# black true amplitude imaging and dmo 1993

TRANSCRIPT

GEOPHYSICS, VOL. 58, NO. 1 (JANUARY 1993), P. 47-66, 8 FIGS., 3 TABLES.

True-amplitude imaging and dip moveout

James L. Black*, Karl L. Schleicher** and Lin

ABSTRACT

True-amplitude seismic imaging produces a threedimensional (3-D) migrated section in which the peakamplitude of each migrated event is proportional to thereflectivity. For a constant-velocity medium, the stan-dard imaging sequence consisting of spherical-diver-gence correction, normal moveout (NMO), dip move-out (DMO), and zero-offset migration produces atrue-amplitude image if the DMO step is done cor-rectly. There are two equivalent ways to derive thecorrect amplitude-preserving DMO. The first is toimprove upon Hale’s derivation of F-K DMO bytaking the reflection-point smear properly into ac-count. This yields a new Jacobian that simply replacesthe Jacobian in Hale’s method. The second way is tocalibrate the filter that appears in integral DMO so asto preserve the amplitude of an arbitrary 3-D dippingreflector. This latter method is based upon the 3-Dacoustic wave equation with constant velocity. The

resulting filter amounts to a simple modification ofexisting integral algorithms. The new F-K and integralDMO algorithms resulting from these two approachesturn out to be equivalent, producing identical outputswhen implemented in nonaliased fashion. As dip in-creases, their output become progressively larger thanthe outputs of either Hale’s F-K method or the integralmethod generally associated with Deregowski andRocca. This trend can be observed both on model dataand field data.

There are two additional results of this analysis,both following from the wave-equation calibration onan arbitrary 3-D dipping reflector. The first is a proofthat the entire imaging sequence (not just the DMOpart) is true-amplitude when the DMO is done cor-rectly. The second result is a handy formula showingexactly how the zero-phase wavelet on the final mi-grated image is a stretched version of the zero-phasedeconvolved source wavelet. This result quantita-tively expresses the loss of vertical resolution due todip and offset.

INTRODUCTION

The goal of seismic processing isto produce a true-amplitude estimate of the earth’s reflectivity in its fullymigrated position. For most interpreters, “true-amplitude”means that each migrated event’s peak amplitude is propor-tional to the reflection coefficient, where we use the term“event” to refer to the processed seismic image correspond-ing to a given reflector. Ideally, the proportionality constantbetween peak amplitude and reflectivity should be the samefor every event on the three-dimensional (3-D) section,regardless of the depth, dip, or final wavelet. In this paper,

we will explicitly show how to accomplish this goal for thecase of constant velocity and point sources/receivers. Thekey to reaching our goal is making dip-moveout (DMO) anamplitude-preserving process.

In this paper, we will assume that the seismic data havebeen processed so as to remove the source signature,instrument response, multiples, ghosts, and noise. The focusof our attention is achieving true-amplitude processing in theseismic imaging steps as shown in Figure 1, when thevelocity is constant. In particular, we will analyze thefollowing familiar 3-D imaging steps, which we call the“standard sequence”:

Presented at the 58th Annual International Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor September 3,1991; revised manuscript received June 16, 1992.*Formerly Halliburton Geophysical Services, Dallas, TX; presently International Business Machines Corp., 1505 LBJ Freeway, Dallas, TX75234.**Halliburton Geophysical Services, P. O. Box 5019, Sugarland, TX 77487

Geophysics Dept., Stanford University, Stanford, CA 94305, presently International Business Machines Corp., 1505 LBJ Freeway,Dallas, TX 75234.© 1993 Society of Exploration Geophysicists. All rights reserved.

47

48 Black et al.

1) Spherical-divergence (spreading loss) correction,2) Normal-moveout correction (NMO),3) 3-D dip-moveout correction (DMO), and4) 3-D zero-offset (exploding-reflector) migration.

We are, of course, not the first to ask how to make thestandard sequence (or something close to it) produce true-amplitude images. The exploding reflector concept (Loe-wenthal et al., 1976) gave an intuitively appealing approachto the kinematics of zero-offset migration, but never fullyaddressed the amplitude question at zero-offset, much less atfinite offset. The introduction of DMO (Deregowski andRocca, 1981) into the sequence in the last decade has finallymade it possible to address the amplitude question forarbitrarily-dipping events at finite offsets. An early attemptwas made by Yilmaz and Claerbout (1980) to decompose thedouble-square-root prestack migration method into NMO,DMO, and zero-offset migration. Likewise Deregowski andRocca (1981) made a preliminary connection between DMOand the double-square-root equation in their landmark pa-per. Finally, Hale (1983) made a much more thorough effortto do the same thing in an unpublished chapter of his Ph.D.thesis. Unfortunately, all three of these efforts fell short ofproducing a definition of DMO that would allow the standardsequence to simultaneously treat both horizontal and dippingevents’ amplitudes properly.

As a consequence, DMO algorithm development retreatedto a more defensible position: guaranteeing that kinematics(i.e., event positioning) were correct for all events, butensuring that amplitudes were correct at most for horizontalevents only. Thus Deregowski and Rocca (1981, pp. 397-8)and Deregowski (1985, 1986, 1987) proposed various calibra-tions of the integral DMO method but could make nodefinitive statement on amplitude preservation for arbitrarilydipping data. Hale’s (1984) F-K algorithm handled ampli-tudes reasonably well, and Berg (1984) developed an integraltechnique based upon it. Hale (1991) has recently derived an

improved integral DMO from this F-K technique that pre-serves amplitudes on horizontal events even when spatialaliasing of the DMO operator is important. Nevertheless thehandling of dipping-event amplitudes differs among all ofthese algorithms. Furthermore it turns out that all of thesealgorithms yield amplitudes that are too small for dippingevents.

In this paper, we establish three results for true-amplitudeseismic imaging. First of all, we produce a revised DMOtechnique that preserves peak amplitudes on horizontal anddipping data equally well. We explicitly demonstrate theform this technique takes in both F-K and integral imple-mentations in three dimensions. Second, we establish thatthe standard processing sequence is a true-amplitude proce-dure when it includes this revised DMO method. Thus weclear up any lingering doubts about the validity of exploding-reflector migration algorithms and spherical-divergence cor-rections. Finally, we compute an explicit expression for thewavelet-stretch factor that controls the vertical resolution onthe migrated image.

The casual reader should skip directly to Table 2 andequations (51) and (52) for a summary of the main results ofthis paper. For the more dedicated reader, the outline of thispaper follows. In the first two sections we define ournotation and what we mean by true-amplitude processing. Inthe next section we reexamine Hale’s (1984) derivation oftwo-dimensional (2-D) F-K DMO and make it amplitude-preserving by including reflection-point smear completely,as has previously been presented by one of us (Zhang, 1988).In the following section, we carry out a complete amplitudeanalysis of a planar reflector with arbitrary dip in threedimensions. Using the solution of the 3-D acoustic waveequation as our starting point, we calibrate the entire stan-dard processing sequence to achieve true-amplitude process-ing, as has been previously presented by one of us (Blackand Egan, 1988; Black and Wason, 1989). [Note that Blackand Egan (1988) contains an error which this work corrects.]

Fig. 1. Processing flow diagram for the standard imaging sequence. showing the wave-equation calibrationprocedure for a dipping reflector.

True-amplitude Imaging and Dip Moveout 49

The major part of this calibration is the 3-D derivation ofamplitude-preserving DMO, this time stated as an integralDMO rather than an F-K DMO. At the end of this section,we show that the entire calibrated standard sequence istrue-amplitude, and we derive an explicit expression for thewavelet-stretch factor. In the section entitled “Summary ofDMO Methods,” we use the connection between integraland F-K DMO in Appendix C to conclude that our F-K andintegral techniques are equivalent to each other. This leadsto Table 2, which summarizes four DMO algorithms: asdescribed in this paper,Hale’s, a generic form ofDeregowski and Rocca’s integral method, and the Bleistein-Liner method (Liner, 1989; Bleistein, 1990). In the finalsections, we examine the outputs produced by three of thesefour DMO algorithms acting on model and field data. Weshow that our amplitude-preserving DMO maintains thepeak amplitudes of both horizontal and dipping data, anecessary condition for the entire standard sequence to betrue-amplitude.

Much of the development of this paper is contained in fourappendices. Appendix A establishes the kinematic identitiesrelating the DMO times and midpoints. Appendix B containsthe mathematics required to evaluate the stationary-phaseintegral that defines our amplitude-preserving integral DMO.Appendix C details the asymptotic relationship between F-KDMO and integral DMO. Finally Appendix D gives themathematical details necessary to perform Stolt zero-offsetmigration for a 3-D dipping reflector.

NOTATIONAL CONVENTIONS

For the sake of continuity with the existing DMO litera-ture, we have followed Hale’s (1984) notation as much aspossible. For example, we use to denote time on thezero-offset section, However, our focus onamplitude preservation, on three dimensions, and on theentire standard sequence has made it necessary to extendHale’s notation. Thus the midpoint position on the zero-offset section is called rather than y, where we useboldface to indicate a vector quantity. Table 1 summarizesour conventions, listing each processing step, the outputsection from each step, and the planar event time after eachstep. For completeness, the table begins with the step“wave-equation generation’’whose output is the seismicsection that is input to the standard sequence. Note that“Latin” times such as are independent variables servingas arguments, whereas “Greek” times such as aredependent variables expressing the traveltimes of a dippingevent as a function of midpoint. For example, the section

Table 1. Notational conventions, showing the output sectionand the event time following each stage in the standardsequence.

contains an event whose peak amplitude occursat = For the sake of conciseness, we havesuppressed the half-offset argument h in all variables.

HOW TO MAKE AMPLITUDE GLOBALLY PROPORTIONALTO REFLECTIVITY

Before going further, we need to clearly define what wemean by “amplitude proportional to reflectivity” in thepresence of band-limited wavelets. This phrase is a littletricky because the zero-phase image wavelet on the finalmigrated section must vary from event to event. This varia-tion is illustrated in Figure 2, which shows correct process-ing of two events that have the same reflectivity and thesame deconvolved zero-phase source wavelet but whichhave different image wavelets. The convolved withreflector 1 has a higher bandwidth than the convolvedwith reflector 2. Why? The answer is that nonverticalraypaths cause some of the vertical resolution in the decon-volved source wavelet w(t) to be traded for lateral resolution(Wu and Toksöz, 1987). This means that the vertical resolu-tion in the image wavelet is generally lower than in thesource wavelet. It also means that the image wavelet de-pends upon whatever factors influence the raypaths, such asdip, velocity, depth, and offset. It turns out that the best

IMAGE OF TWO EVENTSWITH SAME REFLECTIVITY

FIG. 2. Effect of raypath obliquity on the wavelet. Thedipping event with sloping raypath has a more stretched(lower frequency) image wavelet than the event with avertical raypath. Lower part of figure shows same result inthe frequency domain.

50 Black et al.

achievable migrated image is the convolution of theearth’s reflectivity, with a band-limited event-dependent image wavelet, E):

= *

= (1)

where are migrated time and position and where is the reflectivity, which generally depends upon the

angle shown in Figure 3. The quantity is short for showing the wavelet’s dependence on time dip D,

migrated event-time and half-offset h. Extractingthe reflectivity from the migrated image P,, is thus not asimple matter of deconvolving a single, global image waveletfrom equation (1) because is not global.

Despite this fact, we can still define conventions andprocessing sequences to make equation (1) yield a globalconnection between and R(Q). The most popular con-vention is already shown in Figure 2, where we haveprocessed the data so as to obey the following rule: keep thepeak value of the zero-phase image wave/et the same for allevents. In other words, we can process the data so that thepeak value of is a global constant, w(O), independent ofthe values of , and h. This yields the reasonable resultthat two events with the same (q-dependent) reflectivity willhave the same peak amplitude in the migrated image, namely

which is what most interpreters have always expected. Wewill show how to maintain this convention in this paper. Infact our wave-equation analysis of a dipping reflector willestablish a simple wavelet-stretch relationship between and w, namely

= (3)

where is the product of three stretch factors correspondingto NMO, DMO, and zero-offset migration, respectively andis given by equation (51). Note that equation (3) immediatelyimplies that the peak amplitude of (i.e., the t = 0 value)is globally equal to

On the other hand, a Fourier transform of equation (3)clearly shows that the “peak-amplitude” imaging conven-tion has resulted in a spectral density in that is increasedby a factor of relative to the spectral density of w(t). Thisis shown in the lower part of Figure 2, where the spectraldensity of event 2 is higher than that of event 1. This causesno problem unless the interpreter expects the spectral den-sities (rather than the peak amplitudes) of all events to be thesame. If we want an imaging convention and processingsequence that preserve spectral density rather than peakamplitude, we just need to ensure that equation (3) getsmultiplied by which is accomplished by multiplying theNMO, DMO, and migration operations by the stretch factorsof A, and cos defined in equations (30), (40), and (50),respectively. This alternative convention results in weakerevents compared to peak-amplitude imaging when the dip issteep or the offset is large, which is why the peak-amplitudeconvention is usually preferred.

REVISING HALE’S DERIVATION OF F-K DMO

In this section, we will derive a true-amplitude revision ofHale’s (1984) F-K DMO technique. This derivation is sosimilar to what Hale did that we will review his results first.

Hale’s derivation of F-K DMO

Hale’s derivation begins with the two-dimensional raypathgeometry shown in Figure 3. From the geometry of thisfigure, the relation between the zero-offset traveltime,

and the pre-NM0 time at the same midpoint isfound to be (Levin, 1971):

FIG. 3. DMO raypath geometry for finite-offset and zero-offset, with various traveltimes annotated.

True-amplitude Imaging and Dip Moveout 51

=

where is the dip angle of the reflector. Hale’s derivation isa clever heuristic argument in which the distinction betweenindependent variables and dependent variables in Table 1 isblurred. Thus Hale makes the following associations:

t = a

=

=

where

(2

To begin with, Hale defined NM0 according to the stan-dard mapping:

Here JH is the Jacobian for the to and is the phase:

change of variables from

(12)

(13)

where we have used the quantity A that Hale defined:

which satisfies the simple equation

This completes our review of Hale’s derivation of F-KDMO. Equations (l0)-(14a) are a complete description of hisalgorithm.

Corrected derivation of F-K DMO

The subtle flaw in the above derivation is that reflection-point smear has not been completely taken into account.Reflection-point smear (Deregowski, 1982) means that theinput event at location in Figure 3 will be repositioned byDMO to the correct zero-offset location yo. Hale’s methodcorrectly repositions the event, although this fact is notobvious from equation (8) and requires analysis (Hale, 1984)of equations (l0)-(14) to establish. However, the appearanceof the uncorrected position rather than on the left-handside of equation (8) and in equation causes amplitudes ofdipping events to not be preserved, as we show in latersections.

where the connection between and t in the secondequality is obtained from equations (5a), (5b), and (6). Notethat equation (7) preserves peak amplitude rather thanspectral density and so is consistent with the convention ofthis paper.

The crucial step of Hale’s heuristic derivation is thefollowing definition of DMO:

where equations (4), (5b), (5c), and (6) define the mappingbetween and According to this definition, DMO mapseach sample of from time to time without changingits midpoint location, . The lack of midpoint change is thesubtle flaw in Hale’s argument.

Carrying on, equation (8) implicitly requires the quantitysin because of equation (4), but sin is more convenientlycomputed in Fourier space using the well-known identity fortime dip D:

2 sin k

Thus after a double Fourier transform, Hale derived F-KDMO to be:

Making these modifications, the new heuristic definition ofDMO to replace equation (8) is

where equation is changed to

and the connection between and in Figure 3comes from equations (A-6) and (A-4) of Appendix A. Thesecond equality in equation (16) comes from equation (5b),which is still valid. Note the correct form of the relationshipbetween and in equation (16) as opposed to Hale’srelationship in equation (14a). In addition to the connectionbetween and equation (15) requires the relationshipbetween and in Figure 3:

- -

where the first equality comes from andequations (A-ll), (A-6), and (A-4). The second equalitycomes from equations (9) and (5b).

Performing the double Fourier transform as in equation(11) but now using equation (15), we find the new amplitude-preserving DMO integral:

52 Black et al.

where JT is the Jacobian, expressed as the determinant of a2 2 matrix:

I

(19)

obtained by straightforward differentiation of equations (16)and (17), and is the same phase shift as in equation (13).

Comparing equations (18) and (19) with Hale’s DMO, wesee that the phase shifts are identical but that the Jacobiansdiffer by a factor of

22A 1 1 +

JH 1 +

Since this ratio is always larger than 1, our modified F-KDMO always produces outputs that are slightly larger thanthose from Hale’s DMO, as we will confirm in a latersection. We will also show there that our modified DMO isamplitude preserving for all dips, whereas Hale’s method isstrictly amplitude-preserving only for horizontal reflectors.

AMPLITUDE PRESERVATION FOR A DIPPING REFLECTOR

Having just argued for modifying Hale’s well-known F-Kmethod of DMO, we now begin a second attack on theproblem of amplitudes. This will lead to a less heuristicderivation of the same amplitude-preserving DMO. Thissecond approach will also establish that the entire standardprocessing sequence is true-amplitude when it containsamplitude-preserving DMO.

Plan of attack

Figure 1 describes the plan of attack. We begin in threedimensions with an arbitrary dipping reflector of knownreflectivity where is shown in Figure 3. We then usethe acoustic wave equation with deconvolved zero-phasesource wavelet w(t) to generate the seismic response to thisreflector, h). Next we analytically “process”

h) through the standard sequence shown in Figure 1. Weperform all of the processes except 3-D DMO in absolutelystandard fashion, as detailed below. We do the DMO,however, with an extra degree of freedom, in the form of acorrection filter. We will choose this filter S to make the finalmigrated output have amplitude globally proportional to

Specifically, we will make the choice for S that causesDMO to preserve the peak amplitude of an arbitrary event.In fact, with this choice for the DMO correction filter, wewill show that each stage of the standard processing se-quence preserves the peak amplitude of every event. Thusour procedure is a wave-equation calibration technique forconstant velocity. Successful calibration for a reflector witharbitrary dip makes the entire standard sequence amplitude-preserving, as we shall establish below.

For the purposes of this paper, we will keep the offsetvector h fixed at a constant value and guarantee that P isadequately sampled in space and time. In other words, wewill analyze adequately sampled common-offset, common-azimuth subsets of the function P, since we are not con-cerned here with the (important) questions of spatial aliasingand missing input data.

3-D wave-equation response for a dipping reflector

We now begin the wave-equation calibration procedureshown in Figure 1. From the 3-D wave equation, it is known(Aki and Richards, 1980) that the seismic data collected overa 3-D dipping planar reflector is given in high-frequency limitby

=

where w(t) is the fully deconvolved zero-phase sourcewavelet whose peak amplitude is w(O), v is the constantvelocity, and is the sum of the distances from thesource to the reflection point, and then to the receiver, asshown in Figure 3. Note that Figure 3 is still valid in threedimensions since it is a diagram in the plane containing thesource, receiver, and reflection point. As noted earlier, depends upon h as well as but we will suppress the hargument for the sake of concise notation.

Application of spherical divergence and NM0 corrections

We now begin “processing” h) through thestandard sequence with the goal of producing a migratedimage having the true-amplitude property of equation (2).The first step is to apply a constant-velocity sphericaldivergence correction of the standard form:

t h) h) =

Note that equation (22) yields a whose peak amplitude isglobally proportional to with proportionality constantw(0):

Thus we will achieve our goal of true-amplitude processing ifwe guarantee that each subsequent stage of the standardprocessing sequence does not change the peak amplitude onthe section. This is, in fact, what we will do in what follows.

Before proceeding to the NM0 correction, we need tointroduce one additional approximation that will simplify thesubsequent analysis without compromising amplitude pres-ervation. Since the seismic data is assumed to have beendeconvolved prior to entering the standard sequence, thewavelet is zero outside a narrow range of times(50 ms or so) centered at Thus in equation (22), wecan set the ratio to unity without serious effect exceptat the shallowest section times. Even at shallow times, thistype of approximation becomes increasingly accurate as welet t approach to study the peak amplitude behavior of

True-amplitude Imaging and Dip Moveout 53

interest in this work.the approximation:

Thus we will replace equation (22) by

(24)The next step is to apply normal moveout (NMO) using

the velocity v and the offset magnitude = The peak-amplitude-preserving form of this correction takes the formalready seen in equation (7):

To evaluate this, we use Levin’s (1971) 3-D generalization ofequation (4), which relates to the zero-offsettwo-way traveltime at the midpoint location yn:

(26)

where D is the time dip vector defining the slope of

We also introduce the NMO-correction of

Finally we are ready to substitute equation (24) into equation(25) and to use equation (28) to eliminate in favor of :

(29)

(30)

To obtain the second (approximate) equality in equation(29), we employ the same argument used to obtain equation(24), namely that w(t) is sharply peaked near t = 0. Thus thepeak of w occurs when the two square roots are equal, whichoccurs when In the vicinity of this point, it isvalid to approximate the difference of the two square rootsby the first-order term of a Taylor series in powers of

and the first derivative of this difference is thequantity .

Equation (29) says that consists of the reflectivityconvolved with a “stretched” wavelet centered on the time

Note that the wavelet has been stretched by theNMO-stretch factor but that its peak amplitude has notbeen affected. This means that equation (29) exhibits a peakamplitude proportional to with global proportionalityconstant w(O), just as in equation (23). So far, there havebeen no surprises.

Application of integral 3-D DMO

The next, crucial step in the processing of the syntheticdata is to apply 3-D constant-velocity DMO. From earlierwork (Deregowski and Rocca, 1981; Hale, 1983; Hale, 1984,pp. 67-71; Beasley et al., 1988a), we know the form of theintegral 3-D DMO operation required to make the travel-

times in PO be correct: PO must be a line integral along theshot-receiver direction of a filter S applied to a time-mappingof P,. This takes the general form:

=

where the line integral along the shot-receiverparametrized by the coordinate

with

direction is

(32)

Note that the time mapping in equation (31) is the familiar“DMO smile” (Deregowski and Rocca, 1981), shown inFigure 4 and equation (A-l):

is the notation we have chosen to emphasize the similarity to= in equation (16) for F-K DMO. Also note that the

occurrence of in equation (3 1) means that wemap the time samples of prior to applying the filter

What we do not know yet is the correct filter S to be

applied to the data in equation (31). It is this filter that we willdetermine by requiring peak-amplitude preservation on anarbitrary dipping reflector. First we make the assumption (tobe verified later) that we can write S as the product of ato-dependent weight function and a to-independentfilter

= (35a)

Next we introduce a scaled version of PO:

=

Putting equations (35a) and (35b) into equation (31) yields thedouble integral we need to perform:

= dt’

The next step in our calibration is to carry out the doubleintegral in equation (35c), using the dipping-reflector data defined by equation (29). The integral over t is most easilyperformed by Fourier-transforming II0 :

= (36)

We now combine equations (29), (32), (35c), and (36) toaccomplish the t’ integration (using the convolution theo-rem), yielding

54 Black et al.

Equation (37) is very general (Black and Wason, 1989) andcould be carried out numerically, with the integral replacedby a spatial summation, to calibrate DMO amplitudes evenin situations where spatial sampling is marginal (Hale, 1991).However, with the aim of deriving a simple analytic result,we have chosen to evaluate equation (37) by the method ofstationary phase, which is strictly valid only when spatialaliasing is not a serious issue and when offsets are largecompared with where Ax is the CDP interval.This stationary-phase evaluation of equation (37) is carriedout in Appendix B and relies on the kinematic identitiesderived in Appendix A. The result is

and is the value of a($ at the stationary point,given by equation (A-4):

= (40)

where is the 3-D time dip projected onto theshot-receiver axis. Note that is equal to Hale’squantity A, defined in equation (14), when we use the correct3-D form of the dip D l in place of D.

Now comes a key step. Inspection of equations (35b), (38),and (39) suggests the following choices for and

FIG. 4. DMO impulse response superimposed on the traveltime curves in Table 1, for the purpose of carrying out acommon-tangent construction for DMO.

True-amplitude Imaging and Dip Moveout 55

where we have affixed the suffix “T” on and a to denote“true-amplitude.” Examining equation (39), it is clear thatwe have chosen to compensate for the factor of

that appears in G and that we have chosen tocompensate for the remaining factors in G.

Substituting equations (41b) and (39) into equation (38),Fourier-transforming back to the time domain,and then using equations (35b) and (41a) to get yields

where the approximate equality uses a conditionthat is very well enforced by the short time extent of thedeconvolved w(t).

Equation (42) means that PO consists of the reflectioncoefficient centered at the correct zero-offset time and con-volved with the zero-phase source wavelet stretched by thefactor Closer examination confirms that thechoices we made in equations (41a) and (41b) have producedan amplitude-preserving 3-D DMO! For, as in equation (23),the peak amplitude of PO is proportional to with globalproportionality constant :

This, then, is the justification for the choices of and thatwe made in equations (41a) and (41b) and for the form of Sthat we assumed in equation (35a).

Having now confirmed that equations (41a) and (41b) givethe desired behavior of amplitude-preserving DMO, we willuse equation (35a) to find the time-domain filter in the form that is directly applicable to our definition ofintegral DMO in equation (31). This is easily done by aninverse Fourier transform of

defines the “half-derivative” operator, since is theFourier transform of the first-derivative operator, using oursign convention. Note that equations (31) and (44) say toconvolve the mapped with the time-reverse of the halfderivative operator.

Application of 3-D zero-offset migration

The final step in processing our dipping reflector throughthe standard sequence is to apply 3-D zero-offset wave-equation migration to the data set that wasderived in equation (42). We will abide by the conventional“exploding reflector” methodology (Loewenthal et al.,1976) in which the velocity is halved and the seismic energyis assumed to have propagated in one direction only, fromreflectors to receivers. It is natural to have some doubtsabout amplitude preservation under these assumptions,which are ubiquitous in production seismic migration today.Since we have assumed that the velocity is constant, we canemploy the 3-D Stolt migration formula (Stolt, 1978) toobtain the migrated image

In Appendix D, we show that when from equation (42) issubstituted into these equations and equation (27) for isemployed, the result is

Note that the cumulative stretch factor is given by

The inverse Fourier transform of equation (48) leadsfinal time-domain expression for the migrated image:

Equation (52) is the final confirmation that the standardprocessing sequence is true-amplitude! This equation statesthat Pm consists of the reflection coefficient centered atthe correct migated time and convolved with thezero-phase deconvolved source wavelet stretched by thefactor Note that equation (52) confirms that the imagewavelet is of the form shown in equation (3). Finally, itshould come as no surprise by now to observe that the peakamplitude of Pm is proportional to with global propor-tionality constant w( 0) :

56 Black et al.

With equation (53), we have attained our goal of estab-lishing that the standard sequence, when it contains theform of integral DMO defined by equation (44), produces animage whose peak amplitude is proportional to the reflectioncoefficient, with global proportionality constant w(0).

SUMMARY OF DMO METHODS

In this section, we shall summarize the forms that severalimportant DMO algorithms take. Our first task is to establishthe equivalence of our amplitude-preserving F-K and inte-gral DMO equations, which were derived by completely dif-ferent methods in the previous two sections. For this we beginwith the true-amplitude F-K Jacobian JT(A) of equation (19)and substitute it into equation (C-14) from Appendix C. Thislatter equation gives the integral-method filter ST that corre-sponds to JT(A) by the prescription:

It is very pleasing to note that equation (54) is identical toequation (44)! Thus, at least within the limits implied by ourstationary-phase analysis, our F-K and integral DMO meth-ods are equivalent.

Armed with the confidence that we can asymptoticallyrelate F-K and integral methods by equation (C-14), we arenow in a position to summarize the equations defining threesignificant DMO methods in addition to ours. One of these isHale’s (1984) algorithm whose Jacobian JH is given byequation ( 12).

The second DMO method is a very prevalent distillation ofthe integral method of Deregowski and Rocca (1981) andDeregowski (1985, 1986, 1987):

Although Deregowski and Rocca (1981) experimented withvarious filters, steep-dip cutoffs, and weight factors, thecompensating filter of equation (55) was implied by theirFourier analysis of the large-offset response of the integralDMO operator in Section 11, pp. 397-8 of their paper.Subsequently (Deregowski 1985, 1986, 1987) equation (55)became the starting point for widespread implementation ofintegral DMO methods. Thus we will henceforth refer toequation (55) as “Deregowski-Rocca (DR) DMO.” Equation(55) is a generic form that has often been supplemented byvarious weighting factors and cutoffs, many of which havebeen ad hoc and most of which have been unpublished.Since it is impractical to treat such variations here, we willstick to the simple generic form in equation (55) for compar-ison purposes.

The third method of interest is the F-K DMO algorithmderived from Born inversion and zero-offset modeling byLiner (1989) and Bleistein (1990):

= (56)

which is to be inserted into equation (18).

Table 2 summarizes the defining equations for true-ampli-tude DMO (this work), Hale DMO, DR DMO, and Bleistein-Liner DMO, viewing each as both an F-K and an integralmethod. In representing each technique as an F-K method,we have listed its Jacobian, appropriate for insertion intoequation (18). In representing each technique as an integralmethod, we have listed its filter scaled by thecommon factor of and appropriate for insertioninto equation (31). For example we have used equation(C-14) to convert Hale’s F-K method into the correspondingintegral technique. Likewise we have converted theDeregowski-Rocca integral DMO into an F-K technique. Itis convenient to show how the F-K operators depend uponHale’s quantity A defined in equations (14) and (40) and toshow how the integral operators depend on the “smilefunction” a(q) defined in equation (34). Of course A anda($ are equal to each other at the stationary point, as isshown by equation (40).

Before leaving this section, we will establish the connec-tion between Bleistein-Liner DMO and our DMO. Eventhough the two methods obviously differ by a factor of A2

[or equivalently each method is correct in its owncontext. The context of Bleistein-Liner DMO differs fromour context in two important respects:

1) Their imaging philosophy is to preserve the spectraldensity of the image wavelet rather than its peakamplitude.

2) The input and output to their DMO decay with spheri-cal-divergence factors of the form and respec-tively, as opposed to our processing sequence in whichthe loss is removed prior to DMO.

To see that these two differences account for the factor ofA2, we rewrite A2 as follows:

where the last equalityis obtainedwith the help of equations(30) and (A-6). We recognize the first term as the wavelet-stretch factor that is present following theapplication of NM0 and DMO in equation (42). Thus multi-plication by changes from an NMO-DMO system thatpreserves peak amplitudes to one that preserves spectraldensity, as we mentioned in the discussion that followsequation (3). Furthermore, inspection of equation (21) showsthat the second factor in equation (57) is the ratio of thespherical-divergence term on the zero-offset data to the spherical-divergence term on the original data Thus Bleistein-Liner’s DMO operator is com-pletely consistent with ours!

We conclude that there is more than one way to define‘‘true-amplitude DMO,” depending upon which aspect ofthe image wavelet is to be preserved and upon the processingsequence that surrounds DMO. It is not easy, however, tosimilarly interpret that Hale DMO and DR DMO are consis-tent with our DMO and Bleistein-Liner DMO.

True-amplitude imaging and dip moveout 57

CONFIRMATION WITH MODEL DATA

In this section we confirm our theoretical predictions ofthe behavior of various DMO algorithms in preservingamplitudes on dipping model data. Figure 5 shows a series ofevents with dips ranging from 0 to 80 degrees by 20-degreeincrements. Each event is from equation (21) with the samewavelet and value of independent of The top panel,labeled “NMO,” shows the result of applying spherical-divergence and NM0 corrections, and represents the inputto the subsequent DMO panels. The next three panels showthe outputs F-K DMO based upon three of the methods inTable 2. We did not include Bleistein-Liner DMO in thesetests because it is designed for a processing sequence withno spherical-divergence correction. The preservation ofpeak amplitude in true-amplitude (correct) DMO is apparent,as is the loss of amplitude at steep dips in the Hale DMO andDR DMO. Also noticeable on the data is the fact thatwavelet stretch decreases with increasing time and with

FIELD-DATA RESULTS

FIG. 5. Processing of synthetic dipping-reflector data throughthe NM0 and DMO steps of the standard sequence. Param-eters for the data are v = 2000 m/s and offset = 2000 m. Thepanel labelled “NMO” has had only the spherical-diver-gence and NM0 corrections applied. The other panels showDeregowski-Rocca (DR), Hale, and true-amplitude (correct)DMO outputs, all the result of the F-K implementationsshown in Table 2.

increasing dip, consistent with the predicted stretch factor

Figure 6(a) shows the peak amplitudes picked from Figure 5plotted versus dip angle. The plot has been normalized byw(O), the peak of the source wavelet. According to ourtrue-amplitude result in equation (42), the normalized plotshould have a value of unity, independent of dip. It isapparent that our amplitude-preserving DMO algorithmcomes very close to this value. On the other hand, themeasured peak amplitudes of the other two methods fallincreasingly short of equation (42) as dip increases, with theDR DMO amplitude being somewhat weaker than Hale’sDMO. The solid curves in this figure are predictions basedupon Table 2. For example, the prediction for true-amplitudeDMO is unity, and the prediction for Hale DMO is The close agreement between the plotted points and thecurves in Figure 6(a) validates the entire dipping-reflectoranalysis.

Figure 6(b) was derived in the same way as Figure 6(a),except that dip was held constant at 60 degrees and offsetwas varied. This set of three curves illustrates that thedifferences in amplitude response among true, Hale, andDeregowski-Rocca DMO methods increase rapidly withincreasing offset.

Figure 7 is a stack section from a marine survey in the Gulfof Mexico. The steeply dipping salt flanks have been nicelyimaged with the help of the true-amplitude integral DMOmethod described in this paper. Although there are observ-able differences at the stack stage between true-amplitudeDMO and other algorithms, the most dramatic effects areseen on prestack data, as shown in Figure 8. Figure 8contains four panels, each of which is a moved-out CDPgather at CDP 2620 on the right side of the salt dome inFigure 7. The leftmost panel, with NM0 applied but noDMO applied, clearly shows the kinematic problem thatDMO solves. As expected, the steeply dipping event atabout 1.8 seconds is severely overcorrected by the NM0operation. The second panel from the left, which follows theDeregowski-Rocca method given in equation (55) shows howDMO successfully corrects the kinematics on this event.However, the amplitudes are not quite right, as is seen bycomparing with the third panel, which was generated by theintegral formulation of the true-amplitude method describedin equations (31) and (44). It is also instructive to lookat the rightmost panel, showing the difference betweenDeregowski-Rocca and true-amplitude DMO and displayedat the same gain as the other panels. There, it is apparent

58 Black et al.

that true-amplitude DMO produces a stronger image at largeoffsets. This is consistent with what our analysis predictsand with what we saw on the model data in the previoussection.

CONCLUSION

We have analyzed what it takes to make constant-velocityDMO algorithms preserve amplitudes. We have arrived atthe same scheme in two different ways: an F-K analysisbased upon Hale’s (1984) work and a 3-D integral analysis ofthe amplitudes of a dipping reflector. As summarized inTable 2, our prescription is a relatively simple change toother F-K or integral DMO techniques and requires onlythat a modified Jacobian or correction filter be applied to thedata. Applying our prescription to model and field data setsshows a systematic increase in output amplitude with diprelative to the Hale and Deregowski-Rocca algorithms and isconsistent with results derived by Liner and Bleistein. Thetrue-amplitude DMO formulation leads to benefits in notonly the final migrated image for steeply dipping events butalso in the usefulness of DMO-processed amplitudes foramplitude-versus-offset studies.

However, our analysis and prescription are not the com-plete answer to amplitude preservation in DMO processing.In this work we have not, for example, addressed thequestions of adequate sampling of integral DMO operators toprevent the operator aliasing often evident on horizontaldata (Hale, 1991) and the artifacts often observed at smalloffsets and deep times (Beasley et al., 1988b). We have alsorestricted our analysis to the case of constant velocity. Fornonconstant velocity, each stage of the processing sequencein Figure 1 is generally modified to take into account theinhomogeneous velocity field, with the possible exception ofthe DMO step. We believe that our dipping-reflector calibra-tion concept can be applied to such a processing sequence toderive amplitude-preserving operations, but we have notdone so here. Nor have we addressed the important question

of amplitude preservation in the presence of missing inputdata (Black and Schleicher, 1989). Since amplitude-preserv-ing DMO leads to larger amplitudes along the flanks of theDMO “smile” operator than either> Hale DMO orDeregowski-Rocca DMO, we expect such operator samplingproblems to require even more attention when amplitude-preserving DMO is being applied. But full attention to theseissues will have to be addressed elsewhere.

We have established that the concept of true-amplitudeDMO is ruled by context. It makes no sense to talk oftrue-amplitude DMO without first specifying the processingsequence of which it is a part. Thus we have seen that ourDMO operator is correct when a spherical-divergence cor-rection is made prior to DMO whereas the Bleistein-Linerapproach is correct when no such spherical-divergencecorrection is made.

Likewise, the concept of true-amplitude DMO is ruled byconvention. We have observed that the zero-phase imagewavelet has a bandwidth that depends on dip, depth, andshooting geometry. If correct processing causes every eventto have a different wavelet, what do we mean by “amplitude-preserving?” To answer that question here, we have adoptedthe intuitively appealing convention that the peak amplitude ofthe image wavelet should be preserved during processing. Thismeans that the peak of each event will be globally proportionalto its reflectivity on a correctly-processed section. This con-vention has led to the form of our DMO operator, but it is notthe only reasonable convention.Bleistein-Liner andDeregowski et al. (1990) advocate a convention in which thespectral density (e.g., the peak amplitude in the Fourier trans-form) of the image wavelet is preserved in processing. Thisleads to different amplitude-preserving DMO operators,equally as good as ours. The dipping-reflector analysis we havepresented here easily enables us to analyze such DMO meth-ods derived with different processing sequences and imagingconventions in mind. At the very least, we can then tell theinterpreter what he or she is getting.

FIG. 6. Plotted points are peak amplitudes extracted from the processed synthetic data leading to Figure 5. Amplitudes arenormalized by w(O). Solid curves are theoretical predictions based on Table 2. Curves (a) have a constant offset of 2000 m andshow the variation with dip. Curves (b) have a constant dip of 60 degrees and show the variation with offset. In these curves,the circles are for true-amplitude (correct) DMO, the squares for Hale DMO, and the triangles for Deregowski-Rocca DMO.

True-amplitude Imaging and Dip Moveout 59

FIG. 7. Stack section after application the true-amplitude integral DMO step of the standard processing sequence. The data isfrom a marine survey in the Gulf of Mexico. The CDP interval is 41 ft.

60 Black et al.

FIG. 8. Comparison of CDP gathers at CDP 2620 of the dataset of Figure 7. Panel (a) has had only spherical divergence and NM0applied. Panel (b) was processed with Deregowski-Rocca integral DMO. Panel (c) was processed with true-amplitude integralDMO (this work). Panel (d) is the difference between true-amplitude DMO and Deregowski-Rocca DMO, displayed at the samegain as the other panels. In these gathers, the near offset is 863 ft, and the far offset is 10,623 ft.

True-amplitude Imaging and Dip Moveout 61

ACKNOWLEDGMENTS

We wish to thank Cam Wason for valuable comments onour formulation of the calibration technique and for pointingout the importance of the wavelet-stretch issue. We alsowish to thank Mark Egan for arranging the processing of thefield data example. We benefited from careful readings of themanuscript by Paul Fowler and Matt Brzostowski. Thequality of the manuscript was improved by excellent com-ments made during the refereeing process by Dave Hale,Norm Bleistein, and Mark Portney.

REFERENCES

Abramowitz, M., and Stegun, I. A., 1965, Handbook of mathemat-ical functions: Dover Publ., Inc., 295-330.

Aki, K., and Richards, P. G.,1980, Quantitative seismology:Theory and methods (W. H. Freeman & Co.), Section 6.2,200-213.

Beasley, C., Chambers, R., and Jakubowicz, H., 1988a, A methodof processing seismic data: U.S. Patent Number 4,742,497.

Beasley, C. J., and Mobley, E., 1988b, Amplitude and antialiasingtreatment in (x-t) domain DMO: 58th Ann. Internat. Mtg., Soc.Expl. Geophys., Expanded Abstracts, 1113-l116.

Berg, L. E., 1984, Application of dip moveout by Fourier transform:Method overview and presentation of processed data from 2-Dand 3-D surveys: 54th Ann. Internat. Mtg., Soc. Expl. Geophys.,Expanded Abstracts, 796-799.

Black, J. L., and Egan, M. S., 1988, True amplitude DMO in 3-D:58th Ann. Intemat. Mtg., Soc. Expl. Geophys., Expanded Ab-stracts, 1109-l112.

Black, J. L., and Schleicher, K. L., 1989, Effect of irregularsampling on prestack DMO: 59th Ann. Intemat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, 1144-l147.

Black, J. L., and Wason, C. B., 1989, Method of true amplitudedip-moveout correction: U.S. Patent No. 4878204.

Bleistein, N., 1990, Born DMO revisited: 60th Ann. Internat. Mtg.,Soc. Expl. Geophys., Expanded Abstracts, 1366-1369.

Derogowski, S. M., 1982, Dip-moveout and reflector-point dis-persal: Geophys. Prosp., 30, 318-322.

- 1985, An integral method of dip moveout: Presented at the47th Ann. EAEG Mtg.

1986, What is DMO?: First Break, 4, 7-24.- 1987, An integral method of dip moveout: Geophys. Trans.,

33, 1l-22.Deregowski, S. M., Raynaud, B. A., Redshaw, T. C., and Sum-

mers, T. P., 1990, DMO algorithms: A review: Presented at the52nd Ann. EAEG Mtg.

Deregowski, S. M., and Rocca, F., 1981, Geometrical optics andwave theory of constant offset sections in layered media: Geoph.Prosp., 29, 374-406.

Hale, I. D., 1983, Dip moveout by Fourier transform: Ph.D. thesis,Stanford University.

- 1984, Dip moveout by Fourier transform: Geophysics, 49,741-757.

- 1991, A nonaliased integral method for dip moveout: Geo-physics, 56, 795-805.

Liner, C. L., 1989, Mapping reflection seismic data to zero offset:Ph.D. thesis, Center for Wave Phenomena CWP-081, ColoradoSchool of Mines.

Levin, F. K., 1971, Apparent velocity from dipping interfaces:Geophysics, 36, 510-516.

Loewenthal, D., Lu, L., Robertson, R., and Sherwood, J., 1976,The wave equation applied to migration: Geophys. Prosp., 24,380-399.

Messiah, A., 1968, Quantum Mechanics: John Wiley & Sons, Inc.,471-472.

Morse, P. M., and Feshbach, H., 1953, Methods of theoreticalphysics: McGraw-Hill Book Co., 437-441.

Stolt, R. H., 1978, Migration by Fourier transform: Geophysics, 43,23-48.

Wu, R. S., and Toksöz, M. N., 1987, Diffraction tomography andmultisource holography applied to seismic imaging: Geophysics,52, l-11.

Yilmaz, O., and Claerbout, J. F., 1980, Prestack partial migration:Geophysics, 45, 1753-1779.

Zhang, L., 1988, A new Jacobian for DMO: Stanford Explo. Proj.Rep. No. 59, 201-208.

APPENDIX AKINEMATIC IDENTITIES FOR DMO

The purpose of this Appendix is to establish some impor-tant kinematic DMO identities that will simplify the evalua-tion of the stationary phase integral in Appendix B. Thesekinematic identities are summarized in Table 3, and couldhave been derived by a lengthy analysis of the raypaths inFigure 3. However, we choose to derive these identities bycarrying out the common-tangent construction for a planarevent with time dip along the shot-receiver axis. Figure 4plots the traveltime curves (cf. Table 1) for such an eventalong with the DMO impulse response described by thequantity defined from in equation (A-l) below.Singling out the point A on the input curve we knowthat the NM0 operation will move its energy to point B onthe curve We also know that DMO will move itsenergy a distance to some point C on the zero-offset curve

It is the purpose of this appendix to make these shiftsquantitative by computing the relationships among

We wil l do this bydemanding that the DMO impulse response be tangent to thezero-offset traveltime curve at point C.

The traveltime equations (26), (27), and (28) from the bodyof the paper are the basis of this Appendix, so we repeatthem here:

In addition, based upon the DMO kinematics of equations(33) and (34), we introduce a quantity shown in Figure 4,that describes the mapping of a sample B at time (y,) upalong DMO impulse response to an arbitrary output locationy:

where is the unitvector in the source-receiver direction, and is the dis-placement along the DMO ellipse in the el direction from ynto y.

We now begin the common-tangent construction to derivethe DMO shift We first require that the slope of theimpulse response equal the slope of the zero-offset curve

which is D1. This leads to the following condition:

62 Black et al.

where Multiplying both sides of this equation bythe denominator, squaring, and solving for with thecorrect sign yields:

For future use, the denominator can be rewritten usingequation (28):

Now we find the value of the “smile-function” at and call it Using equation (34) and equation (A-2), wefind

(A-4)

which corresponds to the F-K DMO result given in equation(14)

Having determined the value of at which the slopes ofthe impulse-response and zero-offset curves for a given dipin Figure 4 are equal, next we need to confirm that the twocurves do, in fact, intersect at point C. To do this, we need

Table 3. DMO kinematic identities and their equation refer-ences. Note that where A is Hale’s constant,defined by equation (14) in 2-D and by equation (40) in 3-D.

Kinematic IdentityEquation

Reference

to show that the value of the impulse-response mapping at the point

equals To do this, we first use equation (28) to rewriteequation (A-l) as:

Using (A-4) this becomes:

Finally solving (A-3) for Dl, substituting the results inequation (27), and using the previous result proves that thecurves intersect at :

( A - 5 )

Thus the point is indeed the point ofcommon tangency in Figure 4.

Having now established the basic relations in Figure 4, weproceed to derive the additional seven identities(A-6)-(A-12) in Table 3. Equations (A-l) and (A-5the first additional identity:

For the next two identities, we return to (A-2)(A-4) to produce

Then can be eliminated using (A-6), yielding:

isted aslead to

(A-6)

and use

Rearrangement produces the desired quadratic-equationidentity:

(A-7)

which can be solved to yield the next identity:

. ( A - 8 )

Identity (A-9) can be proven starting with equation (28)and using equation (27) to produce:

True-amplitude Imaging and Dip Moveout 63

The term in parenthesis vanishes by equation (A-7), leaving: Finally we derive the last identity (A-12) from (A-4) and

In this Appendix, we will evaluate equation (37) using thestationary phase approximation, which is valid at largevalues of the quantity To this end, we rewrite equation(37) as:

APPENDIX B

STATIONARY-PHASE EVALUATION OF INTEGRAL

The method of stationary phase (or steepest descents)(Morse and Feshbach, 1953) approximates this integral as:

(B-3

where is the value of at the stationary point, where= 0. It will be shown shortly that, as should be

expected, from the kinematic analysis of Appendix A isthe point where vanishes.

The derivatives of and with respect to are required to compute the first derivative of Substi-

tuting (27) into (28) yields:

Differentiation of this expression produces:

Likewise, the derivative of is:

To show that 0, equation (B-7) can be evaluatedat using identities established in Appendix A. Substituting

and equation (A-9) into (B-7) and usingequation (34) produces:

The expression in parenthesis is zero by equation (A-7)completing the proof that is the stationary point:

Now the second derivative will be computed by differen-tiating equation (B-7):

This completes the evaluation of the second derivative.The next step is to evaluate using equations (B-2)

and (A-5):u s i n g :

64 Black et al.

Previous evaluation of has shown that the first term iszero. Continuing theanalysis on the second term only and

produces:

Equation (B-3) can now be computed by substituting thevalues of and from equation (B-8) and (B-9) andrecognizing the Fresnel integral (Abramowitz and Stegun,1965):

Equations (B-5) and (B-6) provide expressions for and that can be substituted in this equation to produce:

Equations (A-6), (A-10), and (A-l1) can be used to eliminate and thereby producing the result:

Substituting expression (A-12) for in this equation andrearranging terms yields:

(B-9)

Equations (B- 10) and (B- 11) are the desired results forthe evaluation of the DMO integral, as stated in equation (38)of the main text.

APPENDIX CCONNECTING F-K DMO WITH INTEGRAL DMO

In this Appendix we establish the mathematical connec-tion between F-K and integral implementations of DMO.This connection is required to check the consistency of ourF-K derivation with our more rigorous integral derivation.Likewise this connection is required to relate F-K andintegral results for other DMO algorithms in Table 2. ThisAppendix is thus a generalization of the work of Berg (1984).

We begin with the general expression for F-K DMO,following equation (18) of the main text:

where is an arbitrary Jacobiansuch as JT in equation (19). For notational simplicity in thisAppendix, we have suppressed the offset argument h thatappears in all quantities. In equation (C-l), we have used thequantity A, whose definition in equation (14) we repeat here:

Our goal is to connect equation (C-l) with the generalexpression for integral DMO, given by equation (31) of themain text:

where is given by equation (34) and The plan of this appendix is to start with equation (C-l)

matical relationship between J and S. We begin by inverse-

where a($ is given by equation (34) and y y The plan of this appendix is to start with equation (C-l)

and derive equation (C-3), thereby establishing the mathe-matical relationship between J and S. We begin by inverse-Fourier-transforming both sides of equation (C-l) with re-spect to and k to obtain

True-amplitude Imaging and Dip Moveout 65

Simple rearrangement and use of the definition of yields:

where we recognize that the term in curly brackets is veryclosely related to the filter S in equation (C-3) since

In fact, if we introduce the dummy variable t’ andmake the change of variables in equation(C-4), we find it strongly resembles equation (C-3) with thefollowing expression for S:

The resemblance would be complete if we could establishthat which we shall do below in equation (C-9).

Next we will use a stationary-phase approximation toevaluate the k integral in equation (C-4). The phase of thisintegral is:

(C-7)

where Differentiation with respect to k anduse of the definition of A(k) yields

y i e l d s

Setting this derivative to zero, using the definition of A(k),and solving for the value of k at the stationary point,yields

Substituting into the definition of A(k) yields the follow-ing expression for the value of A(k) at the stationary point:

This establishes the desired connection between A(k) anda(q) mentioned after equation (C-6). Inserting these values

of and into equation (C-7)the phase at the stationary point:

yields the expression for

The next step in the evaluation of the k-integral in equa-tion (C-5) is to evaluate the second derivative of withrespect to k. Using equations (C-2), (C-7), and (C-9), weobtain

We are now in a position to evaluate the k-integral inequation (C-5) using the stationary-phase method along withequations (C-10) and (C-lla). The result is

If we now examine the phase of the integral, it is clear thatS is very strongly peaked (&function behavior) when thefollowing condition is satisfied:

(C-12)

This is, of course, just the kinematic DMO mapping ofequation (33), as we would hope to obtain. Carrying out thesubstitution of equation (C-6) and using equation (C-9)

Finally, substituting equation (C-12) for under the squareroot gives

where d is the half-derivative operator defined inequation (45).

Equation (C-14) is the principal result of this Appendix. Itconverts any Jacobian for F-K DMO into thecorresponding filter for integral DMO. Animportant part of this conversion is equation (C-9), whichidentifies A(k) at the stationary point with

66 Black et al.

APPENDIX D

3-D ZERO-OFFSET STOLT MIGRATION DETAILS

In this Appendix, we derive equation (48), the result ofzero-offset 3-D constant-velocity migration. Beginning withequations (46)-(47), substituting from theFourier transform of equation (42), and using from equa-tion (27) yields

(D-1)

where

(D-2)

and

(D-3)

We first perform the double integral over by rotating thedummy vector variables k and so that each has onecomponent (e.g., along the dip direction e = D/D and theother component (e.g., along the orthogonal (strike)direction. This leads to

(D-4)

(D-5)

where we have renamed the dummy variable as K andhave defined the argument of the delta-function as thefunction

(D-6)

To evaluate the K-integral, we use the standard expression(Messiah, 1968) for an integral involving a delta-functionwhose argument is a function that has one zero:

(D-7)

where is the zero of Setting and solvingfor yields

(D-8)

where we have used the relationship between and given by equation (50). From equation (D-6), we also get theexpression for the first derivative

(D-9)

In deriving the above equation, note that we have used

(D-10)

Finally, using equations (D-7) through (D-10) to evaluatethe K integral in equation (D-5) gives the result

(D-l 1)

which simplifies easily to equation (48), the desired result.