Migration of Data

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Migration of Seismic DataIENOGAZDAG AND PIERO SGUAZZEROInvited Paper

Reflection seismology seeks to determ ine he structure of heearth from seismic records obtained at the surface. The processingof these data by digital computers is aime d at rende ring them morecomprehensiblegeolog ically. Seismic migration is one of theseprocesses. Its purpo se is to "migrate" the recorded events to theircorrect spatial positions by backward projection or depropagationbased on wave theoretical considerations. During the last 15 yearsseveral method s have appeared on the scene. The pu rpose o f thispaper is to pro vide an overview of the major advances in this field.Mig ratio n me tho ds examined here fall in three major categories: I )integralsolut ions, 2 ) depth extrapolation methods, and 3) t imeextrapolat ion metho ds. Within hese categories, the pert inent eq uations and num erical techniques are discussed in some de tail. Thetop ic of mig ratio n bef ore stacking is treated separately w ith anout l ine of tw o dif ferent approaches to this important p roblem.I . INTRODUCTION

The purpose of migration is to reconstruct the reflectivitymap of the earth from the seismicdata recorded at thesurface.Theseismicsignal recorded by a receiver (geo-phone) is a superposition of seismic waves originating fromall possible directions in the subterrain. Thus the recordedevents most often are not from reflectors directly below thereceiver but from geological formations far away from thepoint of recording. The term migration refers to the move-ment of the observed events to their true spatial positions.Migration i s an inverse process, inwhich the recordedwaves are propagated back to he corresponding reflectorlocations. The concept of migration can be summarized inthe following terms. In the process of seismic data acquisi-ti on the upward traveling waves are recorded at the surface.In hemigration process hese recorded wavesare usedeither as initial conditions or boundary conditions for awavefield governed by the wave equation. As a result, thesewaves are propagated backward and in reverse time, fromthe surface to the reflector locations.Until the 196Os, migration was achieved by graphicalmethods. This was followed by diffraction summation andwavefront migration based on ray theoretical considera-tions. In the 197Os, several important developments tookplace. Based on thepioneeringworkof Jon Claerbout,migration methods based on wave theory were developed.Efficientalgorithms developed fromsimplified finite-dif-ference approximations of the wave equations for down-ward extrapolation. Another important processing advancewas made by the introduction of Fourier transform meth-ods. These frequency-domain approaches proved to be

Manuscript received May 3, 1984; revised May 16, 1984.J. Gazdag i s with IBMScientific Center, Palo Alto, CA 94304, USA.P. Sguazzero is with IBM Scientific Center, 00149 Rome, Italy.

more accurate thaninite-difference methods in thespace-time coordinate frame. At the same time, the diffrac-tion summation migration was improved and modif ied onthe basis of theKirchhoff integral representation of hesolutionof he wave equation. The resultingmigrationprocedure, known as Kirchhoff migration, compares favor-ably with other methods. A relatively recent advance i srepresented by reverse-time migration, which is related towavefront migration.The aim of this paper i s to present the fundamentalconcepts of migration. While it represents a review of thestate of the art, it is intended to be of tutorial nature. Thereader is assumed to have no previous knowledge of seismicprocessing.owever, some fami liar ityith Fouriertransforms and their applications i s assumed. The organiza-tion of the paper is as follows. First, the basic concepts ofseismic data representation are introduced. Next, migrationschemes based on integral or summation methods are dis-cussed. Section IV deals with the derivation of the one-waywave equations. This is followed by the numerical aspectsof migration methods. Migration in object space, known asreverse-time migration, is presented in Section VI, and Sec-tion VI1 is devoted to migration before stacking.The effect of migration on a seismic section depends onthe velocity and the reflectivity structure of the subterrain.Flat, horizontal reflectors in a medium that has no signifi-cant velocity variations may not need to be migrated. Onthe other hand, seismic sections from media having steeplydipping reflectors and strong lateral velocity variations needto be migrated for correct interpretation. In the latter case,migration can have a profound impact on the image, as weshall see in the example ofmodel 2 (Figs. 11-14) to bediscussed in Section V.11. REPRESENTATIONF SEISMIC DATAA. Data Acquisition

Reflection seismology is an echo-ranging technique. Anacoustic source (shot) emits a short pulse and a set ofrecorders (geophones) register the reflected wavesat thesurface. The time series (sampled)dataassociated wi th asingle shot and receiver is known as a trace. In typicalmarineexploration (Fig. I) , a boat tows a source and astreamer of receivers. As it moves one-half receiver intervalalong a seismic line, it fires a shot and records the pressureat each receiver location. A trace i s associated with eachshot and receiver point.Let r be the horizontal coordinate of he receiver and sbe the horizontal coordinate of the source. Both are mea-

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RECEIVERS SHOTCOMMON MIDPOINTOM M ONFFSET

RECEIVER COORDINATE r

RECEIVERS SHOTCOMMON OFFSET

I tRECEIVER COORDINATE rFig. 1. Relationship among the horizontal coordinates r, 5 ,x , and h . Al l axes represent distances measured along theseismic line. Each doton the surface orresponds to aseismic trace.

sured along the seismic line. However, for mathematicalreasoning, it is helpful to represent the receiver coordinaterand the source coordinate s on orthogonal axes, as shownin Fig. 1. We also define the midpoint coordinate betweensource and receiver as x = ( r + s)/2, and the source/re-ceiver half-offset coordinate as h = ( r - s)/2. From heseequations we see that x and h are another set of axesrotated 45" with respect to the axes r and s.To understand the reason for migration et us considerthe ray paths associated with two point reflectors shown inFig.2.Each trace includes two wavelets at times t measured

I T T T T X

Fig. 2. The seismic concept illustrated with wo point re-flectors. The recorded traces form a common-shot gather.The loc i of the recorded wavelets represent two-way traveltime as indicated.

along the raypaths from source to object and from there tothe receiver. This example clearly demonstrates that a traceconsists of a superposition of seismic waves propagatingfrom all possible directions. Consequently, the informationregarding the cross section of reflectivity must be nferredfrom the nterrelationships among an ensemble of traces.To do this, we need efficient and accurate computationaltechniques to map the wavelets onto the ocation of thereflecting objects. This mapping canalso be viewed as a

reconstruction of the wavefield at the time when the reflec-tion ook place. In this process,wave energy i s being"migrated" over the ( r , s) plane shown in Fig. 1 .For practical and economical reasons, conventionalseismic processing techniques have been developed forgroups of traces, called gathers, aligned parallel with one ofthe four axes shown in Fig. 1. A set of traces with the samesource ( s = constant) forma common-shot gather.Thetraces are also earranged into common-midpoint (CMP)gathers or common-offset gathers as shown in Fig. 1. Thereason is that the two standard operations that are responsi-ble for he transferring of wave energy to its proper positionare stacking of the CMPgathersand migration of heresulting stacked CMP section. Stacking shifts energy withrespect to the offset axis, and migration shifts energy withrespect to hemidpoint axis.The eason for performingthese operations separately s economic. Ideally, they shouldbe performed together as discussed in Section VII .B. Stacking

Stacking consists of the summation of the traces of eachCMP gather after correcting them to compensate for theoffset between source and receiver. This i s known as thenormalmoveout (NMO) correction. The NMO process isillustrated on he idealized model shown in Fig. 3. The

L

Fig. 3. Illustration of NMO correction and stacking. Thetraces of the CM P gather are summed along the NMOhyperbola as indicated. The normalized stackedrace i splotted at h = 0.

wavelets indicate reflected events from the horizontal plane.Their two-way travel time i s given byt2 ( h) = t: +-h2l,Jassuming that he velocity of wave propagation v i s con-stant. This equation i s the NMO hyperbola. The differencebetween the reflect ion time observed for an offset 2 h andthat corresponding to h = 0 s the NMO correction, i.e.,

AtNMo t( h) - oWhen this amount of time shift i s applied to the waveletsshown in Fig. 3, apart from a minor distortion effect, theyappear as if they were recorded with coincident source andreceiver, i.e., h = 0. The NMO corrected tracesare thensummed, or stacked. The ensemble of stacked traces along

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the midpoint axis is referred to as the CMP stacked section.A very important benefit of this operation is the significantimprovement n the signal-to-noise ratio of the CMP sec-tion in comparison wi th the unstacked data.C.The Exploding Reflector Model

A CMP section may be egarded as data obtained romcoincident sources and receivers ( h = 0), as shown in Fig.qa). In this zero-offset model, the energy travel path fromsource to reflector is identical to that from reflector tol4

Fig. 4. (a)The CMP section may be regarded as data ob-tained with coincident (zero-offset) sources and receivers.(b) The exploding reflector model of the CMP data formigration.

receiver. The assumption i s that all sourcesare activatedsimultaneously, but each receiver records signalsoriginatingfrom the same source-receiver point.Such zero-offset data do not correspond to any wavefieldresulting from a single experiment. As a result, it i s helpfulto create a hypothetical physical experiment to provide anintui tive picture of zero-offset migration. Suchan experi-ment is known as the exploding reflector model [I]. Inthis model, shown in Fig. qb), the energy sources are not atthe surface, but they are distributed along the eflectingsurfaces. In other words, the reflectors are represented byburied sources, which are activated at the same time t = 0.Therefore, one needs to be concerned only with upwardtraveling waves. Since the record section involves t wo - wa ytravel time, it needs to be converted to oneway traveltime. n practice, the ime scale of CMPsections i s keptunchanged. Instead, the velocity of wave propagation i sscaled down by a factor of two.I\\. N T E G R A L METHODSOR MGRATIONO F CMP DATA

With the help of the exploding reflector model outlinedabove, migration of CMP data can be defined as the map-ping of the wavefields recorded at the surface back to theirorigin at t = 0. The basic theory of wavefield reconstructioni s due to Hagedoorn [2]. Based on hisdeas, graphicalmethods were developed, which were later implementedondigi tal computers. Our objective is to describe twosimple classical methods followed by that of the Kirchhoff

method, which is based on an integral representation of thesolution of the wave equation. First, however, a geometricview of wavefield reconstruction is studied in some detail,though at an introductory level. Our guiding principle i s toconvey he concepts of migration in a simple descriptivemanner. The specific techniques used to implement theseideas in the form of practical and efficient computer pro-grams are not considered in this paper.A . Geometric View of Wave Reconstruction

Let us consider the space-time dist ribution of energy in asimple seismic experiment shown i n Fig. 5. The (x ,z ) objectplane represents the cross section of the earth, and the( x , t ) image plane is the record section. In the exampleshown, there is one point diffractor in theobject plane.When such a point diffractor is excited by a surface source,it sets off outgoing waves in all directions. Using the ex-ploding reflector model, such a diffractor can be approxi-mated by a buried point source. Using v/2 as the velocity

Fig. 5. The space-time distribution of the energy of a pointsource activated at t = 0 forms a cone. The intersection ofthis cone with the image plane z = 0 defines the diffractionhyperbola.

of wave propagation, the waves observed and recorded atz = 0 result in a zero-offset image of the diffractor.Assuming that v = constant, after the point source is setoff at t = 0, the resulting wavefronts are concentric circles,which expand with increasing time, as shown in Fig. 5.From this diagram it i s seen that the space-time ( x , z , t )distribution of he wave energy lies on a cone. The intersec-tionof this cone with the z = 0 plane is a hyperbolaexpressed by

The upward propagating part of the energy from the pointsource lies along a hyperbola in the record section. Theobjective of migration is to recreate from the observed datain the image plane the situation that existed in the objectplane at t = 0. In other words, migration may be regardedas aprocedure of mapping he waveenergy distributedalong diffract ion hyperbolas into the loci of the correspond-ing point sources.Another way to establish the relationship between herecorded image and the object i t represents is to examinethe significance of a point, say a delta function, n he

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inout t r a c e s

Fig. 6. The space-time distribution of the energy of aninward propagating circular wavefront (centered at z = 0)l ies on a cone. A t time Tall the energy is focused in theneighborhood of a point. The reverse of this process ormsthe basis of wavefront migration.

image space. Consider thehalf-circle centered at z = 0shown in Fig. 6. The initia l conditions at t = 0 are arrangedso that the acoustic energy distributed uniformly along thehalf-circle propagates inward, forming smaller concentricsemicircles, until at t = T the energy i s focused in onepoint. Thus the energyobserved atsome t = T on theimage plane may b e thought of as being originated from awavefront lyingon a half-circle centered at z = 0 andhaving a radius r = vT/2, as shown in Fig. 6. Consequently,in the process of reconstructing the wavefront of t = 0, apoint of the image plane must be mapped onto a half-circlein the objectplane. This means that a point diffractor in thesubsurface would cause a hyperbolic seismic event, but apo int on a seismic section must have been causedby ahalf-circle reflector i n the subsurface.B. Classical Methods of Migration

We haveseen that for a constant-velocity medium thezero-offsetecord section of a point reflector is char-acterized by a diff raction hyperbola. Thus to account for allthe energy due to the point source in estimating its in-tensity, weperform he fol lowing summation. Foreach( x , z ) poin t of the object plane, we construct a diffractionhyperbola in the image plane. We determine where thehyperbola intersects each trace. Then we take the value ofeach trace at the point of intersection and sum all thesevalues together. The result of this summation is taken as thevalue of he migrated section a t (x,z) , and this value isplaced in the object plane at that point. This method i scalled diffraction summation (or diffractionstack) migra-tion and represents a straightforward, albeit not theoreti-cally sound, approach. Fig. 7 shows how, in diffractionsummation migration, an output trace is generated from theinput traces. The input traces represent the stacked section,whereas the output races define the depth section.It should be noted that, in practice, depth i s most oftenmeasured in unitsof the two-way vertical travel time,T = 2z/v. Under the constant velocity assumption, T corre-sponds to the apex of the diff raction hyperbola. Thus dif-fract ion summation migration amounts to summing waveamplitudesalong hediffraction hyperbola and mappingthe result at the apex of the same hyperbola.

Fig. 7. Diffraction summation migration.The input tracesare summed along he diffraction hyperbola correspondingto a single scatterer at location x to produce an output tracea t each location x .

A counterpart of diffraction migration is wavefront inter-ference migration, which can be understood from the geo-metrical view shown in Fig. 6. This approach to migration isbased on he observation that each poin t n the imageplane must be mapped into a semi-circle in the objectplane. Thus the wavefront method [3] i s a one to manymapping. For the constant velocity case, the procedure i sthe following. We take a sample of a seismic race in therecord section and distribute it over a semi-circle (equaltravel times) in the object plane as illustrated in Fig. 8.

i n p u t traces

outputracesFig. 8. Wavefrontmigration. Each input trace is mappedonto a semi-circle corresponding to theproper wavefront.

The diffraction summation and the wavefront inter-ference methods had notable success in the late 1960s andearly 1970s [4].They alsohave a number of undesirablecharacteristics [3, p. 1241. The reason for their shortcomingsis rooted in the fact that while these migration proceduresmake good sense and are intuitively obvious, they are notbased on a completely sound theory.C. The Kirchhoff Method

The Kirchhoff integral theorem expresses the value of thewavefield at an arbitrary point in terms of the value of thewavefield and its normal derivative at al l points on anarbitrary closed surface surrounding the point [5,p. 3771. Inpractice, measurements are made only at the surface, there-fore, instead of a closed surface, the integration must be

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limited to the surface of the earth. furthermore, in seismicpractice, only the wavefield is recorded, and its normalderivative is not axtailable.The Kirchhoff integral canbeexpressed more easily for three-dimensional data, i.e., wherethe pressure wavefield i s defined over some area of theearths surface rather than along a line as we assumed in thetwo-dimensional case. Let p(x, y,z = 0, t ) be the recordeddata, where x and yare surface coordinates, z i s depth, andt is time. Then the migrated wavefield atome point(xl, y,, z,, t) is obtained from the fo llowing expression [6],V [8, P. 1251,PI,as:P(xly,,zl,t= 0)=jj=?g (x,y,z= O , t = r/c)dxdy (4)where

r = [(x, - x) +( , - y)2 + z 1 2c =. v/2 i s the half velocity (assuming that t i s the two-waytravel time), and 9 is the angle between the z axis and theline joining (xl, y,,zl) and (x ,y ,z = 0).

To obtain the two-dimensional version of (4), one mustassume that p i s independent of y. After integrating wi threspect to y, one obtains the following asymptotic ap-proximation [8, p. 1261:

p ( x , , ~ ~ , t = O ) - / - ~ a : / ~ p ( x , z = O , t = ~ / c ) d xos9G(5)

where a:/2 denotes the half derivative wi th respect to t. Asimple expression for the operator a:/ can be given in thefrequency domain, where the transfer function of isgiven byIt should be observed that, apart from the half-derivativeoperator and a weighting factor, ( 5 ) represents an integra-tion along he diffraction hyperbola. In this respect it isquite similar to thediffraction summation method. It i simportant to note, however, that the Kirchhoffmethodyields significantly higher quality migration than the diffrac-tion summation approach discussed earlier.When he propagation velocity c varies wi th the spacecoordinates x, y, and z, (4) and (5) must be suitably gener-alized. for instance, in two dimensions we obtain [IO]P(Xl,Zl,t = 0)

=jWa; /*p(x ,z=O,t= T(x, ,z , ,x ,z=~))~x (7)where W i s a weight ing factor, and T( x,, z , , x , z) representsthe wo-way travel time measured along a raypath thatconnects the points(xl,zl) and (x,z) obeying Fermats leasttime principle [ 5 , p. 1281.

arranging these traces into some kind of gathersor con-stant-offset sections. The computer generation of seismicdata calls for the solution of the equations governing thewave propagation in the mediumdefined in themodelunder consideration. It i s important, however, to ensurethat the modeling i s not simply an inverse of the migrationfrom the numerical and algorithmic point of view. Other-wise, errors i n the forward process may be partially canceledin he inverse process, which can lead to incorrect errorestimates. To avoid these pitfalls, the synthetic seismicrecords presented in this paper are obtained by means of analgorithm that permits the computation of two-way reflec-tion times wi th precision limited only by roundoff error.In our approach to forward modeling, the reflecting ob-jects are represented by scattering surfaces in three dimen-sions, or lines in two-dimensions. The util ity of approximat-ing acontinuous eflecting sheetby discrete scatteringpoints.in a scaled seismic test tank was shown by Gardneret a/. [ I l l . The computer mplementation of this method isbased upon the ray theory of diffraction and the superposi-tion principle. The modeling of a complex subsurface re-flectivity structure is done by representing it as a set ofindependent point scatterers.To obtain the zero-offset sec-ti on of an elementary diffractor, .fo llowing the explodingreflector principle, rays are traced in every direction throughthe medium up to the surface. A wavelet whose amplitudeis proportional to the strength of the diffractor i s assigned(after taking into account geometrical spreading) to the gridpoint of the time section corresponding to the arrival of theray at the surface. The contributions of all point diffractorsare then summed to obtain the zero-offset section of hestructure.An example of this modeling approach is Model 1 repre-senting a syncline shown in fig.?a). The synthetic zero-off-set section of Model 1 is shown in Fig. yb), assuming auniform velocity of v = 3 km/s. Fig. 10 illustrates the same

0 2 4 6 8 1 0 1 2rx (km)

:jv

0. ynthetic Seismic RecordsAs we havediscussedearlier, migration is an inverseprocess. To test a migration scheme and evaluate its perfor-mance, we need a set of seismic data,e.g., a zero-offsetsection obtained from an idealized model withnown Fig. 9. (a) Schematic of representinga sync,ine inreflectivi ty and velocities. This i s usually done by simulating a constant velocity medium, (b) Synthetic zero-offset imethe forward process, collecting he results into traces, and section of Mode l 1.

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v. NUMERICAL ETHODSOR ZERO-OFFSET MIGRATIONNIMAGEPACE

Migration of zero-offset seismic data in image space canbe summarized as follows. The zero-offset data areex-trapolated from the surface downward to some depth z =nAz inn computational steps. The wave extrapolation callsfor he numerical solution of (26) or some approximationthereof. The computations are performed in the imagespace, using the zero-offset section as initial condition. Inthis process, as z increases, recorded events are shiftedlaterally toward thier correct posit ion as they move in thenegative t direction.Downward extrapolation to depthzresults in awavefield that would have been recorded, i fboth sources and recordershad been located at depth z.Thus events appearing at t = 0 are at their correct lateralposition. Therefore, the extrapolated zero-offset dataatt = 0 are taken as being the correctly migrated data at thecurrent depth. These data ( t = 0) are then mapped onto thedepth section atz, the depth of extrapolation. This map-ping process is also referred to as imaging. Since imaging isa standard procedure, migration schemes differ from eachother in theirapproach to wave extrapolation, the complex-ityofwhich depends largely on he migrationvelocityfunction.If the migration velocity has no horizontal dependence,the extrapolation of zero-offset seismicdatacanbe ex-pressed by the exact wave-extrapolation equation (26) inthe wavenumber-frequency domain. This equation has asimple analytic solution, whose implementation calls for aphase shift applied to the Fourier coefficients of the zero-offset section. In the presence of lateral velocity variations,the exact wave-extrapolation equation (26) is no longervalid. To circumvent this problem, the exact expression canbe approximated by truncated series expansions [16], 18],which can accommodate horizontal elocity variations.These equations are then solved numerically, either in thespace-time domain or in the space-frequency domain.

+(I - K:)~~] Az)If thedata are defined in the (x , h )domain, i t is desirable toexpress K , and K , as functions of k, and k which are thewavenumbers with respect to x and h, respectively. Usingthe relationship among receiver and source coordinates rand s, and midpoint and half-offset coordinates x and h,that is

x= ( r +s ) / 2 h = ( r - s ) / 2 (19)we find

Once the surface data have been extrapolated by meansof (18) to he entire half-space z > 0, migration can beaccomplished by imaging the wavefield at t = 0, r = s.Thus the correctly migrated zero-offset data are given byp ( x , h = O , t = O , z ) = ~ ~ ~ P ( k , , k h , w , z ) e x p ( i k , x ) .

k z k h *(21)

B. Wave Equation for Zero-O ffset DataIn current seismic practice, chiefly for economic reasons,migration is performed after NMO and stacking on CMPsections. The NMO correction transforms data into zero-offset under the assumption that

p ( x , h , t , z = O) = p ( x , h = O, t , z = O) ( 22 )inwhich t differs rom t by the amount of he NMOcorrection, i.e.,

t = t + AtNMo (23)where AtNMo i s defined by (2) and illustrated i n Fig.3. Ifwe et p denote the NMO corrected version of p, then inview of (22), p can be regarded as being independent of h

P ( x , h , t , z = O ) = p ( x , h = O , t , z = O ) . (24)Therefore, i t s Fourier transform is zero for al l nonzerowavenumbers kh, i.e.,

~(k, ,kh,w,Z=O) IO,Orkh#o. (25)Note that (17) is still the correct extrapolation equation forF. Substituting (20) in to (17) and using (25) we obtain

The one-way wave equation (26) i s the fundamental equa-tion or downward extrapolation of zero-offset data. It isexpressed in the wavenumber-frequency domain (k,,o),and does not havean explicit representation in the mid-point-time domain ( x , t ) . We have obtained it as a specialcase of a general extrapolation equation for multioffsetdata. It can alsobeeasily derived by making use of heexploding reflector model of zero-offset data as shownelsewhere [16], [19]. It should be observed that the velocityvariable that figures in (26) is the half velocity, as one wouldexpect, from the exploding reflector model for zero-offsetdata.

A. Migrat ion in the (k,, 0 ) DomainLetting p ( x , t,z = 0) represent the zero-offset section andP(k,,u,z = 0) its double Fourier transform, the wave-extrapolation equation (26) becomes

The problem i s to reconstruct the wavefield p ( x , = O,z),which existed at t = 0 rom he wavefield p ( x , t , z = 0)observed at z = 0. We shall assume that within one ex-trapolation step, say from depth z o z Az, the velocity isconstant, i.e.,v(S) = constant, z g S < z + Az. (28)

Then the solution of (27) can be expressed asP(k,,o,z+ Az) = P(k,,o,z)

(29)in which P( k a, ) is the zero-offset section extrapolatedto depth z. This analytic solution states that P is extrapo-

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lated from z to z + Az by simply rotating its phase by aspecified amount. Therefore, migration schemesbased onthis principle are referred to as phase-shift methods [19],[20]. From P( k,, o, ) we obtain p ( x , = 0 , z ) by inverseFourier transform wi th respect to k, and a summation withrespect to o, .e.,p ( x , t = 0 , z ) =~~P ( k , , w , z ) exp ( i k , x ) . (30)k 0

Although (29) i s valid only under the assumption expressedin (28), the velocity can vary from one Az step to another.Thus the phase-shift method can accommodate media withvertical velocity variations, i.e., in which v = v(z).Undercertain circumstances, it may be acceptable tomigrate with a velocity that i s constant over theentiredomain (object plane) of interest. This uniform migrationvelocity assumption, if properly exploited, enables one todevelop a very efficient migration algorithm. This migrationscheme, often referred to as the F-K method, was f irstreported by Stolt [21]. Essentially, it i s a fast direct methodfor the evaluation of (29) and (30). Let us rewrite (29) usingz and z = 0 in place of z + Az and z, respectively, i.e.,P(k,,o,z) = P(k,,o,z=O)exp(ik,z) (31)

where

which represents the dispersion relations of the one-direc-tional wave equation (27). The migrated section p ( x , =0,z) can be obtained from P(k,, o ,z = 0) as follows:

p ( x , t = O , ~ ) = / / d o d k , P ( k , , w , z = O )eexp { i [ k( w ) z + k,x]} (33)

which corresponds to (30), except that the summations arereplaced by integrations. In order to take advantage of thefast Fourier transform (FFT) algorithm, we want to integrate(33) with respect to k instead of a.Using (32) we canrewrite (33) asp( x , t = 0 ,z ) =// k, dk, i( k) exp { i [ kz + k,x]}

(34)where

4k,,k,) = Vk2[ k t + k:] P(kx,w,z= 0). (35)In summary, the F -K method consists of three steps:

1 ) the Fourier transformation of p ( x , t , z = 0 ) to obtain2) the nterpolation and scaling of the Fourier coeffi-3) the inverse FFT of the interpolated results as expressed

P(k,,w,z = O),cients as expressed in (35),in (34).

Migration with this fast method i s limited to homogeneousmedia with constant velocity. To overcome this limitation,Stolt [21] suggested coordinate transformations to cast thewave equation in a form that i s approximately velocityinvariant.

8.Migration in the (x, w ) DomainAs we have seen, in the absence of horizontal velocitydependence, the set of independent ordinary differentialequations (27) governs the extrapolation of the zero-offsetseismic data in the (k,,w) domain. The simple analyticsolution expressed by (29) i s not valid for velocity fieldswith lateral variations. In this case, the square-root expres-sion in (27) must be approximated in some form, for in-stance, by a quadratic polynomial (Claerbouts [22] para-

bolic approximation). The expansion can then be formallyconverted into a numerical method in the ( x , o ) domain.After he approximation, the velocity v is allowed to varywith x as well as with z, and a wave-extrapolation equationfor general media i s obtained.Another alternative i s rational approximation by trun-cated continued fractions [23]. In this latter approach, anapproximation of (27) i s

(36)A practical approach to solving extrapolation equations like(36) is by Marchuk splitting. This i s done by decomposing(36) int o two extrapolators

which is know n as the thin lens term, and(37)

which is the Fresnel diffraction term. Advancing to greaterdepths i s done by applying (37) and (38) alternately in smallAz steps. Equation (38) canbeexpressed in the ( x , ~ )domain f i rst eliminating fractions and then Fourier trans-forming it wi th respect to k,. The result is

At each step Az, (37) advances P through the multiplicationby a phasor corresponding to a shift backward in time ofmagnitude AT = 2Az/v. To dvance P with (39), fornumericaltability considerations, an impl icit Crank-Nicolson difference scheme is most often used.The approximationof the square-root operator in (27)withhe rational operator appearing in (36) and thefinite-difference implementation of (39) alter the dispersionrelations of he wave equation andgenerate numericalerrors increasing with the expansion parameter sin9 =vkX/2w, where 9 represents the angle with respect to thevertical axis of the wavefront under consideration. Whenthe dip of the imaging wavefield exceeds 40-45, disper-sion effects separate the low frequencies from hehighfrequencies during the wavefront depropagation. As a re-sult, the image of steeply dipping reflectors may be accom-paniedby ghosts. The phenomenon i s demonstratedthrough a numerical example, themodel ofwhich i s

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depicted in Fig. 11. The synthetic zero-offset section of themodel is shown in Fig. 12. The migrated depth sectionobtained by a finite-difference mplementation of (37) and(39) i s illustrated in Fig. 13. Note that the non-steeplydipp ing events are clear, but the steeply dipping event hasghosts.Better dispersion properties can beobtained includinghigher order terms in the formal expansion [16]. An alterna-tive approach to wavefield extrapolation in general media isdescribed in [24]. A t each Az step, the wave extrapolation is accomplished in two stages. I n the first stage the wavefieldP( x , w ,z) given at depth z is extrapolated to z + Az by thephase-shift method using G reference velocities v,,y;.., v,spanning the velocity range at depth z . This stage generates

0 L 4 6 8 10 12x ( k m )

2 r >v = 2 . 0

v = 3 . 041 v - 4.02 r >

v = 2 . 0

v = 3 . 041 v - 4.0

i6 %-( km )Fig. 11. Schematic of model 2, representing a dipping mul-tilayer example. The thick-line segments ndicate where thereflector segments have been turned on.

0.0 2.0 4.0 6.0 8.0 10.0 12.0xlkrn)

0.0 20 4.0 6.0 8.0 10.02.0x(krnl0.0

2.0

4.0

6.0Fig. 14. Depthmigration section obtained from the timesection shown in Fig. 12 using the phase shift plus interpola-tion method. Al l reflectors are migrated correctly.

G reference extrapolated wavefields at z + A z , namely,4 ,e , - - - ,(. in the second stage, the definitive wavefieldP ( x , w,z + A z ) i s constructed by nterpolationof the Greference wavefields. Thisphase shift plus interpolation(PSPI) method is unconditionally stable and has good dis-persion relation properties worth its relatively high compu-tational cost (for each frequency w , and for eachstep indepth, G+ 1 Fourier transforms in the x direction are re-quired). The synthetic zero-offset section of Fig. 12 migratedwith the PSPI method is shown i n Fig. 14. One can noticesignificant mprovement in themigration of the steeplydipping reflector in comparison with the one shown inFig. 13.C.Migration in th e (x , t ) Domain

Migrat ion as a wave depropagation process is best de-scribed in the transformed domains ( k , , w ) and ( x , @ ) . twas introduced, however, in the early 1970s using paraxialwave equations in the physical space-time ( x , t ) [22],[14]. We will reproduce here those equations, deriving themfrom their frequency-domain counterparts.An inverse Fourier transform of the wavefield P ( x , o , z )in (37) and (39) results in the fol lowing wo equations interms of p ( x , t , z ) :

Fig. 12. Zero-offset ime section of model 2

0.0 2.0 4.0 6.0 8.0 10.0 12.0x(krnl

w z + - - = 0 .Z p a z P4 a 2

Fig. 13. Depth migration obtainedrom the zero-offset timesection of model 2 shown in Fig. 12 . The ( x , o ) domainfinite-difference migration methodusednhisexamplecorresponds to (37) and (39). Note that the steeply dippingreflector is not imaged correctly.

a+--=--.a 3 p v2 a 3 pa t 2 a z 4 ax2& 16 a x 2 a z (41)

For gentle dips, the ight-hand side of (41) an be ne-glected, and it becomes

Equations (40) and (41) or (40) and (42) ar e specialized waveequationshat depropagate waveenergy only within asmall angle about the vertical axis, hence are called paraxialapproximations: details on their numerical integration withfini te-difference methods can be found in 12, pp. 211-2121.Seismic interpreters often ind t useful toworkwithsubsurface migrated sections with depth measured in unitsof time, or ease of comparison with unmigrated CMPsections. Such maps are obtained by introducing a pseudo-depth r (vertical travel time) which is related to true depthz by

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(43)where V(z) s an approximation, independent of x , of hevelocity v ( x , z ) . With the introduction of the vertical coor-dinate T and the help of an additional approximation, (40)and (42) become

Wavefieldextrapolation with (44) nd (45) lead to timemigration as opposed to depth migration obtained with(40) and (42). Time migration schemes depropagate thewavefield as i f the medium were horizontally layered anddo not correctly refract rays at the velocity interfaces, butconstitute a viable and economic tool in most practicalsituations. An excellent example of finite-difference (time)migration of a CMPstack section is illustrated in Figs. 1 5and 16. Notice the decrease in width of the diffractions on

8 8

stack TraceSpacing=33mFig. 15. CMP stacked section from data recorded in theSanta Barbara channel. Steepest dips are approximately 25.(From Hatton et a / . [17]; courtesy of Western Geophysical.)

M(g- TraceSpacing=33rnFig. 16. Finite-difference imemigration of the C M P sec-tion illustrated in Fig.15 . (From Hatton et a /. [17]; courtesy ofWestern Geophysical.)

the ighthand of the migrated section and the develop-ment of synclines at the eft. When a better positioning isrequired, after time migration, a correction is made usingHubrals image ray time-to-depth conversion, [25, p. 1031,[26], [27]. The solution of (44) can be written as p( x , t , T +A T ) = p ( x , + A 7 , 7 ) which representsa uniform translationof the wavefield p along the negative t axis as T increases.There is no need to perform this time shift explicitly.Equation (44) can be accounted for by imaging the data at

t = T , rather than at t = 0. Time migration methods basedon thismagingprinciple are often described as beingsolved in a downward moving coordinate frame.V I. WAVE-EQUATIONIGRATION IN OBJECT SPACE

Migrationof zero-offset data p ( x , ,z = 0) is accom-plished by the methods described in Sections IV and Vthrough a downward extrapolation of surface data. Advanc-ing n depth, image plane sections p ( x , t , z= const) arecomputed and the final migrated section i s given by theextrapolated wavefield at time zero p ( x , = 0,z). An alter-native approach calls for a reversetim e extrapolation of thewavefield using the CMP section as a boundary condition.Marching backward in time, object plane sections (time-slices) p ( x , = const, z ) are computed. Calculations beginat time T, which defines the lastsample of he recordsection, and continue in the negative t direction until timezero. At that time the amplitudes in object space ar e con-sidered as the final migrated section.In this section, dealing with reverse-time migration, forthe sake of clarity, we shall deviate slightly from the nota-tions used throughout this paper. We shall let 9 ( x , ) repre-sent the stacked CMPdata,and p ( x , t , z ) the wavefield.Consequently, p ( x , t = 0, z ) will represent the migratedCMP section.In object space, just as in image space, migration schemesdiffer from each other only in the type of wave equationsbeing used for wave extrapolation. For example, Hemon[28] and McMechan [29] suggested he fu ll wave equation

which corresponds to our (8) or 9) expressed in the mid-point variable x . This equation i s then integrated numeri-cally with the following initia l conditions:p ( x , t = , z ) = 0 , f o r z t o (47)and

q x , t = T,z) = 0.a t (4.8)

p ( x , t , z = O ) = q ( x , t ) . (49)The boundary conditions at the surface are given by

The problem with (46) is that it allows for reflections frominterfaces where he velocity changes rapidly. We recallthat the exploding reflector model shown in Fig. 4 assumesthat he surface record includes only upward propagatingwaves. Therefore, in the depropagation phase all wavecomponents should move downward n reverse time, andnoupward reflected waves ar e admissible. This problemcan be easily avoided i f one uses, instead of (46), a one-directional wave equation. Baysal et a / . [30]and Loewenthalet a / . [31] used an equation for downward propagatingwaves *t = ( ~ i k , [ l +( k x / k , ) 2 ] 1 2 . F ) p (50)where 9 nd 9-re operators representing the doubleFourier transform from the ( x , z ) domain to the ( k , , k , )domain and its inverse. Unfortunately, (50) cannot be writ-ten without the transform operators 9 ands-, ince themultiplication n the Fourier space by ik,[l + ( k x / k , ) 2 ] 1 / 2

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cannotbe expressed in the ( x , z ) space. Further detailsregarding (50) and its numerical solution are found n [32]and [33]. The initial and boundary conditions for reverse-time migration with (50) are given by (47) and (49), respec-tively. Using a hird-order Runge-Kutta algorithm, for ex-ample, to solve (50) on the zero-offset section shown in Fig.12 generates the migrated depth section illustrated in Fig.17.

0.0 2.0 4.0 6.0 8.0 10.0 12.0d k m l0.0

20

4.0

6.0

Fig. 17. Depthmigration section obtained rom the timesection shown in Fig. 1 2 using reverse-time migration basedon (50).

Another approach to reducing reflections was reportedby Baysal et al. [35], who propose the use ofa two-waynonreflecting wave equation. Their approach begins consid-ering the general wave equation [34, p. 1711

where p is the density of the medium. Since p does notplay any role in migration, itsvaluemaybe chosen tominimize reflection. This is done by setting p so that theacoustic impedance

K( x , z )= p ( x , z ) v ( x , z )= constant (52)over the entire object plane.The basic conceptof using CMP data as a boundarycondition and computing its response in the object planeis not new. The heuristic method called wavefront inter-ference migration discussed in Section Il l is based on hesame principle. The only important difference is that thereverse-time migration methods are based on a wave-equa-tion solution rather than the intuitive approach depicted inFig. 8.VII. M I G R A T I O N BEFORESTACKING

The task of imaging the subsurface in conventionalseismic data processing i s partitioned in wo steps: thestacking of the CMP gathers, and the migration of the CMPstacked section. This procedure, however, i s correct only fora simplified subsurface model consisting of horizontal layersof laterally uniform velocity and ofhorizontal reflectors.Levin [36] has shown that the stacking velocity depends notonly on he propagation velocity of the overburden but alsoon the dip angle of the reflectors. in structurally complexformations, the correction for all dip anglesbecomes im-possible, and stacking loses its effectiveness. Under thesecircumstances, it may be advantageous to consider migra-tion before stacking.

The theoretical basis for migration of unstacked (multi -offset) data was discussed in Section IV . in this process, themultioffset data are converted (see, for example, (17) and(21)) into a migrated zero-offset section. This i s a relativelycostly process and does not provide any intermediate re -sults. To overcome these shortcomings of the one-stepmigration of unstacked data, a number of special-purposebefore-stack migration techniques were developed. For ex-ample, the prestack partial migration [IS] and the offsetcontinuation algorithm [37], 38]are applied to common-offset sections after they have been NMO corrected. Thesestack-enhancement [39] techniques share the advantage ofyie lding an unmigrated CMP stack section that helps theinterpreter in resolving spuriousevents generated on thefinal migrated section by inaccurate velocity estimates. Theyare computationally more expensive than the conventionalprocessing since they introduce an additional process be-tween NMO and stacking but leave themigration costunchanged.The methods described in this section belong o theformer category in which the separate components of theconventional processing (NMO, stacking, zero-offset migra-tion) are merged in to a unif ied process. They represent thebest methods available for imaging the subsurface in avariable velocity medium. They are, unfortunately, also themost expensive in terms of computation and data handling.A. Migration of Multioffset Data

The theoretical considerations for migration of multiof f-set data were presented in Section IV . The wavefield ex-trapolation is governed by (13,and the imaging is accom-plished by mplementing (21). The solution by phase shiftof (13,expressed by (18), can accommodate only verticalvelocity variations from one Az step to another. In thepresence of lateral velocity variations, (17) needs to begeneralized as follows [15]:- =p -(I - C y 2 + - ( I1 - C:) Paz v(r,z) v( st .) l*I

(53)where IC,nd IC ,re given by (14). This implies hatwedistinguish between the velocity along he shot axis and thevelocity along the shot axis and thevelocity along thereceiver axis. We hasten to point out, however, that in (53)the algebraic expressions in the operators IC,nd I C ~ re notimmediately defined when v ( r , z ) and v( s,z) have lateralvariations. This is the same limitation as the one we haveencountered with the single-square-root equation (27). For-tunately, most ofhe olution methods for zero-offsetmigration are applicable to (53). In other words, multioffsetdata can be extrapolated by the PSPl method or by finite-difference methods developed in the ( r , s , ~ ) omain or inthe ( r , s, t ) domain.Having decided on a suitable extrapolation scheme, be-fore-stack migration can be summarized as fol lows [ a ] . Thealgorithm operates on the entire data volume p( r , s , t , z = 0)processing it in the alternating directions rand s. For eachdepth level z , in increments of Az, beginning with z = 0,the fol lowing steps are executed:

1) Order the data into common-shot gathers p( r , so , t , z )and extrapolate downward rom z to z + Az each com-mon-source gather s = so using (12).

2) Reorder the data obtained rom step 1 into common-

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receiver gathers r = ro and extrapolate downward by Azeach common-receiver gather r = r, using (13).3) Perform the imaging m( x, z) = p( r = x, s = x, t = 0, z),where m(x,z) is the migrated CMP section.

6. Migration of Common-Shot GathersMigration before stacking canalso be accomplished bymigrating the individualcommon-shot gathers and sum-ming these migrated gathers. The downward extrapolationof the recorders i s governed by (12) or some approximationthereof. The extrapolated wavefield i s focused at the reflec-tor location, not at time zero, but at the propagation timebetween the shot and the reflector. To simplify matters, let us consider a single point reflector, for example Pl in Fig. 2,whose position (r',z') i s specified in the receiver-depthcoordinate system.The event corresponding to he pointreflector 6 is observed at t = t + t 2 , where tl(s; , f ) is the travel time between he source and the reflector, and

t 2 ( c ',z') is the travel time between the reflector and thereceiver. Under these assumptions, the downward extrapo-lat ion focuses the recorded wavefield in the neighborhoodof the point ( r ' , ~ ' )at time r,(s;r',z'), which is the correcttime for maging.We are now in a position o set forth an algorithm ormigration before stack. Forach common-shot gather

p(r, F, t ,z = 0) we execute the fo llowing steps:1) Compute he arrival time of the primary wavefieldtl = t l ($ I ' , z ' ) at all points ( r ' , ~ ' )of the subsurface.2) Extrapolate downward (for instance wi th (12)) in depththe common-shot data p(r,F, t,z = 0), thus regenerating foreach point of thesubsurface the scattered field p ( / , 5, t,z').m ( f , f ; < ) =p( t ,5 , t1 (5 ; t , f ) , f ) . (54)

Then sum over a set of migrated common-source gathers,that is,r n ( f , f ) xm(r',z';


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puted and the imaging mustbe performeddeconvolvingthe scattered wavefield by the primary wavefield.VIII. CONCLUDINGREMARKS

We have presented an overview of the major advances inseismic migration. Our aim was to provide a logical se -quence of development rather than a historical one. Wehave put emphasis on the fundamental concepts of migra-tion and have carefully avoided passing judgement or ap-praising value based upon their current populatiry. Themerit of a migration method is largely decided by economicfactors. These factors change rapidly with the advancementof recording technology, computer architecture, and cost-performance indices of data processing systems.Mainly for reasons of cost, migration in the 1970s wasappliedo stacked CM P sections, and lateral velocitychanges were not correctly taken into account in routinemigration programs. With the continuing trend toward lowercost-performance ratio in advanced computer architec-tures, high-accuracy migration methods can be expected togain acceptance in the near future. Another area that willreceive increased attention is migration before stacking. Inadd ition to better migration, a major benefit resulting frommigrationbefore stacking can be derived from mprovedvelocity information. Ideally, we would ike to have a pro-cess that simultaneously images velocities and migratesdata to correct subsurface locations. This calls for imagingpart of the subsurface recursively step by step, using migra-tion and velocity analysis procedures [41], 42).Seismic exploration methods and processing techniquesare growing rapidly. Innovations can be observedover abroad range from recording techniques to displays. Record-ing techniques have been developed to provide areal cover-age. Acquisition and migrationof seismicdata in threedimensions are becoming widespread. These methods areparticularlyuseful in the presence of complex structures.Migration algorithms for three-dimensional data are naturalextensions of those used for two-dimensional data [43], [MI.Increasing demand for oil and decreasing availability o fhydrocarbon deposits are the motivating forces behind mostinnovative techniques which have one thing n common:they are aimed at improving resolution. A decade ago,geophysicists might have been satisfied with mapping hegross structure of he subterrain. Today, they hope to de-termine rock parameters, hydrocarbon content, as well asentrapment structure. These demands have stimulated re-search in more general and theoretically appealing inversemethods [45]-[52]. These works, when developed into prac-tical processing tools, have the potential of providing in-sight in to long-standing unsolved geophysical problems.ACKNOWLEDGMENT

The authors wish to thank B . Gibson of Western Geo-physical Company for providing Figs. 15 and 16.R EF ER EN C ES[I ] D. Loewenthal, L. Lou, R . Robertson, and J. W. Sherwood,The wave equation applied to migration, Ceophys. Pros-pect., vol. 24, pp. 380-399, 1976.[2] J. C. Hagedoorn, A process of seismic reflection interpreta-tion, Geophys. Prospect., vol. 2, pp. 85-127, 1954.

[3] J . D. Johnson and W. S. French, Migration-The inversemethod, in Concepts and Techniques in Oi l and Cas Ex -ploration, K. C. lain and R. J. P. de Figueiredo, Eds.Tulsa,OK: SOC. Exploration Geophysicists, 1982, pp. 115-157.[4] W. A. Schneider, Developments in seismicdata processingand analysis: 1968-1970, Ceophys., vol. 36, pp. 1043-1073,

1971.[5] M. Born and E. Wolf, Principles of Optics , 3rd ed. New York:Pergamon, 1%5.[6] W. S. French, Computer migration of oblique seismic reflec-tion profiles, Ceophys., voi. 40, pp. 961-980, 1975.[q W. A.Schneider, Integral formulat ion for migration n woand three dimensions, Ceophys., vol. 43, pp. 49-76,1978.[8] A. J. Berkhout, Seismic Migration . Amsterdam, The Nether-lands: Elsevier 1980.[9] A. J. Berkhout,Wave field extrapolation techniques in seismicmigration, a tutorial, Ceophys., vol. 46, pp. 1638-1656, 1981.

[lo] J. A. Carter and L. N. Frazer, Accomodating lateral velocitychanges in Kirchhoff migration by means of Fermats princi-ple, Ceophys., vol. 49, pp. 46-53, 1984.[ I l l C. H. F. Cardner, W. S. French, and T. Matzuk, Elements ofmigration and velocity analysis, Ceophys., vol. 39, pp,

811-825, 1974.

[I 71

J . F. Claerbout, Fundamentals of Geophysical Data Processing.New York: McGraw-Hill, 1976.E . A. Robinson and M. T. Silvia, Digital Foundations of TimeSeries Analysis: V o l2 , Wave Equation Space Time Processing.San Francisco, CA: Holden-Day, 1981.J. F. Claerbout, Toward a unified theory of reflector map-ping, Ceophys., vol. 36, pp. 467-481, 1971.0 . Yilmaz and J. F. Claerbout, Prestack partial migration,Geophys., vol. 45, pp. 1753-1 779, 1980.J. Cazdag, Wave equation migration with he accurate spacederivative method, Ceophys. Prospect., vol. 28, pp. 60-70,1980.L. Hatton, K. L. Larner, and B. S. Gibson, Migrat ion of seismicdata from inhomogeneous media, Ceophys., vol. 46, pp.A. J . Berkhout and D. W. Van W. Palthe, Migration n termsof spatial deconvolution, Ceophys. Prospect., vol. 27, pp.J. Cazdag, Wave equationmigrationwith he phase-shiftmethod, Geophys., vol. 43, pp. 1342-1351, 1978.C. Bolondi, F. Rocca, and A. Savelli, A frequency domainapproach to two-dimensional migration, Ceophys. Prospect.,R . H. Stolt, Migration by Fourier transform, Ceophys., vol.J. F. Claerbout, Coarse grid calculation of waves in inhomo-geneous media with application to the delineation of com-1970.plicated seismic structures, Ceophys., vol. 35, pp. 407-418,R . Clayton and B. Engquist, Absorbing boundary conditionsfor acoustic and elastic wave equations, Bull. Seis. Soc.Amer., vol. 67, pp. 1529-1540,1977,J. Cazdag and P. Sguazzero, Migration of seismicdata by1984.phase shift plus interpolation, Ceophys., vol. 49, pp. 124-131,P. Hubral and T. Krey, Interval Velocities from Seism ic Reflec-t ion Time Measurements. Tulsa, OK: Soc. Exploration Ceo-physicists, 1980.P. Hubral, Time migration-Some ray theoretical aspects,Ceophys. Prospect., vol. 25, pp. 738-745, 1977.K. L. Larner, L. Hatton, B. S. Gibson, and I-C. Hsu, Depth

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migration of imaged time sections, Ceophys., vol. 46, pp,734-750, 1981,

[28] C. Hemon, Equations donde et modeles, Ceophys. Pros-pect., vol. 26, pp. 790-821, 1978.[29] C. A. McMechan, Migration by extrapolation of time-depen-dent oundary values, Ceophys. Prospect., vol. 31, pp.[30] E. Baysal, D.D. Kosloff, and J. Sherwood, Reverse timemigration, Ceophys., vol. 48, pp. 1514-1524, 1983.[31] D. Loewenthal and I . R. Mufti, Reversed timemigration inspatial frequency domain, Ceophys., vol. 48, pp. 627-635,

1983.[32] J. Cazdag, Extrapolation of seismic waveforms by Fourier

413-420, 1983.

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methods, ISM1.Res. Develop., vol. 22, pp. 481-486, 1978.-, Modeling of the acoustic wave equation withtransform methods, Ceophys., vol. 46, pp. 854459,1981.L. M. Brekhovskikh, Waves in Layered Media. New York:Academic Press, 1 W .E. Baysal, D. D. Kosloff, and J. Sherwood, A two-way nonre-flecting wave equation, Ceophys., vol. 49, pp. 132-141,1984.F. K. Levin, Apparent velocity from dipping interface reflec-tions, Ceophys., vol. 36, pp. 510-516, 1971.G . Bolondi, E. Loinger, and F. Rocca, Offset continuation ofseismic sections,Ceophys.Prospect., vol. 30, pp. 813-828,1982.S. M. Deregowski and F. Rocca, Geometrical optics andwave theory of constant offset sections in layered media,Ceophys. Prospect., vol. 29, pp. 374-406, 1981.P. Hood, Migration, in Developments in Geophysical Ex-plo ration Methods-2, A. A. Fitch, Ed. London, England:Applied Science Publ. 1981, pp. 151-230.P. S.Schultz and J. W. C. Shenvood, Depth migration beforestack, Ceophys., vol. 45, pp. 376-393, 1980.P. S . Schultz, A method for direct estimation of intervalvelocities, Ceophys., vol. 47, pp. 1657-1671, 1982.J. Gazdag and P. Sguazzero, Interval analysis by wave ex-trapolation, Ceophys. Prospect., vol. 32, pp. 454479,1984,6. Gibson, K. Larner, and S. Levin, Efficient 3-D migration intwo steps, Ceophy. Prospect., vol. 31, pp. 1-33, 1983.

[441 H. Jakubowicz and S. Levin, A simple exact method of 3 - 01983.migration-Theory, Geophys.Prospect., vol. 31, pp. 34-56,J. K. Cohen and N. Bleistein, An inverse method or de-termining small variations in propagation speed, SIAM ].Appl. Math., vol. 32, pp. 784-799, 1977.-, Velocit y inversion procedure for acoustic waves,Ceophys., vol. 44 , pp. 1077-1087,1979.S. Raz, Direct reconstruction of velocity and density fromscattered field data, Ceophys., vol. 46, pp. 832-836,1981.R. W. Clayton and R. H. Stolt, A Born-WKBJ nversion methodfor acoustic ref lection data, Ceophys., vol. 46, pp. 1559-1567,1981.M. ahlou, 1. K . Cohen, and N. Bleistein, Highly accurateinversion methods for three-dimensional stratified media,SIAM J. Appl. Math., vol. 43, pp. 726-758, 1983.F. C. Hagin and J. K. Cohen, Refinements to the linearveloci ty nversion theory,Ceophys., vol. 49, pp. 112-118,1984.A. Bamberger, C. Chavent, and P. Lailly, About the stabilityof the inverse problem in 1-D wave equations-Applicationto the interpretation of seismic profiles, Appl. Math. Optim.,P. Lailly, The seismicnverse problem as a sequence ofbefore stack migrations, presented at the Conf. on InverseScattering, Univ. Tulsa, Tulsa, OK, M a y 1983.

VOI., pp. 1-47, 1979.

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