ΥCT (N ) =OD(η)

OF (η)

OF (ξR) cos βRhB(ξR;MR)

LFLF

KCC(ξR, rR; NR) , (H-11)

where we have used equation (5.6.9a). As before, ξR = ξ∗(rR) and rR = r∗(η).

Using expressions (7.2.26) for KDS(ξ;M) (computed for the input configuration) and (9.1.23)for KIS(r;N) (computed for the output configuration), we have derived expression (9.2.9) for thekernel function KCC(ξ, r; N) from its definition (9.2.7b). Taking equation (9.2.9) at ξR and rR,and substituting it into formula (H-11), we finally arrive at

ΥCT (N) =LL. (H-12)

This result is obtained using the relationship (7.3.3) between the obliquity factors OF and ODS

as well as the decomposition formula (6.2.14) for both geometrical-spreading factors L and L.Hence, the output of the cascaded CT solution at NR with coordinates (η, TR(η)), as representedin equation (9.2.6), is

U(η, τ) =LLU0(ξR) F [

mD(ξR;MR)

mD(η;MR)(τ − TR(η))] . (H-13)

It describes the simulated primary reflection event pertaining to the output configuration. Thesimulated event in equation (H-13) is kinematically equivalent to the true primary reflection eventthat was really recorded at G(η) in the output configuration. Also, it is correctly rescaled withthe ratio of the two different geometrical-spreading factors pertaining to the input and outputconfigurations. However, the length of event as described by equation (H-13) is stretched by afactor mD(ξR;MR)/mD(η;MR), when compared to the true event.

H.2 Cascaded remigration

In this section, we prove that the cascaded operation in equation (9.2.24) indeed yields theamplitude-preserving remigration (RM) result of equation (9.2.28), provided the TA kernels

316 APPENDIX H. EVALUATION OF CHAINED INTEGRALS

KDS(ξ;M) and KIS(r;N) specified in equations (7.2.26) and (9.1.23) are used. For that pur-pose, we assume again that the migrated section to be remigrated can be represented in form ofequation (9.1.3). By application of the chain rule to the z-derivatives, equation (9.2.24) can thenbe recast into the form

Φ(M ) =−1

4π2

∫

A

∫

d2ξ

∫

E

∫

d2r BCR Fm[m(r)(ZCR(ξ, r; M) −ZR(r))] , (H-14)

As before, the double dot denotes the second derivative with respect to the argument. Moreover,we have introduced the notation

BCR(ξ, r; M) = KCR(ξ, r; M) m2(r) Φ0(r) . (H-15)

Now, let us denote by Ir(ξ; M) the inner integral in equation (H-14), viz.,

Ir(ξ; M ) =1

2π

∫

E

∫

d2r BCR Fm[m(r)(ZCR(ξ, r; M ) −ZR(r))] . (H-16)

Taking into account the definition (9.2.25a) of the chained isochron function ZCR(ξ, r; M) =ZI(r;ND), a comparison of equation (H-16) with expression (9.1.4) immediately reveals thatIr(ξ; M ) represents a isochron-stack integral, with respect to the point ND on the diffraction curvet = TD(ξ; M), however with a different kernel or weight function KCR(ξ, r; M ). The wavelet inthis case is the first derivative of the source wavelet Fm[t]. Because of this observation, we knowthat integral (H-16) can, in the same way as we have done this before with integral (9.1.4), beasymptotically evaluated. The result is [compare with equation (9.1.12)]

Ir(ξ; M) ' ΥCR(ξ; M )Φ0(r∗) Fm[

m(r∗)

mD(ξ;MΣ)(TD(ξ; M ) − TR(ξ))] , (H-17)

where r∗ = r∗(ξ) denotes the stationary point of the function

δCR(r, ξ; M) = m(r)(ZCR(ξ, r; M ) −ZR(r))

= m(r)(ZI(r;ND) −ZR(r)) . (H-18)

Since m(r) > 0, this implies that r∗ determines the point where the gradient of the differenceZI(r;ND) −ZR(r) with respect to r vanishes, viz.,

∇r(ZI(r;ND) −ZR(r))

r∗= 0 . (H-19)

It is supposed that to each ξ in A, there exists a point NΓ with coordinates (ξ, TR(ξ)) on thereflection time surface ΓR in the vicinity of which integral (H-16) is evaluated. This point NΓ isdual to the point MΣ with coordinates (r∗,ZR(r∗)) in the input space, where the isochron ΣN ofNΓ is tangent to the reflector ΣR.

The Hessian matrix of δCR(r, ξ; M ) with respect to r is closely related to the already com-puted Hessian matrix H

˜IS. The reason is that the r-dependence of δCR(r, ξ; M ) in equation (H-18)

is equivalent to that of δIS(r;NR) in equation (9.1.6). The amplitude factor ΥCR(ξ; M ) is thusgiven correspondingly to equation (9.1.20) by

ΥCR(ξ; M ) =OD

OFKCR(ξ, r∗; M )LF (MΣ)m(r) cos2 βR . (H-20)

H.2. CASCADED REMIGRATION 317

The additional factorm(r) in the above formula appears because of the additionalm(r) in the kernelof integral (H-16) when compared to that of integral (9.1.4). Inserting the asymptotic evaluationresult (H-17) of Ir(ξ; M ) into the chained integral (H-14), we obtain the asymptotic expression

Φ(M ) ' −1

2π

∫

A

∫

d2ξ ΥCR(ξ; M) Φ0(r∗) Fm[

m(r∗)

mD(ξ;MΣ)(TD(ξ; M ) − TR(ξ))] . (H-21)

We recognize that the Huygens surface t = TD(ξ; M ), constructed for the output point M , rep-resents the migration inplanat as described in Chapter 2. Therefore, the asymptotic evaluationof integral (H-21) can be readily performed in full analogy to that of the diffraction-stack inte-gral (7.1.4). We find at M , slightly displaced in ζ-direction from the point MR with coordinates(ρ, ZR(ρ)) on the reflector ΣR, that

Φ(M ) ' ΥRM (M ) Φ0(rR) F [m(rR)mD(ξR; M )

mD(ξR;MR)(ζ − ZR(ρ))] . (H-22)

Here, ζ = ZR(ρ) represents the reflector image ΣR observed in the output space, i.e., after remi-gration, and ξR = ξR(ρ) is the stationary point of the difference function

δRM (ξ; M) =m(r∗)

mD(ξ;MΣ)(TD(ξ; M) − TR(ξ)) , (H-23)

where r∗ = r∗(ξ). In other words, ξR determines the tangency point between the Huygens surfacet = TD(ξ; M) and the reflection traveltime surface ΓR given by t = TR(ξ). Moreover, rR = r∗(ξR)and NR = N(ξR, TR(ξR)). Point NR is the dual point to MR with coordinates (ρ, ZR(ρ)), thelatter being the point on the reflector image ΣR that has the same horizontal coordinates ρ as M .Correspondingly, the same point NR on ΓR is also dual to point MR with coordinates (rR,ZR(rR))on the reflector image ΣR.

Equation (H-22) represents the RM output at M . All that remains to be done is to computea convenient expression for ΥRM (M). Similar to equation (7.3.2), we may express it as

ΥRM (M ) =OF (ξR)OD(ξR)

hB(ξR; MR)

mD(ξR;MR)

m(rR)

ΥCR(ξR; M )

LF (MR). (H-24a)

The additional ratio of stretch factors appears due to the corresponding stretch of the argument inintegral (H-21). We now insert equation (H-20) into expression (H-24a) to obtain

ΥRM (M ) =OF (ξR)

OF (ξR)

OD(ξR)OD(ξR)

hB(ξR; MR)mD(ξR;MR) cos2 βR

LF (MR)

LF (MR)KCR(ξR, rR; M ) .

(H-24b)Using the definition (9.2.25b) of the kernel function KCR(ξ, r; M), and observing from the dualityof MR and NR that TD(ξ; M ) and TR(ξ) are tangent at NR, one can express KCR(ξR, rR; M) as

KCR(ξR, rR; M) =KIS(rR;NR) KDS(ξR; M)

mD(ξR;MR). (H-25)

Combining equations (H-24b) and (H-25) and considering expressions (7.2.26) for KDS(ξ;M) (herecomputed for the output model) and (9.1.23) for KIS(r;N) (here computed for the input model),we finally arrive at

ΥRM (M) =LL . (H-26)

318 APPENDIX H. EVALUATION OF CHAINED INTEGRALS

This result is obtained using the relationship (7.3.3) between the obliquity factors OF and ODS aswell as the decomposition formula (6.2.14) for both geometrical-spreading factors L and L. Hence,the output of the cascaded RM solution at MR as represented in equation (9.2.24) is

Φ(M ) =LL Φ0(rR) F [m(rR)

mD(ξR; M )

mD(ξR;MR)(ζ − ZR(ρ))] . (H-27)

It describes the remigrated primary reflection event pertaining to the output velocity model. Wesee that remigration performs the relocation of the migrated event from ΣR to ΣR. Moreover,the amplitude is divided by the geometrical-spreading factor L as computed in the input modeland multiplies with the factor L as computed in the output model. Finally, the migrated pulse isunstretched by mD(ξR;MR) and restretched by mD(ξR; M ).

H.3 Single-stack remigration

In this section, we derive the single-stack RM solution as stated in equation (9.2.30) from thecascaded RM solution in formula (9.2.24). After interchanging the order of integration in equation(9.2.24), substituting equation (9.1.3) for the original migrated section Φ(r, z), and applying thechain rule to the z-derivatives, we have [compare to equation (H-14)]

Φ(M) =−1

4π2

∫

E

∫

d2r

∫

A

∫

d2ξ BCR(ξ, r; M) Fm[m(r)(ZCR(ξ, r; M ) −ZR(r))] , (H-28)

where we have again used notation (H-15). We must now asymptotically evaluate the inner integral

I(r, M ) =−1

2π

∫

A

∫

d2ξ BCR(ξ, r; M ) Fm[m(r)(ZCR(ξ, r; M ) −ZR(r))] , (H-29)

to find the amplitude-preserving single stack solution. Integral (H-29) can be interpreted as adiffraction stack for the output model, where the Huygens surface of the input model plays the roleof the reflection-time surface. The asymptotic evaluation of integral (H-29) is readily given by

I(r; M) = BCR(ξRM , r; M )exp−iπ2 (1 − Sgn(Z

˜CR)/2)

m(r)|detZ˜CR|

1

2

×

× Fm[m(r)(ZCR(ξRM , r; M) −ZR(r))] , (H-30)

where ξRM is the stationary point of integral (H-29). It is defined by the condition

∇ξ

[

m(r)(ZCR(ξ, r; M ) −ZR(r))]

ξRM

= ∇ξZCR(ξ, r; M)

ξRM

= 0 . (H-31)

Note that ξRM depends on both, r and ρ. Moreover, Z˜CR is the Hessian matrix of ZCR(ξ, r; M)

with respect to ξ. This matrix can be expressed as [see Appendix I, equations (I-12) and (I-19)]

Z˜CR =

1

mD(ξRM ;MRM )

(

H˜D − H

˜D

)

, (H-32)

where H˜D and H

˜D are the Hessian matrices of the diffraction traveltimes TD(ξ;M) and TD(ξ; M),

respectively, evaluated at the stationary point ξRM defined in equation (H-31). Point MRM is

H.3. SINGLE-STACK REMIGRATION 319

the dual point in the input model to NRM defined by ξRM on ΓR. Substituting this into equation(H-30) and taking into account all quantities that make up the factor BCR(ξRM , r; M ) [see equation(H-15)], we find, after some algebra, the simpler expression

I(r, M) ' KRM (r; M )∂Φ(r, z)

∂zz = ZRM (r; M)

, (H-33)

where ZRM (r; M) and KRM (r; M ) are given by equations (9.2.31) in the text. The insertion ofequation (H-33) into integral (H-28) finally provides the single-stack RM operator as representedin equation (9.2.30).

320 APPENDIX H. EVALUATION OF CHAINED INTEGRALS

Appendix I

Hessian matrices

To derive the single-stack solutions from the cascaded two-step solutions for the configurationtransform (CT) and remigration (RM) problems, we had to interchange the order of integrations.In this Appendix, we determine suitable expressions for the Hessian matrices H

˜CC and Z

˜CR that

appear in the asymptotic evaluations of the inner integrals. In this way, we show that the TAweights for the single-stack CT and the single-stack RM do not depend on the reflector itself.

I.1 Configuration transform Hessian matrix

Let the point N with coordinates (η, τ) in the output space be fixed and let us consider a given,fixed vector ξ in A. It is our aim to compute the Hessian matrix

H˜CC =

(

∂2TCC(ξ, r; N )

∂ri∂rj

)

(I-1)

at the stationary point rCT = rCT (ξ) determined by equation (9.2.17). The composite traveltimefunction t = TCC(ξ, r; N ) is [cf. equation (9.2.7a)],

TCC(ξ, r; N) = TD(ξ; MI) , (I-2)

where, as before, MI denotes a generic point with coordinates (r, ZI(r; N )) on the isochron z =ZI(r; N) pertaining to the output configuration. In other words, TCC(ξ, r; N) describes the ensembleof Huygens surfaces for all points MI on z = ZI(r; N).

We start by differentiating TCC(ξ, r; N) with respect to ri. Using the chain rule, we get

∂TCC(ξ, r; N)

∂ri=

∂TD(ξ, r, z)

∂ri+

∂TD(ξ, r, z)

∂z

∂ZI(r; N )

∂ri, (I-3)

where the two derivatives of TD(ξ, r, z) are taken at z = ZI(r; N)). Using equation (9.2.17), weobtain at the stationary point rCT

∂TD(ξ, r, z)

∂ri+

∂TD(ξ, r, z)

∂z

∂ZI(r; N)

∂ri= 0 . (I-4)

321

322 APPENDIX I. HESSIAN MATRICES

Note that in this appendix, we have switched from notation TD(ξ;M) to TD(ξ, r, z), which is moreconvenient for the following derivations. By differentiating equation (I-3) once again, this time withrespect to rj , we find by application of the product and chain rules

∂2TCC(ξ, r; N )

∂ri∂rj=

∂2TD(ξ, r, z)

∂ri∂rj+

∂2TD(ξ, r, z)

∂ri∂z

∂ZI(r; N)

∂rj+

+∂ZI(r; N )

∂ri

[

∂2TD(ξ, r, z)

∂z∂rj+

∂2TD(ξ, r, z)

∂z2

∂ZI(r; N)

∂rj

]

+

+∂TD(ξ, r, z)

∂z

∂2ZI(r; N)

∂ri∂rj. (I-5)

This expression for the elements of H˜CC can be simplified by means of an auxiliary construc-

tion similar to the one used in the proof of statement IIb of the duality theorems in Chapter 5. Asa first step, we recall that the isochron z = ZI(r; N) for a point N with coordinates (η, τ) in theoutput time domain is formed by all points MI in the depth domain for which

TD(η; r; ZI(r; N )) = τ . (I-6)

As a second step, let us now construct the corresponding auxiliary isochron z = ZI(r;NI) that istangent to z = ZI(r; N ) at MCT with coordinates (rCT , ZI(rCT ; N )), i.e., the particular point onZI(r; N) specified by rCT (ξ). This isochron has to be constructed for the point NI with coordinates(ξ, tI) in the input time domain that is defined by the given, fixed vector ξ mentioned above and bythe sum of traveltimes tI along the ray segments from S(ξ) to MCT and from there to G(ξ). Observethat, because of the stationarity condition, the ray SMCTG would belong to a specular reflectionif the isochron surface z = ZI(r; N) was a reflector. Note that all points MI with coordinates(r,ZI(r;NI)) on the isochron z = ZI(r;NI) specified by NI with coordinates (ξ, tI) satisfy

TD(ξ, r,ZI(r;NI)) = tI = const. (I-7)

Differentiation of the above equation, at first with respect to ri and then with respect to rj , togetherwith a straightforward use of the chain rule, yields [compare with equation (I-4)]

∂TD(ξ, r,ZI(r;NI))

∂ri=

∂TD(ξ, r, z)

∂ri+

∂TD(ξ, r, z)

∂z

∂ZI(r;NI)

∂ri= 0 (I-8a)

and [compare with equation (I-5)]

∂2TD(ξ, r,ZI(r;NI))

∂ri∂rj=

∂2TD(ξ, r, z)

∂ri∂rj+

∂2TD(ξ, r, z)

∂ri∂z

∂ZI(r;NI)

∂rj+

+∂ZI(r;NI)

∂ri

[

∂2TD(ξ, r, z)

∂z∂rj+

∂2TD(ξ, r, z)

∂z2

∂ZI(r;NI)

∂rj

]

+

+∂TD(ξ, r, z)

∂z

∂2ZI(r;NI)

∂ri∂rj= 0 . (I-8b)

By construction, the point MCT belongs to both isochrons specified by N and NI , i.e.,

ZI(rCT ; N) = ZI(rCT ;NI) . (I-9a)

I.2. REMIGRATION HESSIAN MATRIX 323

We have, moreover [see equations (I-4) and (I-8a)]

∂ZI(r;NI)

∂rir= rCT

=∂ZI(r; N)

∂rir= rCT

, (I-9b)

i.e., both isochrons are tangent at MCT .

Taking the difference of equations (I-5) and (I-8b), we find, using the above tangency relationof equation (I-9), at rCT = rCT (ξ)

∂2TCC(ξ, r; N)

∂ri∂rj=

∂2TCC(ξ, r; N)

∂ri∂rj− ∂2TD(ξ, r,ZI(r;NI))

∂ri∂rj

=∂TD(ξ, r, z)

∂z

(

∂2ZI(r; N )

∂ri∂rj− ∂2ZI(r;NI)

∂ri∂rj

)

(I-10)

or, in matrix notation,

H˜CC = mD(ξ;MCT )

(

Z˜I − Z

˜I

)

(I-11)

with the obvious meaning of the quantities involved.

Note the close relationship of equation (I-11) to equation (5.4.2b). In fact, the above derivationis identical to the one of equation (5.4.2b), except that the role of the reflector z = ZR(r) is playedby the isochron of the output configuration, z = ZI(r; N).

I.2 Remigration Hessian matrix

We consider a point M with coordinates (ρ, ζ) in the output depth domain, i.e., pertaining to theoutput macro-velocity model, and a given, fixed coordinate vector r within the aperture E of theinput depth domain. The Hessian matrix Z

˜CR is then defined by

Z˜CR =

(

∂2ZCR(ξ, r; M)

∂ξi∂ξj

)

, (I-12)

taken at the stationary point ξRM = ξRM (r) defined by equation (H-31), where the functionZCR(ξ, r; M ) is given by equation (9.2.25a).

For the derivation of an appropriate expression for Z˜CR, we consider at first the Huygens

surface t = TD(ξ; M ) for a point M with coordinates (ρ, ζ). For each point ND with coordinates(ξ, TD(ξ; M)) on t = TD(ξ; M ), we construct the isochron ZI(r;ND) = ZI(r, ξ, TD(ξ; M)) in theinput space. Each of these isochrons is defined by application of the definition of the isochron,equation (5.3.2), with a specific value of t for each given ξ, given by t = TD(ξ; M). Thus, introducingthe set of pointsMID with coordinates (r,ZI(r;ND)), i.e., all points on the isochrons z = ZI(r;ND)for all points ND, we can write, for any fixed r and all ξ,

TD(ξ, r,ZI(r;ND)) = TD(ξ;MID) = TD(ξ; M) . (I-13)

According to equation (9.2.25a), the ensemble of these isochrons z = ZI(r;ND) for all points ND

defines the set of input surfaces z = ZCR(ξ, r; M) for the cascaded remigration. Thus,

TD(ξ, r,ZCR(ξ, r; M )) = TD(ξ, r,ZI(r;ND)) = TD(ξ; M) . (I-14)

324 APPENDIX I. HESSIAN MATRICES

By calculating the total derivative of equation (I-14) with respect to ξi , we find with the chain rule

∂TD(ξ, r, z)

∂ξi+

∂TD(ξ, r, z)

∂z

∂ZCR(ξ, r; M )

∂ξi=

∂TD(ξ; M )

∂ξi, (I-15)

where all derivatives on the left-hand-side are taken at MID, i.e., at z = ZCR(ξ, r; M). At thestationary point ξRM of ZCR(ξ, r; M ) defined by equation (H-31), it follows from equation (I-15)that

∂TD(ξ, r, z)

∂ξi z = ZCR(ξ, r; M)

=∂TD(ξ; M )

∂ξi. (I-16)

This means that the Huygens surfaces t = TD(ξ;MRM ) and t = TD(ξ; M) are tangent at ξRM in thetime domain. Here, we have introduced the notation MRM for that particular point MID with coor-dinates (r,ZI(r;NRM )) on the isochron z = ZI(r;NRM ) of NRM . In other words, MRM is the dualpoint of NRM with respect to the input model, where NRM with coordinates (ξRM , TD(ξRM ; M ))is the point on t = TD(ξ; M ) defined by ξRM .

To find the desired expression for the second derivative of ZCR(ξ, r; M), we next differentiateequation (I-15) further with respect to ξj . Applying the chain and product rules, we arrive at

∂2TD(ξ, r, z)

∂ξi∂ξj+

∂2TD(ξ, r, z)

∂ξi ∂z

∂ZCR(ξ, r; M )

∂ξj+

+∂ZCR(ξ, r; M )

∂ξi

[

∂2TD(ξ, r, z)

∂z∂ξj+

∂2TD(ξ, r, z)

∂z2

∂ZCR(ξ, r; M )

∂ξj

]

+

+∂TD(ξ, r, z)

∂z

∂2ZCR(ξ, r; M )

∂ξi∂ξj=

∂2TD(ξ; M )

∂ξi∂ξj, (I-17)

where again all derivatives are taken at z = ZCR(ξ, r; M ). At the stationary point ξRM , i.e., at thepoint NRM , we find by inserting the stationarity condition, equation (H-31), into expression (I-17)

∂2ZCR(ξ, r; M)

∂ξi∂ξj=

(

∂2TD(ξ; M )

∂ξi∂ξj− ∂2TD(ξ, r, z)

∂ξi∂ξj

)/

∂TD(ξ, r, z)

∂z(I-18)

or, in matrix notation,

Z˜CR =

1

mD(ξRM ;MRM )

(

H˜D − H

˜D

)

(I-19)

with the obvious meaning of the quantities involved.

Note the close relationship of equation (I-19) to equation (5.4.3a). In fact, the above derivationis identical to the one of equation (5.4.3a), except that the role of the reflection-time surfacet = TR(ξ) is played by the Huygens surface of the output model, t = TD(ξ; M).

Index

aliasing, 204amplitude

best possible, 15preserving, 16, 141true, see true amplitude

amplitude factoracoustic, 59configuration transform

cascaded, 314chained, 314, 315

diffraction stack, 192, 213generic, 69Green’s function, 165, 170isochron stack, 242Kirchhoff-Helmholtz integral, 167ray, 98, 100, 168remigration

cascaded, 316chained, 317

scalarP-wave, 63S-wave, 64

vectorial, 57, 58complex, 58

analytic signal, 58, 180anterior surface, 92, 109aperture, 141, 143, 181, 184, 204

minimum, 126, 204, 207aplanat, 14arclength, 66–68, 70, 75, 76, 78, 87, 107auxiliary surfaces, 144, 176

Beylkin determinant, 157, 192, 293boundary conditions

Dirichlet, 306Neumann, 306

boundary effects, 206, 233bulk modulus, 54

cascaded solution, 240caustic, 73

first order, 73

number of, 74

phase shift, 74, 172

second order, 73

chaining, see integral chaining

characteristics, 65

method of, 66

rays, 66

Christoffel matrix, 62

component

additional, 63, 64

principal, 63, 64

compressibility, 54

configuration transform, 14, 19, 34, 47, 240

asymptotic evaluation, 314

cascaded solution, 248, 313

Hessian matrix, 321

single-stack solution, 253

stack result, 254

stacking integral, 255

conversion coefficient, 195, 283

coordinates

Cartesian

global, 20, 53, 66, 69, 74, 76, 89, 119,154, 289

local, 85, 87, 89, 154, 190, 289

ray, 68, 77

ray-centered, 72, 75, 87

curvature, 144

matrix, 148, 153, 155, 156, 289

data space, 26

diffraction stack, 1, 7, 144, 175, 181

integral, 161, 175, 178, 181, 225

asymptotic evaluation, 192

result, 192

multiple, 224

time-dependent, 182

vector, 228

diffraction traveltime, 7, 13, 29, 166

325

326 INDEX

divergence theorem, 297, 307domain

depth, 27time-trace, 27

DSI, see diffraction stack integraldual

points, 143surfaces, 143

duality, 14, 139, 141, 142of curvature, 156theorem, 148

first, 139, 148, 149second, 139, 148, 151

dynamic, 3, 44ray-tracing system, see ray tracing, sy-

stem, dynamic

earth model, 20, 56eikonal equation, 60

acoustic, 61elastodynamic, 62, 65generic, 295solution of, 66

elastic tensor, 307symmetry, 308

emergence direction, 284energy law

P-wave, 64S-wave, 65

Fermant’s principle, 219fermat’s principle, 149Fourier transform, 57free surface, 180, 194, 283

acoustic waves, 286P waves, 283S waves, 285

Fresnelgeometrical spreading, see geometrical

spreading, Fresnel factorobliquity factor, see obliquity factor, Fres-

nelvolume, 103, 123zone, 105, 123, 219, 221

determination, 130matrix, 125, 153, 187paraxial, 125projected, 126projection matrix, 128

time-domain, 126

geometrical spreading, 3, 9, 71, 103, 122, 187decomposition, 130, 131, 132, 169, 171,

178, 188Fresnel factor, 4, 132, 139, 153, 156, 170normalized, 72, 168, 180

Green’s theorem, 298

Hamiltonian, 66Helmholtz equation

anisotropic, 307scalar elastic, 295

Hessian matrix, 145, 183, 186, 289, 321

configuration transform, 321decomposition, 186remigration, 323

Huygenssource, 8, 159, 161

reconstruction, 161, 193surface, 7, 8, 13, 29, 44, 45, 139, 141, 143,

148, 161, 176, 182, 185wave, 7

image demigration, 45image migration, 44imaging, 44

conditionsgeneralized, 33Hagedoorn’s, 29

mapping vs., 30, 34problems, 14, 240, 260surfaces

generalized, 32Hagedoorn’s, 27

true-amplitude, 1, 12unified theory, 1, 2, 12, 46, 239

inplanat, 14, 32, 35, 37, 41, 44, 45, 239configuration transform, 254demigration, 314remigration, 259

integral chaining, 247, 313integral kernel, see weight function

interferenceconstructive, 168, 178destructive, 178

ISI, see isochron stack integralisochron, 9, 14, 29, 44, 45, 139, 141, 143, 148

auxiliary, 145

INDEX 327

isochron stack, 1, 9, 144, 239, 241integral, 241

asymptotic evaluation, 242result, 243, 245

KHI, see Kirchhoff-Helmholtz integralkinematic, 2Kirchhoff

demigration, see isochron stack-Helmholtz approximation, 159, 160, 164,

303anisotropic, 309

-Helmholtz integral, 8, 159, 160, 165anisotropic, 310isotropic, 311scalar elastic, 166, 307, 312

-Helmholtz theory, 159imaging, 44

integral, 7, 159, 161anisotropic, 307direct waves, 298reflected waves, 301scalar elastic, 295transmitted waves, 299

migration, see diffraction stackKMAH index, 74, 172

decomposition, 172

Lame parameters, 54Lame’s Theorem, 55length scale, 59

macrovelocity model, 6, 20map

demigration, 31migration, 31, 141

measurementconfiguration, 22, 114, 119, 178, 195

common midpoint, 25common offset, 23common receiver, 23common shot, 23common-midpoint offset, 25

cross profile, 25cross spread, 26matrix, 26

surface, 21metric tensor, 76

scale factors of the, 76

migration to zero offset, 35, 47minimal data set, 7

model data, 144monotypic reflection, 5, 100, 101, 130, 132,

133, 146, 156, 192, 306

NIP-wave theorem, 105, 137extended, 133

NMO stretch, 208

obliquity factor, 166depth, 155

diffraction stack, 192Fresnel, 154

observation data, 144

optical length, 66outplanat, 32, 35, 37, 40, 44, 45

paraxialapproximation, 82Fresnel zone, see Fresnel, zone, paraxial

ray-tracing system, see ray tracing, sy-stem, paraxial

physical inverse, 161, 193physical optics approximation, 160, 306plane wave, 303

local, 305point source

solution, 71, 74

for homogeneous media, 56, 74for inhomogeneous media, 75transient, 100

Poisson ratio, 54

polarization vector, 310posterior surface, 92, 109primary reflection, 20, 27, 31, 35, 47, 56, 96,

180

simulated, 47principal component, 101, 178, 295projection matrix, see Fresnel zone, projection

matrixpropagator matrix, see ray, propagator matrix

pulsedistortion, see source pulse, distortionstretch, see source pulse, distortion

rayamplitude, 71

at the geophone, 97

328 INDEX

point source, 73

ansatztime-harmonic, 57

transient, 58

at a surface, 85at an interface, 93

boundary conditions, 94

central, 80–83, 85, 88, 91, 93, 102, 105

coordinates, see coordinates, rayfield, 67, 75

Jacobian, 69

across an interface, 96in ray-centered coordinates, 78

paraxial, 75, 80, 82, 85, 103

traveltime, 105

propagator matrix, 103, 104chain rule, 105, 115–117

meaning, 120

ray-centered, 84surface-to-surface, 92, 93, 109

symplecticity, 111

reverse, 111segment, 108, 112, 115, 116, 119, 122, 125,

132, 135, 146, 166, 170, 171

decomposition, see ray, propagator ma-trix, chain rule

traveltime, 106, 118tangent vector, 67

theory, 5, 53

validity of, 5, 59

tube, 70, 73ray tracing

initial conditions, 83

general, 84plane wave, 83

point source, 83

matricesdynamic, 95

system, 67, 107

dynamic, 75, 80–83

in ray-centered coordinates, 78paraxial, 80

two point, 85

Hamilton’s equation, 108ray-centered, see coordinates, ray-centered

reciprocity, 111, 180

reflection coefficient, 95, 98–101, 160, 161, 181,185

amplitude normalized, 98, 277

energy normalized, 100

linearized, 98

textit, 278

reciprocal, 100, 168

reflection traveltime, 142, 180

reflection-signal strip, 31, 44, 208, 216, 239

demigrated, 45, 243

migrated, 45, 208, 216, 239, 242

remigration, 15, 19, 38, 48, 240, 256

cascaded solution, 256, 315

Hessian matrix, 323

single-stack solution, 258, 318

stack result, 257

reproducible, 26, 180

resolution, 218

horizontal, 218

definition, 219

example, 222

vertical, 218

rigidity, 54

scattered field, 163, 164, 173, 300, 301, 308,311

seismic system, 104, 113

shear modulus, 54

signature, 132, 168, 183

slowness vector, 66

component, 85

projection, 89, 116

cascaded, 87

smearing, 29

smearing surface, 46

Snell’s law, 94, 171

source pulse, 57, 184

distortion, 185, 208, 212

example, 216

length of, 31, 45, 126, 131, 180, 204, 206,211, 216, 217, 223

source strength, 72

stacking, 29

stacking surface, 7, 9, 46

stationarity condition, 183

stationary phase method, 167, 178, 183, 242

stationary point, 183, 184

configuration transform, 253, 313

isochron stack, 242

remigration, 258, 316–318

INDEX 329

stretch factor, 146, 148, 181, 211–216configuration transform, 315isochron stack, 243

surface curvature matrix, 87surface of

equal reflection time, 29maximum convexity, 7, 13, 29

symplecticity, see ray, propagator matrix,symplecticity

tangency, 144, 148, 178taper, 204, 233target, 49

reflector, 142, 176, 182transmission coefficient, 98

amplitude normalized, 98, 281energy normalized, 100reciprocal, 100

transmission loss, 99, 170, 180acoustic, 100

transport equation, 60acoustic, 61elastodynamic

P-wave, 63S-wave, 64

generic, 65, 295solution of, 69

traveltime, 57, 66approximation

second-order, 113difference, 106, 186

Hessian matrix of the, 186functions, 144, 185paraxial, 112ray segment, see ray, segment, traveltime

true amplitude, 2, 9, 15, 141, 176, 178, 181,245

demigration, 241, 245result, 245

event, 4, 181, 181, 184kernel, see weight functionmigration, 175, 178

implementational forms, 199procedure, 198result, 192

uniquenessof reflection point, 142of reflection ray, 142

of source-receiver pair, 142

vector diffraction stack, 228vertical component, 195

wave equationacoustic, 55elastodynamic, 54

for homogeneous media, 55scalar elastic, 295

wave velocityacoustic, 54P-, 54, 63S-, 54, 63

wavefieldseparation, 55

wavelet, see source pulsewavemode, 20

decoupling, 55imaging, 29

weight function, 49, 166, 167, 176, 178, 184,185, 240

alternative expressions, 191Bleistein’s, 194configuration transform

cascaded, 249, 251chained, 254

final, 190isochron stack, 244, 246multiple, 224phase of, 191ratio of, 225remigration

cascaded, 256, 257chained, 258

vector, 228example, 229

Young modulus, 54