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4 Migration• Introduction • Exploding Reflectors • Migration Strategies • Migration Algorithms • Migration Parameters • As-pects of Input data • Migration Velocities • Migration Principles • Kirchhoff Migration • Diffraction Summation• Amplitude and Phase Factors • Kirchhoff Summation • Finite-Difference Migration • Downward Continuation •Differencing Schemes • Rational Approximations for Implicit Schemes • Reverse Time Migration • Frequency-SpaceImplicit Schemes • Frequency-Space Explicit Schemes • Frequency-Wavenumber Migration • Phase-Shift Migra-tion • Stolt Migration • Summary of Domains of Migration Algorithms • Kirchhoff Migration in Practice •Aperture Width • Maximum Dip to Migrate • Velocity Errors • Finite-Difference Migration in Practice •Depth Step Size • Velocity Errors • Cascaded Migration • Reverse Time Migration • Frequency-Space Migra-

tion in Practice • Steep-Dip Implicit Methods • Depth Step Size • Velocity Errors • Steep-Dip Explicit Methods

•Dip Limits of Extrapolation Filters

•Velocity Errors

•Frequency-Wavenumber Migration in Practice

•Maximum Dip to Migrate • Depth Step Size • Velocity Errors • Stolt Stretch Factor • Wraparound • Residual Mi-gration • Further Aspects of Migration in Practice • Migration and Spatial Aliasing • Migration and RandomNoise • Migration and Line Length • Migration from Topography • Exercises • Appendix D: Mathematical

Foundation of Migration • Wavefield Extrapolation and Migration • Stationary Phase Approximations • TheParabolic Approximation • Frequency-Space Implicit Schemes • Stable Explicit Extrapolation • Optimum DepthStep • Frequency-Wavenumber Migration • Residual Migration • References

4.0 INTRODUCTION

Migration moves dipping reflections to their true sub-surface positions and collapses diffractions, thus increas-

ing spatial resolution and yielding a seismic image of the subsurface. Figure 4.0-1 shows a CMP-stacked sec-tion before and after migration. The stacked section in-dicates the presence of a salt dome flanked by gentlydipping strata. Figure 4.0-1 also shows a sketch of twoprominent features — the diffraction hyperbola D thatoriginates at the tip of the salt dome, and the reflectionB off the flank of the salt dome. After migration, notethat the diffraction has collapsed to its apex P and thedipping event has moved to a subsurface location A,

which is at or near the salt dome flank. In contrast, re-flections associated with the gently dipping strata have

moved little after migration.Figure 4.0-2 is an example with a different typeof structural feature. The stack contains a zone of near-horizontal reflections down to 1 s. After migration, theseevents are virtually unchanged. Note the prominent un-conformity that represents an ancient erosional surface

just below 1 s. On the stacked section, the unconfor-mity appears complex, while on the migrated section,it becomes interpretable. The bowties on the stackedsection are untied and turned into synclines on the mi-



464 Seismic Data Analysis

FIG. 4.0-1.A CMP stack (a) before, (b) after migration; (c) sketch of a prominent diffraction D and a dipping event before(B) and after (A) migration. Migration moves the dipping event B to its assumed true subsurface position A and collapses

the diffraction D to its apex P. The dotted line indicates the boundary of a salt dome.

FIG. 4.0-2. A CMP stack (a) before, (b) after migration.Migration unties the bowties on the stacked section andturns them into synclines (Taner and Koehler, 1977).

grated section. The deeper event in the neighborhoodof 3 s is the multiple associated with the unconformityabove. When treated as a primary and migrated withthe primary velocity, it is overmigrated.

Figure 4.0-3a shows a stacked section that con-tains fault-plane reflections conflicting with the shal-low gently-dipping reflections. Note the accurate posi-tioning of the fault planes and delineation of the faultblocks on the migrated section in Figure 4.0-3b. Fromthe three examples shown in Figures 4.0-1, 4.0-2, and4.0-3, note that migration moves dipping events in theupdip direction and collapses diffractions, thus enablingus to delineate faults while retaining horizontal eventsin their original positions.

The goal of migration is to make the stacked sec-tion appear similar to the geologic cross-section in depthalong a seismic traverse. The migrated section, however,commonly is displayed in time. One reason for this isthat velocity estimation based on seismic and other dataalways is limited in accuracy. Therefore, depth conver-sion is not completely accurate. Another reason is thatinterpreters prefer to evaluate the validity of migratedsections by comparing them to the unmigrated data.Therefore, it is preferable to have both sections dis-played in time. The migration process that producesa migrated time section is called time migration . Timemigration, the main theme of Chapter 4, is appropriateas long as lateral velocity variations are mild to moder-ate.

When the lateral velocity gradients are significant,time migration does not produce the true subsurface im-age. Instead, we need to use depth migration , the output



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FIG. 4.0-3. A CMP stack (a) before, (b) after migration. Migration collapses subtle diffractions associated with the growthfaults, moves the fault-plane reflections to the fault positions, and thus makes detailed structural interpretation easier.




FIG. 4.0-4. A CMP stack (a) before, (b) after time migration. Time migration is adequate for accurate imaging of thetop-salt boundary, whereas depth migration is imperative for accurate imaging of the base-salt boundary (B).



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of which is a depth section. Consider the data from anarea with intense salt tectonics in Figure 4.0-4. Timemigration has produced an acceptable image of the re-gion above the salt. However, note the crossing of eventsthat is a manifestation of overmigration of the reflection

associated with the base-salt boundary (denoted by B in Figure 4.0-4b). The improper migration is the resultof inadequate treatment by the time migration of theeffects of severe raypath bendings at the top-salt bound-ary caused by the strong velocity contrast between thesalt layer and the overlying rocks.

Complex structures associated with salt diapirism,overthrust tectonics and irregular water-bottom topog-raphy usually are three dimensional (3-D) in character.A stacked section really is the seismic response of a 3-D subsurface on a two-dimensional (2-D) plane of pro-file. Therefore, 2-D migration is not completely validfor 3-D data from areas with complex 3-D structures.

Figure 4.0-5a is an inline stacked section from a land3-D survey. Figure 4.0-5b is a 2-D migration of thissection, while Figure 4.0-5c is the same section after3-D migration of the entire 3-D survey data. In particu-lar, note the significant difference in the imaging of thetop of salt T and base of salt B . In 2-D migration, weassume that the stacked section does not contain anyenergy that comes from outside the plane of recording(sideswipe). Three-dimensional imaging of the subsur-face is discussed in Section 7.3.

Exploding Reflectors

When a stacked section is migrated, we use the migra-tion theory applicable to data recorded with a coin-cident source and receiver (zero-offset). To develop aconceptual framework for discussing migration of zero-offset data, we now examine two types of recordingschemes.

A zero-offset section is recorded by moving a sin-gle source and a single receiver along the line with noseparation between them (Figure 4.0-6). The recordedenergy follows raypaths that are normal incidence toreflecting interfaces. This recording geometry obviously

is not realizable in practice.Now consider an alternative geometry (Figure 4.0-6) that will produce the same seismic section. Imag-ine exploding sources that are located along the reflect-ing interfaces (Loewenthal et al., 1976). Also, considerone receiver located on the surface at each CMP lo-cation along the line. The sources explode in unisonand send out waves that propagate upward. The wavesare recorded by the receivers at the surface. The earthmodel described by this experiment is referred to as theexploding reflectors model .

FIG. 4.0-5. A 2-D CMP stack (a) represents a 2-D cross-section of a 3-D wavefield. Thus, it can contain energy fromoutside the plane of the 2-D line traverse. A 2-D migra-tion (b) is inadequate when this kind of energy is presenton the 2-D CMP-stacked section. (c) Clear imaging of thesalt structure requires both 3-D data collection and 3-D mi-gration (Section 7.3). (Data courtesy Nederlandse AardolieMaatschappij B.V.)




FIG. 4.0-6. Geometry of zero-offset recording (left), and hypothetical simulation of the zero-offset experiment using explodingreflectors (right) (Claerbout, 1985).

The seismic section that results from the explodingreflectors model is largely equivalent to the zero-offsetsection, with one important distinction. The zero-offsetsection is recorded as two-way traveltime (from sourceto reflection point to receiver), while the exploding re-flectors model is recorded as one-way traveltime (fromthe reflection point at which the source is located to

the receiver). To make the sections compatible, we canimagine that the velocity of propagation is half the truemedium velocity for the exploding reflectors model.

The equivalence between the zero-offset section andthe exploding reflectors model is not quite exact, par-ticularly in the presence of strong lateral velocity vari-ations (Kjartansson and Rocca, 1979).

These concepts now are applied to the velocity-depth model in Figure 4.0-7. Consider source-receiverpairs placed along the earth’s surface at every tenthmidpoint. In this case, a zero-offset section is mod-eled. At midpoint 130, five different arrivals are asso-ciated with rays that are normal incidence to the firstinterface. Alternatively, imagine receivers placed alongthe earth’s surface at every tenth midpoint and sourcesplaced along the interface where the rays emerge at theright angle to the interface (equivalent to the normal-incidence rays of the zero-offset section). In the lattercase, the velocities indicated in Figure 4.0-7 must behalved to match the time axis with that associated withthe zero-offset section.

The interface can be sampled more densely by plac-ing receivers and sources at closer spacing (Figure 4.0-8a). The next deeper interface can be modeled; that is,

FIG. 4.0-7. A velocity-depth model (top) and the zero-offset traveltime response (bottom) of the water-bottom re-flector. Shown also are the normal-incidence rays used tocompute the zero-offset traveltime trajectory. Note the fivearrivals A, B, C, D , and E at CMP 130 — all from the waterbottom.



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FIG. 4.0-8. Exploding-reflector modeling of zero-offset traveltimes associated with (a) a water bottom, (b) a flat, and (c)a dipping reflector. (d) The superposition of the normal-incidence traveltime responses in (a), (b), and (c). Shown on thevelocity-depth models in the left-hand column are the normal-incidence rays used to compute the traveltime trajectories.The time sections shown on the right-hand column are equivalent to a zero-offset traveltime section with the vertical axis intwo-way time.




the traveltime trajectory can be computed by placingsources along this interface and leaving the receiverswhere they were on the surface (Figure 4.0-8b). Finally,the same experiment can be repeated for the third inter-face (Figure 4.0-8c). To derive the composite response

from the velocity-depth model in Figure 4.0-8d (the left-hand column), individual responses shown in Figures4.0-8a, 4.0-8b and 4.0-8c from each interface are su-perimposed. The result is shown in Figure 4.0-8d (theright-hand column). We can imagine that sources wereplaced at all three interfaces and turned on simultane-ously. Such an experiment would cause the rays emerg-ing from the three interfaces to be recorded at receiversplaced on the surface, along the line.

Actually, Figure 4.0-8d (the right-hand column)represents the modeled zero-offset traveltime section.Seismic wavefields , however, are represented not onlyby wave traveltimes but also by wave amplitudes . Figure

4.0-9a shows the modeled zero-offset wavefield sectionbased on the same velocity-depth model in Figure 4.0-8d (the left-hand column). The shallow complex inter-face (horizon 1 in Figure 4.0-8a) caused the complicatedresponse of the two simple interfaces (horizons 2 and 3)in this zero-offset traveltime section.

How valid is the assumption that a stacked sec-tion is equivalent to a zero-offset section? The conven-tional CMP recording geometry provides the wavefieldat nonzero offsets. During processing, we collapse theoffset axis by stacking the data onto the midpoint-timeplane at zero offset. For CMP stacking, we normally as-sume hyperbolic moveout. Figure 4.0-10 shows selected

CMP gathers modeled from the velocity-depth modelin Figure 4.0-8d (the left-hand column). Because of thepresence of strong lateral velocity variations, the hy-perbolic assumption may not be appropriate for somereflections on some CMP gathers (Figure 4.0-10a); how-ever, it may be valid for others (Figure 4.0-10b). We ob-tain a stacked section (Figure 4.0-9b) that resembles thezero-offset section (Figure 4.0-9a) to the extent that thehyperbolic moveout assumption is valid. The assump-tion that a conventional stacked section is equivalent toa zero-offset section also is violated to varying degreesin the presence of strong multiples and conflicting dipswith different stacking velocities (Chapter 5). While mi-

gration of unstacked data is discussed in Chapter 5, ourmain focus in this chapter is on migration after stack.

Migration Strategies

In practice, migration of seismic data requires decisionmaking with regards to:

(a) an appropriate migration strategy,

(b) a migration algorithm compatible with the strat-egy,

(c) appropriate parameters for the algorithm,(d) issues concerning the input data, and(e) migration velocities.

Migration strategies include:

(a) 2-D versus 3-D migration,(b) post- versus prestack migration, and(c) time versus depth migration.

The spectrum of migration strategies extend from 2-Dpoststack time migration to 3-D prestack depth migra-tion. Depending on the nature of the subsurface geol-ogy, any other in-between combination can be selected.In practice, 2-D/3-D poststack time migration is used

most often for a good reason — it is the least sensitiveto velocity errors, and it often yields results acceptablefor a reliable interpretation. Table 4-1 is an overview of different migration strategies applied to different typesof seismic data (2-D, 3-D, stacked, and unstacked).

Choice of an appropriate migration strategy re-quires input from the interpreter as to the structuralgeology and stratigraphy in an area. Dipping events on astacked section call for time migration. Conflicting dipswith different stacking velocities is one case in which aconventional stacked section differs from a zero-offsetsection. Thus, strictly speaking, poststack migration

Table 4-1. Migration strategies.

Case Migration

dipping events time migration

conflicting dips with prestack migrationdifferent stacking velocities

3-D behavior of 3-D migrationfault planes and salt flanks

Case Migration

strong lateral velocity depth migrationvariations associated withcomplex overburden structures

complex nonhyperbolic moveout prestack migration

3-D structures 3-D migration



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FIG. 4.0-9. (a) The zero-offset wavefield section equivalentto the zero-offset traveltime section in Figure 4.0-8d (theright-hand column); (b) the CMP stack generated from theCMP gathers as in Figure 4.0-10. (Modeling by Deregowskiand Barley, 1981.)

which assumes that the stacked section is equivalent toa zero-offset section is not valid to handle the case of conflicting dips. Instead, one needs to do prestack timemigration.

Conflicting dips often are associated with saltflanks and fault planes, which can have 3-D characteris-tics. This then requires 3-D prestack time migration. InSection 5.3, we shall discuss a practical alternative to2-D prestack time migration strategies. The alternativesequence includes the application of normal-moveout(NMO) correction using velocities appropriate for flatevents followed by 2-D dip moveout correction (DMO)to correct for the dip and source-receiver azimuth ef-fects on stacking velocities. As a result, conflicting dipsare preserved during stacking, and thus, imaging can bedeferred until after stacking using 2-D poststack timemigration strategies. This series of processing steps islargely equivalent to 2-D prestack time migration andresults often are comparable. The same workflow also isapplicable to 3-D prestack time migration (Section 7.4).

Accurate imaging of targets beneath complexstructures with strong lateral velocity variations re-quires depth migration. Aside from the problem of con-flicting dips with different stacking velocities, strong lat-eral velocity variations associated with complex over-burden structures usually cause conventional stackingbased on the hyperbolic moveout assumption to fail.

Therefore, a case of complex overburden structures callsfor depth migration before stacking the data.

Furthermore, complex overburden structures, en-countered in areas with salt tectonics, overthrust tec-tonics and irregular water-bottom topographies can

often exhibit 3-D characteristics. Thus, imaging suchstructures may require 3-D prestack depth migration.

Field surveys are designed such that line orienta-tions are, as much as possible, along the dominant strikeand dip directions, so as to minimize 3-D effects. Un-der these circumstances, the 2-D assumption for migra-tion can be acceptable. However, if the subsurface hasa truly 3-D geometry, without a dominant dip or strikedirection, then it is imperative to do 3-D migration of 3-D data. In such cases, 2-D migration (whether post-stack or prestack, time or depth) can lead to potentialproblems in interpretation.

A practical alternative to 2-D prestack depth mi-

gration can be a prestack layer replacement to cor-rect for the complex nonhyperbolic moveout followedby time migration after stack. This, however, is appli-cable to situations involving a single overburden layer,such as irregular water-bottom topography for it to bereasonably practical.

Migration Algorithms

The one-way-in-depth scalar wave equation is the ba-sis for common migration algorithms. These algorithms

do not explicitly model multiple reflections, convertedwaves, surface waves, or noise. Any such energy presentin data input to migration is treated as primary re-flections. Migration algorithms can be classified underthree main categories:

(a) those that are based on the integral solution to thescalar wave equation,

(b) those that are based on the finite-difference solu-tions, and

(c) those that are based on frequency-wavenumber im-plementations.

Whatever the algorithm, it should desirably:

(a) handle steep dips with sufficient accuracy,(b) handle lateral and vertical velocity variations, and(c) be implemented, efficiently.

Figure 4.0-11 is a migrated CMP stacked sectionwith a major unconformity. The undermigration — in-complete imaging of the unconformity, is not because of




FIG. 4.0-10. Selected CMP gathers modeled from the velocity-depth model in Figure 4.0-8d (the left-hand column). (a)Gathers from the complex part of the velocity-depth model, (b) gathers from the simpler part of the velocity-depth model.CMP locations are indicated in Figure 4.0-8d. The CMP stack is shown in Figure 4.0-9b. (Modeling by Deregowski and Barley,1981.)

erroneously too low velocities. Although we should al-ways be aware of velocity errors when migrating seismicdata, the undermigration in Figure 4.0-11 is the resultof using a dip-limited algorithm. By using a steep-dipalgorithm, we can achieve a more accurate imaging of the unconformity (Figure I-9).

The three principle migration techniques are dis-cussed in this chapter in their historical order of devel-opment as outlined below. The first migration techniquedeveloped was the semicircle superposition method thatwas used before the age of computers. Then came

the diffraction-summation technique, which is based onsumming the seismic amplitudes along a diffraction hy-perbola whose curvature is governed by the medium ve-locity. The Kirchhoff summation technique introducedlater (Schneider, 1978), but actually in use earlier, basi-cally is the same as the diffraction summation techniquewith added amplitude and phase corrections applied tothe data before summation. These corrections make thesummation consistent with the wave equation in thatthey account for spherical spreading (Section 1.4), the

obliquity factor (angle-dependency of amplitudes), andthe phase shift inherent in Huygens’ secondary sources(Section 4.1).

Another migration technique (Claerbout and Do-herty, 1972) is based on the idea that a stacked sectioncan be modeled as an upcoming zero-offset wavefieldgenerated by exploding reflectors. Using the explod-ing reflectors model, migration can be conceptualizedas consisting of wavefield extrapolation in the form of downward continuation followed by imaging. To under-stand imaging, consider the shape of a wavefield at ob-

servation time t = 0 generated by an exploding reflec-tor. Since no time has elapsed and, thus, no propagationhas occurred, the wavefront shape must be the same asthe reflector shape that generated the wavefront. Thefact that the wavefront shape at t = 0 corresponds tothe reflector shape is called the imaging principle . Todefine the reflector geometry from a wavefield recordedat the surface, we only need to extrapolate the wave-field back in depth then monitor the energy arriving att = 0. The reflector shape at any particular extrapola-



Migration 473

FIG. 4.0-11. A CMP stack (a) before, and (b) after migration. Note the undermigration of the unconformity event (U)caused by the use of a dip-limited algorithm.

tion depth directly corresponds to the wavefront shape

at t = 0.Downward continuation of wavefields can be im-

plemented conveniently using finite-difference solutionsto the scalar wave equation. Migration methods basedon such implementations are called finite-difference mi-gration. Many different differencing schemes applied tothe differential operators in the scalar wave equationexist both in time-space and frequency-space domains.Claerbout (1985) provides a comprehensive theoreticalfoundation of finite-difference migration and its practi-cal aspects.

After the developments on Kirchhoff summationand finite-difference migrations, Stolt (1978) introduced

migration by Fourier transform. This method involvesa coordinate transformation from frequency (the trans-form variable associated with the input time axis) tovertical wavenumber axis (the transform variable asso-ciated with the output depth axis), while keeping thehorizontal wavenumber unchanged. The Stolt methodis based on a constant-velocity assumption. However,Stolt modified his method by introducing stretchingin the time direction to handle the types of velocityvariations for which time migration is acceptable. Stolt

and Benson (1986) combine theory with practice in the

field of migration with an emphasis on the frequency-wavenumber methods.

Another frequency-wavenumber migration is thephase-shift method (Gazdag, 1978). This method isbased on the equivalence of downward continuation to aphase shift in the f requency-wavenumber domain. Theimaging principle is invoked by summing over the fre-quency components of the extrapolated wavefield ateach depth step to obtain the image at t = 0.

A reason for the wide range of migration algorithmsused in the industry today is that none of the algorithmsfully meets the important criteria of handling all dipsand velocity variations while still being cost-effective.

Migration algorithms based on the integral solu-tion to the scalar wave equation, commonly known asKirchhoff migration, can handle all dips up to 90 de-grees, but they can be cumbersome in handling lateralvelocity variations.

Finite-difference algorithms can handle all types of velocity variations, but they have different degrees of dip approximations. Furthermore, differencing schemes,if carelessly designed, can severely degrade the intendeddip approximation.




FIG. 4.0-12. A CMP stack (a) before, and (b) after migration. Lack of any event to the right of the dotted line on themigrated section is a result of the finite line length.

Finally, frequency-wavenumber algorithms havelimited ability in handling velocity variations, partic-ularly in the lateral direction. As a result of limi-tations of the three main categories of migration al-gorithms — integral, finite-difference, and frequency-wavenumber methods, migration software has expandedfurther to additional extensions and combinations of thebasic algorithms. Residual migration — phase-shift orconstant-velocity Stolt migration followed by the appli-cation of a dip-limited algorithm is one example.

Migration Parameters

After deciding on the migration strategy and the ap-propriate algorithm, the analyst then needs to decideon the migration parameters. Migration aperture width

is the critical parameter in Kirchhoff migration. A smallaperture causes removal of steep dips; it generates spu-rious horizontal events and organizes the random noiseuncorrelated from trace to trace.

Depth step size in downward continuation is thecritical parameter in finite-difference methods. An opti-

mum depth step size is the largest depth step with theminimum tolerable phase errors. It depends on tempo-ral and spatial samplings, dip, velocity, and frequency.It also depends on the type of differencing scheme usedin the algorithm.

Finally, the stretch factor is the critical parameterfor Stolt migration. A constant-velocity medium impliesa stretch factor of 1. In general, the larger the verticalvelocity gradient, the smaller the stretch factor needsto be.



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Aspects of Input Data

When migrating seismic data, one needs to be con-cerned with various aspects of the input data set itself:

(a) line length or areal extent,(b) signal-to-noise ratio, and(c) spatial aliasing.

The line length must be sufficient to allow a steeplydipping event to migrate to its true subsurface location.It is a fatal error to record short profiles in areas withcomplex geology. Also, for 3-D migration, the surfaceareal extent of a 3-D survey is almost always larger thanthe target subsurface areal extent.

Random noise at late times on a stacked section,when migrated, potentially can be hazardous for shal-lower data. One may have to compromise on migration

aperture for deep data to prevent this problem to occur.Trace spacing must be small enough to prevent

spatial aliasing of steep dips at high frequencies. Al-though this is not an issue for modern prestack data, acoarse shot-receiver spacing can degrade the fidelity of prestack migration severely. Old data and 3-D marinedata in the crossline direction often are trace interpo-lated prior to poststack migration.

Figure 4.0-12 is a CMP stacked section before andafter migration. From an interpretation viewpoint, thereliable part of this migrated section is confined to theupper central part. Lack of any reflection energy to theright of the dotted line does not mean that there is a

structural discontinuity there. It only means that thereflections associated with the imbricate structure havebeen migrated in the updip direction from right to left.As a result, a zone with no reflectors to the right of thedotted line is left behind because of the truncation of the wavefield represented by the right-hand edge of thestacked section. The deeper part is useless, because thenoise dominates the section.

Migration Velocities

Horizontal displacement during migration is propor-tional to migration velocity squared (equation 4-1).Since velocities generally increase with depth, errors inmigration are usually larger for deep events than shal-low events. Also, the steeper the dip, the more accuratethe migration velocities need to be, since displacementis proportional to dip.

Figure 4.0-13 shows a portion of a CMP-stackedsection after 3-D poststack time migration using a per-cent range of stacking velocities. Note the subtle under-and overmigration effects on dipping events below the

major unconformity represented by the strong, near-horizontal reflection. Events A dipping up to the leftand B dipping up to the right cross over one anotheron sections that correspond to 90 percent and 95 per-cent of stacking velocities, indicating undermigration.

The same events are split away from one another inopposite directions on sections that correspond to 105percent and 110 percent of stacking velocities. The mostoverall acceptable image is seen on the section that cor-responds to the 100 percent of stacking velocities.

Accuracy in event positioning after migration actu-ally depends on the combined effects of the performanceof the migration algorithm used and the velocity errors.For example, the inherently undermigrating characterof a 45-degree finite-difference algorithm can be, for anevent with a specific dip, coincidentally counterbalancedby the overmigration effect of erroneously too high ve-locities. In the presence of large vertical velocity gradi-ents, a two-pass 3-D migration can also cause overmi-gration of steep dips (in the form of lateral translation)even with the correct migration velocities.

Figure 4.0-14a shows a portion of a migratedstacked section. Although this section does not containsteep dips, accurate imaging of the faults along the low-relief structures can be important to the interpreter.Note the slight undermigration, which may be causedby any of the following: (a) error in migration veloci-ties, (b) a dip-limited algorithm that failed to focus thediffraction energy adequately, or (c) a possible 3-D be-havior of the geometry of the reflector. The section in

Figure 4.0-14a has been migrated with a dip-limited al-gorithm. Using a proper algorithm, with the same veloc-ities, we get the migrated section in Figure 4.0-14b. Theresulting section shows slight overmigration, which canbe attributed to errors in migration velocities. Loweringthe velocities gives the improved, but not completely ac-curate, image in Figure 4.0-14c. Perhaps, the remainingissues in imaging may be attributed to 3-D effects.

As demonstrated by the example in Figure 4.0-14,migration results generally are self-evident — under-and overmigration often can be recognized on a mi-grated section. Problems in imaging often are traced toaccuracy in migration velocities. I consider migrationvelocities as the weak link between the seismic sectionand the geologic cross-section.

In the next section, basic principles of migrationare presented and the Kirchhoff summation, finite-difference, frequency-space, and frequency-wavenumberalgorithms are reviewed. Practical aspects of the migra-tion algorithms are expounded in Sections 4.2 through4.5. Specifically, key parameters for each category of the migration algorithms are analyzed using appropri-ate synthetic and field data examples. Further aspects of




FIG. 4.0-13. A portion of a CMP-stacked section after 3-D poststack time migration using, from top to bottom, 90,95, 100, 105 and 110 percent of stacking velocities. Note thesubtle under- and overmigration effects on dipping eventsbelow the major unconformity represented by the strong,near-horizontal reflection. Note the effect of velocities usedin migration on the positioning of the event A dipping upto the left and event B dipping up to the right.

migration in practice, including spatial aliasing, migra-tion response to random noise, line length, and irregulartopography are discussed in Section 4.6. The problemof conflicting dips with different stacking velocities thatrequires dip-moveout (DMO) correction and prestacktime migration, and the accompanying topic on migra-tion velocity analysis are deferred until Chapter 5. Theproblem of imaging beneath complex structures that re-quires earth imaging and modeling in depth is discussedin Chapters 8 and 9, respectively.

FIG. 4.0-14. (a) A portion of a migrated CMP stack; notethe subtle undermigration at fault locations A and B causedby the use of a dip-limited algorithm; (b) same data set butmigrated with an algorithm with no dip limitation; notethe subtle overmigration most likely due to erroneously toohigh velocities; (c) same data set migrated with the same

algorithm as in (b) but with velocities adjusted to preventovermigration.

4.1 MIGRATION PRINCIPLES

Consider the dipping reflector CD of the simple ge-ologic section in Figure 4.1-1a. We want to obtain azero-offset section along the profile Ox . As we move thesource-receiver pair (s, g) along Ox , the first normal-incidence arrival from the dipping reflector is recorded

at location A. In this discussion, we assume a normal-ized constant-velocity medium v = 1 so that time anddepth coordinates become interchangeable. The reflec-tion arrival at location A is indicated by point C

on thezero-offset time section in Figure 4.1-1b. As we movefrom location A to the right, normal-incidence arrivalsare recorded from the dipping reflector CD . The lastarrival is recorded at location B, which is indicated bypoint D

in Figure 4.1-1b. In this experiment, diffrac-tions off the edges of reflector CD are excluded to sim-plify the discussion.



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FIG. 4.1-1. Migration principles: The reflection segment

C

D

in the time section as in (b), when migrated, is movedupdip, steepened, shortened, and mapped onto its true sub-surface location CD as in (a). (Adapted from Chun andJacewitz, 1981.)

Compare the geologic section in Figure 4.1-1a,which is in depth, with the zero-offset seismic sectionin Figure 4.1-1b, which is in time. The true subsurfaceposition of reflector CD is superimposed onto the timesection for comparison. Clearly the true geologic posi-tion of reflector CD is not the same as the reflectionevent position C D.

From this simple geometric construction, note thatthe reflection in the time section C D must be migrated

to its true subsurface position CD in the depth section.The following observations can be made from the geo-metric description of migration in Figure 4.1-1:

(a) The dip angle of the reflector in the geologic sectionis greater than in the time section; thus, migrationsteepens reflectors.

(b) The length of the reflector, as seen in the geologicsection, is shorter than in the time section; thus,migration shortens reflectors.

(c) Migration moves reflectors in the updip direction.

The example in Figure 4.0-1 demonstrates theabove observations. In particular, the dipping event (B)has moved in the updip direction, become shorter, and

steepened after migration (A).As mentioned in the previous section, conventional

migration output is displayed in time, as is the inputstacked section. To distinguish the two time axes, wewill denote the time axis on the stacked section as t— event time in the unmigrated position, and the timeaxis on the migrated section as τ — event time in themigrated position.

We shall now examine the horizontal and verti-cal displacements as seen on the migrated time section.From Figure 4.1-2, consider a reflector segment CD . As-sume that CD migrates to C D and that point E onC D migrates to point E on CD. The horizontal andvertical (time) displacements — dx and dt, and the dip∆τ /∆x, all measured on the migrated time section (Fig-ure 4.1-2), can be expressed in terms of medium velocityv, traveltime t, and apparent dip ∆t/∆x as measuredon the unmigrated time section (Figure 4.1-2). Chunand Jacewitz (1981) derived the following formulas:

dx =v2 t

4

∆t

∆x, (4 − 1)

dt = t

1 −

1 −

v∆t

2∆x

2, (4 − 2)

∆τ

∆x=

∆t

∆x

1 1 −

v∆t

2∆x

2

. (4 − 3)

To gain a quantitative insight into these expres-sions, we consider a numerical example. For a realis-tic velocity function that increases with depth, considerfive reflecting segments at various depths. For simplic-ity, assume that quantity ∆t/∆x is the same for all (10ms per 25-m trace spacing). From equations (4-1), (4-2), and (4-3), compute the horizontal and vertical dis-placements dx and dt and the dips (in ms/trace) after

migration. The results are summarized in Table 4-2.Refer to Table 4-2 and equations (4-1), (4-2), and

(4.3) and make the following observations:

(a) The time dip ∆τ /∆x on the migrated section isalways greater than the time dip ∆t/∆x on theunmigrated section.

(b) The horizontal displacement dx increases withevent time t in the unmigrated position. At 4 s,the horizontal displacement is more than 6 km.




FIG. 4.1-2. Quantitative analysis of the migration process. Dipping event AB on the unmigrated section (left) is moved toAB on the migrated section (right). The event after migration also is superimposed on the unmigrated section to compare theposition of the event before and after migration. Point C on dipping reflector AB is moved to C after migration. The amountof horizontal displacement dx, vertical displacement dt, and the dip ∆τ/∆x after migration is calculated from equations (4-1),(4-2) and (4-3).

Table 4-2. Horizontal and vertical displacements as aresult of migration of a series of dipping reflections with

the same apparent dip (10 ms/trace) as measured onthe unmigrated stacked section at various depths, andchanges in dip angle as measured on the migrated sec-tion in time.

t v dx dt ∆t/∆x ∆τ /∆x(s) (m/s) (m) (s) (ms/trace) (ms/trace)

1 2500 625 0.134 10 11.52 3000 1800 0.400 10 12.53 3500 3675 0.858 10 14.04 4000 6400 1.600 10 16.75 4500 10125 2.820 10 23.0

(c) The horizontal displacement dx is a function of thevelocity squared. If there is a 20 percent error inthe velocity used in migration, then the event ismisplaced by an error of 44 percent.

(d) The vertical displacement dt also increases withtime and velocity.

(e) The steeper the event dip, the more the horizontaland vertical displacements after migration.

In Figure 4.1-1a, assume that the zero-offset sec-tion was recorded only between surface locations A and

B. The time section would include the event C

D

, butwhen migrated, the event would migrate out of the sec-tion, resulting in a blank migrated section (Figure 4.1-1b). Therefore, the data on a stacked section are notnecessarily confined to the subsurface below the seismicline. The converse is even more important; the structurebelow the seismic line may not be recorded on the seis-mic section. Suppose that the data were recorded onlybetween surface locations O and A. This time, the re-sulting time section would be blank. So, we should notrecord between A and B, and neither should we recordbetween O and A. Instead, we should record between Oand B in order to record the reflector of interest prop-

erly and also to migrate it properly.In areas with a structural dip, line length must be

chosen by considering the horizontal displacements of dipping events from the structures causing the events.This is an important consideration, especially in 3-Dseismic work. The areal surface coverage of a surveyusually is larger than the areal subsurface coverage of interest.

To achieve a complete image of a dipping reflec-tor, also, the recording time must be long enough. For




FIG. 4.1-5. Curved reflecting interfaces (synclines and an-ticlines) (a) before and (b) after migration. See text for de-tails. (Modeling courtesy Union Oil Company.)

FIG. 4.1-6. (a) A velocity-depth model consisting of a syn-clinal reflector; (b) selected normal-incidence arrivals on thezero-offset section. Trace the bowtie in the time section.

example, if only OE seconds were recorded (Figure 4.1-1), then the recorded segment C D would yield onlypart of the complete image CD . An excellent example of recording deeper in time and with longer line length forsteeper dips is shown in Figure 4.0-1. Proper imaging of the salt dome boundary required that data be recordedfor more than 6 s.

The migration concepts described above aredemonstrated further by the dipping events model in

Figure 4.1-3. The edge diffractions are included here.The dipping reflectors on the zero-offset section aresteepened, shortened, and moved in the updip direc-tion as a result of migration. A field data example of aseries of dipping events on a stacked section before andafter migration is shown in Figure 4.1-4. Note that thesteeper the dip, the more the event moves after migra-tion.

So far, only linear reflectors were considered. Wenow consider a more realistic geologic situation thatinvolves curved reflecting interfaces. Figure 4.1-5 shows



Migration 481

FIG. 4.1-7. A portion of a CMP stack (a) before and (b) after migration. Anticlines appear bigger, while synclines appearsmaller than their actual sizes on the unmigrated section (a).

three synclines and a small anticlinal feature. The syn-clines appear as bowties on the zero-offset section. Byusing the principles deduced from the geometry of Fig-ure 4.1-1, note that as a result of migration, segmentA of the bow tie moves in the updip direction to the

left. Similarly, segment B moves to the right, while flat-topped segment C does not move much at all. Conse-quently, after migration the flanks of bow ties associatedwith synclines are opened up. On the other hand, thesmall anticline seems to be broader on the zero-offsetsection than it is on the migrated section. Again notethat segment D moves updip to the right, while segmentE moves updip to the left as a result of migration. Thus,synclines broaden and anticlines compress as a result of migration. Migration velocities also affect the apparentsize of the structure; higher velocities mean more mi-gration and, hence, smaller anticlinal structure.

Why does a syncline look like a bowtie on thestacked section? The answer is in Figure 4.1-6, wherea symmetric syncline is modeled. Given the subsurfacemodel in Figure 4.1-6a, the normal-incidence rays canbe computed to derive the zero-offset traveltime sectionin Figure 4.1-6b. Only five CMP locations are shownfor clarity. At locations 2 and 4, there are two distinctarrivals, while at location 3, there are three distinct ar-rivals. By filling in the intermediate raypaths, the bowtie character of the syncline can be constructed on the

time section. Complete the procedure by tracing thetraveltime trajectory in Figure 4.1-6b.

Two field data examples containing synclinal andanticlinal structures are shown in Figures 4.1-7 and4.1-8. In Figure 4.1-7, note that the synclinal feature

broadens and the anticlinal feature narrows as a resultof migration. In Figure 4.1-8, the bow ties associatedwith two small synclinal basins A and B grow larger indepth. After migration, the bowties are untied and thesynclines are delineated.

Kirchhoff Migration

Claerbout (1985) uses the harbor example in Figure 4.1-9 to describe the physical principles of migration. As-sume that a storm barrier exists at some distance z3from the beach and that there is a gap in the barrier.

Imagine a calm afternoon breeze that comes from theocean as a plane incident wave. The wavefront is par-allel to the storm barrier. As we walk along the beachline, we see a wavefront different from a plane wave.The gap on the storm barrier has acted as a secondarysource and generated the semicircular wavefront that ispropagating toward the beach.

If we did not know about the storm barrier and thegap, we might want to lay out a receiver cable along thebeach to record the approaching waves. This experiment




FIG. 4.1-8. A portion of a CMP stack (a) before and (b) after migration. Migration unties the bowties and turns them intosynclines below A and B .

is illustrated in Figure 4.1-10 with the recorded timesection. Physicists call the gap on the barrier a point aperture . It is somewhat similar to a point source , sinceboth generate circular wavefronts. However, the ampli-tudes on the wavefront that propagate outward froma point source are isotropic, while those from a pointaperture are angle-dependent. The point aperture onthe barrier acts as a Huygens’ secondary source .

From the beach experiment, we find that Huygens’secondary source responds to a plane incident wave andgenerates a semicircular wavefront in the x − z plane.The response in the x− t plane is the diffraction hyper-bola shown in Figure 4.1-11.

Imagine that the subsurface consists of points alongeach reflecting horizon that behave much as the gapon the storm barrier. From Figure 4.1-12, these pointsact as Huygens’ secondary sources and produce hyper-bolic traveltime trajectories. Moreover, as the sources

(the points on the reflecting interface) get closer to eachother, superposition of the hyperbolas produces the re-sponse of the actual reflecting interface (Figure 4.1-13).In terms of the harbor example, this is like assumingthat the barrier is wiped out by a storm so that theprimary incident plane wave reaches the beach with-out modification. The diffraction hyperbolas, which arecaused by sharp discontinuities at both ends of the re-flector in Figure 4.1-13, remain. These hyperbolas areequivalent to diffractions seen at fault boundaries onstacked sections.

In summary, we find that reflectors in the subsur-face can be visualized as being made up of many pointsthat act as Huygens’ secondary sources. We also findthat the zero-offset section consists of a superpositionof the many hyperbolic traveltime responses. Moreover,when there are discontinuities (faults) along the reflec-tor, diffraction hyperbolas often stand out.



Migration 483

FIG. 4.1-9. The gap in the barrier acts as Huygens’ sec-

ondary source, causing the circular wavefronts that approach

the beach line. (Adapted from Claerbout, 1985.)

FIG. 4.1-10. Waves recorded along the beach generated by

Huygens’ secondary source (the gap in the barrier in Figure

4.1-9) have a hyperbolic traveltime trajectory.

FIG. 4.1-11. A point that represents a Huygens’ secondarysource (a) produces a diffraction hyperbola on the zero-offsettime section (b). The vertical axis in this section is two-waytime, while the vertical axis in the time section in Figure4.1-10 is one-way time.

FIG. 4.1-12. Superposition of the zero-offset responses (b)of a discrete number of Huygens’ secondary sources as in(a).

FIG. 4.1-13. Superposition of the zero-offset responses (b)of a continuum of Huygens’ secondary sources as in (a).




Diffraction Summation

Huygens’ secondary source signature is a semicircle inthe x − z plane and a hyperbola in the x − t plane.This characterization of point sources in the subsurface

leads to two practical migration schemes. Figure 4.1-14a shows a zero-offset section that consists of a singlearrival at a single trace. This event migrates to a semi-circle (Figure 4.1-14b). From Figure 4.1-14, note thatthe zero-offset section recorded over a constant-velocityearth model consisting of a semicircular reflecting inter-face contains a single blip of energy at a single trace asin Figure 4.1-14a. Since this recorded section consists of an impulse, the migrated section in Figure 4.1-14b canbe called the migration impulse response . An alternatescheme for migration results from the observation thata zero-offset section consisting of a single diffraction hy-perbola migrates to a single point (Figure 4.1-15b).

The first method of migration is based on the su-perposition of semicircles, while the second method isbased on the summation of amplitudes along hyperbolicpaths. The first method was used before the age of dig-ital computers. The second method, which is known asthe diffraction summation method , was the first com-puter implementation of migration.

The migration scheme based on the semicircle su-perposition consists of mapping the amplitude at a sam-ple in the input x− t plane of the unmigrated time sec-tion onto a semicircle in the output x − z plane. Themigrated section is formed as a result of the superposi-

tion of the many semicircles.The migration scheme based on diffraction sum-

mation consists of searching the input data in the x− tplane for energy that would have resulted if a diffract-ing source (Huygens’ secondary source) were located ata particular point in the output x−z plane. This searchis carried out by summing the amplitudes in the x − tplane along the diffraction curve that corresponds toHuygens’ secondary source at each point in the x − zplane. The result of this summation then is mappedonto the corresponding point in the x − z plane. Asnoted early in this section, within the context of timemigration, however, the summation result actually ismapped onto the x− τ plane, where τ is the event timein the migrated position.

The curvature of the hyperbolic trajectory for am-plitude summation is governed by the velocity function.The equation for this trajectory can be derived from thegeometry of Figure 4.1-15. A formal derivation also isprovided in Section D.2. Assuming a horizontally lay-ered velocity-depth model, the velocity function usedto compute the traveltime trajectory is the rms veloc-ity at the apex of the hyperbola at time τ (Section 3.1).

FIG. 4.1-14. Principles of migration based on semicirclesuperposition. (a) Zero-offset section (trace interval, 25 m;constant velocity, 2500 m/s), (b) migration.

FIG. 4.1-15. Principles of migration based on diffractionsummation. (a) Zero-offset section (trace interval, 25 m; con-stant velocity, 2500 m/s), (b) migration. The amplitude atinput trace location B along the flank of the traveltime hy-perbola is mapped onto output trace location A at the apexof the hyperbola by equation (4-4).



Migration 485

From the triangle CO A in Figure 4.1-15a, we note that

t2 = τ 2 +4x2

v2rms

. (4 − 4)

Having computed the input time t, the amplitude

at input location B is placed on the output section atlocation A, corresponding to the output time τ at theapex of the hyperbola.

From Section 3.1, reflection traveltimes in a layeredearth approximate small-spread hyperbolas. This mayseem to impose a serious restriction on the aperturewidth — the lateral extent of the diffraction hyperbola,in the summation process. However, the small-spreadapproximation is valid even at large distances from theapex, and the errors associated with it are insignificantat late times. In practice, this approximation is not usu-ally an issue.

Amplitude and Phase Factors

Now consider several factors associated with the am-plitude and phase behavior of the waveform along thediffraction hyperbola. From Figure 4.1-9, given the al-ternative of standing at location A or B, we intuitivelythink that it is safer to stand at location B. This isbecause the wave amplitude at location A, which is onthe z-axis, is stronger than the wave amplitude at loca-tion B, which is at an oblique angle from the z-axis. Asmentioned earlier, this is one difference between a pointsource with uniform amplitude response at all anglesand the point aperture that produces a wavefront with

angle-dependent amplitudes. This angle dependence of amplitudes, which is described by the obliquity factor ,should be considered before summation. To correct forthe obliquity factor, the amplitude at location B in Fig-ure 4.1-15a is scaled by the cosine of the angle betweenBC and CA before it is placed at output location A.

Another factor is the spherical divergence of waveamplitudes. Again, from Figure 4.1-9, given the alterna-tive of standing at location B or C , we prefer to stand atlocation C . The reason for this is that the wave ampli-tude along the wavefront at location C , which is fartherfrom the point aperture source, is weaker than the wave

amplitude at location B. Wave energy decays as (1/r

2

),where r is the distance from the source to the wavefront,while amplitudes decay as (1/r). Thus, amplitudes mustbe scaled by factor (1/r) before summation for wavepropagation in three dimensions.

Finally, there is a third factor that involves theinherent property of Huygens’ secondary source wave-form. This factor is difficult to explain from a physicalviewpoint. Nevertheless, it is obvious from Figure 4.1-13 that Huygens’ secondary sources must respond as awavelet along the hyperbolic paths with a unique phase

and frequency characteristic. Otherwise, there would beno amplitude cancelation when they are close to one an-other. The waveform that results from the summationmust be restored in both phase and amplitude.

In summary, we must consider the following three

factors before diffraction summation:

(a) The obliquity factor or the directivity factor, whichdescribes the angle dependence of amplitudes andis given by the cosine of the angle between the di-rection of propagation and the vertical axis z (Fig-ure 4.1-15).

(b) The spherical spreading factor, which is propor-tional to

1/vr for 2-D wave propagation, and

(1/vr) for 3-D wave propagation.(c) The wavelet shaping factor, which is designed with

a 45-degree constant phase spectrum and an am-plitude spectrum proportional to the square root of

the frequency for 2-D migration. For 3-D migration,the phase shift is 90 degrees and the amplitude isproportional to frequency.

Kirchhoff Summation

The diffraction summation that incorporates the obliq-uity, spherical spreading and wavelet shaping factorsis called the Kirchhoff summation, and the migrationmethod based on this summation is called the Kirchhoff migration. To perform this method, multiply the inputdata by the obliquity and spherical spreading factors.Then apply the filter with the above specifications andsum along the hyperbolic path that is defined by equa-tion (4-4). Place the result on the migrated section attime τ corresponding to the apex of the hyperbola. Inpractice, the order of the filter application, specified byfactor (c), and summation can be interchanged withoutsacrificing accuracy because the summation is a linearprocess and the filter is independent of time and space.

The velocity used in equation (4-4) is taken as therms velocity, which can be allowed to vary laterally.However, lateral variation in velocity distorts the hy-perbolic nature of the diffraction pattern and somehowmust be considered. The value for the rms velocity typi-

cally is that of the output time sample; that is, the apextime τ of the hyperbola.What was determined from a physical point of view

in the preceding discussion can be rigorously describedby the integral solution to the scalar wave equation.Schneider (1978), Berryhill (1979) and Berkhout (1980)are excellent references for the mathematical treatmentof the Kirchhoff migration method. The integral solu-tion of the scalar wave equation yields three terms; the

far-field term which is proportional to (1/r), and twoother terms which are proportional to (1/r2). Hence,




it is the far-field term that makes the most contribu-tion to the summation that is used in practical imple-mentation of Kirchhoff migration. The output imageP out(x0, z = vτ /2, t = 0) at a subsurface location (x0, z)using only the far-field term is computed from the 2-D

zero-offset wavefield P in(x, z = 0, t), which is measuredat the surface (z = 0), by the following summation overa spatial aperture

P out =∆x

2π

x

cos θ

√ vrmsr

ρ(t) ∗ P in

, (4 − 5)

where vrms is the rms velocity at the output point(x0, z) and r =

(x − x0)2 + z2, which is the distance

between the input (x, z = 0) and the output (x0, z)points. The asterisk denotes convolution of the rho fil-ter ρ(t) with the input wavefield P in.

The rho filter ρ(t) corresponds to the time deriva-tive of the measured wavefield, which yields the 90-

degree phase shift and adjustment of the amplitudespectrum by the ramp function ω of frequency (Ta-ble A-1 of Appendix A). For 2-D migration, the half-derivative of the wavefield is used. This is equivalentto the 45-degree phase shift and adjustment of the am-plitude spectrum by a function of frequency defined as√

ω. Since the rho filter is independent of the spatialvariables, it actually can be applied to the output of thesummation in equation (4-5). Finally, the far-field termin equation (4-5) is proportional to the cosine of the an-gle of propagation (the directivity term or the obliquityfactor) and is inversely proportional to vr (the sphericalspreading term) in three dimensions. In two dimensions,

the spherical spreading term is √ vr.Equation (4-5) can be used to compute the wave-

field at any depth z. The ouput image P out is computedat (x0, z = vτ /2, t = 0) using the input wavefield P inat (x, z = 0, t− r/v). To obtain the migrated section atan output time τ , equation (4-5) must be evaluated atz = vτ /2 and the imaging principle must be invoked bymapping amplitudes of the resulting wavefield at t = 0onto the migrated section at output time τ . The com-plete migrated section is obtained by performing thesummation in equation (4-5) and setting t = 0 for eachoutput location. The range of the summation is calledthe migration aperture .

Finite-Difference Migration

To describe the physical basis of finite-difference migra-tion, recall the harbor example of Figure 4.1-9. Insteadof taking the section recorded along the beach, whichcontains the diffraction hyperbola, then collapsing it toget the migrated section in Figure 4.1-15, consider thefollowing alternative procedure. Again, start with the

wavefield recorded along the beach (Figure 4.1-16a). As-sume that the barrier is 1250 m from the beach. Nowmove the recording cable into the water, 250 m from thebeach. Start recording at the instant the plane wave hitsthe barrier. The recorded section is shown in Figure 4.1-

16b. Move the cable 500 m from the beach and recordthe section in Figure 4.1-16c, followed by a recording750 m from the beach to obtain the section in Figure4.1-16d. Finally, 1000 m from the beach, record the sec-tion shown in Figure 4.1-16e.

Note that each recording yields a hyperbola inwhich the apex moves closer to zero time. The actualextent of the recording cable is denoted by the solid lineon top of each frame. Had we recorded at the barrier(1250 m from the beach), the apex of the hyperbolawould be positioned at t = 0.

In Kirchhoff migration, the diffraction hyperbolais collapsed by summing the amplitudes, then placing

them at the apex. The alternative approach impliedby the result of the experiment shown in Figure 4.1-16 is to use the hyperbola recorded a distance awayfrom the beach to construct the hyperbola that wouldbe recorded at another distance closer to the source of the diffraction hyperbola. The process is stopped whenthe hyperbola collapses to its apex. In the harbor ex-periment, this collapse occurs when the receiver cablecoincides with the barrier, or, equivalently, when t = 0.As stated in the introductory section, this is called theimaging principle .

Downward Continuation

The harbor experiment described above can be simu-lated in the computer. Pretend that moving the receivercable from the beach into the water closer to the barrieris like moving the receiver cable from the surface downinto the earth closer to the reflectors. Think of the gapon the barrier as equivalent to a point diffractor on a re-flecting interface causing the diffraction hyperbola (Fig-ure 4.1-17a). Start with the wavefield recorded at thesurface and move the receivers down to depth levels at

finite intervals. Downward continuation of the upcom-ing wavefield at the surface, therefore, can be consideredequivalent to lowering the receivers into the earth.

The computer-simulated wavefields at these differ-ent depths are shown in Figure 4.1-17. By applying theimaging principle at each depth, the entire wavefield isimaged. The final output from this process is the mi-grated section. The last section (panel f ) at 1250 mhas only one arrival at t = 0. The recording cable ison the storm barrier and the arrival from the gap oc-curs at t = 0. As the cable moved into the ocean and



Migration 487

FIG. 4.1-16. Moving the receiver cable in the harbor experiment (Figure 4.1-9) from the beach into the water at discreteintervals parallel to the beach line. Numbers on top indicate the distance of the receiver cable from the beach line.

FIG. 4.1-17. Computer simulation of the experiment illustrated in Figure 4.1-16. Here, we downward continue the receiversat discrete depth intervals. The numbers on top indicate the distance of the receiver cable from the surface, z = 0.

recorded closer to the barrier, the recorded diffractionhyperbola arrived earlier, and became shorter and morecompressed. It collapsed to a point when the receiverscoincided with the storm barrier over which the sourcepoint forms a gap.

There is one important difference between thephysical experiment in Figure 4.1-16 and the computer-simulated downward-continuation experiment in Figure4.1-17. The receiver cable is the same at each step inFigure 4.1-16, whereas the effective cable length getsshorter and shorter toward the source (the gap in thebarrier) in Figure 4.1-17. This is because we started byrecording the wavefield at the surface (Figure 4.1-16a)with a finite cable length. The recorded information is

confined to within the two raypaths depicted on thesection in Figure 4.1-17a. As the cable moves closer tothe source, the effective receiver cable containing theinformation is confined to smaller and smaller lengths.Although receivers are lowered vertically, energy movesdown along raypaths it originally took on the way up.

To relate these recordings at different depths (Fig-ure 4.1-17), we superimpose them as shown in Figure4.1-18a. Moreover, the recordings can be shifted so thatthe apexes of the hyperbolas coincide and are positioned

at a time that is equivalent to the distance from the sur-face to the diffractor as shown in Figure 4.1-18b. Thisis called time retardation .

Reconsider the results from the computer simula-tion of the harbor experiment in Figure 4.1-17. Sup-pose we stopped recording at a depth of 1000 m before

FIG. 4.1-18. (a) Superposition of the time sections in Fig-ure 4.1-17; (b) removing the translational effect by retar-dation to place the energy at the apex of the hyperbolaobtained initially along the beach line.



Migration 489

Table 4-4. Application of two-point implicit and ex-plicit operators to extrapolate data P from t to t + ∆t.

Explicit Implicit DataOperator Operator Column

−1− a −1− a/2 P (t)

1 1− a/2 P (t + ∆t)

of wave amplitudes growing from one extrapolation stepto another, can be an issue with this type of operator(Section D.6) An implicit operator produces stable re-sults because of averaging on the right side of equation(4-9b), known as the Crank-Nicolson scheme. For thedifferential equations used in finite-difference migrationalgorithms, such as the parabolic equation described inSection D.3, scalar a becomes a matrix coefficient. Im-plicit schemes require inversion of this matrix. However,no inversion is needed with explicit schemes, since fu-ture values can be written explicitly in terms of onlypast values.

Equation (4-9a) is rewritten by redefining scalar aas a ∆t to obtain

P (t + ∆t)− P (t)

∆t= a P (t). (4− 10)

The left side of equation (4-10) is the discrete repre-sentation of the continuous derivative of P with respectto time, dP/dt. Therefore, equation (4-10) is the finite-difference equation that corresponds to the differential equation

dP

dt= a P (t). (4− 11)

We have derived the differential equation that de-scribes the inflation of money (equation 4-11). Now con-sider the analysis in reverse order. We start with the dif-ferential equation (4-11), and write the correspondingdifference equation (4-10), which is the equation thatis solved in the computer. This equation is written ineither the explicit (equation 4-9a) or implicit (equation4-9b) form to extrapolate the present value of P to thefuture.

This example illustrates how finite-differenceschemes can solve differential equations in the com-puter. The scalar wave equation can be treated in asimilar, but more complicated manner. Complicationsarise because it is a partial differential equation thatcontains the second derivatives of the wavefield with re-spect to depth, time, and spatial axes. Setting up thecomputer algorithm is more involved and is not dis-cussed here. Claerbout (1976, 1985) provides details of

various aspects of the finite-difference migration meth-ods.

Rational Approximations for Implicit Schemes

The scalar wave equation is a two-way wave equation indepth that describes propogation of both upcoming anddowngoing waves. If we consider the resulting wavefieldfrom the exploding reflectors model as the upcomingwaves, then we are really interested in a one-way wave equation to downward continue the upcoming waves. Infact, we normally use some rational approximation tothe one-way wave equation in finite-difference imple-mentations.

To get the actual differential equation to be usedin downward extrapolation of the upcoming waves, and

therefore to perform a finite-difference migration, thegeneral strategy is as follows:

(a) Start with the two-way scalar wave equation:

∂ 2P

∂x2+

∂ 2P

∂z2−

1

v2(x, z)

∂ 2P

∂t2= 0, (4− 12)

where x and z are the space variables, t is the timevariable, v is the velocity of wave propagation, andP (x, z, t) is the pressure wavefield.

(b) Assume constant velocity and perform 3-D Fouriertransform of the pressure wavefield. This is equiv-

alent to substituting the plane-wave solutionexp(ikxx + ikzz− iωt) to equation (4-12). The sub-stitution yields the dispersion relation between thetransform variables

kz = ∓

ω2

v2− k2

x, (4− 13a)

where kx and kz are the wavenumbers in the xand z directions, and ω is the angular temporalfrequency.

(c) We are interested in upcoming waves, hence weonly need one of the two solutions. We also want toinvoke the exploding reflector model by replacing v

by v/2 to obtain the following paraxial dispersionrelation

kz =2ω

v

1−

vkx2ω

2

, (4− 13b)

where the horizontal wavenumber kx has been nor-malized with respect to 2ω/v.

(d) Make a rational approximation to the square-rootexpression in equation (4-13b) so as to derive adifferential equation (Sections D.3 and D.4). This




approximation imposes a dip limitation to the dif-ferential equation. One approximation to the dis-persion relation given by equation (4-13b) is ob-tained by Taylor expansion of the square root andretaining the first two terms in the series (Section

D.3)

kz =2ω

v

1−

1

2

vkx2ω

2. (4− 14a)

This dispersion equation is known as the 15-degreeapproximation and is the basis for the first finite-difference time migration algorithm developed byClaerbout and Doherty (1972). Albeit no longer inuse, we shall review the 15-degree finite-differencealgorithm for its historical significance.

(e) Operate on the pressure wavefield P (kx, kz, ω) withthe approximate form of the dispersion relationgiven by equation (4-14a), and inverse Fourier

transform in the z direction to get the differentialform of the approximate one-way wave equation.

∂

∂zP (kx, z , ω) = −i

2ω

v

1−

1

2

vkx2ω

2P (kx, z , ω).

(4− 14b)(f) Recall from Figure 4.1-17 that, after each

downward-continuation step, we retard the wave-field by translating it in time so that after migra-tion, events appear in their correct depth locations.The time retardation is done by applying a linearphase shift to the pressure wavefield P

Q = P exp(−iωτ ), (4− 15a)

where the retarded time is

τ =

z

0

dz

v(z), (4− 15b)

and Q is the retarded wavefield. The velocity v(z) isthe horizontally averaged v(x, z). Substitute equa-tion (4-15a) into (4-14b) to obtain the differen-tial equation associated with the 15-degree finite-difference algorithm in two parts

∂ 2Q

∂z∂t

=v

4

∂ 2Q

∂x2

, (4− 16a)

and

∂Q

∂z= 2

1

v(z)−

1

v(x, z)

∂Q

∂t, (4− 16b)

where Q is the retarded wavefield. Derivationof equations (4-16a,b) is based on the assump-tion that velocity varies vertically. Nevertheless,in practice, the velocity function in equations (4-16a,b) can be varied laterally, provided the varia-tion is smooth. Equation (4-16a) accounts for col-

lapsing diffraction energy to the apex of the trav-eltime curve only. Hence, it is referred to as thediffraction term . When lateral velocity variationsare significant, the diffraction curve is somewhatlike a skewed hyperbola with its apex shifted later-

ally away from the diffraction source. This lateralshift is accounted for by the thin-lens term givenby equation (4-16b) (Section D.3). If the lateralvelocity variations are significant, then the thin-lens term is not negligible. Migration algorithmsthat implement both the diffraction and thin-lensterms represented by equations (4-16a,b) generallyare two-step schemes that alternately solve thesetwo terms. To propagate one depth step, first applythe diffraction term on wavefield Q. The thin-lensterm then is applied to the output from the diffrac-tion calculation. A migration method that includesthe effects of the thin-lens term is called depth mi-

gration , since the output section is in depth. Depthmigration is warranted if there are strong lateralvariations in velocity; in this case, the coefficientof the thin-lens term cannot be negligible. If weassume that velocity varies only in the vertical di-rection, then v(z) = v(x, z). This makes the thin-lens term of equation (4-16b) vanish, and we areleft with the diffraction term of equation (4-16a).A migration method that implements the diffrac-tion term (equation 4-16a), only, is known as time migration, the output of which is in time τ of equa-tion (4-15b). When recast in terms of the τ vari-able, equation (4-16a) takes the form

∂ 2Q

∂τ∂t=

v2

8

∂ 2Q

∂x2. (4− 17)

This is the parabolic equation for time migration.(g) Finally, write down the difference forms of the dif-

ferential operators either in implicit form to be usedin finite-difference solution of the parabolic equa-tion (4-17) for migration.

Boundary and initial conditions are needed to solvethe differential equations. The initial condition for mi-gration is the recorded wavefield at the surface z = 0.

Also, in migration we assume that the wavefield is zeroafter a maximum observation time, typically the endtime of the recorded trace. Then there are the sideboundaries, beyond which assumptions must be madeabout the form of the wavefield.

In the (x, z, t) coordinates, the seismic section isrepresented by the x− t plane, while the migrated sec-tion (earth) is represented by the x − z plane. Finite-difference migration, as discussed here, extrapolates thex − t plane in finite increments of z and outputs thewavefield at t = 0 at each step (Figure 4.1-19).



Migration 491

FIG. 4.1-19. The seismic section represented by the x− t

plane at the surface z = 0 is downward continued to obtainthe time sections at discrete depth levels. The direction of extrapolation is indicated by the thick arrow. The migratedsection is represented by the x− z plane at t = 0.

There are two ways to downward continue thewavefield recorded at the surface in the computer (Fig-ure 4.1-20). Starting with the wavefield at the surfacez = 0 represented by the vectors in x — s1, s2, s3, . . .,which are perpendicular to the page, we can computethe wavefield at different depth levels using the order of the computation shown in Figure 4.1-20a. Assume zerovalue for the bottom of the extrapolated wavefield ateach depth step. So, for example, using s7, s8, and 0,compute the wavefield at position 1. Then use s6, s7,and the wavefield already computed at position 1, com-pute that at position 2, and so on. Notice that, in thisscheme, we compute the wavefield at all times for onedepth step, then compute the wavefield at all times forthe next depth step, followed by the next depth step,and so on. Hence, this is called the z-outer computa-tional scheme.

The alternate scheme involves a different order of computation (Figure 4.1-20b). First, compute the wave-field at one time for all depths, then using those al-ready computed values, compute the wavefield at thenext shallower time for all depths, and so on. Hencethis is called the t-outer computational scheme.

In both schemes, the output migrated section is ob-tained by collecting the diagonal elements. Dependingon the depth step size, which can be conveniently de-fined as the number of time samples, one collects one ormore samples at each depth level. In the example shown

FIG. 4.1-20. Two algorithmic schemes to downward con-tinue wavefields in the computer: (a) z -outer, and (b) t-

outer. The midpoint axis is perpendicular to the plane of the paper. In b oth schemes, the CMP-stacked data are rep-resented by the s-column at z = 0 (Claerbout, 1976).

in Figure 4.1-20, there are two samples from each depthstep collected into the migrated section.

Reverse Time Migration

Another migration method, known as reverse time mi-

gration (Baysal et al., 1983), extrapolates an initiallyzero x− z plane backward in time , bringing in the seis-mic data P (x, z = 0, t) as a boundary condition z = 0at each time step to compute snapshots of the x − zplane at different times. At time t = 0, this x− z planecontains the migration result P (x, z, t = 0) (Figure 4.1-21).

The algorithmic structure for the reverse time mi-gration is illustrated schematically in Figure 4.1-22.Start with the x− t section at the surface, z = 0. Also,consider an x−z frame at tmax. This frame is blank ex-cept for the first row which is equal to the bottom rowof the x−t section at t

max

. Extrapolate this snapshot att = tmax to t = tmax−∆t by using the phase-shift oper-ator exp(iω∆t). This yields a new snapshot of the x−zframe at t = tmax−∆t. The first row of numbers in thisframe is identical to the row in the x − t plane — theoriginal unmigrated section, at t = tmax − ∆t. Hence,replace the first row in the snapshot at t = tmax −∆twith the row of the x − t section at t = tmax − ∆tand continue the extrapolation back in time. The lastsnapshot is at t = 0 that represents the final migratedsection.




FIG. 4.1-21. Reverse time migration: Start with an all-zerox− z plane at the bottom of the data cube and extrapolatebackward in time toward t = 0 to compute snapshots of thex− z plane at different times. These snapshots of the sub-surface are indicated by the horizontal planes; the directionof extrapolation — reverse in time, is indicated by the thickarrow. At each time level, include the boundary value (x-slice at z = 0, indicated by the dotted lines) into the x− z

plane from the seismic section. The migrated section is thex− z plane at t = 0 (the top horizontal plane).

Frequency-Space Implicit Schemes

As discussed in Section 4.3, in practice the 15-degreefinite-difference migration can handle dips up to 35 de-grees with sufficient accuracy. A steep-dip approxima-tion to equation (4-13b) is achieved by continued frac-tions expansion (Section D.4) as

kz =2ω

v

1−

v2k2x

8ω2

1

1−v2k2

x

16ω2

. (4− 18)

This dispersion equation is known as the 45-degree ap-proximation and is the basis of the most common imple-

mentation of steep-dip implicit finite-difference schemes(Kjartansson, 1979).Refer to the steps described earlier and replace the

Taylor expansion given by equation (4-14a) with thecontinued fractions expansion given by equation (4-18).Follow the subsequent steps to derive the correspond-ing differential equation associated with the 45-degreediffraction term (Section D.4):

iv

4ω

∂ 3Q

∂z∂x2−

∂ 2Q

∂x2+ i

4ω

v

∂Q

∂z= 0, (4− 19a)

FIG. 4.1-22. An algorithmic description of reverse timemigration.

where Q(x, z, ω) is the retarded wavefield in thefrequency-space domain.

When recast for time migration, equation (4-19a)

becomes (Section D.4):

i1

2ω

∂ 3Q

∂τ∂x2−

∂ 2Q

∂x2+ i

8ω

v2∂Q

∂τ = 0, (4− 19b)

where τ is the time variable associated with the mi-grated data.

Note that dropping the first term in equation (4-19a) and inverse Fourier transforming in time yields the15-degree diffraction equation (4-16a). Similarly, drop-ping the first term in equation (4-19b) yileds the 15-degree equation (4-17) for time migration.

As for the 15-degree equation, the thin-lens equa-tion (4-16b) also applies for the 45-degree equation.When implemented in the frequency-space domain, thethin-lens term is represented by the phase-shift oper-ator of equation (4-15a). Again, the final step in theprocedure is to write down the difference forms of thedifferential operators in implicit form to be used infinite-difference solution of the 45-degree equation (4-19) for migration. Kjartansson (1979) provides an im-plicit scheme in which the extrapolation is in z. Never-theless, as for the 15-degree equation (4-17), it is trivialto adapt his scheme for time migration with the extrap-



Migration 493

FIG. 4.1-23.An algorithmic description of frequency-spacemigration.

olation in τ of equation (4-15b). The phase-shift op-erator of equation (4-15a) is velocity-dependent whenimplemented for depth migration, and it is velocity-independent when implemented for time migration.

The 45-degree approximation given by equation (4-19b) actually is fairly accurate in practice up to 60 de-grees. As described in Section D.4, the basic 45-degreeequation (4-19b) also can be adapted to obtain extrapo-lation schemes for imaging steeper dips up to 90 degrees.Nevertheless, a penalty is paid for steep-dip accuracy in

terms of dispersive noise incurred by implicit schemes(Section 4.3).Steep-dip finite-difference algorithms may be more

conveniently implemented in the frequency-space do-main than in the time-space domain. A general frame-work for implementing such algorithms involve a loopover the depth step z, and a loop over the frequency ω(Figure 4.1-23). For each depth step:

(a) Apply the shift term (equation 4-15a).(b) Apply the diffraction term (equation 4-19) by per-

forming implicit extrapolation of each of the fre-quency components of the wavefield.

(c) Sum over the frequencies to invoke the imagingprinciple which is equivalent to setting t = 0.

(d) Repeat the computation for all the depth steps tocomplete the imaging.

Frequency-Space Explicit Schemes

Start with the paraxial dispersion relation given byequation (4-13b) adapted to the exploding reflectors

model. Operate on the pressure wavefield P and in-verse Fourier transform in z to obtain the differentialequation

∂

∂zP (kx, z , ω) = −ikzP (kx, z , ω), (4− 20)

whose solution can be used to extrapolate the wavefieldat the surface down in depth

P (kx, z , ω) = P (kx, 0, ω)exp(−ikzz). (4− 21)

For a discrete depth step ∆z, equation (4-21) takesthe form

P (z + ∆z) = P (z)exp(−ikz∆z), (4− 22a)

where, for convenience, the variables kx and ω have beenomitted from P .

When designing extrapolation operators, whateverthe differencing scheme, the objective must be to en-

sure that the phase and amplitude of the actual op-erator closely resembles those of the desired operatorexp(−ikz∆z).

Discretize the one-way wave equation (4-20) andapply differencing approximation using an explicitscheme such as

P (z + ∆z) = P (z)

1− ikz∆z

, (4− 22b)

and an implicit scheme (Section D.6):

P (z + ∆z) = P (z)

1− ikz∆z/2

1 + ikz∆z/2

. (4− 22c)

The explicit extrapolation operator (1 − kz∆z) of

equation (4-22b) actually is the first two terms of theTaylor expansion of the exact operator exp(−ikz∆z) of equation (4-22a). Table 4-5 provides the amplitude andphase of the exact, explicit and implicit operators usedin equations (4-22a,b,c) for wavefield exptrapolation indepth.

A desired property of an extrapolation operator isthat it must be stable — its amplitude should be less orequal to unity. The implicit operator defined by equa-tion (4-22c) is stable, while the explicit operator definedby equation (4-22b) causes amplitudes of the extrapo-lated wavefield grow with depth (Section D.6). In fact,the larger the depth step ∆z, the more unstable arethe results of extrapolation. Another desired propertyof an extrapolation operator is that it should yield theleast phase error. The inherently stable nature of im-plicit schemes has been the compelling reason for theiruse in practice. Recent developments in the design of stable explicit schemes, however, now have made themwidely accepted (Holberg, 1988; Hale, 1991; Soubaras,1992).

In principle, the exact extrapolation operator inequation (4-22a) can be inverse Fourier transformed to




Table 4-5. Amplitude and phase of the exact, explicitand implicit extrapolation operators used in equations(4-22a,b,c). See Section D.6 for details.

Operator Amplitude Phase

Exactexp(−ikz∆z) 1 kz∆z

Explicit1− ikz∆z

1 + (kz∆z)2 tan−1(kz∆z)

Implicit1− ikz∆z/2

1 + ikz∆z/21 tan−1

kz∆z

1− (kz∆z/2)2

frequency-space (ω− x) domain and applied to P (z) in

an explicit manner. Each output sample of P (z + ∆z)at some x location for a frequency ω and velocity v iscomputed independently by convolving an explicit fil-ter operator of a specified length centered at the out-put location x with the input data array P (z) in the xdirection. In contrast, implicit schemes require solvinga set of linear equations to obtain the output samplesof P (z + ∆z) — computationally more intensive thanconvolution. Efficiency is an advantage of the explicitschemes over the implicit schemes.

Another attractive property of stable explicitschemes is their extension to 3-D extrapolation that pre-serves circular symmetry — a feature that is relatively

more difficult to attain with implicit schemes (Section7.3).Whether an explicit filter is computed by inverse

transforming the exact filter exp(−ikz∆z) of equation(4-22a) to the frequency-space domain or by Taylor ex-pansion as in equation (4-22b), the problem is that nei-ther approach yields a stable filter operator. A stable ex-plicit extrapolation filter in the frequency-space domaincan be designed using a constrained least-squares tech-nique (Holberg, 1988), or by a modified Taylor seriesexpansion of the exact extrapolation filter exp(−ikz∆z)(Hale, 1991). Another method of explicit operator de-sign based on on an alternative stability criterion is pre-sented by Soubaras (1992).

By substituting for kz from equation (4-13b), theexact extrapolation filter exp(−ikz∆z) of equation (4-22a) is expressed in the frequency-wavenumber domainas

D(kx) = exp

−i

2ω

v

1−

vkx2ω

2

∆z

. (4− 23)

The objective is to find, for a specific frequency ωand velocity v, a symmetric explicit filter with com-plex coefficients h(x) in the frequency-space domain

such that, when Fourier transformed to the frequency-wavenumber domain, the difference between the actualtransform H (kx) and the desired transform D(kx) of equation (4-23) is minimum, subject to the stabilityconstraint that the amplitude of H (kx) is never greater

than unity within the propagation region kx ≤ (2ω/v).Details of the method of modified Taylor expansionbased on this design criterion by Hale (1991) are de-scribed in Section D.5.

As for the implicit schemes (Figure 4.1-23), a mi-gration algorithm based on an explicit extrapolation fil-ter design involves a loop over the depth step z and aloop over frequency ω. For each depth step:

(a) Convolve the explicit extrapolation filter h(x) witheach of the frequency components of the wavefield.

(b) Sum over the frequencies to invoke the imaging

principle which is equivalent to setting t = 0.(c) Repeat the computation for all the depth steps tocomplete the imaging.

The length of the filter coefficients h(x) determinesthe dip accuracy of the explicit operator. The largerthe number of filter coefficients 2N + 1, the steeper thedip accuracy. In practice, extrapolation filter lengths 7,11, and 25 are often associated with 30-, 50-, and 70-degree dip accuracies. Phase error of the extrapolationoperator at steep dips may be reduced by increasing thenumber of coefficients. Also, lateral velocity variationscan be accommodated by varying the velocity at eachx location of the filter coefficient.

Frequency-Wavenumber Migration

Frequency-wavenumber (f −k) migration is not as easilyexplained as the Kirchhoff or finite-difference migrationfrom a physical point of view. Chun and Jacewitz (1981)provide practical insight into the principles of f − kmigration.

In Section 1.2, we learned that dipping events inthe t− x domain map onto the f − k domain along ra-dial lines. The steeper the dip, the closer the radial lineis to the wavenumber axis. Figure 4.1-24 shows dippingevents before and after migration in the t−x and f −kdomains. The Nyquist wavenumber is 20 cycles/km andthe bandwidth is given by the corner frequencies 6, 12- 36, 48 Hz for the passband region of the spectrum.(See Figure 1.1-26 for the definition of corner frequen-cies.) The red is associated with the flat part of thepassband region and the blue is associated with the ta-per zone. Note that migration rotates the radial lines



Migration 495




FIG. 4.1-25. Migration in the f −k domain. (Migration in

the t − x domain is illustrated in Figure 4.1-1.) (a) A dip-ping reflector is represented by a radial line OB in the f −k

plane. (b) After migration, the radial line OB maps onto an-other radial line OB, while B maps onto B. The horizontalwavenumber is invariant under migration. For comparison,the f −k response of the dipping event before migration (a)has been superimposed on the f −k response after migration(b). (Adapted from Chun and Jacewitz, 1981.)

in the 2-D amplitude spectrum outward and away fromthe frequency axis. The steepest event represented byradial line A maps onto radial line B after migration.The feather-like energy especially prominent in the leftquadrant of the f − k spectrum is associated with theflanks of the diffraction hyperbolas in the t−x domain.The energy associated with the left flanks which aredipping opposite to the dipping reflections maps ontothe left quadrant of the f − k plane. And the energyassociated with the right flanks of the diffraction hy-perbolas that are dipping in the same direction as thatof the dipping reflections maps onto the right quadrant

of the f − k plane and is superimposed on the energyassociated with the dipping reflections themselves.

Migration of a dipping event in the f −k domain issketched in Figure 4.1-25. Note that this figure is the f −k equivalent of Figure 4.1-1. In both figures, we assume

velocity equal to 1. The vertical axis in Figure 4.1-25represents the temporal frequency ω for the event in itsunmigrated position B, and the vertical wavenumber kzfor the event in its migrated position B.

Migration in the frequency-wavenumber domain in-volves mapping the lines of constant frequency AB inthe ω − kx plane to circles AB in the kz − kx plane.Therefore, migration maps point B vertically onto pointB. Note that in this process, the horizontal wavenum-ber kx does not change as a result of mapping. Whenthis mapping is completed, the dipping event OB ismapped along OB after migration; thus, the dip angleθ after migration is greater than the dip angle θ beforemigration. For comparison, these two radial lines areshown on the same plane kz − kx.

We now examine the diffraction hyperbola and itscollapse to the apex after migration in the f −k domain.A diffraction hyperbola is represented by an invertedtriangular area in the frequency domain as shown inFigure 4.1-26. The Nyquist wavenumber is 40 cycles/kmand the bandwidth is given by the corner frequencies 6,12 - 36, 48 Hz for the passband region of the spectrum.As for the dipping events model in Figure 4.1-24, the redis associated with the flat part of the passband regionand the blue is associated with the taper zone. The

two edges in the right and left quadrant of the f − kplane correspond to the asymptotes of the flanks of thediffraction hyperbola, the base of the inverted trianglecorresponds to the high-frequency end of the passband,and the tip of the triangle in the proximity of the originof the f −k plane corresponds to the low-frequency endof the passband. Migration turns the triangular areainto a circular shape as shown in Figure 4.1-26.

The f − k analysis of the diffraction hyperbolashown in Figure 4.1-26 is based on the representationof the hyperbola as a series of discrete dipping seg-ments. Figure 4.1-27 depicts a diffraction hyperbola inthe t− x and f − k domains. We imagine that the hy-perbola is made up of a series of dipping segments, suchas A , B, C, D and E . The zero-dip segment A is at theapex, while the steepest dip segment E is along theasymptotes. In the f − k domain, the zero-dip segmentA maps along the frequency axis, while the dipping seg-ments B, C and D map along the radial lines, increas-ingly further away from the frequency axis. Finally, theasymptotic tail E maps along the radial line that rep-resents the boundary between the propagation and theevanescent region. The evanescent region corresponds



Migration 497




FIG. 4.1-27. A hyperbola on the t−x plane maps onto an inverted triangular area on the f

−k plane. (See text for details.)

to the energy that is located at or greater than 90 de-grees from the vertical. The opposite side of the hyper-bola maps onto the second quadrant (negative kx) inthe f − k domain. In the continuous case, a diffractionhyperbola is represented by a series of continuous radiallines that constitute an inverted triangular area in thef − k domain (Figure 4.1-26).

A curious fact emerges from the f − k spectrum of the migrated section in Figure 4.1-26. We expect migra-tion to collapse the diffraction hyperbola to a point at

the apex. The spectrum of this migrated section reallyshould be more like that in Figure 4.1-28 — a rectangle.Why is there a difference between this spectrum and thespectrum after migration in Figure 4.1-26?

If you start with a point and model it, you get thediffraction hyperbola in Figure 4.1-28. However, in re-ality we deal with a diffraction hyperbola as shown inFigure 4.1-26. The hyperbolas do not look different inthe t− x domain, but note the difference in their f − kspectra. The f − k spectrum of the real-life diffraction,which is always subjected to bandpass filtering (Figure4.1-26), is missing the energy above the 48-Hz line thatis present in the f − k spectrum of the modeled diffrac-

tion curve (Figure 4.1-28). These missing high frequen-cies cause the difference between the spectra after mi-gration.

Phase-Shift Migration

Theory of the frequency-wavenumber (f −k) migrationtechniques is left to Section D.7. For now, we brieflyreview the f − k migration algorithms as follows:

(a) Just as any other migration algorithm, start withthe two-way scalar wave equation (4-12).

(b) Assume constant velocity and perform 3-D Fouriertransform and obtain the dispersion relation be-tween the transform variables (equation 4-13a).

(c) Then, adapt the dispersion relation to the explod-ing reflectors model by halving the velocity for theupcoming waves (equation 4.13b).

(d) Operate on the pressure wavefield P and inversetransform in z to obtain the differential equation

(4-20).(e) Obtain the solution given by equation (4-21).

The discrete form of this solution given by equation(4-22) is the basis for phase-shift migration in whichvelocity can be varied at each depth step in the verticaldirection.

The phase-shift method involves the followingsteps:

(a) Start with the stacked section — an approximationto the zero-offset section P (x, z = 0, t), and per-form 2-D Fourier transform to get P (kx, z = 0, ω).

(b) By using equation (4-22), for each frequency ω, ex-trapolate the transformed wavefield P (kx, z , ω) atdepth z with a phase-shift operator exp (−ikz∆z)to get the wavefield P (kx, z + ∆z, ω) at depthz + ∆z. At each z step, a new extrapolation op-erator with the velocity defined for that z value iscomputed.

(c) As for any other migration, invoke the imagingprinciple t = 0 at each extrapolation step to ob-tain the migrated section P (kx, z , t = 0) in the



Migration 499

4.1-28. (a) A diffraction hyperbola modeled from an impulse in the t−x domain, (b) the f −k spectrum of (a), (c) migrationof (a), (d) the f − k spectrum of (c); (e) bandpass-filtered version of (a), (f) the f − k spectrum of (e), (g) migration of (e),and (h) the f − k spectrum of (g).




transform domain. The imaging condition t = 0is met by summing over all frequency componentsof the extrapolated wavefield at each depth step.This is easily shown from the integral representingthe inverse Fourier transform of the extrapolated

wavefield (equation D-84).(d) Repeat steps (b) and (c) for downward continua-tion and imaging, respectively, for all depth steps toget the migrated section in the transform domainP (kx, z , t = 0).

(e) Final step involves inverse transforming in the xdirection to get the migrated section P (x, z, t = 0).

Figure 4.1-29 shows a flowchart of the phase-shiftmethod.

The phase-shift method (Gazdag, 1978) can onlyhandle vertically varying velocities. A way to accommo-date lateral velocity variations judged to be acceptable

for time migration is to first stretch the CMP-stackedsection in the time direction so as to make it correspondto a velocity field v(z) that only varies vertically. Thisvelocity field is obtained by averaging the original veloc-ity field associated with the unstretched CMP-stackedsection in the x direction. Following the stretching op-eration, the stacked section is migrated using the veloc-ity function v(z) in the standard phase-shift migrationscheme. Finally, the migrated section is unstretched.

Gazdag and Squazzero (1984) extended the phase-shift method to handle lateral velocity variations. Toachieve this, first the input wavefield is extrapolated bythe phase-shift method using a multiple number of lat-

erally constant velocity functions and a series of refer-ence wavefields are created. The imaged wavefield thenis computed by interpolation from the reference wave-fields. This migration method is known as phase-shift-plus-interpolation. An alternative extension of phase-shift migration to handle lateral velocity variations ispresented by Kosloff and Kessler (1987).

Stolt Migration

If the medium velocity is constant, migration canbe expressed as a direct mapping (Stolt, 1978) from

temporal frequency ω to vertical wavenumber kz(Figure 4.1-25). Figure 4.1-30 is a flowchart of theStolt algorithm; the mathematical details are leftto Section D.7. The equation for Stolt mapping is

P (kx, kz, t = 0) =

v

2

kz k2x

+ k2z

P

kx, 0, ω =

v

2

k2y

+ k2z

,

(4− 24a)

FIG. 4.1-29. Flowchart for Gazdag’s phase-shift method of migration.

FIG. 4.1-30. Flowchart for Stolt’s constant-velocity migra-tion method in the f − k domain.

where P (kx, z = 0, ω) is the zero-offset section andP (kx, kz, t = 0) is the migrated section in the frequency-

wavenumber domain.

Note that Stolt migration involves, first, mapping

from ω to kz for a specific kx by using the dispersion

relation of equation (4-13a) recast as



Migration 501

ω =v

2

k2x

+ k2z

. (4− 24b)

The output of mapping is then scaled by the quantityS

S =v

2

kz k2x

+ k2z

. (4− 24c)

Stolt’s algorithm for constant velocity thus involvesthe following steps:

(a) Start with the input wavefield P (x, z = 0, t) ap-proximated by the CMP stack, and apply 2-DFourier transform to get P (kx, z = 0, ω).

(b) Map the wavefield from ω to kz using the dispersionrelation given by equation (4-24b).

(c) Apply the scaling factor S of equation (4-24c) aspart of the mapping procedure (Section D.7).

(d) Invoke the imaging principle by setting t = 0 andobtain P (kx, kz, t = 0).

(e) Finally, apply 2-D inverse transform to get the mi-grated section P (x, z, t = 0).

It may be questionable as to whether the constant-velocity Stolt method has value on its own as a practicalmigration algorithm. Nevertheless, Stolt’s method canbe used efficiently to perform a constant-velocity mi-gration as the first step in a residual migration scheme

(Section 4.5). Additionally, the method constitutes anessential procedural step for migration velocity analysisas described in Section 5.4.

Stolt extended his method to handle velocity vari-ations (Section D.7). For the variable-velocity case,Stolt’s extension consists of

(a) modifying the input wavefield to make it appear asif it were the response of a constant-velocity earth,

(b) applying the constant-velocity algorithm outlinedin Figure 4.1-30, and

(c) reversing the original modification of the inputwavefield.

This modification essentially is a type of stretchingof the time axis (Section D.7) to make the reflectiontimes approximately equivalent to those recorded for aconstant-velocity earth. The nature of stretching is de-scribed by the stretch factor W . The constant-velocitycase is equivalent to W = 1.

Note that the phase-shift and Stolt migration out-puts normally are displayed in two-way vertical zero-offset time τ = 2z / v, as are the outputs from the

finite-difference and Kirchhoff migrations. In practice,mapping in the f − k domain really is from ω − kx toωτ − kx rather than to kz − kx, where ωτ is the Fourierdual of τ and is simply kz of equation (4-13b) scaled byv/2 (Section D.3):

ωτ = ω

1−

vkx2ω

2

. (4− 25)

One important concept must be pointed out fromequation (4-25). Note that for a constant kx, ωτ < ω;thus, migration shifts the bandwidth to lower frequen-cies. This is analogous to the conclusion derived in rela-tion to the NMO correction, since the latter also causesdata stretching to lower frequencies (Section 3.1). Theimplication from equation (4-25) is demonstrated bythe dipping events model in Figure 4.1-24. While thebandwidth of the zero-dip event is retained after mi-gration, the bandwidth of the event with steepest diphas shifted from approximately 40 Hz to 36 Hz at thehigh-frequency end of the spectrum. In fact, the shift inbandwidth is dip-dependent; events with different dipswhich have the same bandwidth before migration willhave different bandwidths after migration.

Summary of Domains of Migration Algorithms

Migration algorithms described in this section are basedon the assumption that the input stacked section rep-resents a zero-offset acoustic wavefield. As such, thesealgorithms are all based on the scalar wave equation(4-12). Table 4-6 provides a list of the migration al-gorithms described in this section with the associateddesign and application domains. While there exist sev-eral other migration algorithms, those listed in Table4-6 are the most widely used in practice.

Also included in Table 4-6 are the types of velocityfields and dips each migration algorithm can accommo-date. It is clear that each algorithm is limited by either

the type of velocity variations or dip ranges. Therefore,each has an appropriate usage in practice dependingon field data and velocity characteristics (Section 4.0).Note that the lateral velocity variations implied by thevelocity fields in Table 4-6 are mild to moderate andare within the bounds of time migration. Additionally,the choice of migration algorithm depends on whetheryour objective is imaging or migration velocity analysis.While imaging is the subject of this chapter, migrationvelocity analysis is discussed in Chapter 5.




Table 4-6. Domains of migration algorithms.

Algorithm Domain Dips andVelocities

Kirchhoff t− x up to 90 deg

Summation time-space rms v(x, τ )

Finite-Difference t− x up to 35 deg15-deg Implicit time-space int v(x, τ )

Finite-Difference ω − x up to 65 deg45-deg Implicit frequency-space int v(x, τ )

Finite-Difference ω − x up to 80 deg70-deg Explicit frequency-space int v(x, τ )

Phase-Shift ω − kx up to 90 degfreq.-wavenumber int v(τ )

Stolt Method ω − kx up to 90 degwith Stretch freq.-wavenumber rms v(x, τ )

4.2 KIRCHHOFF MIGRATION

IN PRACTICE

In this and the following three sections, the parametersthat affect performance of Kirchhoff summation, finite-difference, and f −k migration methods are discussed. InKirchhoff migration, the important parameters are theaperture width used in summation and the maximumdip to migrate. In finite-difference and phase-shift mi-grations, the depth step size needs to be selected prop-erly. The stretch factor is important in Stolt migration.The responses of these methods to velocity errors alsoare examined. All practical aspects are discussed usingsynthetic models of two zero-offset sections — a modelof dipping-events and a model of a diffraction hyper-

bola. Real data examples also are used to evaluate thechoice of optimum parameters.In Sections 4.2, 4.3, 4.4, and 4.5, migration re-

sults of different algorithms using various parametersare compared with a desired migration. In all cases,this desired migration was obtained using the phase-shift method with appropriate parameters and veloci-ties. This does not imply that the phase-shift methodalways provides a desirable output; it only means thatthe data examples in this section were chosen so thatthe phase-shift algorithm is appropriate. The choice of

the phase-shift method was a compromise; it handlesdips of up to 90 degrees and velocities that can onlyvary vertically.

Before a migration algorithm is used on field data,its impulse response must be tested. A band-limited im-

pulse response is generated by using an input that con-tains an isolated wavelet on one trace only To also limitthe spatial bandwidth, this trace is replicated on eitherside with the wavelet amplitude halved. The ideal mi-gration algorithm should produce an impulse responsethat has the shape of a semicircle. Kirchhoff migrationproduces the section shown in Figure 4.2-1d. The im-pulse response indicates that Kirchhoff migration canaccurately handle dips up to 90 degrees. The dip ona migration impulse response is measured as the angleθ between the vertical and a specified radial direction.Note that migration can be limited to smaller dips (Fig-ure 4.2-1).

Aperture Width

From the previous section, we know that Kirchhoff migration involves a summation of amplitudes alongdiffraction hyperbolas. Given the rms velocity at a par-ticular time sample of a particular input trace, a hyper-bolic traveltime trajectory associated with a fictitiousdiffractor is overlaid on the input section with its apexat that time sample. In theory, a diffraction hyperbolaextends to infinite time and distance. In practice, wehave to deal with a truncated hyperbolic summationpath. The spatial extent that the actual summationpath spans, called the migration aperture , is measuredin terms of the number of traces the hyperbolic pathspans.

FIG. 4.2-1. Migration can be confined to a range of dipspresent on a seismic section. The impulse response for thedip-limited migration operator is a truncated semicircle. Dipangle θ is measured from the vertical axis.



Migration 503

FIG. 4.2-2. Summation paths for Kirchhoff migration in a medium with (a) low velocity (2000 m/s), (b) high velocity (4000m/s), and (c) vertically varying velocity. Migration aperture is small for low velocities and large for high velocities.

The curvature of the diffraction hyperbola is gov-erned by the velocity function. Figure 4.2-2a showsa number of low-velocity diffraction hyperbolas, whileFigure 4.2-2b shows a number of high-velocity hyper-bolas. A low-velocity hyperbola has a narrower aper-ture when compared to a high-velocity hyperbola. Thisagrees with our intuition — high velocity means moremigration. In practice, we deal with a velocity function

that at least varies with depth. The diffraction hyperbo-las can have different curvatures depending on the ve-locity value at a given time sample (Figure 4.2-2c). Be-cause of the vertical variation in velocity, aperture widthgenerally is time variant. For the usual case in whichvelocity increases with depth, migration of the shallowpart of the section requires a narrow aperture, whilemigration of the deep portion requires a wider aperture(Figure 4.2-2c). This implies that, given the same dip,deep events migrate farther than shallow events.

Figure 4.2-3 shows a zero-offset diffraction hyper-bola (8 ms/trace dip along the asymptotes) and migra-tions using four different aperture widths. The smaller

the aperture, the less capable the migration is in col-lapsing the diffraction hyperbola. In this case, use of anaperture width that is equal to the width of the inputsection (half aperture, 256 traces) yields the best result.

Figure 4.2-4 shows a synthetic zero-offset sectionthat consists of a number of dipping events ranging from0 to 45 degrees in increments of 5 degrees. Aperturewidth is related closely to the horizontal displacementdx that takes place in migration as defined by equa-tion (4-1). The number of traces an event migrates is

nx = dx/∆x, where ∆x is the CMP interval. Therefore,the aperture width that is required is 2nx+1. Figure 4.2-4 also shows Kirchhoff migrations of the dipping eventsusing four different aperture widths. Small-aperture mi-gration eliminates steeply dipping events on the outputsection. Increasing the aperture width allows proper mi-gration of the steeply dipping events. From this we seethat using too small an aperture width causes a dip fil-

tering action during migration, because a small apertureexcludes the steeper flanks of the diffraction hyperbolafrom the summation.

For any given event position in time t before mi-gration, the optimal value for the aperture width is de-fined by twice the maximum horizontal displacement inmigration for the steepest dip of interest in the inputsection. In this case, the horizontal displacement asso-ciated with the 45-degree dipping event is computed bysubstituting the values for v = 3500 m/s, ∆x = 25 m,∆t / ∆x = 12 ms/trace, where t = 2 s in equation (4-1).The value for the horizontal displacement is 118 traces,giving an aperture width of 237 traces. Typically, we

consider somewhat larger values to allow for velocityerrors.

A good way to determine aperture width is to gen-erate diffraction hyperbolas as shown in Figure 4.2-2cusing the regionally averaged, vertically varying veloc-ity. Clearly, the larger the aperture width, the moretraces are used in the summation. For the dippingevents in Figure 4.2-4, the optimal value of the half-aperture width is 150 traces; increasing the width to300 traces resulted in no further improvement.




FIG. 4.2-3. Tests for aperture width in Kirchhoff migration: (a) a zero-offset section that contains a diffraction hyperbola with2500-m/s velocity, (b) desired migration using the phase-shift method; Kirchhoff migration using (c) 35-trace, (d) 70-trace,(e) 150-trace, and (f) 256-trace half-aperture width.



Migration 505

FIG. 4.2-4. Tests for aperture width in Kirchhoff migration: (a) a zero-offset section that contains dipping events with3500-m/s velocity, (b) desired migration using the phase-shift method; Kirchhoff migration using (c) 35-trace, (d) 70-trace,(e) 150-trace, and (f ) 256-trace half-aperture width.




FIG. 4.2-5. Tests for aperture width in Kirchhoff migration: Insufficient aperture width causes removal of steeply dippingevents.



Migration 507

FIG. 4.2-6. Tests for aperture width in Kirchhoff migration: Insufficient aperture width causes spurious horizontal events indeep, noisy part of a stacked section.



Migration 509

action passes flat or nearly flat events — those hori-zontal wavenumber components that are zero or nearlyzero.

In conclusion, the following assessments are madeconcerning the choice of aperture width.

(a) Excessively small aperture width causes destruc-tion of steeply dipping events and rapidly varyingamplitude changes.

(b) Excessively small aperture width organizes randomnoise, especially in the deeper part of the section,as horizontally dominant spurious events.

(c) Excessively large aperture means more computertime. More importantly, large apertures can de-grade the migration quality in poor signal-to-noiseratio conditions. Use of large aperture will causerandom noise at late times to creep into the goodshallow data. Aperture width always is a compro-

mise with noise.(d) Sometimes it is better to use a smaller aperture

than would theoretically be required to avoid theadverse effect of noise on the migrated event. Noiseconsiderations may even require a time-dependentaperture width.

(e) It is recommended that the aperture width be keptconstant in migrating all lines from a particularsurvey so that an overall uniformity in amplitudecharacteristics on the migrated sections is main-tained.

In practice, a regional velocity function and the steepestdip in a survey area are used to compute the optimal

aperture width that can be used over the entire set of data from the area (equation 4-1).

Maximum Dip to Migrate

During migration, we can specify the maximum dipwe want migrated in the section. This may be usefulwhen we need to suppress the steeply dipping coher-ent noise. Figure 4.2-8 shows migrations of the dippingevents with four different maximum allowable dips. Fora 4 ms/trace dip limit, events with dips greater than

this value are suppressed. Similarly, for an 8 ms/tracedip value, events with dips greater than this value aresuppressed. When the dip value is 12 ms/trace, no sup-pression occurs, since all events in the input section havedips less than this value. Limiting the dip parameter isa way to reduce computational cost, since it is relatedto aperture width (equation 4-1), which determines thecost.

From Figure 4.2-1, note that the Kirchhoff migra-tion impulse response can be limited to various max-

imum dips. The smaller the maximum allowable dip,the smaller the aperture. This combination of maximumaperture width and maximum dip limit determines theactual effective aperture width used in migration. Inparticular, diffraction hyperbolas along which summa-

tion is done are truncated beyond the specified maxi-mum dip limit.

A field data example of testing the maximum dipparameter is shown in Figure 4.2-9. Some steep dips arelost on the section that corresponds to the 2 ms/tracemaximum allowable dip. The 8 ms/trace dip appearsto be optimum. The maximum dip parameter must bechosen carefully so that the steep dips of interest in theinput section are preserved. Finally, dip value can bechanged spatially and in time; however, practical im-plementation can be cumbersome.

Velocity Errors

We now examine the response of Kirchhoff migrationto velocity errors. Figure 4.2-10 shows the diffractionhyperbola and migrations using the 2500 m/s mediumvelocity, and 5, 10, and 20 percent lower velocities. Withincreasingly lower velocities, the diffraction hyperbola iscollapsed less and less taking the shape of a frown — itis undermigrated.

Figure 4.2-11 shows the same diffraction hyperbolaand migrations using the 2500 m/s medium velocity,and 5, 10, and 20 percent higher velocities. With in-

creasingly higher velocities, the the diffraction hyper-bola is inverted more and more taking the shape of asmile — it is overmigrated.

The under-and overmigration effects resulting fromthe use of erroneously low or high velocities on the dip-ping events model are seen in Figures 4.2-12 and 4.2-13, respectively. Label the correct position of the eventwith the steepest dip from the desired migration on theresults of migrations with different velocities and notethe event mispositioning caused by erroneously low andhigh velocities. Compare with the desired migration andalso note that the steeper the dip the more the under-

and overmigration effect. Sensitivity of migration to ve-locity errors can be measured quantitatively via equa-tions 4-1 and 4-2.

From the migrated sections in Figure 4.2-14, notethat the bow tie becomes increasingly less resolved atlower velocities; this indicates undermigration.

Figure 4.2-15 shows a CMP-stacked section andthe desired migration. The steep left flank of the saltdome has been imaged with acceptable accuracy. Theaccuracy of the imaging of the slightly overturned right

(text continues on p. 520)




FIG. 4.2-8. Tests for maximum dip to migrate in Kirchhoff migration: (a) a zero-offset section that contains a diffractionhyperbola with 2500-m/s velocity, (b) desired migration; Kirchhoff migration using (c) 4-ms/trace, (d) 8-ms/trace, (e) 12-ms/trace, and (f) 24-ms/trace maximum dip.



Migration 511

FIG. 4.2-9. Tests for maximum dip to migrate in Kirchhoff migration: A low value for maximum dip to migrate can behazardous. All dips of interest must be preserved during migration.




FIG. 4.2-10. Tests for velocity errors in Kirchhoff migration: (a) a zero-offset section that contains a diffraction hyperbolawith 2500-m/s velocity, (b) desired migration; Kirchhoff migration using (c) the medium velocity of 2500 m/s, (d) 5 percentlower, (e) 10 percent lower, and (f) 20 percent lower velocity.



Migration 513

FIG. 4.2-11. Tests for velocity errors in Kirchhoff migration: (a) a zero-offset section that contains a diffraction hyperbolawith 2500-m/s velocity, (b) desired migration; Kirchhoff migration using (c) the medium velocity of 2500 m/s, (d) 5 percenthigher, (e) 10 percent higher, and (f) 20 percent higher velocity.




FIG. 4.2-12. Tests for velocity errors in Kirchhoff migration: (a) a zero-offset section that contains dipping events with3500-m/s velocity, (b) desired migration; Kirchhoff migration using (c) the medium velocity of 3500 m/s, (d) 5 percent lower,(e) 10 percent lower, and (f) 20 percent lower velocity.



Migration 515

FIG. 4.2-13. Tests for velocity errors in Kirchhoff migration: (a) a zero-offset section that contains dipping events with3500-m/s velocity, (b) desired migration; Kirchhoff migration using (c) the medium velocity of 3500 m/s, (d) 5 percent higher,(e) 10 percent higher, and (f) 20 percent higher velocity.




FIG. 4.2-14. Tests for velocity errors in Kirchhoff migration: Undermigration manifested as inadequate handling of thebowtie is caused by the use of velocities lower than those considered to be medium velocities.



Migration 517

FIG. 4.2-15. (a) A CMP-stacked section, and (b) migration using the phase-shift method. Event A is the water-bottommultiple, and Event B is the peg-leg multiple associated with the top-salt boundary. These multiples are respectively denotedby Events C and D on the migrated section.




FIG. 4.2-16. Tests for velocity errors in Kirchhoff migration using, from top to bottom, 100, 95, 90, and 80 percent of rmsvelocities. The input stacked section is shown in Figure 4.2-15a and the desired migration using the phase-shift method isshown in Figure 4.2-15b.



Migration 519

FIG. 4.2-17. Tests for velocity errors in Kirchhoff migration using, from top to bottom, 100, 105, 110, and 120 percent of rms velocities. The input stacked section is shown in Figure 4.2-15a and the desired migration using the phase-shift methodis shown in Figure 4.2-15b.




flank can only be inferred by the lateral positioning of the gently dipping reflections in the vicinity of the saltflank.

Figure 4.2-16 shows results of Kirchhoff migrationof the stacked section in Figure 4.2-15 using velocities

lower than what may be optimum for imaging. Whilethe undermigration of the left flank of the salt domeis not so evident, the steeply dipping reflection off theright flank intersects the gently dipping reflections as-sociated with the surrounding strata, thus providing aclue for undermigration.

Figure 4.2-17 shows results of Kirchhoff migrationof the stacked section in Figure 4.2-15 using velocitieshigher than what may be optimum for imaging. Whileovermigration effects may be marginal on the sectionwith a small velocity error (105 percent of optimum ve-locities), migration with higher velocity errors (110 and120 percent of optimum velocities) shows signs of over-

migration in the form of crossing events along the leftflank of the salt diapir. Under- and overmigration effectscaused by large velocity errors often are detectable; nev-ertheless, small velocity errors can cause subtle effectsmaking it difficult to judge whether there is under- orovermigration. Uncertainties in migration velocities in-evitably cause uncertainties in the interpretation madefrom migrated sections or volumes of data. For instance,the shape of the salt diapir inferred from the resultsshown in Figures 4.2-16 and 4.2-17 varies significantlydepending on the percent velocity errors.

4.3 FINITE-DIFFERENCE MIGRATION

IN PRACTICE

As described in Section 4.1, finite-difference migrationis implemented using implicit and explicit schemes. Inthis section, we shall include in our discussion the 15-degree finite-difference algorithm because of its histori-cal significance. Nevertheless, we shall primarily discusspractical aspects of the steep-dip implicit and explicitschemes in the frequency-space domain. Specifically, weshall deal with the impulse responses, depth step sizeand response to velocity errors in implicit and explicitschemes.

The first finite-difference migration algorithm thatwas introduced to the seismic industry was based onthe parabolic approximation to the scalar wave equation(Claerbout and Doherty, 1972). The algorithm was im-plemented in the time-space domain and designed usingan implicit scheme. The parabolic approximation theo-retically limits the algorithm to handling dips up to 15degrees (Section D.3). Nevertheless, in practice, it canhandle dips up to 35 degrees with sufficient accuracy

due to the bandlimited nature of seismic data. Steeperdips, in principle, can be migrated by a cascaded appli-cation of the 15-degree algorithm (Larner and Beasley,1990).

Finite-difference migration of stacked data cur-

rently is performed using steep-dip algorithms basedon the continued fractions expansion to the scalar waveequation. This approximation provides a theoretical dipaccuracy up to 45 degrees. The basic 45-degree schemecan be improved to handle steeper dips up to 80 de-grees with reasonable accuracy (Section D.4). The 45-degree finite-difference algorithm commonly is imple-mented using an implicit scheme in the frequency-spacedomain.First, as we did for the Kirchhoff migration, we ex-amine the impulse response of the 15-degree implicitscheme. The shape of the impulse response of a desiredmigration algorithm with no dip limitation is a semicir-

cle. The shape of the impulse response of the 15-degreeequation is, in theory, an ellipse (Claerbout, 1985) asseen in Figure 4.3-1. The nature of the dispersive noisepattern inside the ellipse is discussed in the next sectionon depth step size. Isolated noise spikes in field data canintroduce such noise patterns on migrated sections.

The parts of the responses above the small circles inFigure 4.3-1 correspond to the evanescent energy, whilethe parts below the circles correspond to the propagat-ing energy (Claerbout, 1985). The parts below the cir-cles are the useful part of the response. The evanescent

FIG. 4.3-1. Desired impulse response of a 90-degree mi-gration algorithm is a semicircle (top), while the impulseresponse of the 15-degree migration algorithm is an ellipse(bottom). The propagation zone is defined by the portion of the ellipse below the small circles, and the evanescent zoneis defined by the portions above the small circles. For com-parison, the desired response has been superimposed on theimpulse response of the 15-degree equation.



Migration 521

FIG. 4.3-2. (a) CMP stack, (b) desired migration by phase-shift method, (c) 15-degree finite-difference migration. Thefinite-difference migration based on the parabolic equation has the inherent property of undermigrating the steep flank of thediffraction and the steeply dipping event. See Figure 4.3-3 for a sketch of the migration results.

energy travels horizontally and is characterized by imag-inary wavenumbers kz, which occur when the quantityin the square root in equation (4-13b) becomes nega-tive. This means that the evanscent region correspondsto horizontal wavenumbers kx > 2ω/v. For negative kz,the exact extrapolator exp(−ikzz) is no longer a wavepropagator; instead, it causes waves to decay rapidly indepth. Thus, evanescent energy is not expected to bepresent in recorded wavefields. The impulse response of the 15-degree finite-difference algorithm, however, sug-gests propagation in the region of evanescence. This isnot desirable; the parts of the elliptical wavefront above

the small circles should be removed. Use of a depth stepsize that is greater than the input sampling rate tendsto suppress the response in the evanescent region. Ex-cessively large depth steps, however, cause truncation of the wavefront further into the propagating zone belowthe small circles in Figure 4.3-1.

The impulse response (Figure 4.3-1) is used toestimate the maximum dip that the implicit finite-difference algorithm can handle without serious ampli-tude distortions or phase errors. This is done by super-

imposing the desired semicircular response and measur-ing the angle between the indicated lines. Note from themeasured angle in Figure 4.3-1 that the 15-degree im-plicit scheme can be used to migrate dips up to approx-imately 35 degrees with sufficient accuracy. This is pri-marily because errors associated with finite-differenceapproximations used in particular implementations of the 15-degree equation usually are adjusted to cancelsome of the theoretical error associated with the 15-degree differential equation.

The dip-limited nature of the parabolic equationcauses undermigration of steeply dipping events and

steep flanks of diffractions. This is demonstrated by thefield data example in Figure 4.3-2. The two prominentfeatures, diffraction D and dipping event B, are locatedas shown in Figure 4.3-3 before and after migration.

Depth Step Size

Finite-difference migration involves downward continu-ation of the wavefield at the surface, such as a stacked




FIG. 4.3-3. A sketch of the diffraction D and steeply dip-ping event before (B) and after (A) desired migration fromthe sections in Figure 4.3-2. The diffraction and the dippingevent after finite-difference migration using the parabolicequation are denoted by FD−D and FD−B, respectively.

section, and invoking the imaging principle so as to cre-ate an image of the subsurface at t = 0. The downwardcontinuation takes place in the computer at discretedepth intervals (Section 4.1). Depth step size governsperformance of finite-difference migration. Inappropri-ate specification of this parameter can cause artifacts inthe migrated section. We want to choose an optimumdepth step size that is large for computational savings,yet yields a tolerable error in positioning the events af-ter migration.

Figure 4.3-4 shows the constant-velocity diffractionhyperbola and the 15-degree implicit finite-difference

migrations using four different depth steps. Large depthsteps cause severe undermigration as well as kinks alongthe flank of the diffraction curve (especially apparent inthe 60- and 80-ms cases). At smaller depth steps, suchas the 20- and 40-ms cases, more energy collapses to theapex, but the 15-degree scheme fails to achieve a com-plete focusing of the energy at the apex of the hyper-bola. Note also the dispersive noise that trails the unfo-cused energy. The dip-limited nature of the 15-degree al-gorithm, however, causes undermigration whatever thedepth step size (Figure 4.3-5).

Undermigration of the diffraction energy along thesteep flanks of the hyperbola is caused by the parabolicapproximation to the scalar wave equation. The dis-persive noise that accompanies the undermigrated en-ergy is an effect of approximating differential operators

with difference operators. The accuracy of this approxi-mation decreases at large frequencies and wavenumbers(Claerbout, 1985). Thus, the dispersive noise becomesless with smaller trace spacing and sampling in depthand time. For example, the difference operator of equa-tion (4-10) becomes an increasingly better approxima-tion to the differential operator of equation (4-11) as ∆tis made smaller. To emphasize more strongly the pres-ence of the dispersive noise, migrated sections in Fig-ures 4.3-4 and 4.3-5 have been displayed with the samedisplay gain level as the input section. The dispersionnormally is much less pronounced on field data.

Figure 4.3-6 shows the dipping events model and

the 15-degree implicit finite-difference migration resultsusing four different depth step sizes. We can make thefollowing inferences:

(a) Increasing depth step size causes more and moreundermigration at increasingly steep dips.

(b) The waveform along reflectors is dispersed at steepdips and large depth steps.

(c) Kinks occur along reflectors at discrete intervalsthat correspond to the depth step size. Kinks aremore pronounced at increasingly steeper dips.

The first inference results from the parabolic ap-proximation, the second from differencing approxima-tions, and the third from gradual undermigration to-ward the base of each depth step. The kinks are goodfor diagnostics; their presence indicates that the depthstep size that is used is too coarse for the dips present inthe data. In that case, smaller depth step size should beused; then the kinks disappear altogether (Figure 4.3-7). Nevertheless, kinks that characterize undermigra-tion can be eliminated by a local adjustment of migra-tion velocities or interpolation between the wavefieldsat the adjacent depth steps.

It is apparent from Figure 4.3-6 that migrationwith 20-ms depth step, which corresponds to one-half of the typical dominant period of recorded seismic waves,has the least dispersion with the least undermigration— optimum accuracy in event positioning. Further de-creasing depth step size does not improve migration sig-nificantly (Figure 4.3-7). The 15-degree implicit schemecauses precursive dispersion at large depth steps greaterthan 20 ms (Figure 4.3-6) and postcursive dispersion atsmall depth steps less than 20 ms (Figure 4.3-7). Hence,taking smaller depth steps does not necessarily mean a



Migration 523

FIG. 4.3-4. Tests for extrapolation depth step size in 15-degree finite-difference migration: (a) a zero-offset sectionthat contains a diffraction hyperbola with 2500-m/s velocity,(b) desired migration; 15-degree finite-difference migrationusing (c) 20-ms (d) 40-ms, (e) 60-ms, and (f) 80-ms depthstep.

FIG. 4.3-5. Tests for extrapolation depth step size in 15-degree finite-difference migration: (a) a zero-offset sectionthat contains a diffraction hyperbola with 2500-m/s velocity,(b) desired migration; 15-degree finite-difference migrationusing (c) 4-ms (d) 8-ms, (e) 12-ms, and (f) 16-ms depth step.




FIG. 4.3-6. Tests for extrapolation depth step size in 15-

degree finite-difference migration: (a) a zero-offset section

that contains dipping events with 3500-m/s velocity, (b) de-

sired migration; 15-degree finite-difference migration using

(c) 20-ms (d) 40-ms, (e) 60-ms, and (f) 80-ms depth step.

FIG. 4.3-7. Tests for extrapolation depth step size in 15-degree finite-difference migration: (a) a zero-offset sectionthat contains dipping events with 3500-m/s velocity, (b) de-sired migration; 15-degree finite-difference migration using(c) 4-ms (d) 8-ms, (e) 12-ms, and (f) 16-ms depth step.



Migration 525

better quality migration free of the artifacts that occurwith the finite-difference method.

Figures 4.3-8 and 4.3-9 show the migrations of thestacked section in Figure 4.3-2a with five different depthsteps using the 15-degree implicit scheme. Note that

as the depth step size is increased, the dipping eventoff the flank of the salt diapir is more undermigratedand the diffraction off the tip of the diapir is less col-lapsed. Dispersion along the diffraction hyperbola is ap-parent at larger depth steps (Figure 4.3-9). Again, thisphenomenon is caused by the differencing approxima-tions to the differential operators used in the design of a finite-difference algorithm.

Velocity Errors

Figure 4.3-10 shows the diffraction hyperbola and its

migration using the 2500 m/s medium velocity and 5,10, and 20 percent lower velocities. When velocitieslower than medium velocity are used, the diffractionhyperbola gets undermigrated by the 15-degree algo-rithm more than it would be by an algorithm with nodip limitation (compare with Figure 4.2-10).

Figure 4.3-11 shows the diffraction hyperbola andits migration using the 2500 m/s medium velocity and5, 10, and 20 percent higher velocities. When velocitieshigher than medium velocity are used, the diffractionhyperbola gets overmigrated less by the 15-degree al-gorithm than it would be by an algorithm with no diplimitation (compare with Figure 4.2-11). Moreover, notethe increase in dispersive noise as a result of overmigra-tion.

Figure 4.3-12 is the dipping events model with mi-grations using the 3500 m/s medium velocity, and 5,10, and 20 percent lower velocities. For comparison, la-bel the correct position of the event with the steepestdip from the desired migration on the results of migra-tion with different velocities. As in any other migrationmethod, velocity errors cause events to be mispositionedat increasingly steeper dips. The undermigration effectof lower velocities is reinforced by the inherently un-dermigrating nature of the 15-degree algorithm. As aresult, dipping events are undermigrated more by the15-degree algorithm in contrast than by an algorithmwith no dip limitation (compare with Figure 4.2-12).

Figure 4.3-13 is the dipping events model with mi-grations using the 3500 m/s medium velocity, and 5,10, and 20 percent higher velocities. For comparison,again, label the the correct position of the event withthe steepest dip from the desired migration on the re-sults of migration with different velocities. The overmi-gration effect of higher velocities is counteracted by theinherently undermigrating nature of the 15-degree algo-rithm. As a result, dipping events are not overmigrated

by the 15-degree algorithm as much as they would be

by an algorithm with no dip limitation (compare withFigure 4.2-13).

Velocity error test results on field data using the

15-degree implicit scheme are shown in Figures 4.3-14

and 4.3-15. Figure 4.3-16 is a sketch of the under- andovermigration effects. As noted above with the dippingevents model (Figures 4.3-12 and 4.3-13), when using

velocities greater than medium velocities, overmigrationis not as pronounced with the 15-degree finite-differencemigration as it is with a 90-degree algorithm, such as

the Kirchhoff or phase-shift method. On the other hand,when using velocities lower than medium velocities, theundermigration effect is more pronounced with the 15-

degree finite-difference migration in comparison with a90-degree algorithm (compare Figure 4.3-16 with Figure4.5-13).

At first, it may appear to be sensible to compensatefor the undermigration caused by a low-dip algorithmby adjusting migration velocities. For example, the best

match between the desired migration and the 15-degreefinite-difference results for the dipping event in Figure4.3-16 occurs when 10 percent higher velocities are usedin the finite-difference migration. While for one dip this

adjustment may be acceptable, for another dip it maynot be. Therefore, deficiencies of a migration algorithm should not be compensated for by making modifications

to the velocity field for migration.

Cascaded Migration

To compensate for the inherent undermigration bythe 15-degree finite-difference migration, Larner andBeasley (1990) proposed performing migration using

the 15-degree equation, repeatedly — the input to thenext migration stage being the output from the previ-ous stage. Such cascaded application of the 15-degreemigration is demonstrated in Figure 4.3-17. Migration

of the constant-velocity diffraction hyperbola using the15-degree equation only once yields the familiar re-

sult of unfocused energy accompanied with dispersivenoise (Figure 4.3-17c). A cascaded application of the15-degree equation produces improved focusing of theenergy at the apex of the hyperbola. The more the num-

ber of stages in the cascaded migration the more the im-provement in focusing the energy (Figures 4.3-17d,e,f ).

An interesting theoretical observation is that cas-

caded migration using a dip-limited algorithm, such asthe 15-degree finite-difference scheme, actually requiresa depth step size that is greater than the optimal depth

size for a single-stage application of the algorithm. In





FIG. 4.3-8. Tests for extrapolation depth step size in 15-degree finite-difference migration: (a) Desired migration using thephase-shift method, (b) 4-ms depth step, and (c) 20-ms depth step. The input CMP stack is shown in Figure 4.3-2a.

FIG. 4.3-9. Tests for extrapolation depth step size in 15-degree finite-difference migration: (a) 40-ms depth step, (b) 60-msdepth step, and (c) 80-ms depth step. The input CMP stack is shown in Figure 4.3-2a and the desired migration is shown inFigure 4.3-2b.



Migration 529

FIG. 4.3-14. Tests for velocity errors in 15-degree finite-difference migration using (a) 95 percent, (b) 90 percent, and (c) 80percent of rms velocities. The input stacked section is shown in Figure 4.3-2a and the desired migration using the phase-shiftmethod is shown in Figure 4.3-2b. Depth step size is 20 ms.

FIG. 4.3-15. Tests for velocity errors in 15-degree finite-difference migration using (a) 105 percent, (b) 110 percent, and(c) 120 percent of rms velocities. The input stacked section is shown in Figure 4.3-2a and the desired migration using thephase-shift method is shown in Figure 4.3-2b. Depth step size is 20 ms.




FIG. 4.3-16. The undermigration and overmigration ef-fects from Figures 4.3-14 and 4.3-15. B = dipping eventbefore and A = dipping event after desired migration, D =diffraction before and D = diffraction after 15-degree finite-difference migrations, and L = percent lower velocities, andH = percent higher velocities.

fact, the more the number of stages in the cascade, thecoarser the depth step size to achieve better focusing of the energy (Figure 4.3-17).

Cascaded migration of the constant-velocity zero-offset section that contains dipping events is shownin Figure 4.3-18. Again, the 15-degree finite-differencemigration yields the familiar result of undermigratedsteeply dipping events accompanied by the dispersivenoise. When applied in a cascaded manner, the algo-rithm positions the steeply dipping events more accu-rately. With a sufficient number of cascades (Figure 4.3-18f ), the 15-degree algorithm can actually position theevents as accurately as a 90-degree algorithm appliedonly once (Figure 4.3-18b). For comparison, the eventwith the steepest dip (AB) is labeled on the desiredmigration (Figure 4.3-18b) and the cascaded migration(Figure 4.3-18f ).

Unfortunately, the encouraging results from cas-caded migration using the 15-degree algorithm shownin Figures 4.3-17 and 4.3-18 are only attainable for aconstant-velocity medium. In case of a medium withvertically varying velocity, the cascaded application of the 15-degree algorithm causes overmigration (Figure

4.3-19). While this observation can be verified by the-ory, the situation can also be remedied by a cleverlyimplemented form of cascaded migration with a con-stant velocity used in each stage (Larner and Beasley,1990).

Actually, it turns out that cascaded migration the-ory dictates constant velocity to be used in each stage.If a variable velocity is used, then a 90-degree algorithmsuch as the phase-shift method is required in lieu of adip-limited algorithm for each stage. While the cascadedapplication of a dip-limited finite-difference algorithmwith a variable velocity causes overmigration (Figure4.3-19c), the cascaded application of the phase-shift al-gorithm with no dip limit yields an accurate image (Fig-ure 4.3-20e).

Since the advancements made in practical im-plementation of steep-dip implicit and explicit finite-difference schemes, practical use of cascaded migration,

however, has been limited.

Reverse Time Migration

In Section 4.1, a migration algorithm based on extrap-olation back in time while using the stacked section tobe the boundary condition at z = 0 was discussed. Theimpulse response of this algorithm, which is known asreverse time migration, is shown in Figure 4.3-21. Notethat the algorithm can handle dips up to 90 degreeswith the accuracy of phase-shift migration. The impor-tant consideration is that the extrapolation step ∆t in

reverse time migration must be taken quite small, usu-ally a fraction of the input temporal sampling interval.This then makes the algorithm computationally inten-sive.

Figure 4.3-22 shows a portion of a CMP-stackedsection and its reverse time migration. The steep flanksof the salt diapirs have been imaged accurately, enablingdelineation of the geometry of the top-salt boundarywith confidence. Reverse time migration, albeit its sim-ple and elegant implementation (Section 4.1), has notbeen used widely in practice. Again, this is primar-ily because it requires very small extrapolation step intime, which increases the computational cost of the al-

gorithm.

4.4 FREQUENCY-SPACE MIGRATION

IN PRACTICE

The basis of the steep-dip implicit algorithms is thedispersion relation of equation (4-18). Finite-differenceschemes with steep-dip accuracy are implemented con-veniently in the frequency-space domain. An important



Migration 535

FIG. 4.3-21. Impulse response of a reverse time migrationalgorithm.

advantage of the implicit method is its exceptional abil-ity to handle velocity variations, whether vertical orlateral. Its accuracy for the lateral velocity problem re-sults from the fact that the time shift associated withthe thin-lens term (equation 4-16b) can be implementedexactly in the frequency domain. For these reasons, thealgorithm is most appropriate for depth migration toimage targets beneath complex structures (Chapter 8).

The frequency-space, sometimes referred to as ω−x

or f − x, migration also has the important operationaladvantage that each frequency can be processed sep-arately. This property can reduce computer memoryrequirements significantly and, thus, decrease input-output operations for large data sets. Also, in frequency-space migration, some accuracy features can be conve-niently implemented. For example, wave extrapolationcan be limited to a specified signal bandwidth. Each fre-

quency component can, in principle, be downward con-tinued using an optimum depth step size that yields aminimum acceptable phase error, leading to a minimumamount of dispersive noise on the migrated section.

Steep-Dip Implicit Methods

Figure 4.4-1 shows the impulse responses of a series of implicit frequency-space finite-difference schemes withdifferent degrees of dip accuracy. Whether it is im-plemented in the time-space domain (Figure 4.3-1) or

frequency-space domain (Figure 4.4-1), the 15-degreealgorithm yields an elliptic impulse response. The 45-degree algorithm yields an impulse response in theshape of a heart.

The 15-degree equation is derived from the Tay-lor expansion of the dispersion relation (equation 4-14a). The 45-degre equation is based on the continuedfractions expansion (equation 4-18), which allows widerangle approximations. Kjartansson (1979) implementedthe 45-degree equation for migration of stacked data.

The 45-degree equation (4-18) can be upgraded to

be accurate for dips up to 65 degrees by tuning some co-efficients (Section D.4). Higher-order operators can beobtained by the successive application of a number of operators like the 45-degree operator (Ma, 1981) witha different set of coefficients (Lee and Suh, 1985). As

shown in Figure 4.4-1, with increasing dip accuracy,the impulse responses of the algorithms approach theshape of a semicircle. However, branches in the impulseresponse associated with evanescent energy remain.

Figure 4.4-2 shows migration of a constant-velocitydiffraction hyperbola using the 65-, 80-, 87-, and 90-degree implicit schemes (Section D.4). While the fo-cusing is better than that achieved by the 15-degreeimplicit scheme (Figure 4.3-4c), note that the disper-sive noise still persists in the image obtained from the65-degree implicit scheme (Figure 4.4-2c). It is evidentthat the quality of focusing from the 80-degree im-plicit scheme is superior (Figure 4.4-2d). The 87- and90-degree schemes have caused overmigration of thediffraction hyperbola (Figures 4.4-2e,f ).

The overmigration effect also can be observed onthe results from the constant-velocity dipping eventsmodel in Figure 4.4-3. In fact, the dispersive noise thataccompanies the steeply dipping events is present in allcases. The response of an implicit scheme is the productof a complicated interplay of various parameters (Sec-tion D.6) — depth step size, sampling intervals in spaceand time, dip angle, velocity and frequency. Dispersivenoise, and under- or overmigration characteristics of im-plicit schemes depend on the specific implementation.

Figure 4.4-4 shows the stacked data migrated us-ing three different approximations in frequency-spacedomain — 15, 45, and 65 degrees. Note that by higher-degree approximation, the collapse of the diffraction be-comes complete, and the steeply dipping event is mi-grated more accurately. Compare these results with thedesired migration in Figure 4.3-2b. Also note the similarresults obtained from the 15-degree time-space (t − x)algorithm (Figure 4.3-2c) and the 15-degree frequency-space (ω − x) algorithm (Figure 4.4-4a).

Figure 4.4-5 shows a field data example of frequency-space implicit finite-difference migrationswith different degrees of accuracy. Compare the re-sults with the desired migration in Figure 4.2-15b andnote that the 80-degree scheme probably produces themost preferred image of the salt dome as comparedto that from the 65-degree scheme. The schemes withsteeper dip accuracy (87- and 90-degree schemes), how-ever, yield marginal improvements over the 80-degreescheme. Often the 65-degree scheme produces accept-able results, and the 80-degree scheme, which requirestwice the computational effort, is used occasionally inpractice.



Migration 537

FIG. 4.4-1. Impulse responses of the frequency-space implicit schemes with various degrees of approximations to the one-wayscalar wave equation. (See Section D.4 for the theoretical basis of these responses.)

Depth Step Size

Figure 4.4-6 shows the constant-velocity diffraction hy-

perbola and the steep-dip implicit 65-degree finite-difference migrations using four different depth steps.Large depth steps cause undermigration as well as kinksalong the flank of the diffraction curve (especially ap-parent in 60- and 80-ms cases). The dispersive noise,

typical of finite-difference schemes, persists to varying

degrees irrespective of the depth step size. At smaller

depth steps, more energy collapses to the apex (Figure

4.4-7f ).

As in the case of the parabolic approximation (Fig-

ure 4.3-5), the dispersive noise that accompanies the

undermigrated energy (Figures 4.4-6 and 4.4-7) is an

effect of approximating differential operators with dif-




FIG. 4.4-2. (a) A zero-offset section that contains a diffraction hyperbola with 2500-m/s velocity, (b) desired migration usingthe phase-shift method; frequency-space finite-difference migrations using (c) the 65-degree, (d) 80-degree, (e) 87-degree, and(f) 90-degree implicit scheme.



Migration 539

FIG. 4.4-3. (a) A zero-offset section that contains dipping events with 3500-m/s velocity, (b) desired migration using thephase-shift method; frequency-space finite-difference migrations using (c) the 65-degree, (d) 80-degree, (e) 87-degree, and (f)90-degree implicit scheme.




FIG. 4.4-6. Tests for extrapolation depth step size in 65-degree frequency-space implicit finite-difference migration:(a) a zero-offset section that contains a diffraction hyperbolawith 2500-m/s velocity, (b) desired migration; 65-degreefinite-difference migrations using (c) 32-ms, (d) 40-ms, (e)60-ms, and (f) 80-ms depth step.

FIG. 4.4-7. Tests for extrapolation depth step size in 65-degree frequency-space implicit finite-difference migration:(a) a zero-offset section that contains a diffraction hyperbolawith 2500-m/s velocity, (b) desired migration; 65-degreefinite-difference migrations using (c) 8-ms, (d) 12-ms, (e)16-ms, and (f) 20-ms depth step.



Migration 545

FIG. 4.4-10. Tests for extrapolation depth step size in 65-degree frequency-space implicit finite-difference migration using,from top to bottom, 32-ms, 40-ms, 60-ms, and 80-ms depth steps. The input CMP stack is shown in Figure 4.2-15a, and thedesired migration using the phase-shift method is shown in Figure 4.2-15b.




FIG. 4.4-11. Tests for extrapolation depth step size in 65-degree frequency-space implicit finite-difference migration using,from top to bottom, 8-ms, 12-ms, 16-ms, and 20-ms depth steps. The input CMP stack is shown in Figure 4.2-15a, and thedesired migration using the phase-shift method is shown in Figure 4.2-15b.




FIG. 4.4-16. Tests for velocity errors in 65-degree frequency-space implicit finite-difference migration using interval velocitiesderived from, from top to bottom, 100, 95, 90, and 80 percent of rms velocities. The input stacked section is shown in Figure4.2-15a, and the desired migration using the phase-shift method is shown in Figure 4.2-15b. Depth step size is 20 ms.



Migration 551

FIG. 4.4-17. Tests for velocity errors in 65-degree frequency-space implicit finite-difference migration using interval velocitiesderived from, from top to bottom, 100, 105, 110, and 120 percent of rms velocities. The input stacked section is shown inFigure 4.2-15a, and the desired migration using the phase-shift method is shown in Figure 4.2-15b. Depth step size is 20 ms.




FIG. 4.4-18. (a) Impulse response of a desired migration algorithm using the phase-shift method; impulse responses of (b)a 30-degree, (c) 50-degree, and (d) 70-degree frequency-space explicit scheme for migration.

A field data example of a stacked section which

has been migrated using the 30-degree, 50-degree and70-degree extrapoaltion filters is shown in Figure 4.4-23.The steep flanks of the salt diapirs are clearly better im-aged by the steep-dip extrapolation filter. For compari-son, Figure 4.4-24 shows the desired migration using thephase-shift method. Although the length of the extrap-olation filter for a steep-dip algorithm is much longerthan that for a low-dip extrapolation filter, the benefitof using the former is indisputably demonstrated by thefield data example shown in Figure 4.4-23.

Velocity Errors

Figure 4.4-25 shows migrations of a zero-offset sectionthat contains a diffraction hyperbola using a frequency-space explicit algorithm based on 30-degree, 50-degreeand 70-degree extrapolation filters and a velocity thatis 90 percent of the medium velocity. For comparison,desired migration using the phase-shift method with themedium velocity and 90 percent of the medium velocityare also shown in the same figure. Migration with an



Migration 553

FIG. 4.4-19. (a) A zero-offset section that contains adiffraction hyperbola with 2500-m/s velocity, (b) desiredmigration using the phase-shift method; migrations usingfrequency-space explicit schemes with (c) 30-degree, (d) 50-degree, and (e) 70-degree accuracy.

FIG. 4.4-20. The f −k spectra of the sections in Figure 4.4-19: (a) A zero-offset section that contains a diffraction hy-perbola with 2500-m/s velocity, (b) desired migration usingthe phase-shift method; migrations using frequency-spaceexplicit schemes with (c) 30-degree, (d) 50-degree, and (e)70-degree accuracy.




FIG. 4.4-21. (a) A zero-offset section that contains dippingevents with 3500-m/s velocity, (b) desired migration usingthe phase-shift method; migrations using frequency-spaceexplicit schemes with (c) 30-degree, (d) 50-degree, and (e)70-degree accuracy.

erroneously low velocity yields the undermigrated formof the diffraction hyperbola as seen in Figure 4.4-25c.Note, however, the dip-limited explicit schemes appear

to cause less undermigration compared to the phase-shift method with 90-degree accuracy. This behavior isin contradiction to intuition — the undermigration ef-fect of an erroneously low velocity is reinforced by adip-limited algorithm. In fact, this intuitive effect wasdemonstrated by the steep-dip implicit scheme (Figure4.4-12).

The deceptive behavior of the explicit schemesthat contradicts our intuition can be explained by thefact that these schemes filter out the energy at highwavenumbers (Figures 4.4-19 and 4.4-20). As a result,

FIG. 4.4-22. The f − k spectra of the sections in Fig-ure 4.4-21: (a) A zero-offset section that contains dippingevents with 3500-m/s velocity, (b) desired migration usingthe phase-shift method; migrations using frequency-spaceexplicit schemes with (c) 30-degree, (d) 50-degree, and (e)70-degree accuracy.

the steep limbs of the undermigrated diffraction hyper-bola are truncated (Figure 4.4-25). This in turn makesthe result of migration using lower velocity appear lessundermigrated in case of an explicit scheme comparedto the case of the phase-shift method with 90-degree ac-curacy. In fact, if we apply a wavenumber filter to rejectthe high wavenumbers from the output of phase-shiftmigration (Figure 4.4-25c), the result would resemble



Migration 555



Migration 559

Migration with an erroneously high velocity yields theovermigrated form of the diffraction hyperbola as seenin Figure 4.4-26c.

We make the following observations from Figure4.4-26:

(a) The dip-limited explicit schemes, much like the im-plicit schemes (Figure 4.4-13), cause less overmi-gration compared to the phase-shift method with90-degree accuracy.

(b) The low-dip explicit scheme manifests the effect of overmigration much less than the steep-dip explicitscheme.

(c) The wavenumber filtering effect (Figure 4.4-20) fur-ther truncates the steep limbs of the overmigratedhyperbola.

The interplay of the three factors results in the

response to velocity errors by the explicit schemes asshown in Figure 4.4-26.

Tests for velocity errors are repeated for a zero-offset section that contains a set of dipping events asshown in Figures 4.4-27 and 4.4-28. For comparison, la-bel the correct position of the event with the steepestdip from the desired migration on the results of migra-tion with different velocities. The residual diffractionsoff the end of the dipping reflectors on the migratedsections from the explicit schemes have been truncatedby the wavenumber filtering effect of the extrapolationfilters. This filtering effect is most prominent in the caseof the explicit scheme with the low-dip limit and erro-neously high velocity (Figure 4.4-28d).

Field data examples for tests of velocity errors forthe explicit schemes are shown in Figures 4.4-29, 4.4-30, and 4.4-31. First, note the better imaging of thesalt flanks by the steep-dip explicit scheme comparedto the low-dip explicit scheme (Figure 4.4-29). The un-dermigration effect of erroneously low velocities (Figure4.4-30) and the overmigration effect of erroneously highvelocities (Figure 4.4-31) may be compared with theresults of migration using optimum velocities (Figure4.4-29).

We shall complete this section by reviewing theperformance of Kirchhoff summation, finite-differencefrequency-space implicit, and frequency-space explicitschemes with various dip limits. Figure 4.4-32 shows thecompilation of the results of migration of a zero-offsetsection that contains a diffraction hyperbola. For com-parison, desired migration using the phase-shift methodis included in the panel. The dip-limited nature of theimplicit and explicit schemes is manifested by the in-complete focusing of the energy at the apex of thediffraction hyperbola (Figures 4.4-32c,g,h). The under-migration effect caused by the dip limitation is allevi-

ated by using a steep-dip explicit scheme (Figure 4.4-32j). A steep-dip implicit scheme, on the other hand,can actually overshoot in the opposite direction andcause overmigration (Figure 4.4-32f ). The differencingapproximations are manifested by the dispersive noise

(Figure 4.4-32c).Figure 4.4-33 shows the compilation of the resultsof migration of a zero-offset section that contains a setof dipping events. Again, for comparison, desired mi-gration using the phase-shift method is included in thepanel. The dip-limited nature of the implicit and ex-plicit schemes is manifested by the undermigration of the steeply dipping events (Figures 4.4-33c,g,h). Thiseffect is alleviated by using a steep-dip explicit scheme(Figure 4.4-33j). A steep-dip implicit scheme, on theother hand, can actually overshoot in the opposite di-rection and cause overmigration (Figure 4.4-33f ).Thedifferencing approximations are manifested by the dis-

persive noise accompanying the steeply dipping events(Figure 4.4-33c,d,e,f ).

4.5 FREQUENCY-WAVENUMBER

MIGRATION IN PRACTICE

Two different methods of migration are implemented inthe frequency-wavenumber domain. The Stolt method isexact for a constant-velocity medium, while the phase-shift method is exact for a medium with vertical ve-locity variations. The Stolt method can be extended

to the case of a medium with lateral velocity varia-tions judged to be acceptable for time migration. Thephase-shift method also can be extended to handlelateral velocity variations. One extension, phase-shift-plus-interpolation scheme, accommodates lateral varia-tions in velocity by interpolating between the resultsof migration using a group of vertically-varying ve-locity functions. Another extension, phase-shift-plus-correction, applies an additional extrapolation opera-tor at each depth step to account for differences be-tween the laterally varying velocity field and verticallyvarying-only velocity function used for phase-shift mi-gration. The phase-shift method also has been extendedto image turning waves associated with salt overhangstructures.

Maximum Dip to Migrate

The phase-shift method of migration (Section 4.1 andSection D.7) allows vertical variations in velocity andis accurate for up to dips of 90 degrees. Figure 4.5-1shows the impulse response of the phase-shift algorithm.





FIG. 4.5-4. Tests for maximum dip to migrate in phase-shift migration: A low value for maximum dip to migrate can behazardous. All dips of interest must be preserved during migration.

Depth Step Size

Figure 4.5-5 shows a zero-offset section that contains

a set of dipping events migrated with the phase-shift

method using different depth step sizes. Since the phase-

shift method is based on the dispersion relation given by

equation (4-13b) of the exact one-way wave equation,

we do not expect undermigration. However, we see dis-

continuities along the reflectors at intervals equal to the

depth step size, which is similar to the finite-difference

results (Figure 4.4-8). As with the finite-difference algo-rithms, the problem occurs along the steeper dips first;therefore, the steep dips require smaller depth steps(Figure 4.5-5).

Because of the band-limited nature of seismic data,very small depth steps are not needed. From Figure4.5-5, note that migration with a 20-ms depth step pro-duces a section without spurious kinks along the reflec-tors; this is comparable to the desired migration usinga depth step equal to the temporal sampling interval.



Migration 567

FIG. 4.5-5. Tests for extrapolation depth step size in phase-shift migration: (a) a zero-offset section that contains dip-ping events with 3500-m/s velocity, (b) desired migration;phase-shift migrations using (c) 20-ms (d) 40-ms, (e) 60-ms,and (f) 80-ms depth step.

Depth step size tests on field data are shown inFigure 4.5-6. Unlike the finite-difference results (Figure4.4-10), the phase-shift migration with different depthstep sizes produces equally adequate results in terms of

the positioning of events. The only problem with largedepth steps is the kinks along the steep dips. In prin-ciple, as long as there is no aliasing in the z-direction,the depth step kinks can be eliminated by a local inter-polation process. In practice, as for the implicit finite-

difference methods, the depth step size used in migra-tion with the phase-shift method typically is taken be-tween the half-and full-dominant period of the wave-field — 20 to 40 ms, depending on steepness of the dipspresent in the section.

Velocity Errors

We now examine the response of phase-shift migrationto velocity errors. Figure 4.5-7 shows the diffraction hy-perbola and phase-shift migrations using the 2500 m/smedium velocity, and 5, 10, and 20 percent lower ve-locities. The lower the velocity, the more the diffrac-tion hyperbola is undermigrated. These results are usedas a benchmark to evaluate the response of the othermigration algorithms discussed in this chapter. Specif-ically, compare Figure 4.5-7 with Figures 4.2-10 (thecase of Kirchhoff migration), 4.3-10 (the case of implicitfinite-difference migration), 4.4-12 (the case of implicitfrequency-space migration), and 4.4-25 (the case of ex-plicit frequency-space migration), note how the variousalgorithms respond to velocity errors with significantdifferences.

Figure 4.5-8 shows the same diffraction hyperbolaand phase-shift migrations using the 2500 m/s mediumvelocity, and 5, 10, and 20 percent higher velocities.The higher the velocity, the more the diffraction hyper-bola is overmigrated. These results are used to bench-mark the response to velocity errors by the other migra-tion algorithms — Figures 4.2-11 (the case of Kirchhoff migration), 4.3-11 (the case of implicit finite-differencemigration), 4.4-13 (the case of implicit frequency-spacemigration), and 4.4-26 (the case of explicit frequency-space migration).

The under- and overmigration effects caused by theuse of erroneously low or high velocities on the dip-ping events model are seen in Figures 4.5-9 and 4.5-

10, respectively. Label the correct position of the eventwith the steepest dip from the desired migration onthe results of migrations with different velocities, andnote the event mispositioning caused by erroneously lowand high velocities. Recall that sensitivity of migrationto velocity errors can be measured quantitatively viaequations 4-1 and 4-2. The test results shown in Fig-ures 4.5-9 and 4.5-10 are used to benchmark the re-sponse to velocity errors by the other migration algo-rithms — Figures 4.2-12 and 4.2-13 (the cases of Kirch-hoff migration), 4.3-12 and 4.3-13 (the cases of implicit




FIG. 4.5-6. Tests for extrapolation depth step size in phase-shift migration: Note the kinks along steep dips with large depthsteps.

finite-difference migration), 4.4-14 and 4.4-15 (the casesof implicit frequency-space migration), and 4.4-27 and4.4-28 (the cases of explicit frequency-space migration).

An aspect of phase-shift migration uniquely dif-ferent from others is its exceptional quality of output.As noted from Figures 4.5-7, 4.5-8, 4.5-9, and 4.5-10,phase-shift migration produces no dispersive noise sinceit does not involve any differencing of differential oper-ators. Instead, the entire design of extrapolation oper-ator and application of migration are in the frquency-wavenumber domain. The results do not suffer from anydip limitation since the phase-shift method is based onan extrapolation filter that is exact for all dips up to 90

degrees (Figure 4.5-1). Of course, we must also remindourselves of the fact that the phase-shift method is lim-ited to velocities that vary only in the vertical direction.

Figures 4.5-11 and 4.5-12 are field data examplesof phase-shift migration using erroneously low and highvelocities, respectively. Figure 4.5-13 shows a sketch of the combined results of these migrations. Clearly, ve-locities that are too low cause undermigration of thesteeply dipping event that defines the flank of the saltdiapir and incomplete collapse of the diffraction off thetip of the salt diapir. Velocities that are too high causeovermigration as manifested by the crossing events atthe vicinity of the crest of the salt diapir.



Migration 569

FIG. 4.5-7. Tests for velocity errors in phase-shift migra-

tion: (a) a zero-offset section that contains a diffraction

hyperbola with 2500-m/s velocity, (b) desired migration;

phase-shift migration using (c) the medium velocity of 2500

m/s, (d) 5 percent lower, (e) 10 percent lower, and (f) 20

percent lower velocity.

Figure 4.5-14 shows the results of phase-shift mi-

gration of the stacked section in Figure 4.2-15 using

velocities lower than what may be optimum for imag-

ing. The under-migration of the left flank of the salt

dome is not so evident. However, the steeply dipping

reflection off the right flank intersects the gently dip-

ping reflections associated with the surrounding strata

— an indication of undermigration.

FIG. 4.5-8. Tests for velocity errors in phase-shift migra-tion: (a) a zero-offset section that contains a diffractionhyperbola with 2500-m/s velocity, (b) desired migration;phase-shift migration using (c) the medium velocity of 2500m/s, (d) 5 percent higher, (e) 10 percent higher, and (f) 20percent higher velocity.

Figure 4.5-15 shows results of phase-shift migrationof the stacked section in Figure 4.2-15 using velocitieshigher than what may be optimum for imaging. Migra-tion with erroneously high velocities (110 and 120 per-cent of optimum velocities) shows signs of overmigrationin the form of crossing events along the left flank of thesalt diapir.




FIG. 4.5-9. Tests for velocity errors in phase-shift migra-tion: (a) a zero-offset section that contains dipping eventswith 3500-m/s velocity, (b) desired migration; phase-shiftmigration using (c) the medium velocity of 3500 m/s, (d) 5percent lower, (e) 10 percent lower, and (f) 20 percent lowervelocity.

FIG. 4.5-10. Tests for velocity errors in phase-shift migra-tion: (a) a zero-offset section that contains dipping eventswith 3500-m/s velocity, (b) desired migration; phase-shiftmigration using (c) the medium velocity of 3500 m/s, (d)5 percent higher, (e) 10 percent higher, and (f) 20 percenthigher velocity.



Migration 571

FIG. 4.5-11. Tests for velocity errors in phase-shift migration using interval velocities derived from (a) 95 percent, (b) 90percent, and (c) 80 percent of rms velocities. The input stacked section is shown in Figure 4.3-2a, and the desired migrationusing the phase-shift method is shown in Figure 4.3-2b.

FIG. 4.5-12. Tests for velocity errors in phase-shift migration using interval velocities derived from (a) 105 percent, (b) 110percent, and (c) 120 percent of rms velocities. The input stacked section is shown in Figure 4.3-2a, and the desired migrationusing the phase-shift method is shown in Figure 4.3-2b.




FIG. 4.5-13. The combined results of migrations from Fig-ures 4.5-11 and 4.5-12. B = dipping event before and A =dipping event after desired migration; D = diffraction, andL = percent lower velocities and H = percent higher veloc-

ities.

Stolt Stretch Factor

As discussed in Section 4.1, the generalized Stoltmethod of migration involves converting the time sec-tion to an approximately constant-velocity section,which then is migrated by the constant-velocity Stoltalgorithm. This conversion is essentially stretching inthe vertical (time) direction. Once the section is mi-

grated in the stretched domain, it is converted back tothe original time domain. The generalized Stolt methodmust be distinguished from the constant-velocity algo-rithm. The constant-velocity algorithm is accurate fordips up to 90 degrees for a constant-velocity medium.The generalized method approximately accounts for ve-locity variations by stretching the section.

Stretching is defined by the stretch factor W . In hisoriginal paper, Stolt (1978) discusses implementation of the W factor. Although W is a complicated function of velocity and stretch coordinate variables, it often is set

to a scalar (Section D.7). The theoretical range of W isbetween 0 and 2.

To understand the effects of the stretch factor, re-fer to the impulse responses in Figure 4.5-16, where asingle, isolated wavelet on a single trace is migrated us-

ing various stretch factors. Here,W

= 1 correspondsto the exact constant-velocity Stolt algorithm. So, set-ting W < 1 compresses the impulse response inwardalong its steep flanks, while setting W > 1 opens it up.Thus, the value of W partially controls the aperture of the generalized Stolt algorithm. The farther W is from1, the more limited the aperture becomes. A value of W < 1 implies undermigration at steeper dips, while avalue of W > 1 implies overmigration at steeper dips,if the medium velocity is constant.

Although not strictly implied by the impulse re-sponses in Figure 4.5-16, when using a stretch factordifferent from 1, the Stolt algorithm tries to emulate a

wavefront in a variable velocity medium (Stolt, 1978),while compromising on the ability to migrate steeperdips. Experience has proven that the Stolt migrationwith stretch produces acceptable results provided ve-locity variations are within limits of time migration.

Consider the zero-offset section and the migrationresults in Figure 4.5-17. Stretch factor W = 1 producesthe best migrated section because the zero-offset sec-tion was modeled using a constant-velocity value. For0 < W < 1, the algorithm produces an undermigratedsection, while for 1 < W < 2, it produces an over-migrated section. These observations are in agreementwith the impulse responses in Figure 4.5-16. The near-

vertical streaks in the section withW

= 1.95 representwraparound artifacts.

The generalized Stolt algorithm produces the bestresult when W = 1, provided the medium velocity isconstant. Since this is never the case, we should examinethe algorithm for a vertically varying velocity medium.Figure 4.5-18 shows the impulse responses for differentvalues of W . Velocity varies linearly from t = 0 to t = 4 sbetween 2000 and 4000 m/s. For different W values, theportions of the wavefronts that best match the desiredmigration using the phase-shift method are between thesolid lines. For a vertically varying velocity medium,W = 1 is no longer the desired factor. In Figure 4.5-18,

accuracy over the widest range of dip angles with theStolt method is attained when W = 0.6. In general, thelarger the velocity gradient, the farther the optimum W is from 1. Strictly speaking, the optimum value for W is even different at different times.

In practice, wavefront plots, like those in Figure4.5-18, can be generated using both the phase-shift andStolt methods for a vertically varying regional velocityfunction. The W factor that yields the best fit at thelargest angular aperture is used then to migrate thedata with the Stolt method.



Migration 573

FIG. 4.5-14. Tests for velocity errors in phase-shift migration using interval velocities derived from 100, 95, 90, and 80 percentof rms velocities (from top to bottom). The input stacked section is shown in Figure 4.2-15a, and the desired migration usingthe phase-shift method is shown in Figure 4.2-15b.




FIG. 4.5-15. Tests for velocity errors in phase-shift migration using interval velocities derived from 100, 105, 110, and 120percent of rms velocities (from top to bottom). The input stacked section is shown in Figure 4.2-15a, and the desired migrationusing the phase-shift method is shown in Figure 4.2-15b.



Migration 575

FIG. 4.5-16. Tests for the stretch factor in Stolt migration:By varying the stretch factor W , the impulse response of theexact 90-degree migration operator (semicircle) is modified.For comparison, the desired response has been superimposed

on the Stolt migration impulse responses.

To circumvent the difficulty of defining an optimumstretch factor W , Beasley and Lynn (1992) suggestedapplying the constant-velocity Stolt migration in a cas-caded manner. The idea is based on a clever represen-tation of a vertically varying velocity function by a setof constant velocities, which are then used to performcascaded migration (Section 4.3). Since each migrationstage is done by using a constant velocity, the stretchfactor W is by default set to 1. Of course, representa-tion of a vertically varying velocity function by a set of

constant velocities is only an approximation that canbe valid for small vertical gradients.Figures 4.5-19 and 4.5-20 show Stolt migrations of

the CMP stack in Figure 4.3-2a using different valuesof the W factor. Migration velocities are varied onlyin the vertical direction. Figure 4.5-21 is a sketch of the migration results for the diffraction D off the tipof the salt diapir and steeply dipping event B off theflank of the salt diapir. The best match between thedesired migration and the Stolt method with stretch isfor W = 0.5.

Wraparound

Wraparound is the effect of finite data length in timeand space on a migration algorithm implemented inthe Fourier transform domain. A migration algorithmimplemented in the time-space domain does not sufferfrom wraparound effect. But a migration algorithm im-plemented in the frequency-space domain suffers fromwraparound along the time axis. Similarly, a migrationalgorithm implemented in the frequency-wavenumber

domain suffers from wraparound effects both along thetime and space axes.

Figure 4.5-22 shows a zero-offset section that con-tains a diffraction hyperbola and its migration usingfrequency-wavenumber migration based on the phase-

shift method. When plotted with a very high displaygain, we observe the energy in the migrated sectionbouncing off the edges of the section both in the timeand space directions.

A field data example of the wraparound effect onfrequency-wavenumber migration is shown in Figure4.5-23. This is the same section as in Figure 4.2-15b,except that it has been displayed using a very high gain.The energy above the water bottom is associated withthe wraparound effect in the time and space directions.The wraparound noise actually exists within the im-age portion of the section, also. A way to reduce thewraparound effect is to pad the data with zeros along

the axis of Fourier transformation. For frequency-spacemigration, data must be padded along the time axis;and for frequency-wavenumber migration, data must bepadded along both the time and space axes.

A field data example of the wraparound effect onfrequency-space migration is shown in Figure 4.5-24.This is the same section as in Figure 4.4-11d, exceptthat it has been displayed using a very high gain. Theenergy above the water bottom is associated with thewraparound effect in the time direction.

Residual Migration

The constant-velocity Stolt algorithm has useful ap-plications in residual migration as described here andmigration velocity analysis as described in Chapter 5.Consider a zero-offset section that contains a set of dip-ping events as shown in Figure 4.5-25a. The desiredmigration is obtained by using the medium velocityof v = 3500 m/s as shown in Figure 4.5-25b. Sup-pose, instead, that you migrate using a velocity of 3000m/s. The resulting migrated section is shown in Fig-ure 4.5-25c. Label the event with the steepest dip (AB)from the desired migration on the migrated section withv1 = 3000 m/s, and note the undermigration of the dip-ping events. By migrating the already migrated sectionusing a velocity of v2 = v2 − v21 = 1802 m/s (SectionD.8), we get the section shown in Figure 4.5-25d. Notethat this section obtained from the two-stage migrationusing velocities 3500 m/s and 1802 m/s is equivalent tothe one-stage migration using a velocity of 3500 m/s.The second stage of the two-stage migration using a ve-locity of 1802 m/s is called residual migration (Rothmanet al., 1985).




FIG. 4.5-17. Tests for the stretch factor in Stolt migration: W < 1 causes undermigration, and W > 1 causes overmigration.(Modeling courtesy Union Oil Company.)

So, what is the practical use of residual migration?It can improve upon the result of migration using a dip-

limited finite-difference algorithm. Figure 4.5-26a showsa zero-offset section that consists of three point scatter-ers in a layered earth model with vertically varying ve-locity field. A 15-degree dip-limited finite-difference mi-gration has difficulty collapsing these diffractions (Fig-ure 4.5-26b). Now, first migrate the zero-offset sectionwith the constant-velocity Stolt algorithm using thelowest value, 2000 m/s, in the vertically varying ve-locity function. The result is shown in Figure 4.5-26d.Then, take this section and migrate it again (Figure

4.5-26e) using the appropriate residual velocity (Sec-tion D.8) and the 15-degree finite-difference algorithm.

When compared with the single-stage finite-differencemigration (Figure 4.5-26b), note the superior perfor-mance of the residual migration. Also compare thiswith the desired migration using the phase-shift method(Figure 4.5-26c). The important point to keep in mindis that input to residual migration (the second stage)must be data which have been migrated (first stage)using a constant velocity (Rothman et al., 1985).

A field data example is shown in Figure 4.5-27with a sketch of the migration results in Figure 4.5-28.



Migration 577

FIG. 4.5-18. Tests for the stretch factor in Stolt migration: The medium velocity varies vertically from 2000 m/s at t = 0to 4000 m/s at t = 4 s.

The single-stage 15-degree finite-difference result showsthe typical undermigrated character (Figure 4.3-3). The1500-m/s constant-velocity Stolt migration followed bythe finite-difference migration seems to produce an out-put that is reasonably close to the desired migration.

A limitation of residual migration is that an ade-quate migration is not always achieved since the first-stage migration requires constant velocity which may befar off from the velocity field associated with the data.This is the case in Figure 4.5-27, since after residual mi-gration, the dipping event still is slightly undermigrated(see the sketch in Figure 4.5-28). Undermigration occursbecause the apparent dip perceived by the second-stagemigration still may be too large to be handled accu-rately. From equation (D-8c) note that the lower thevelocity used in migration, the smaller the dip that isperceived by migration. If the residual velocity func-tion given by equation (D-96b) is not too different fromthe original velocity function because of a large verticalgradient, then residual migration may not be adequate.

Residual migration is different from cascaded mi-gration that is discussed in Section 4.3. The latter in-

volves application of a dip-limited algorithm, such as animplicit finite-difference scheme, repeatedly. Whereasresidual migration involves the application of a dip-limited algorithm only once to data which already havebeen migrated using a constant-velocity Stolt migra-tion. In practice, Stolt migration can be replaced withphase-shift migration and a vertically varying velocityfunction with a gently varying gradient to accommodatethe constant-velocity requirement for the first-stage mi-gration.

Whether it is residual or cascaded migration, thetheoretical requirement is for constant velocity to beused at each stage preceding the last stage. Departurefrom this restriction will always limit the implementa-tion of residual and cascaded migration. Figure 4.5-29shows a zero-offset section that contains a set of dippingevents with 3500-m/s constant velocity. First, migratewith a constant velocity of 2500 m/s; then, perform acascade of four migrations with appropriate residual ve-locities (Section D.8) to obtain the accurate image thatis equivalent to the result from the desired migration us-ing the phase-shift method applied only once with the



Migration 581

FIG. 4.5-25. Principle of residual migration: (a) a zero-offset section that contains dipping events with 3500-m/svelocity, (b) desired migration using the phase-shift methodwith the medium velocity of 3500 m/s, (c) first-pass mi-gration of (a) using constant-velocity Stolt migration with3000-m/s velocity, and (d) second-pass migration of the out-put from the first-pass migration as in (c) using a 15-degreefinite-difference migration with the residual velocity of 1802m/s (equation D-96b).

are generally higher. This results in random noise or-ganized along wavefront arches, commonly referred to

as smiles. This organized noise corrupts the migratedprimary energy not just in the deep part of the sec-tion but also has detrimental effect on shallow data ina migrated section.

Line length and location of the line traverse at thesurface relative to the location of the target in the sub-surface have a direct effect on the useability of the a mi-grated section. Usually a line traverse longer than thespatial extent of the subusrface target is needed (Figure4.1-1). Keep in mind that your target does not neces-sarily lie directly beneath the CMP location where the

reflection from that target appears on your unmigratedstacked section.

Irregular topography associated with areas subjected to overthurst tectonics has to be accounted forduring migration if surface elevation changes are rapid

along the line traverse. Migration algorithms, with theexception of the Kirchhoff summation and the constant-velocity Stolt method, are all based on wave extrapo-lation from one flat depth level to another. A CMP-stacked section is assumed to be equivalent to a zero-offset wavefield and usually is referenced to a flat da-tum. In the presence of severe topography, one needsto account for the difference between the elevation pro-file and the reference datum. Otherwise, if the referencedatum is above the surface elevation, to a migration al-gorithm, events appear deeper than they are, and thusare overmigrated. If, on the other hand, the referencedatum is below the surface elevation, events appear to

a migration algorithm shallower than they actually are,and thus are undermigrated.

Migration and Spatial Aliasing

The concept of spatial aliasing is presented in Section1.2. Here, we shall examine the effect of spatial aliasingon migration. Figure 4.6-1 shows a zero-offset sectionthat contains a diffraction hyperbola with 2500-m/s ve-locity and 12.5-m trace spacing. By discarding everyother trace, obtain another zero-offset section with 25-

m trace spacing. Repeat the procedure to obtain thezero-offset sections with 50-m and 100-m trace spacings(Figure 4.6-1).

The f − k spectra of the zero-offset sections withthe four different trace spacings are displayed in Fig-ure 4.6-2. The diffraction hyperbola with 12.5-m tracespacing maps onto an inverted triangular area in thef − k plane (Section 4.1). The Nyquist wavenumber is40 cycles/km and the bandwidth is given by the cornerfrequencies 6, 12 - 36, 48 Hz for the passband regionof the spectrum. (See Figure 1.1-26 for the definition of corner frequencies.) The red is associated with the flatpart of the passband region and the blue is associated

with the taper zone.The f − k spectrum of the zero-offset section with25-m trace spacing (Figure 4.6-1) indicates spatial alias-ing beyond approximately 24 Hz (Figure 4.6-2). Conse-quently, the triangular shape of the passband regionin the f − k plane that defines the diffraction hyper-bola is corrupted around the edges near the Nyquistwavenumber of 20 cycles/km. At a coarser trace spacingof 50 m, which corresponds to a Nyquist wavenumberof 10 cycles/km, the triangular shape in the spectrumis preserved below the threshold frequency for aliasing,




FIG. 4.5-26. Principle of residual migration: (a) a zero-offset section with vertically varying velocities, (b) 15-degree finite-

difference migration, (c) desired migration using the phase-shift method, (d) first-pass migration of (a) using constant-velocityStolt migration with 2000-m/s velocity, and (e) second-pass migration of the output from the first-pass migration as in (d)using a 15-degree finite-difference migration with the residual velocity function computed by equation (D-96b).

approximately 12 Hz, only. Finally, at trace spacing of 100 m, which correpsonds to a Nyquist wavenumber of 5 cycles/km, the triangular shape is obliterated, com-pletely (Figure 4.6-2).

Figure 4.6-3 shows the results of Kirchhoff migra-tion of the zero-offset sections in Figure 4.6-1. Fre-quency components that are spatially aliased are per-ceived by migration with a dip different from the actual

dip along the diffraction hyperbola. Normally, energy ismoved in the up-dip direction along the diffraction hy-perbola and is mapped onto the apex. However, in eachmigrated section, the spatially aliased part of the energyis split away from the flanks of the diffraction hyperbolaand mapped onto the regions to the left and right of theflanks. The aliased energy is dispersed. Since each fre-quency component of the aliased energy is perceived tohave a different dip by migration, the displacement of the energy after migration is frequency dependent. The

unaliased portion of the energy is of course mapped onto

the apex. The more frequency components are spatially

aliased, the less energy at lower frequencies is mapped

onto the apex.

As discussed in Section 4.1, the triangular area on

the f −k plane associated with the diffraction hyperbola

(Figure 4.6-2) is mapped onto the circular area on the

f −

k plane associated with the migrated section (Figure4.6-4). Ideally the area in the f −k plane of the migrated

section should be semicircular in shape. Because the

diffraction hyperbola is defined within a finite spatial

aperture (Figure 4.6-1), there is an implicitly imposed

dip limit on migration. As a result, the semicircular area

is notched on either side (Figure 4.6-4).

Spatial aliasing corrupts the semicircular shape of

the f −k spectrum on both ends of the spectrum at the

vicinity of the Nyquist wavenumber. In case of severe



Migration 583

FIG. 4.5-27. Residual migration applied to field data: first pass using constant-velocity Stolt migration (1500 m/s), andsecond pass using a 15-degree finite-difference migration of the result from the first pass.




FIG. 4.5-28. The combined migration results from Figure4.5-27. B = dipping event before and A = dipping eventafter desired migration; D = diffraction; A1, D1 = after thefirst pass; A2, D2 = after the second pass.

undersampling, spatially aliased frequency componentsinvade much of the f −k plane as shown in Figure 4.6-4.

Aside from the spatial aliasing noise, dispersivenoise also is seen on data migrated with a dip-limitedfinite-difference algorithm (Section 4.3). Figure 4.6-5shows migration of a zero-offset section that containsa diffraction hyperbola using the 15-degree implicitfinite-difference method. Note the undermigration of the diffraction hyperbola that is caused by the 15-degreedip limitation, the dispersive noise A that is caused bythe finite-difference approximations, and the spatiallyaliased energy B that splits away from the unaliasedpart that collapses to the apex.

Figure 4.6-6 shows the results of migration of thezero-offset sections in Figure 4.6-1 using an implicitfrequency-space finite-difference scheme (Section 4.4).Note the dispersive noise caused by the finite-differenceapproximations in the section with 12.5-m trace spac-ing. The dispersive noise in the sections with 25-m,50-m, and 100-m trace spacings, however, is attributedlargely to spatial aliasing.

It is instructive to note that the diffraction energyappears slightly undermigrated with 12.5-m trace spac-ing, but is overmigrated with 25-m and 50-m trace spac-ings. As discussed in Sections 4.3 and 4.4, the fidelity of migration by finite-difference schemes is dictated by an

intricately complex interplay between the various pa-rameters — spatial and temporal sampling intervals,dip, frequency, and velocity. Depending on the valuesof these parameters, one scheme may cause undermi-gration in one case and overmigration in another case.

Figure 4.6-7 shows the f −k spectra of the migratedsections in Figure 4.6-6. Note that implicit frequency-space migration can create high-frequency noise beyondthe passband of the input data. Note also that spatialaliasing combined with the inherent dispersive effect of finite-difference schemes corrupt the semicircular shapeof the f − k spectrum on both ends of the spectrum.

Figure 4.6-8 shows the results of the migration of

the zero-offset sections in Figure 4.6-1 using an explicitfrequency-space finite-difference scheme (Section 4.4).There appears to be no aliasing noise in either sectionswith 12.5-m and 25-m trace spacings. Also note that,compared to the results of Kirchhoff migration (Figure4.6-2), there is less aliasing noise in the sections with 50-m and 100-m trace spacings. These observations can beverified by referring to the f − k spectra shown in Fig-ure 4.6-9. Explicit schemes, by the design criterion, at-tenuate energy associated with wavenumbers kx abovea specified cutoff wavenumber defined by a fraction of the Nyquist wavenumber. This effectively removes partof the aliased energy that maps onto the spectral re-gion above the cutoff wavenumber associated with theextrapolation filter for the explicit scheme. Note fromthe f −k spectrum in Figure 4.6-9 that almost all of thealiased energy has been filtered out for the case of the25-m trace spacing. This is why aliasing noise is absentin the corresponding migrated section in Figure 4.6-8.Despite the wavenumber filtering effect of the explicitscheme, however, much of the aliased noise remains inthe sections with the 50-m and 100-m trace spacings.

Figure 4.6-10 shows the results of phase-shift mi-gration of the zero-offset sections in Figure 4.6-1, andFigure 4.6-11 shows the corresponding f − k spectra.

These results are used as a benchmark to evaluate theresults obtained from the other migration algorithms(Figures 4.6-3 through 4.6-9). Except for the aliasingnoise, phase-shift migration produces no artifacts.

The experiments described above clearly demon-strate that all migration algorithms suffer from saptialaliasing. We now examine the effect of spatial alias-ing on migration using a dipping events model. Figure4.6-12 shows a zero-offset section that contains a set of dipping events with 3500-m/s velocity and 25-m trace




Migration 585

FIG. 4.5-29. Multiple passes of residual migration: (a) a zero-offset section that contains dipping events with 3500-m/svelocity, (b) desired migration using the phase-shift method with the medium velocity of 3500 m/s, (c) first-pass migrationof the zero-offset section in (a) using the phase-shift method with a constant velocity of 2500 m/s, (d) second-pass migrationof the output from the first-pass migration as in (c) using the phase-shift method with a constant residual velocity of 1145m/s, (e) third-pass migration of the output from the second-pass migration as in (d) using the phase-shift method with aconstant residual velocity of 1198 m/s, (f) fourth-pass migration of the output from the third-pass migration as in (e) usingthe phase-shift method with a constant residual velocity of 1250 m/s, and (g) fifth-pass migration of the output from thefourth-pass migration as in (f) using the phase-shift method with a constant residual velocity of 1299 m/s.




FIG. 4.5-30.

Multiple passes of residual migration: (a) a zero-offset section that contains dipping events with 3500-m/svelocity, (b) desired migration using the phase-shift method with the medium velocity of 3500 m/s, (c) first-pass migrationof the zero-offset section in (a) using the 65-degree implicit method with a constant velocity of 2500 m/s, (d) second-passmigration of the output from the first-pass migration as in (c) using the 65-degree implicit method with a constant residualvelocity of 1145 m/s, (e) third-pass migration of the output from the second-pass migration as in (d) using the 65-degreeimplicit method with a constant residual velocity of 1198 m/s, (f) fourth-pass migration of the output from the third-passmigration as in (e) using the 65-degree implicit method with a constant residual velocity of 1250 m/s, and (g) fifth-passmigration of the output from the fourth-pass migration as in (f) using the 65-degree implicit method with a constant residualvelocity of 1299 m/s.



Migration 587

FIG. 4.6-1. Zero-offset sections, which contain a diffraction hyperbola with 2500-m/s velocity, with trace spacings, from topto bottom, 12.5 m, 25 m, 50 m, and 100 m.



Migration 589

FIG. 4.6-3. Kirchhoff migrations of the zero-offset sections in Figure 4.6-1 with trace spacings, from top to bottom, 12.5 m,25 m, 50 m, and 100 m.




FIG. 4.6-4. The f − k spectra of the outputs from the Kirchhoff migrations in Figure 4.6-3 with trace spacings, from top tobottom, 12.5 m, 25 m, 50 m, and 100 m.



Migration 591

FIG. 4.6-5. 15-degree finite-difference migrations of the zero-offset sections in Figure 4.6-1 with trace spacings, from top tobottom, 12.5 m, 25 m, 50 m, and 100 m.




FIG. 4.6-6. 65-degree implicit frequency-space migrations of the zero-offset sections in Figure 4.6-1 with trace spacings, fromtop to bottom, 12.5 m, 25 m, 50 m, and 100 m.



Migration 593

FIG. 4.6-7. The f − k spectra of the outputs from the 65-degree implicit frequency-space migrations in Figure 4.6-6 withtrace spacings, from top to bottom, 12.5 m, 25 m, 50 m, and 100 m.




FIG. 4.6-8. 70-degree explicit frequency-space migrations of the zero-offset sections in Figure 4.6-1 with trace spacings, fromtop to bottom, 12.5 m, 25 m, 50 m, and 100 m.



Migration 595

FIG. 4.6-9. The f − k spectra of the outputs from the steep-dip explicit frequency-space migrations in Figure 4.6-8 withtrace spacings, from top to bottom, 12.5 m, 25 m, 50 m, and 100 m.




FIG. 4.6-10. Phase-shift migrations of the zero-offset sections in Figure 4.6-1 with trace spacings, from top to bottom,12.5 m, 25 m, 50 m, and 100 m.



Migration 597

-40 40

20

40

60

Hz

0 cycles/km

-20 20

20

40

60

Hz

0

-10 10

20

40

60

Hz

0

- 5 5

20

40

60

Hz

0

FIG. 4.6-11. The f − k spectra of the outputs from the phase-shift migrations in Figure 4.6-10 with trace spacings, fromtop to bottom, 12.5 m, 25 m, 50 m, and 100 m.




FIG. 4.6-12. Zero-offset sections, which contain dipping events with 3500-m/s velocity, with trace spacings, from top tobottom, 25 m, 50 m, and 100 m.




FIG. 4.6-14. Kirchhoff migrations of the zero-offset sections in Figure 4.6-12 with trace spacings, from top to bottom, 25 m,50 m, and 100 m.




FIG. 4.6-16. 15-degree finite-difference migrations of the zero-offset sections in Figure 4.6-12 with trace spacings, from topto bottom, 25 m, 50 m, and 100 m.



Migration 603

FIG. 4.6-17. 65-degree implicit frequency-space migrations of the zero-offset sections in Figure 4.6-12 with trace spacings,from top to bottom, 25 m, 50 m, and 100 m.




FIG. 4.6-18. The f − k spectra of the outputs from the 65-degree implicit frequency-space migrations in Figure 4.6-17 withtrace spacings, from top to bottom, 25 m, 50 m, and 100 m.



Migration 605

FIG. 4.6-19. 70-degree explicit frequency-space migrations of the zero-offset sections in Figure 4.6-12 with trace spacings,from top to bottom, 25 m, 50 m, and 100 m.




FIG. 4.6-20. The f − k spectra of the outputs from the steep-dip explicit frequency-space migrations in Figure 4.6-19 withtrace spacings, from top to bottom, 25 m, 50 m, and 100 m.



Migration 607

FIG. 4.6-21. Phase-shift migrations of the zero-offset sections in Figure 4.6-12 with trace spacings, from top to bottom,25 m, 50 m, and 100 m.




FIG. 4.6-22. The f − k spectra of the outputs from the phase-shift migrations in Figure 4.6-21 with trace spacings, fromtop to bottom, 25 m, 50 m, and 100 m.



Migration 609

spacing. By discarding every other trace, obtain anotherzero-offset section with 50-m trace spacing. Repeat theprocedure to obtain the zero-offset section with 100-mtrace spacing (Figure 4.6-12).

The f − k spectra of the zero-offset sections with

the three different trace spacings are displayed in Figure4.6-13. The dipping events with 25-m trace spacing maponto a series of radial lines in the f − k plane (Section4.1). The Nyquist wavenumber is 20 cycles/km and thebandwidth is given by the corner frequencies 6, 12 - 36,48 Hz for the passband region of the spectrum. As forthe diffraction hyperbola model (Figure 4.6-3), the redis associated with the flat part of the passband regionand the blue is associated with the taper zone.

The f − k spectrum of the zero-offset section with50-m trace spacing (Figure 4.6-12), which correspondsto a Nyquist wavenumber of 10 cycles/km, indicatesspatial aliasing beyond approximately 24 Hz (Figure

4.6-13). Consequently, the aliased segments of the radiallines map onto the left quadrant of the f − k spectrum.At a coarser trace spacing of 100 m, which correspondsto a Nyquist wavenumber of 5 cycles/km, spatial alias-ing occurs first at approximately 12 Hz. Then, someof the energy already aliased becomes aliased for thesecond time at approximately 36 Hz. Moreover, part of the aliased energy is remapped onto the right quadrant(Figure 4.6-13).

Figure 4.6-14 shows the results of Kirchhoff mi-gration of the zero-offset sections in Figure 4.6-12. Noaliasing noise is present on the migrated section with25-m trace spacing. Next, consider the migrated sec-

tion with 50-m trace spacing. Frequency componentsthat are spatially aliased are perceived by migration todip in the direction opposite to the actual dips of theevents. Normally, energy is moved in the up-dip direc-tion, in this case from right to left as seen in the mi-grated section with 25-m trace spacing. However, in themigrated section with 50-m trace spacing, the spatiallyaliased part of the energy is split away from the dip-ping events and moved from left to right. Note that thealiased energy is dispersed. As for the diffraction hyper-bola model (Figure 4.6-2), each frequency component of the aliased energy is perceived to have a different dip

by migration, the displacement of the energy after mi-gration is frequency dependent. The unaliased portionof the energy is of course moved from right to left andpositioned accurately. The more frequency componentsare spatially aliased, the less energy at lower frequenciesis mapped to the correct position.

Finally, consider the case of the migrated sectionwith 100-m trace spacing (Figure 4.6-14). Note thatthere exists aliasing noise not only to the right of thedipping events but also in the left-most portion of thesection. The latter is associated with the energy that has

been aliased twice (Figure 4.6-3). Because of the com-plexity of aliasing, the noise essentially disperses overthe whole of the section.

An interesting observation on the migrated sectionsin Figure 4.6-14 relates to the energy in the region above

1 s. In Kirchhoff migration, amplitudes in the input sec-tion are summed along a hyperbolic summation pathand placed at the apex of the hyperbola. Imagine a sum-mation path whose apex is situated at a time less than 1s. There will be some energy placed at this apex locationsince the flanks of the summation path under consider-ation will intersect through traces with nonzero samplevalues. This situation is encountered when migratingmarine data using Kirchhoff summation. Normally, themigrated section is muted above the water bottom toremove the noise created by migration within the waterlayer.

Besides data aliasing, there is also the problem of

operator aliasing. In particular, for a low-velocity hy-perbola or for a hyperbola with its apex situated atshallow times, Kirchhoff summation may require morethan one sample per trace. This results in some energyin the form of precursors above the migrated sea-bottomreflection, when only one point per trace is included inthe summation.

Figure 4.6-15 shows the f − k spectra of the mi-grated sections in Figure 4.6-14. As discussed in Section4.1, migration rotates the radial lines on the f −k planeassociated with the dipping events.

Figure 4.6-16 shows migration of a zero-offset sec-tion that contains a set of dipping events using the 15-

degree implicit finite-difference method. Note the un-dermigration of the steeply dipping events caused by the15-degree dip limitation, the dispersive noise A causedby the finite-difference approximations, and the spa-tially aliased energy B that splits away from the un-aliased part and moves in the opposite direction.

Figure 4.6-17 shows the results of migration of thezero-offset sections in Figure 4.6-12 using an implicitfrequency-space finite-difference scheme (Section 4.4).Note the dispersive noise caused by the finite-differenceapproximations in the section with 25-m trace spacing.In the sections with 50-m and 100-m trace spacings, twosets of dispersive noise can be distinguished — one that

is caused by the finite-difference approximations andthe other caused by spatial aliasing. At coarse spatialsampling, the steeply dipping events are faintly detectedon the migrated section.

Figure 4.6-18 shows the f − k spectra of the mi-grated sections in Figure 4.6-17. As in the case of thediffraction hyperbola (Figure 4.6-7), implicit frequency-space migration can create high-frequency noise beyondthe passband of the input data.

Figure 4.6-19 shows the results of the migration of the zero-offset sections in Figure 4.6-12 using an explicit




frequency-space finite-difference scheme (Section 4.4).There is no aliasing noise in the section with 25-m tracespacing. But there is precursive dispersion along thesteeply dipping events because of the inherent natureof the explicit scheme used here. This dispersion is not

as severe as that observed on the result from the implicitscheme (Figure 4.6-17).The corresponding f − k spectra shown in Figure

4.6-20 explains why there is less aliasing noise on themigrated sections in Figure 4.6-19 compared to thosefrom Kirchhoff summation (Figure 4.6-14). As for thecase of the diffraction hyperbola (Figure 4.6-9), explicitschemes attenuate energy associated with wavenumberskx above a specified cutoff wavenumber. This effectivelyremoves part of the aliased energy that maps onto thespectral region above the cutoff wavenumber associatedwith the extrapolation filter for the explicit scheme.Note from the f − k spectrum in Figure 4.6-20 that

a significant portion of the aliased energy in the leftquadrant has been filtered out for the case of the 50-mtrace spacing. Despite the wavenumber filtering effect of the explicit scheme, however, much of the aliased noiseremains in the section with the 100-m trace spacing.

Figure 4.6-21 shows the results of the phase-shiftmigration of the zero-offset sections in Figure 4.6-12,and Figure 4.6-22 shows the corresponding f −k spectra.These results are used as a benchmark to evaluate theresults obtained from the other migration algorithms(Figures 4.6-14 through 4.6-20). Except for the aliasingnoise, phase-shift migration produces no artifacts.

The effect of spatial aliasing on migration of field

data, to begin with, is demonstrated in Figures 4.6-23and 4.6-24. We see the original stacked section and itsresampled versions at coarser trace spacings. From themigrations of these four stacked sections with coarsertrace spacings, note the loss of spatial resolution. Thenearly flat events are not adversely affected by spatialaliasing, while the steeply dipping reflection off the rightflank of the salt diapir can only be detected on themigrated section with very coarse sampling by a low-frequency, weak-amplitude event. The diffraction en-ergy off the tip of the salt diapir is largely dispersedinto the region with nearly flat events to the right of the diapir.

We now examine the response of the various migra-tion algorithms to spatial alaising using the data shownin Figures 4.6-25 and 4.6-26. The stacked data are asso-ciated with the same line sampled at four different tracespacings — 12.5, 25, 50, and 100 m. Figure 4.6-27 showsmigrations of the stacked sections using Kirchhoff sum-mation. The section with 12.5-m trace spacing providesa crisp image of the salt diapir, while the sections withcoarser trace spacings degrade gradually. Specifically, itis almost impossible to delineate the salt boundary on

the section with 50-m trace spacing, and the sectionwith 100-m trace spacing does not even provide an im-age of the gently dipping reflections. This is because of the aliasing noise associated with the steep flanks of thesalt diapir corrupting the surrounding reflections. Spa-

tial aliasing not only adversely affects the quality of theimage associated with a dipping event that is aliased,but it also can obliterate other nonaliased events in thedata.

Figure 4.6-28 through 4.6-30 show migrations of the stacked sections in Figures 4.6-25 and 4.6-26 usingsteep-dip frequency-space implicit and explicit schemes,and the phase-shift method. Similar conclusions aredrawn for the Kirchhoff summation results shown inFigure 4.6-27. Differences in terms of delineation of the salt boundary and the surrounding strata are at-tributable to the manner in which these algorithms areimplemented and how they treat the velocity field for

migration. For instance, when examining the resultsfrom the frequency-space implicit scheme (Figure 4.6-28), one must keep in mind the effect of spatial aliasingcombined with the effect of undermigration caused bythe dip-limited nature of the algorithm and the effect of dispersion caused by finite-difference approximations.

What is the remedy for spatial aliasing noise inmigration? Arrange the sequence of the sections in Fig-ures 4.6-27 through 4.6-30 from a coarser to a finer tracespacing. Note that the deleterious effect of spatial alias-ing in migration disappears as we go to finer trace spac-ings. To avoid spatial aliasing, we must record with suf-

ficiently fine CMP trace interval or interpolate the datathat have been recorded with coarse spatial sampling.Most modern surveys are conducted using spatial

sampling rates that are perfectly adequate to meet ex-ploration and development objectives. If we are dealingwith vintage data with coarse spatial sampling, thereare two ways to circumvent the effect of spatial alias-ing. The first approach would be to filter out the aliasedfrequencies. This is undesirable, since it severely limitsvertical and lateral resolutions (Section 11.1). The sec-ond approach would be to do trace interpolation beforemigration. In Sections 7.2 and G.5, we discuss interpo-lation of aliased data. Note from section 1.3 that the

smaller the trace interval, the higher the Nyquist inthe spatial wavenumber direction (Figures 1.3-10 and1.3-11), and thus, the less likelihood of aliasing high-frequency data.

A schematic illustration of the spatial aliasing phe-nomenon is shown in Figure 4.6-31. Start with thespectral bandwidth that spans COA in the spatialwavenumber axis, where A is the location of the Nyquistwavenumber, and ON in the temporal frequency axis,where N is the location of the Nyquist frequency. Dip




Migration 611

FIG. 4.6-23. A portion of a CMP-stacked section spatially sampled at four different trace spacings.




FIG. 4.6-24. Phase-shift migrations of the stacked sections in Figure 4.6-23 sampled at four different trace spacings.



Migration 613

FIG. 4.6-25. A CMP-stacked section spatially sampled at 12.5-m (top) and 25-m (bottom) trace spacings.




FIG. 4.6-26. A CMP-stacked section as in Figure 4.6-25 spatially sampled at 50-m (top) and 100-m (bottom) trace spacings.



Migration 615

FIG. 4.6-27. Kirchhoff migrations of the stacked sections in Figure 4.6-25 and 4.6-26 with trace spacings, from top to bottom,12.5 m, 25 m, 50 m, and 100 m.




FIG. 4.6-28. 65-degree implicit frequency-space migrations of the stacked sections in Figure 4.6-25 and 4.6-26 with tracespacings, from top to bottom, 12.5 m, 25 m, 50 m, and 100 m.



Migration 617

FIG. 4.6-29. 70-degree explicit frequency-space migrations of the stacked sections in Figure 4.6-25 and 4.6-26 with tracespacings, from top to bottom, 12.5 m, 25 m, 50 m, and 100 m.




FIG. 4.6-30. Phase-shift migrations of the stacked sections in Figure 4.6-25 and 4.6-26 with trace spacings, from top tobottom, 12.5 m, 25 m, 50 m, and 100 m.



Migration 619

FIG. 4.6-31. Two dipping events in the f − k domain. Seetext for details.

components 1 and 2 are aliased beyond frequency val-

ues AT and AS , respectively. Extend the wavenumberaxis to DOB by making the trace interval half of theoriginal. Event 1 no longer is spatially aliased withinthe frequency bandwidth ON . Event 2 still is aliasedbeyond the frequency value BV . However, at this pointand beyond, there may be no significant energy, so fur-ther extension of the wavenumber axis may not be nec-essary. Another important point is that if the temporalfrequency band only extended up to OG to start with,extension of the wavenumber axis to DOB also wouldresult in Event 2 being unaliased. Thus, the amountof trace interpolation that is required also depends ontemporal bandwidth as well as on structural dip.

Trace interpolation often is necessary when dealingwith 3-D data and old data recorded with a large groupinterval. In a typical 3-D survey, the inline trace intervalmay be as little as 12.5 m, while the trace interval in thecrossline direction, for some old data, may be as muchas 100 m. Therefore, interpolation is required before mi-gration in the crossline direction. You do not necessarilyinterpolate down to the inline trace spacing; instead, de-pending on the maximum structural dip and velocity inthe area, the optimum trace spacing for interpolation inthe crossline direction can be computed using equation(1-8). Section 7.2 provides more information on traceinterpolation in relation to 3-D migration.

Migration and Random Noise

Figure 4.6-32 shows a section that contains band-limited random noise uncorrelated from trace to traceand its migration using the phase-shift method. Veloc-ity increases linearly from 2000 m/s at the top to 4000m/s at the bottom of the section. The amplitude andfrequency characteristics of the input section in Figure

FIG. 4.6-32. Response of migration to random noise: (a)zero-offset section with random noise, only, (b) frequency-wavenumber migration.

FIG. 4.6-33. (a) A deeper portion of a CMP-stacked sec-tion with significant noise level, (b) the same portion aftermigration.




FIG. 4.6-34. Migrations of a portion of a CMP stack with different lateral extents.

FIG. 4.6-35. Edge effects in migration: (a) a portion of a CMP stack, (b) the same portion after migration, and (c) a sketchthat illustrates the edge effect problem using the semicircle superposition method of migration. See text for A, B, C, and D.



Migration 621

FIG. 4.6-36. A CMP stack (a) before and (b) after migration.

4.6-32 are virtually unchanged in the interior portionof the migrated section. However, note the smearingof amplitudes at the bottom and side boundaries aftermigration.

Ambient noise commonly dominates the deep por-tion of a stacked section where velocities are high.Therefore, organization of random noise caused by mi-gration generally is more severe in the deeper part of astacked section. A field data example is shown in Fig-ure 4.6-33. In addition to smearing effects, the migratedsection also has smiles , which are caused by sparsely dis-tributed bursts of amplitude in the input section. Keepin mind that a single spike on the time section migrates

to a semicircle on the depth section.We already have seen the adverse effect of an im-

proper choice of aperture width in Kirchhoff summation(Figure 4.2-7). A narrow aperture can introduce strongsmearing as spurious, nearly horizontal events. A similareffect occurs for all types of migration algorithms if themaximum dip to migrate is severely restricted (Figure4.5-4). It is a misconception to imagine that migrationattenuates random noise and improves signal-to-noiseratio. Instead, one must keep in mind that migration

organizes random noise, it does not attenuate it. A dip-limited migration algorithm acts upon the random noiselike a dip filter and removes the noise energy beyondthe dip limit much like shown in Figure 6.2-2. The dip-limited algorithm also attenuates unaliased linear noisewith a dip steeper than the dip limit.

Migration and Line Length

For one reason or another, a seismic line may have tobe recorded in the field with a shorter length than de-sired. To see the effect of line length on migration, we

will examine the migrations of the decreasing lengths of the same CMP stack (Figure 4.6-34). Migration of thesmaller portions, BD and CD , results in an increasinglysmeared section, particularly in the deeper parts. Weconclude that short seismic lines really are not suitedfor migration.

If the line traverse is too short, two effects occur.First, there is not enough space in the section for dip-ping events to move during migration. This problemmay be alleviated by padding the stacked section with



Migration 623




FIG. 4.6-39. A sketch of the events after migration from the flat reference datum level as in Figure 4.6-37 denoted by thedotted segments and migration from the floating datum as in Figure 4.6-38 denoted by the solid segments. The area coveredcorresponds to the upper central portions of the sections in Figures 4.6-37 and 4.6-38 between midpoints A and B.

zero traces on both sides b efore migration. Second, sideboundary effects contaminate a significant portion of the migrated section. The real solution to circumventthe boundary effects is to record data with sufficientline length.

With a general idea of structural dip in an area,the geophysicist must consider the additional spatialextent that is required by migration. (Refer to the dis-cussion on Figure 4.1-1.) This is especially important in

3-D surveys in which the surface areal coverage must beextended beyond the subsurface areal coverage so thatsteep dips and structural discontinuities can be recordedand imaged properly (Section 7.1). The problem with3-D is that cost increases as the square of the surveydimension, so that temptation to record too small asurvey is great.

Regardless of line lengths, there are additionalproblems associated with the side boundaries of thestacked section input to migration. All migration al-gorithms implicitly make some assumption about thenature of data outside the side boundaries of the inputstacked section. The simple assumptions, zero ampli-

tude or zero gradient at the side boundaries of the sec-tion, cause data that should migrate past the edge to bereflected back into the section. To prevent this, traces of zero amplitude often are appended to the edges of theinput section. This allows the dipping events to movefreely into the zero-amplitude region during migration.If the events that would migrate off the input section arenot needed, they often are suppressed by using absorb-ing side boundary conditions (Clayton and Engquist,1980).

Figure 4.6-35 shows a section with significantsmearing caused by side boundary effects. The wave-front character that dominates the left boundary of themigrated section down to the bottom of the mute zonecan be explained using the principle of semicircle super-position for migration. Consider a dipping event A thatextends down to the edge of the section as in the sketchin Figure 4.6-35. After migration B , note the remainderof the semicircular wavefront C on the left side. This

wavefront did not cancel out during superposition be-cause no data were available beyond the left boundaryof the section.

Another source of edge effects is the presence of am-plitude bursts at or near the edge of the stacked sectionassociated with a low signal-to-noise ratio that resultsfrom low fold. The edge effects on the left boundarybelow the mute zone in the migrated section (Figure4.6-35) probably stem from the lack of amplitude bal-ance on the CMP stacked section. The latter is causedby changes in fold at the end of the line.

Figure 4.6-36 shows a CMP stack with an imbri-cate structure associated with overthrust tectonics. Af-

ter migration, note that there are two zones with noreflections. The zone to the left of CMP 100 resultedfrom finite line length. Specifically, the events on theleft flank of the imbricate structure are migrated to theright in the up-dip direction, thus leaving behind a zoneof no events into which no energy is moved since the lineends to the left of the structure. The zone of no eventsbetween CMP 200 and 300 is a direct consequence of the overthrusting that has given rise to a culminatingstructure with very steep, almost overturned events. Mi-



Migration 625

FIG. 4.6-40. Principles of migration from topography. See text for details.




gration of such steep dips is possible only if they existin the recorded data and are imaged using algorithmswhich handle dips beyond 90 degrees.

Migration from Topography

A CMP-stacked section is assumed to be equivalent toa zero-offset wavefield and usually is referenced to a flatdatum. Figure 4.6-37 shows a migrated CMP stackedsection associated with a seismic line that follows a tra-verse with severe topography from an area with over-thrust tectonics. Migration was done from a flat refer-ence datum. When migrating data recorded over suchan irregular and severe topography, however, one needsto account for the difference between the elevation pro-file and the reference datum. Otherwise, events appearto a migration algorithm shallower than they actuallyare if the flat reference datum is below the elevationprofile, and thus are undermigrated. If the flat referencedatum is above the elevation profile, then events appearto a migration algorithm deeper than they actually are,and thus are overmigrated.

Figure 4.6-38 shows the same data as in Figure4.6-37 with migration from the floating datum whichis a smooth form the elevation profile. A sketch of keyevents between midpoints A and B from both sectionsis provided in Figure 4.6-39. The flat reference datum inthis case is above the elevation profile. Hence, migrationfrom the flat datum causes overmigration as denoted bythe dotted interpretation segments. By migrating thedata from the floating datum and interpreting the re-sulting section, we obtain the solid segments in Figure4.6-39.

Migration algorithms, with the exception of Kirch-hoff summation and the constant-velocity Stolt method,are all based on wave extrapolation from one flat depthlevel to another. To accommodate an irregular topog-raphy, the following formal approach can be used:

(a) Stack the data referenced to the floating datum andassume it to be the zero-offset wavefield recordedalong the floating datum profile.

(b) Apply wave-equation datuming (Section 8.1) to ex-trapolate the zero-offset wavefield as defined in (a)from the floating datum to a flat datum above usinga velocity the same as that just below the floatingdatum.

(c) Migrate the output wavefield from (b) using a pre-ferred migration algorithm.

The stacking velocity field required in step (a) isreferenced to the floating datum. The migration ve-locity field required in step (c) would need to be de-rived by redefining the stacking velocities with respect

to the flat datum. To circumvent this tedious task, mi-gration from an irregular topography is done either bythe zero-velocity trick (Beasley and Lynn, 1992) or thezero-wavefield trick (Reshef, 1991).

In the first approach by Beasley and Lynn (1992),

a zero velocity value is assigned to the region betweenthe floating datum and the flat datum, which is specifiedabove the floating datum. Just as it should be in con-ventional processing, the stacked section is referenced tothe flat datum, and the velocities are referenced to thefloating datum. Extrapolate the stacked section downone depth step using the zero velocity as part of themigration process. This amounts to a simple verticaltime shift. If the depth level intersects the floating da-tum profile, then invoke the diffraction term (equation4-16a) for the traces in the stacked section that coin-cide with the intersection points. Continue the extrap-olation process from one depth level to the next while

turning on the diffraction term for those traces whichcoincide with the intersection points of the depth levelsand floating datum profile.

In the second approach by Reshef (1991), to startwith, a zero wavefield is assigned to the flat datum levelz0 in Figure 4.6-40. The stacked section and the veloc-ities are referenced to the floating datum. Extrapolatethis wavefield down one depth level to z1 using a ve-locity the same as that just below the floating datumprofile. Import the traces 1 and 2 from the stacked sec-tion at the intersection points of the depth level and thefloating datum profile, and insert them to the section

referenced to the depth level z1. Now, extrapolate downto the next depth level z2, and import traces 3, 4, 5, and6 from the stacked section, and insert them into sectionreferenced to the depth level z2. At each depth level,while new traces are imported from the original stackedsection, the previously imported traces are subsjectedto wave extrapolation as symbolized by the small arcsassociated with each time sample.

EXERCISES

Exercise 4-1. Consider the special case of a 90-degree dipping reflector in Figure 4.1-1a. Sketch the cor-responding zero-offset time section.

Exercise 4-2. Consider Huygens’ secondarysources along a dipping reflector. Sketch the zero-offsetsection by superimposing the individual responses fromthese sources. Remember, to do zero-offset modeling,you must map a point in the x− z plane to a hyperbolain the x − t plane with its apex as the input point.

Exercise 4-3. For which case is spatial aliasing amore serious problem, the low-velocity or high-velocitymedium?



Migration 627

FIG. 4.E-1. See Exercise 4-4.

Exercise 4-4. Locate the dipping event AA on

the migrated section in Figure 4.E-1.

Exercise 4-5. A point in the x−t plane is mapped

onto a semicircle in the x− z plane. Where does it map

on the x−

τ plane, where τ = 2z/ v.Exercise 4-6. Refer to Figure 4.1-14. It suggests

that if the subsurface consisted of a semicircular reflec-

tor (b), then the zero-offset response would be as in

(a). What would the subsurface be like if you obtained

(a) using a source-receiver pair with a finite separation

between them? (See Figures D-5 and D-6.)

Exercise 4-7. Suppose you specified the wrongtrace spacing in your migration. What effect does ithave, overmigration or undermigration? Assume thatyou supplied the wrong sampling rate in time. Whateffect does it have on migration output?

Exercise 4-8. Suppose you want to do zero-offsetrecording of the steep flank of a salt dome. Which casewould require a longer line length when the medium ve-locity along the raypath is (a) constant or (b) verticallyincreasing?

Exercise 4-9. How would Figure 4.3-6 look if youused a 15-degree phase-shift migration algorithm?



Appendix D

MATHEMATICAL FOUNDATION OF MIGRATION

D.1 Wavefield Extrapolation and Migration

A fundamental equation of reflection seismology is the double-square-root (DSR) equation. This

equation describes downward continuation of both shots and receivers into the earth. It is exact

for all dips and offsets. Neglecting the velocity gradient dv(z)/dz makes the DSR equation also

applicable to a stratified earth. The DSR equation can be extended, with some approximation,

to treat weak lateral velocity variations. A comprehensive mathematical treatise of the DSR

equation is found in Claerbout (1985).

The basic 2-D theory for wavefield extrapolation is presented here. Then, using the DSR

equation, a rigorous analysis of conventional seismic data processing is made. We show that

conventional implementation of the DSR equation requires zero-dip and zero-offset assumptions.

Before discussing the DSR equation, we review the basic theory for wavefield extrapolation.

Once the extrapolation equations are developed, they can be used with the imaging principle to

migrate 2-D or 3-D prestack and poststack data.Start with the 2-D scalar wave equation, which describes propagation of a compressional

wavefield P (x, z, t) in a medium with constant material density and compressional wave velocityv(x, z):

∂ 2

∂x2+

∂ 2

∂z2− 1

v2∂ 2

∂t2

P (x, z, t) = 0, (D − 1)

where x is the horizontal spatial axis, z is the depth axis (positive downward), and t is time.

Given the upcoming seismic wavefield P (x, 0, t), which is recorded at the surface, we want to

determine reflectivity P (x,z, 0). This requires extrapolating the surface wavefield to depth z,

then collecting it at t = 0. The process of obtaining the earth’s reflectivity P (x, z, t = 0) from

the observed wavefield P (x, z = 0, t) at the surface z = 0 is called migration , and the reverse

process is called modeling (Figure D-1).

It is advantageous to decompose the wavefield into monochromatic plane waves with dif-

ferent angles of propagation from the vertical. Therefore, we will work in the Fourier transform

domain whenever possible. The wavefield can always be Fourier transformed over time t. If

there is no lateral velocity variation, then the wavefield also can be Fourier transformed over

the horizontal axis x. Thus,

P (kx, z , ω) =

P (x, z, t) exp(ikx − iωt) dx dt, (D − 2a)

and inversely,

P (x, z, t) = P (kx, z , ω) exp(−ikx + iωt) dkx dω. (D − 2b)

When the differential operator in equation (D-1) is applied to equation (D-2b), we get

∂ 2

∂z2P (kx, z , ω) +

ω2

v2− k2x

P (kx, z , ω) = 0. (D − 3)

Although v can be varied with depth z in equation (D-3), for now we assume a constant

velocity case. The stratified earth case is considered later in this appendix. Equation (D-3) has

two solutions, one for upcoming waves, the other for downgoing waves. The upcoming wave

solution to equation (D-3) is recognized as



Mathematical Foundation of Migration 629

FIG. D-1. Relationship between migration and wavefield modeling (see Section D.1).

P (kx, z , ω) = P (kx, z = 0, ω) exp−i ω2

v2 − k2

xz. (D − 4)

Equation (D-4) also is the solution to the following one-way wave equation:

∂

∂zP (kx, z , ω) = −i

ω2

v2− k2xP (kx, z , ω). (D − 5)

This solution can be verified by substituting equation (D-4) into equation (D-5).We define the vertical wavenumber as

kz =ω

v

1 −

vkx

ω

2. (D − 6)

Equation (D-6) often is called the dispersion relation of the one-way scalar wave equation. Byusing this expression, equation (D-4) takes the simple form

P (kx, z , ω) = P (kx, z = 0, ω) exp(−ikzz). (D − 7)

To determine the reflectivity P (x,z, 0) from the wavefield recorded at the earth’s surfaceP (x, 0, t), proceed as follows:

(a) Perform a 2-D Fourier transform over x and t to get P (kx, 0, ω).(b) Multiply by the all-pass filter exp(−ikzz) to obtain the wavefield P (kx, z , ω) at depth z.(c) Perform summation over ω to obtain P (kx, z, 0).(d) Finally, inverse Fourier tranform over kx to obtain the earth’s image P (x,z, 0) at that depth.

For the constant velocity case P (kx, kz, 0) can be computed by a direct mapping in thetransform domain from (kx, ω) to (kx, kz) using equation (D-6) (Stolt, 1978).

The main objective here is to interpret equation (D-7) as a tool for downward extrapolatingwavefields given at the surface. While the mathematical development of the process presentedis simple, its physical basis is not obvious. In an effort to develop a physical motivation forequation (D-7), a simpler derivation follows.

Given the upcoming wavefield P (x, 0, t) recorded at the surface, we can decompose it intomonochromatic plane waves, each traveling at a different angle from the vertical. We identifythese plane waves by attaching each one to a unique (kx, ω) pair. This plane-wave decompositionis equivalent to Fourier transforming the wavefield to yield P (kx, 0, ω).

Now consider one of these plane waves as shown in Figure D-2. Imagine that this plane wavepassed point P at t = 0, traveled upward, and was recorded by a receiver at surface point G at




FIG. D-2. Geometry for wavefield extrapolation (see Section D.1) (Yilmaz, 1979).

time t. For reflector mapping, we need to take the energy located at point G on the wavefront attime t, back to its position at t = 0 — to reflection point P . To return the energy to P , it makessense to follow the same raypath used for outward propagation from P . The fact that the samepath is used means that downward continuation does not alter the horizontal wavenumber kx.

Suppose that the wavefront is moved to a depth ∆z = GG

beneath the receiver at G sothat the waveform at G now is at G

. If a receiver were buried at G

, it would have recordedthe plane wave at t−∆t, where ∆t is the traveltime between G and G

. In other words, movingthe receiver at G vertically down a distance ∆z to a new location G

changes the traveltime

along the raypath from G to P by −∆t.From the geometry of Figure D-2, we have

∆t =∆z

vcos θ, (D − 8a)

where v/ cos θ is the vertical phase velocity. We know the kx and ω values for the plane wave.Suppose the distance between G and G

is one wavelength λ. At time t − ∆t, the wavefrontintersects the x-axis at distance λx from G. From the geometric relation in Figure D-2, we have

λ

λx= sin θ. (D − 8b)

By using the definitions λ = 2π/ (ω/v), λx = 2π/ kx, and equation (D-8b), we obtain

sin θ =

vkx

ω (D − 8c)and

cos θ =

1 −

vkx

ω

2, (D − 8d)

where ω/v is the wavenumber along the raypath. By substituting equation (D-8d) into equation(D-8a), we have

∆t =1

v

1 −

vkx

ω

2∆z. (D − 8e)




As we move down, we do not want to change the amplitude of the plane wave. Given thechange in traveltime, ∆t by equation (D-8e), the corresponding phase shift is −ω∆t. At each∆z step of descent, we may propagate the plane wave with a different velocity v(z). The totalphase shift to which the waveform was subjected with arrival at P is − ω dt.

To compute the wavefield at P , we use equation (D-8e) and multiply the transformed surface

wavefield P (kx, 0, ω) by

exp

−i

P G

ω dt

= exp

−i

z0

ω

v(z)

1 −

v(z) kx

ω

2dz

. (D − 9)

Equation (D-9) is the same operator used in equation (D-7), except that equation (D-7) wasderived for constant v.

We now return to the more mathematical discussion and consider the stratified earth with avelocity v(z). Since we have not Fourier transformed P (x, z, t) over z, the one-way wave equation(D-5) also is valid for v(z):

∂

∂zP (kx, z , ω) = −i

ω2

v2(z)− k2x P (kx, z , ω), (D − 10)

in which case equation (D-6) becomes

kz(z) =ω

v(z)

1 −

v(z)kx

ω

2. (D − 11)

Substitution verifies that equation (D-10) has the following solution:

P (kx, z , ω) = P (kx, 0, ω) exp

−i

z0

kz(z) dz

. (D − 12)

We must check whether this solution satisfies the two-way scalar wave equation (D-3). Bydifferentiating equation (D-10) and using equation (D-11), we have

∂ 2

∂z2

P =

−i

dkz(z)

dv

dv(z)

dz

P

−ikz(z)

∂

∂z

P, (D

−13a)

where P = P (kx, z , ω). By substituting equation (D-10) for ∂P/dz, we obtain

∂ 2

∂z2P = −i

dkz(z)

dv

dv(z)

dzP − k2z(z) P. (D − 13b)

If the velocity gradient dv(z)/ dz is ignored, then the first term on the right side drops out. Thefinal expression then is

∂ 2

∂z2P + k2z(z) P = 0. (D − 13c)

When equation (D-11) is substituted into this expression, we get:

∂ 2

∂z2

P + ω2

v2

(z) −k2xP = 0, (D

−14)

which is identical to equation (D-3) where velocity can be varied with depth z.So far, we have shown that a 2-D wavefield recorded at the earth’s surface can be extrap-

olated downward using the phase-shift operator given by equation (D-9). Wave extrapolationcan be done through either a constant-velocity medium (equation D-7) or a vertically varyingvelocity medium (equation D-12). Seismic imaging is not complete until a stopping condition isimposed on downward continuation. The process of downward continuation is terminated whenthe clock, which measures t − dt, reads zero traveltime.

The concepts described above can be used to downward continue a complete seismic exper-iment that involves many shots and receivers. The vertical wavenumber given by equation (D-6)




FIG. D-3. A flow diagram of shot-geophone migration (see Section D.1).

By substituting equations (D-21a) and (D-21b) into equation (D-16), the DSR equationtakes the following form in midpoint-offset coordinates:

DSR(Y, H ) =

1 − (Y + H )2 +

1 − (Y − H )2. (D − 22)

The vertical wavenumber (equation D-15) now is expressed in terms of normalized midpoint-offset wavenumbers Y and H :

kz =ω

vDSR(Y, H ). (D − 23)

The newly defined vertical wavenumber in equation (D-23) is inserted into the extrapolationequation (D-18):

P (ky, kh, z , ω) = P (ky, kh, 0, ω) exp(

−ikzz), (D

−24)

where P (ky, kh, 0, ω) is the Fourier transform of the prestack data P (y, h , z = 0, t) in midpoint-offset coordinates.

Figure D-4 shows the ω −ky plane for a specific value of z and h, and the ky− z plane for aspecific value of ω and h. The radial line A corresponds to ky = 2ω/v. The region in the ω − kyplane below the radial line corresponds to the evanescent energy and that above the radial linecorresponds to the propagating energy. The same transition between the two regions also arenoted in the ky−z plane. The zero-offset case (Figure D-4c) clearly shows the evanescent energyto the right of the point on the ky axis labeled as ky = 2ω/v dying off rapidly with depth.The width of the propagation region stays constant with depth. The nonzero-offset case shownin Figure D-4d, however, indicates that the width of the propagation region varies with depth— zero at the surface z = 0 and approaching rapidly to the zero-offset case immediately atshallow depths. The physical interpretation of this depth-dependency is quite intuitive — the

normalized offset wavenumber H becomes increasingly less significant at greater depths, and thenormalized midpoint wavenumber Y becomes the dominating wavenumber. More specifically,on a CMP gather, moveout decreases with depth which implies nearly zero H .

Figure D-5 shows the response characteristics of the dispersion relation defined by equation(D-23). Note the semi-elliptical wavefronts in the y − z plane for a single frequency ω; while inthe y − t plane, note the table-top traveltime trajectories. The equations for the wavefront andtraveltime trajectories are derived in Section D.2 using stationary phase approximations.

Note that in equation (D-16), the terms with different spatial wavenumbers are separable.However, we have lost the property of separation in equation (D-22) because the operators inY and H are strongly coupled. As a result, the Taylor series expansions of the square roots in




FIG. D-4. Real part of the ω − ky plane at z = 200 m: (a) DSR (equation D-23) with h = 0 m, and(b) DSR with h = 400 m. Real part of the ky − z plane at ω = 16 m: (c) DSR (equation D-23) withh = 0 m, and (d) DSR with h = 400 m (Yilmaz, 1979).

equation (D-22) yield terms that contain cross-products of the two wavenumbers. The penaltyfor processing in the conventional coordinate system (y, h , t) is that strong coupling in theextrapolation operator requires the entire prestack data set to be handled at the same time foreach depth step.

Conventional processing comprises two important steps. First, the data are organized intocommon-midpoint (CMP) gathers, and normal-moveout (NMO) correction is applied to each




FIG. D-5. The response characteristics of the DSR operator (equation D-23) (Yilmaz, 1979). (a) Realpart of the y− z plane at 16 Hz and h = 400 m. Note the semielliptical wavefronts. (b) Real part of they − z plane at t = 1024 ms, h = 400 m. Because of the wraparound in h, we observe two wavefronts,one for h = 400 m and one for h = 0. (c) Real part of the y − t plane at z = 200, 400, 600, and 800m superimposed. These are the table-top trajectories for h = 400 m. The loci of the arrival times aredetermined by a stationary-phase approximation to DSR (see Section D.2) (Clayton, 1978). Periodicityin y and t result from approximating Fourier integrals by sums.

gather. The time shift ∆t = t(h) − t0 associated with the NMO correction is given by

∆t = t0 1 + 2hvt02 − 1, (D − 25)

where t(h) is the two-way traveltime for a given (half) offset h, and t0 is the corresponding two-way zero-offset time. Here, v is the root-mean-square (rms) velocity at t0. Equation (D-25) isbased on the stratified earth (zero-dip) assumption. After NMO correction, traces of the CMPgather are stacked. This not only reduces data volume, but also enhances the signal-to-noiseratio.

Second, the CMP stack is migrated as if it were the zero-offset wavefield generated byexploding reflectors (Section 4.0). The equation used for the downward extrapolation portion of migration is the solution to the one-way wave equation (equation D-12).




FIG. D-6. The response characteristics of the exploding reflectors operator ER(Y ) (equation D-29)(Yilmaz, 1979). (a) Real part of the y − z plane at 16 Hz. Note the circular wavefronts. (b) Real partof the y − z plane at t = 1024 ms. (c) Real part of the y − t plane at z = 200, 400, 600, and 800 msuperimposed. These are the hyperbolic trajectories. The loci of the arrival times are determined viathe stationary-phase approximation to ER(Y ) (see section D.2) (Clayton, 1978). Periodicity in y and t

result from approximating Fourier integrals by sums.

which is identical to equation (D-26). We conclude that the zero-offset migration operator ER(Y )(equation D-28) derived from the DSR equation (D-22) is identical to the migration operatorthat is based on the exploding reflectors model of conventional processing.

Figure D-6 shows the response characteristics of the dispersion relation defined by equation(D-30). Note the semicircular wavefronts in the y−z plane and the hyperbolic traveltime curvesin the y − t plane. Compare these with the response of the complete DSR operator for thenonzero-offset case in Figure D-5. The equations for the wavefront and traveltime trajectoriesare derived in Section D.2 using stationary phase approximations.




D.2 Stationary Phase Approximations

In this section, we shall use the method of stationary phase to derive the traveltime equa-tion inferred by the double square root equation for nonzero-offset source-receiver separation.Consider the double square root operator defined by equation (D-22) with the wavenumbers

Y and H defined by equations (D-21a,b). We want to operate on the transformed wavefieldP (ky, kh, z = 0, ω) with the DSR operator. Subsequent inverse Fourier transformation will yieldthe wavefield P (y, h , z, t):

P (y, h , z, t) =

P (ky, kh, z = 0, ω) exp (iΦz) dky dkh dω, (D − 31)

where the total phase Φ, normalized with respect to z, is given by

Φ = −ω

vDSR(Y, H ) − ky

y

z− kh

h

z+ ω

t

z. (D − 32)

The main contribution to integration in equation (D-31) occurs when the phase stays nearlyconstant. We therefore determine the variation of the phase with respect to variables ky, kh, andω

∂ Φ∂ky

= −ωv

∂DSR∂Y

∂Y ∂ky

− yz

, (D − 33a)

∂ Φ

∂kh= −ω

v

∂DSR

∂H

∂H

∂kh− h

z, (D − 33b)

and

∂ Φ

∂ω= −1

vDSR − ω

v

∂DSR

∂H

∂H

∂ω+

∂DSR

∂Y

∂Y

∂ω

+

t

z, (D − 33c)

and set each variation to zero. Substitute equation (D-32) and carry out the differentiations inequations (D-33a,b,c) to obtain

1

2

G

√ 1 − G2 +

1

2

S

√ 1 − S 2 =

y

z , (D − 34a)

1

2

G√ 1 − G2

− 1

2

S √ 1 − S 2

=h

z, (D − 34b)

and

1√ 1 − G2

+1√

1 − S 2=

vt

z, (D − 34c)

where G and S are defined by equations (D-20a,b).Now, eliminate G and S amongst equations (D-34a,b,c) to get the final expression from

stationary phase approximation to the double square root equation as

y + h2 + z2

+ y − h2 + z2

= vt. (D − 35)This is the equation of an ellipse in the y−z plane at constant t (Section E.5). Figure D-5 showsthe elliptic wavefront and the table-top traveltime trajectory described by equation (D-35).

When equation (D-35) is specialized to the zero-offset case, h = 0, we obtain y2 + z2 =

vt

2, (D − 36)

which is a circle in the y − z plane at constant t and a hyperbola in the y − t plane at constantz. Figure D-6 shows the circular wavefront and the hyperbolic traveltime trajectory describedby equation (D-36).




We now consider the stacking operator defined by equation (D-27). The total phase is givenby

Φ = −ω

vSt(H ) − kh

h

z+ ω

t

z. (D − 37)

Differentiate equation (D-37) with respect to kh

and ω and set the results to zero to obtain

H √ 1 − H 2

=h

z(D − 38a)

and

1√ 1 − H 2

=vt

2z. (D − 38b)

Now, eliminate H between equations (D-38a) and (D-38b) to obtain the stationary phase ap-proximation to the stacking operator:

h2 + z2 =vt

2. (D − 39a)

Define the zero-offset time as t0 = 2z/v and substitute into equation (D-39a) to get h2 +

vt0

2

2=

vt

2. (D − 39b)

Finally, rearrange to get the equation for normal moveout:

∆tNMO = t0

1 +

2h

vt0

2− 1

, (D − 40)

where ∆tNMO = t− t0. This is the same equation as equation (3-2b) in the main text with offsetdefined as x = 2h.

D.3 The Parabolic Approximation

Start with the dispersion relation defined by equation (D-6) recast for the exploding reflectorsmodel for which v is replaced with v/2 to obtain

kz =2ω

v

1 −

vkx2ω

2. (D − 41)

Then apply Taylor series expansion and retain the first two terms:

kz =2ω

v

1 − 1

2

vkx2ω

2. (D − 42a)

By simplifying, we get the dispersion relation associated with the parabolic equation

kz =2ω

v− vk2x

4ω. (D − 42b)

By operating on the wavefield P (x, z, t) and replacing −ikzP with ∂P/∂z, we write the corre-sponding differential equation as

∂P

∂z= −i

2ω

v− vk2x

4ω

P. (D − 43)

Derivation of equation (D-43) is based on the constant-velocity assumption. Nevertheless, just as we did for the 90-degree one-way wave equation (D-10), the 15-degree one-way wave




equation (D-43) can be recast using a vertically varying velocity function v(z). Going one stepfurther, once equation (D-43) is inverse Fourier transformed from the horizontal wavenumber kxto the horizontal axis x, we will replace v(z) with a laterally varying velocity function v(x, z).Theoretically, this may not be permissible, but in practice, its validity is widely accepted.

The effect of translation is removed by retardation (Figure 4.1-18):

τ = 2 z0

dzv(z)

, (D − 44)

where v(z) generally is chosen to be a horizontal average of v(x, z). The time shift defined byequation (D-44) is equivalent to a phase shift in the frequency domain. Therefore, the actualwavefield P is related to the time-shifted wavefield Q by

P = Q exp(−iωτ ). (D − 45)

By differentiating with respect to z, we get

∂P

∂z=

∂

∂z− i

2ω

v(z)

Q exp(−iωτ ). (D − 46)

Finally, by substituting equations (D-45) and (D-46) into equation (D-43), we obtain

∂Q

∂z= i

vk2x4ω

Q + i2ω 1

v(z)− 1

v

Q. (D − 47a)

The first term on the right side of this equation is called the diffraction term and the secondterm is called the thin-lens term .

Consider the special case of v = v(z). The thin-lens term then vanishes and we are left with

∂Q

∂z= i

vk2x4ω

Q. (D − 47b)

After inverse Fourier transforming, we obtain the parabolic differential equation

∂ 2Q

∂z∂t=

v

4

∂ 2Q

∂x2, (D − 48)

where, in practice, velocity can be varied horizontally as well as vertically.In some practical implementations of equation (D-48), downward continuation is performed

in τ rather than z. The migrated section then is displayed in time τ and the process is called timemigration. The two variables z and τ are related by equation (D-44). The dispersion relationin equation (D-41) for the scalar wave equation in terms of ωτ = vkz/ 2, the Fourier dual of τ ,takes the form

ωτ = ω

1 −

vkx2ω

2. (D − 49a)

By squaring both sides, we get the equation of an ellipse in the ωτ − kx plane.The dispersion relation in equation (D-42b) for the parabolic equation, also expressed in

terms of ωτ , takes the form

ωτ = ω − v2k2x8ω

, (D − 49b)

which is the equation of a parabola in the ωτ − kx, plane. The first term is associated with avertical time shift that can be removed by retardation. To obtain the differential equation interms of τ , equivalent to equation (D-48), apply the second term on the right side of equation(D-49b) to the retarded wavefield Q as defined by equation (D-45) and inverse Fourier transform

∂ 2Q

∂τ∂t=

v2

8

∂ 2Q

∂x2. (D − 50)

This equation is the basis for the 15-degree time migration algorithms.




The dispersion relations given by equations (D-49a) and (D-49b) are plotted in Figure D-7for constant velocity v and input frequency ω = F E . Curve 1 is associated with the 90-degreescalar wave equation and is the same as shown in Figure 4.1-25, except that the latter is in termsof kz. Point A is mapped onto point B after migration with the 90-degree equation (D-49a).The same point A is mapped onto point C after migration with the 15-degree equation (D-49b).

The dispersion curve 2 for the parabolic equation increasingly departs from the dispersion curve1 for the exact wave equation as the dip gets steeper. (The dip before migration is measured asthe angle between the vertical axis FE and radial direction F A.) Thus, the parabolic equationcauses more and more undermigration as dip increases.

D.4 Frequency-Space Implicit Schemes

Higher-order approximations to the dispersion relation (equation D-41) of the one-way equa-tion can be found by continued fraction expansion (Claerbout, 1985). When equation (D-41) isrewritten, we have

kz =

2ω

v R, (D − 51a)where

R =

1 − X 2, (D − 51b)

with

X =vkx2ω

. (D − 51c)

The various orders of approximations to equation (D-51b) are defined by the followingrecurrence relation

Rn+1 = 1 − X 2

1 + Rn, (D − 52)

with the initial value R0 = 1. By setting n = 0 in equation (D-52), we have

R1 = 1 − X 2

2. (D − 53)

Then, substitution of equation (D-53) into equation (D-51a) yields

kz =2ω

v

1 − X 2

2

, (D − 54)

which is the same equation obtained with the parabolic approximation, equation (D-42a).The next higher-order expansion is obtained by setting n = 1 in equation (D-52) and using

equation (D-53) to obtain

R2 = 1−

X 2

2 − X 2

2

, (D

−55)

This is referred to as the 45-degree approximation to equation (D-41).Ma (1981) discovered that the recurrence relation for the continuous fraction expansion can

be expressed as ratios of two polynomials for the even-ordered expansions. He also showed thatthe expression can be split into the following partial fractions:

R2n = 1 −ni=1

αiX 2

1 − β iX 2. (D − 56)




FIG. D-7. Dispersion relations for the 90-degree equation (D-49a), 15-degree equation (D-49b) andthe 45-degree equation (D-60a) with α1 = 0.5 and β 1 = 0.25, plotted on the ωτ −

kx plane for a giveninput frequency, where ω = FE and velocity is v. Input frequency ω = FE = AP is mapped on tooutput frequency ωτ which is PB, PC, and PD for the 90-degree, 15-degree, and 45-degree equations,respectively. The wavenumber at the intersection of the curves along the kx axis is 2ω/v, 2

√ 2ω/v, and

(4ω)/(√

3v) for the 90-degree, 15-degree, and 45-degree equations, respectively.

For example, when n = 1,

R2 = 1 − α1X 2

1 − β 1X 2, (D − 57)

which is equivalent to the 45-degree expansion in equation (D-55) when α1 = 0.5 and β 1 = 0.25.Note that expansions R4, R6, R8, · · · are made up of sums of the 45-degree term, each witha different set of coefficients, αi and β i. Lee and Suh (1985) minimized the difference in the

least-squares sense between R of equation (D-51a) and R2n of equation (D-56) for a specifieddip angle and derived optimal coefficients (αi, β i) for up to the 10th order (Table D-1).

Table D-1. Coefficients of optimized, fractioned, one-way wave equations (Lee and Suh, 1985).

Order, 2n Accuracy Deg. αi β i

2 45 0.5 0.252 65 0.478242060 0.3763695274 80 0.040315157 0.873981642

0.457289566 0.2226919836 87 0.004210420 0.972926132

0.081312882 0.7444180590.414236605 0.150843924

8 90- 0.000523275 0.9940650880.014853510 0.9194326610.117592008 0.6145206760.367013245 0.105756624

8 90 0.000153427 0.9973702360.004172967 0.9648279920.033860918 0.8249185650.143798076 0.4833407570.318013812 0.073588213




We now derive the differential equation associated with the 45-degree dispersion relation.By substituting equation (D-57) and the definition for X given by equation (51c) into equation(D-51a), we obtain

kz =2ω

v 1 − α1v2k2x4ω2

1

1 −β 1v2k2

x4ω2 . (D − 58)

The first term is a vertical shift that can be removed by retardation in the same manner asfor the 15-degree approximation (equations D-42 through D-46). By simplifying the remainingterms of equation (D-58), we obtain

β 1α1

v

2ωkz k2x − k2x −

1

α1

2ω

vkz = 0. (D − 59a)

Finally, by operating on the retarded wavefield Q(x, ω, z) (equation D-45), we obtain

iβ 1v

2α1ω

∂ 3Q

∂z∂x2− ∂ 2Q

∂x2+ i

2ω

α1v

∂Q

∂z= 0. (D − 59b)

Kjartansson (1979) implemented equation (D-59b) for 45-degree modeling and migration in the

frequency-space domain. Migration in the frequency-space domain (commonly known as theω − x algorithm) involves two interleaved operations:

(a) a time shift based on equation (D-45), which is velocity-independent for time migration andvelocity-dependent for depth migration, and

(b) focusing the diffraction energy using equation (D-59b).

Once you have a code for the basic 45-degree operator, it is easy to implement the higher-order approximations that are given by equation (D-56) with the associated coefficients in TableD-1. Note that the difference between the 45-degree and 65-degree algorithms is the values usedfor coefficients (α1, β 1). Also note that the 15-degree equation (D-47b) is obtained from equation(D-59b) by setting α1 = 0.5 and β 1 = 0.

The dispersion relation in equation (D-58) also can be expressed in terms of ωτ = vkz/2,

the Fourier dual of the time variable τ associated with the migrated data:

ωτ = ω − α1ωv2k2x4ω2 − β 1v2k2x

. (D − 60a)

The first term is associated with a vertical time shift that can be removed by retardation as forthe 15-degree equation (D-49b).

To obtain the differential equation in terms of τ , equivalent to equation (D-59b), apply thesecond term on the right side of equation (D-60a) to the retarded wavefield Q as defined byequation (D-45) and inverse Fourier transform:

iβ 1

α1ω

∂ 3Q

∂τ∂x2− ∂ 2Q

∂x2+ i

4ω

α1v2∂Q

∂τ = 0. (D − 60b)

This equation is the basis for the 45-degree and related steep-dip implicit finite-difference

frequency-space time migration algorithms.The dispersion relation given by equation (D-60a) for the 45-degree equation, in which

α1 = 0.5 and β 1 = 0.25, is plotted in Figure D-7 for constant velocity v and input frequencyω = F E . Point A is mapped onto point D after migration with the 45-degree equation (D-60a).The dispersion curve 3 for the 45-degree equation lies somewhere between those of the 90-degreeequation (D-49a) (curve 1) and the 15-degree equation (D-49b) (curve 2).

Figure 4.4-1 shows the impulse responses of various approximations to the one-way disper-sion relation based on equation (D-56). Note that the wavefronts become increasingly closer toa semicircle as the higher-order terms are included in equation (D-56). The 15-degree approxi-mation yields an elliptical wavefront. The 45-degree approximation yields an impulse response




of a heart shape. Refer to Section 4.4 for the practical aspects of 2-D frequency-space steep-diptime migration, and Section 7.3 for its application to 3-D migration.

D.5 Stable Explicit Extrapolation

The exact extrapolation filter for a specific frequency ω and velocity v is expressed in thefrequency-wavenumber domain as

D(kx) = exp

−i

2ω

v

1 −

vkx2ω

2∆z

. (D − 61)

The objective is to find, again for a specific frequency ω and velocity v, an explicit filter withcomplex coefficients h(x) in the frequency-space domain such that, when Fourier transformedto the frequency-wavenumber domain, the difference between the actual transform H (kx) andthe desired transform D(kx) of equation (D-61) is minimum. To ensure stability, we impose theconstraint that the amplitude of H (kx) is never greater than unity within the propagation regionkx ≤

(2ω/v).Consider the Fourier transform H (kx) of the actual extrapolation filter h(x), which we will

write in discrete form as hn:

H (kx) = h0 + 2

N n=1

hn cos(nkx), (D − 62)

where 2N + 1 is the length of the symmetric filter hn.To determine the filter coefficients hn, perform Taylor series expansion of the exact D(kx)

and the actual H (kx) extrapolators given by equations (D-61) and (D-62), respectively, andmatch the coefficients of the terms in each series at kx = 0 (Holberg, 1988). Such a direct matchof the coefficients in the Taylor series results in a response H (kx) whose amplitude is greaterthan unity beyond a certain kx, thus violating the stability constraint that the amplitude of

H (kx) is never greater than unity within the propagation region kx ≤ (2ω/v).A way to circumvent the unstable response of the conventional Taylor series method is tomatch the first M < N coefficients in the series expansion and set the remaining N −M to zero(Hale, 1991). This modified Taylor series method proceeds as follows.

Let the filter coefficients hn be defined by the series

hn = c0 + 2M m=1

cmbmn, (D − 63)

where

bmn = 2 cos

2πmn

N

. (D − 64)

By way of equations (D-63) and (D-64), equation (D-62) takes the form

H (kx) = c0

1 + 2

n

cos(nkx)

+ 2

m

cm

1 + 2

n

cos

2πmn

N

cos(nkx)

. (D − 65)

Equation (D-65) can be written as a single summation

H (kx) =

M m=0

cmBm(kx), (D − 66)




where

B0(kx) = 1 + 2n

cos(nkx), (D − 67a)

and

Bm(kx) = 1 + 2n

cos2πmnN

cos(nkx). (D − 67b)

Now, perform the Taylor series expansion of the exact extrapolator given by equation (D-

61):

D(kx) = D(0) + kx∂D(kx)

∂kx+

k2x2

∂ 2D(kx)

∂k2x+ · · · , (D − 68)

where the derivatives are evaluated at kx = 0, and thus the terms associated with the odd

derivatives vanish.

Matching the terms of the actual extrapolator H (kx) with those of the desired extrapolatorD(kx) is equivalent to matching their even derivatives

M m=0

cmB2jm (0) = D2j(0), j = 0, 1, 2, . . . , M , (D − 69)

where the derivative terms B2jm are

B2j0 (0) = (−1)j

1 + 2

n=1

n2j

, (D − 70a)

and

B2jm (0) = 2(−1)j

1 + 2

n=1

cos

2πmn

N

n2j

. (D − 70b)

Equation (D-69) represents a system of M linear equations that can be solved for the

coefficients cm. The extrapolation filter coefficients hn then can be computed by substituting

the solution of equation (D-69) into equation (D-63).

The modified Taylor expansion yields filter coefficients hn with its response H (kx) that

vanishes beyond a cutoff value for kx in the wavenumber domain that depends on the scalarM . Also, the response of the extrapolator based on the modified Taylor expansion satisfies the

stability constraint H (kx) < 1 in the passband region of the filter. The cutoff wavenumber kxdetermines the maximum dip accuracy of the extrapolation filter. The larger the number of filter

coefficients 2N + 1, the steeper the dip accuracy. In practice, extrapolation filter lengths 7, 11

and 25 are often associated with 30-, 50- and 70-degree dip accuracies.

An alternative method for computing the filter coefficients hn with more accuracy at large

wavenumbers kx is based on the Remez exchange algorithm (Soubaras, 1996). The objective is

to minimize the error function

E (kx) = W (kx)[D(kx) − H (kx)], (D − 71)

where W (kx) is a weighting function, such that

||E ||∞ = max|E (kx)| (D − 72)

is minimum. The criterion ||E ||∞ means that the actual response H (kx) is equiripple. Specifically,

the response H (kx) has extrema of the same absolute value with alternating signs within the

passband region specified by a cutoff wavenumber kx. The objective of the error function is to

minimize the deviation of the extrema from the desired value of unity.




D.6 Optimum Depth Step

The objective is to specify an optimum depth step size that yields minimum-phase errors inwave extrapolation as part of the design of finite-difference migration algorithms. First, we shallreview implicit and explicit schemes. Then, we shall derive equations for optimum depth size

for frequency-wavenumber implicit schemes.Start with the one-way wave equation (D-5):

∂

∂zP (kx, z , ω) = −ikzP (kx, z = 0, ω), (D − 73)

whose solution is

P (z + ∆z) = P (z) exp−ikz∆z

, (D − 74)

where, for simplicity, kx and ω variables are omitted from P , and

kz =2ω

v

1 −

vkx2ω

2. (D − 75)

The phase of the exact extrapolation operator

exp(−ikz∆z) (D − 76a)

is

Φ = kz∆z, (D − 76b)

and its amplitude is

A = 1. (D − 76c)

Discretize the one-way wave equation (D-73):

δP

δz= −ikzP . (D − 77)

Apply differencing approximation using the explicit scheme

P (z + ∆z) − P (z)

∆z= −ikzP (z), (D − 78a)

and the implicit scheme

P (z + ∆z) − P (z)

∆z= −ikz

P (z + ∆z) − P (z)

2. (D − 78b)

Rewrite the explicit equation (D-78a) as

P (z + ∆z) = P (z)

1 − ikz∆z

. (D − 79a)

The phase of the explicit extrapolation operator

(1 − ikz∆z) (D − 79b)

is Φ = tan−1

kz∆z

, (D − 79c)


A =

1 +

kz∆z

2. (D − 79d)

Note that the amplitude of the explicit operator is greater than unity and grows from oneextrapolation step to the next. In fact, large depth steps cause the operator to yield unstableresults early in the extrapolation process. In general, explicit schemes tend to be unstable unlessspecial design considerations are made (Section D.5).




Rewrite the implicit equation (D-78b) as

P (z + ∆z) = P (z)

1 − ikz∆z/2

1 + ikz∆z/2

. (D − 80a)

The phase of the implicit extrapolation operator

1 − ikz∆z/21 + ikz∆z/2

(D − 80b)

is

Φ = tan−1

kz∆z

1 − kz∆z/22 , (D − 80c)


A = 1. (D − 80d)

Note that the amplitude of the implicit scheme is unity. This means that implicit schemes areunconditionally stable.

We now specialize the phase of the frequency-space implicit scheme given by equation (D-

80b) for the 65-degree dispersion relation as in equation (D-58) and obtain Φ = tan−1

kz∆z

1 − kz∆z/22 , (D − 81a)

where

kz =αk2x

2ω

v− β

v

2ωk2x

, (D − 81b)

with α and β specified as in Table D-1 for the 65-degree scheme. Equation (D-81a) is the time-retarded form of equation (D-58) and hence corresponds to the second term of the latter. Thediscrete forms of the transform variables are (Claerbout, 1976):

k2x =4

∆x2

sin2ωv ∆x sin θ1 − 4bsin2

ω

v∆x sin θ

, (D − 81c)

and

ω =2

∆ttan

ω∆t

2

. (D − 81d)

The scalar b in equation (D-81c) is set to a value between 1/12 and 1/6. The phase error for the65-degree implicit scheme then is given by

∆Φ =

Φ − Φ, (D − 82)

where Φ and Φ are given by equations (D-81a) and (D-76a), respectively. Refer to the equations

(D-81a,b,c,d) and note that the phase error depends on ∆x, ∆t , v, θ, ω, and the depth step ∆z.Figure D-8 shows contour plots of the phase error defined by equation (D-82) for three

ω/v values as a function of dip angle and depth step size. Note the complicated behavior of the contours which indicates that the optimum depth step size is associated with a complicatedinterdependence of the various parameters — dip, frequency, velocity, spatial, and temporalsampling rates. Therefore, in practice, migration algoirthms that require wave extrapolation atdiscrete depth steps usually do not incorporate an automated estimation of optimum depth steps.Instead, a constant value between one-half and one dominant period, 20 to 40 ms, dependingon maximum reflector dip is specified in practice. See Section 4.4 for the practical aspects of frequency-space migration.




FIG. D-8. Phase error contours in degrees associated with the 65-degree implicit extrapolator (see

Section D.6) for a range of dip angles θ versus depth step sizes ∆z , and for specific ratios of ω/v, from

top to bottom, 0.001 (low frequency or high velocity), 0.06 (medium frequency and velocity) and 0.2

(high frequency or low velocity). The parameter b in equation (D-81c) was set to 0.14 for all three cases.

(Computation by Dave Nichols.)




D.7 Frequency-Wavenumber Migration

We start with the solution of the scalar wave equation for the zero-offset wavefield as givenby equation (D-7) and assume a horizontally layered earth model associated with a verticallyvarying velocity function v(z). By inverse Fourier transforming equation (D-7), we have

P (x, z, t) =

P (kx, 0, ω) exp(−ikzz) exp(−ikxx + iωt) dkx dω, (D − 83a)

where kz is defined by equation (D-6) adapted to the exploding reflectors model by replacing vwith v/2:

kz =2ω

v

1 −

vkx2ω

2. (D − 83b)

The imaging principle t = 0 then is applied to get the migrated section P (x, z, t = 0),

P (x, z, t = 0) =

P (kx, 0, ω) exp(−ikxx − ikzz) dkx dω. (D − 84)

This is the equation for the phase-shift method (Gazdag, 1978). Equation (D-84) involves inte-gration over frequency and inverse Fourier transformation along midpoint axis x. Refer to Figure4.1-29 for a flow diagram of phase-shift migration.

We now consider the special case of constant velocity v. Stolt (1978) devised a migrationtechnique that involves an efficient mapping in the 2-D Fourier transform domain from temporalfrequency ω to vertical wavenumber kz. We rewrite equation (D-83b) to get

ω =v

2

k2x + k2z . (D − 85)

By keeping the horizontal wavenumber kx unchanged and differentiating, we obtain

dω =v

2

kz

k2x + k2z

dkz. (D − 86)

When equations (D-85) and (D-86) are substituted into equation (D-84), we get

P (x, z, t = 0) =

v

2

kz k2x + k2z

P

kx, 0,

v

2

k2x + k2z

exp(−ikxx − ikzz) dkx dkz.

(D − 87)

This is the equation for constant-velocity Stolt migration. It involves two operations in thef − k domain. First, the temporal frequency ω is mapped onto the vertical wavenumber kz viaequation (D-85). This is the same as mapping point B onto point B in Figure 4.1-25. Second,the amplitudes are scaled by the quantity

S =v

2

kz

k2x + k2z

,

which is equivalent to the obliquity factor associated with Kirchhoff migration (Section 4.1).Refer to Figure 4.1-30 for a flow diagram of constant-velocity Stolt migration.

To extend the algorithm to the variable-velocity case, yet retain efficiency, Stolt (1978)did a coordinate transformation that involves stretching the time axis to make the scalar waveequation velocity independent. A summary of the theoretical procedure is given here. Considerwavefield P (x, z, t) and the transformed wavefield P (x,d,T ):

P (x, z, t) = P (x,d,T ),

where T is the stretched time axis and d is the output variable (equivalent of z) for migrationin the stretched coordinate system. Here, the x variable is identical in both coordinate systems.




This coordinate transformation basically is equivalent to stretching the data using the rmsvelocities

T (t) =1

c

2

t0

dt v2rms(t) t1/2

, (D − 88)

where

v2rms(t) =1

t

t0

v2(t) dt, (D − 89)

and c is an arbitrary reference velocity used to maintain the vertical axis as time after thecoordinate transformation from t to T . After some tedious algebra, the time-retarded scalarwave equation takes the following form in the stretched coordinates (Stolt, 1978):

∂ 2P

∂x2+ W

∂ 2P

∂d2+

2

c

∂ 2P

∂d∂T , (D − 90)

where W is a complicated function of velocity and coordinate variables. In practice, it normallyis set to a constant between 0 and 1. The procedure for Stolt migration with stretch follows:(a) Start with the stacked section, which is assumed to be a zero-offset section P (x, z = 0, t).

(b) Convert this time section to stretched section P (x, d = 0, T ) by the coordinate transforma-tion (equation D-88).

(c) 2-D Fourier transform the stretched section P (kx, d = 0, ωT ).(d) Apply the following mapping function to perform migration:

kd =

1 − 1

W

ωT

c− 1

W

ω2T

c2− W k2x. (D − 91)

This equation is based on the dispersion relation of the retarded wave equation in thestretched coordinates (equation D-90). The expression for the output from migration is(omitting the heavy algebra)

P (kx, kd, 0) = c

2−

W (1 − W +

1

K ) P kx, 0,

ckd2−

W (1 − W + K ), (D − 92a)

where

K =1

1 + (2 − W )k2xk2d

. (D − 92b)

(e) 2-D inverse Fourier transform the migrated section in the stretched coordinates P (x,d,T =0).

(f) Convert back to the familiar space-time coordinates P (x, z, t = 0). This is the final migratedsection.To derive the equations for Stolt migration with the output in time τ = 2z/v, use the

dispersion relation as in equation (D-49a) in lieu of equation (D-83b) and replace the vertical

wavenumber kz with the output frequency ωτ = vkz/2 in equations (D-84) through (D-88).When W = 1, equation (D-91) takes the simple form

kd =

ω2T

c2− k2x, (D − 93)

which makes mapping of equation (D-91) equivalent to the constant-velocity Stolt algorithm.Note that Stolt migration with stretch tries to handle velocity variations. However, it is no

substitute for depth migration; it only accommodates velocity variations that can be handled bytime migration. Figure D-9 shows the flow diagram for Stolt migration with stretch. See Section4.5 for the practical aspects of frequency-wavenumber migration.




FIG. D-9. Flowchart for Stolt migration with stretch.

D.8 Residual Migration

The relationship between the output and input temporal frequencies for a 90-degree time mi-gration operator (rewritten from equation D-49a) is given by

ωτ =

ω2 − v2k2x

4, (D − 94)

where ωτ = vkz/ 2 is the output frequency, ω is the input frequency, v is the true mediumvelocity, and kx is the midpoint wavenumber. Equation (D-94) refers to single-pass migration if the migration velocity is the same as the medium velocity.

Consider a two-pass migration, first with velocity v1, followed by a second migration withvelocity v2. The output frequency from the first pass is

ω1 = ω2 − v2

1k2x

4, (D − 95a)

which is the input frequency for the second pass

ω2 =

ω21− v2

2k2x

4, (D − 95b)

The horizontal wavenumber kx is fixed, since it is invariant under migration.If the output frequency from the single-pass migration given by equation (D-94) is set equal

to that from the two-pass migration given by equation (D-95b), then the necessary relationshipbetween the residual migration velocity v2 and the medium velocity v can be established as

v2 = v21 + v22. (D − 96a)




One practical scheme for residual migration is implemented as follows:(a) The constant-velocity Stolt migration with a stretch factor of W = 1 is used for the first

pass. Velocity v1 in equation (D-96a) is chosen as the minimum value in the actual velocityfield.

(b) For the second pass, a dip-limited finite-difference migration can be used. The first-pass

migration brings the dips down to within the range that the second-pass finite-differencemigration can accommodate, accurately. Remember that the velocity field for the secondpass is computed by the the relation

v2 =

v2 − v21

, (D − 96b)

where v, v1, and v2 are the original, first- and second-pass migration velocities, respectively. Amultiple-pass application of residual migration is referred to as cascaded migration. Refer toSections 4.3 and 4.5 for the practical aspects of cascaded migration and residual migration,respectively.

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4. migration

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