GEOPHYSICS, VOL. 66, NO. 3 (MAY-JUNE 2001); P. 939–952, 19 FIGS., 1 TABLE.

Poststack migration deconvolution

Jianxing Hu∗, Gerard T. Schuster‡, and Paul A. Valasek∗∗

ABSTRACT

Reflectivity images of the earth are calculated by mi-grating discrete grids of seismic traces. Typically, suchtraces are spatially undersampled on a recording gridwith limited aperture width and so give rise to migrationnoise sometimes referred to as the acquisition footprint.For poststack migration images, we show how to partlydeconvolve the acquisition footprint by applying a de-blurring filter to the migration section, where the filter isthe approximate inverse to the migration Green’s func-tion. Results with synthetic and field data show that post-stack migration deconvolution can noticeably improvethe spatial resolution of migration images, decrease thestrength of migration artifacts, and improve the quality ofthe migration image. We conclude that migration decon-volution can be a viable alternative to some of the otherpostmigration processing procedures based on statisticsand ad hoc parameter choices.

INTRODUCTION

Astronomers use powerful telescopes to probe the oldestparts of the universe. In the visible spectrum, an optical lensrefocuses the incident light from hot stardust to reconstructimages of stellar objects. Unfortunately, lenses are limited inaperture size and can be imperfect in construction so that a stel-lar image is a blurred version of the original star. This blurringoperation can be mathematically represented by

m(r)′ =∫

�

G(r | ro)m(ro) dro, (1)

where m(r) represents the stellar object, m(r)′ is the blurredimage, G(r | ro) is the blurring function, and the integrationis over the area � of the image plane. Engineers sometimescall G(r | ro) the point-spread function, or PSF (Jansson, 1997),where the ideal PSF of a lens is a delta function.

Manuscript received by the Editor January 26, 1999; revised manuscript received October 11, 2000.∗Formerly University of Utah, Department of Geology and Geophysics, Salt Lake City, Utah; presently GX Technology Corporation, 5847 SanFelipe, Suite 3500, Houston, Texas 77057. E-mail: jhu@gxt.com.‡University of Utah, Department of Geology and Geophysics, 719 W.B.B., Salt Lake City, Utah 84112-0111. E-mail: schuster@mines.utah.edu.∗∗Phillips Petroleum Company, 560 Plaza Office Building, Bartlesville, Oklahoma 74004. E-mail: pav@ppco.com.c© 2001 Society of Exploration Geophysicists. All rights reserved.

An example of an imperfect PSF is the original lens of theHubble space telescope. Soon after its 1990 launch, a mechan-ical flaw was discovered in the Hubble’s lens so that its PSFwas blurred; consequently, so were the associated astronomi-cal images (upper-left image, Figure 1). Since the mechanicalflaw or blurring function G(r | ro) was known, then the inverseor deconvolution operator was computed (Hanisch et al., 1997)and applied to m(r)′ to give the deblurred image m(r) (upper-right image, Figure 1). Deblurring yielded stellar images withmuch better resolution and less noise, significantly alleviatingthe blurring problem in Hubble images.

A similar blurring problem exists in exploration seismology.Instead of using optical telescopes, seismologists use large ar-rays of geophones as a seismic lens to probe thick piles of coldstardust, otherwise known as sedimentary rock. Here, the seis-mic lens first records the seismic data; then a migration algo-rithm refocuses this energy into an image of the earth’s re-flectivity distribution. Similar to the original Hubble telescope,most seismic lenses are flawed. The flaws include coarse spatialsampling, inaccurate migration operators, and limited aperturewidths, all of which lead to a blurred PSF and a blurred migra-tion image.

In this paper we show that the migrated seismic image m(r)′

is a blurred representation of the earth’s actual reflectivity dis-tribution m(r), where the blurring operation can also be repre-sented by equation (1). In the seismic case, the blurring takesthe form of migration artifacts, or the acquisition footprint. Weshow that this migration PSF (also denoted as the migrationGreen’s function in Schuster and Hu, 2000) is mostly knownand can be used to design the inverse or migration deconvolu-tion operator that deblurs the migration image. As an example,the lower-left image in Figure 1 shows the depth slice of the mi-gration image of a point scatterer obtained from a coarse gridof synthetic seismic data. To the right of it is the image decon-volved of the migration PSF, where there is better resolutionand less noise.

We first present the theory of migration deconvolution, thenshow the deconvolution of poststack migration images usingboth synthetic and field data. The synthetic data are computed

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from point-scatterer, meandering-stream, and overthrust mod-els; the field data are from the Gulf of Mexico and the NorthSea. The last section discusses these results and their implica-tions and indicates future directions in research.

PRINCIPLE OF MIGRATION DECONVOLUTION

Seismic data d can be generated by applying the forwardmodeling operator L to the reflectivity model m:

d = Lm. (2)

In contrast, the geophysicist attempts to reconstruct the re-flectivity model by applying the adjoint LT of the modelingoperator to the data to get the migrated section m′:

m′ = LT d,

= [LT L]m, (3)

where the last step follows by use of equation (2). Here, LT Lrepresents the integral blurring operator in equation (1), andm′ represents the blurred migration image of the reflectivity.For a poststack migration image, the blurring partly resultsfrom a limited recording aperture, poor trace sampling, andthe fact that the adjoint operator LT is not the inverse of L inequation (2).

Equation (3) suggests that the migration image can be de-blurred by applying the inverse of LT L to give

m = [LT L]−1m′. (4)

FIG. 1. Hubble space telescope images (Hanisch et al., 1997)of stars before (upper left) and after (upper right) deblur-ring by a deconvolution filter. (Lower left) Depth slice of apoint-scatterer image obtained by migrating poststack syn-thetic 3-D seismic data. (Lower right) Deconvolved image us-ing migration deconvolution algorithm. Whiter shading corre-spond to larger intensities.

An expensive means for computing the above deblurred imageis least-squares migration (Nemeth et al., 1999), where m =[LT L]−1LT d = [LT L]−1m′ could be approximated by 15 or soiterations of a conjugate gradient method. Since each iterationcosts about two migrations, this is still too expensive for manydata sets.

To alleviate this cost, we assume a uniform poststack trace ge-ometry and a 1-D velocity v(z) model. We show in Appendix Athat these assumptions reduce the 3-D integral in equation (1)[or equation (3)] to a 1-D integration in z and a 2-D convolutionin the horizontal coordinates:

m(r)′ =∫

�

G(x − x0, y − y0, z | 0, 0, z0)m(ro) dro, (5)

which, by a Fourier transform in the x and y variables, gives

m̃(kx , ky, z)′

=∫ zlimit

0G̃(kx , kz, z | 0, 0, z0)m̃(kx , ky, z0) dz0, (6)

where zlimit is the maximum depth of integration.The above 1-D integral can be discretized into a small system

of linear equations that can be inexpensively solved by regu-larized Gaussian elimination for each wavenumber. Inversetransforming m̃(kx , ky, z) then yields the deconvolved migra-tion image. The total cost of this procedure is less than onemigration. (Implementation details are given in Appendix A.)

Physical picture of migration deconvolution

To understand the physical meaning of migration deconvo-lution, we focus our attention on the kernel G(r | ro) in equa-tion (1), namely, the migration Green’s function. Here, G(r | ro)is the migration image at r for a point scatterer at ro. Schusterand Hu (2000) show for a 2-D recording array and a homoge-neous medium the far-field migration response of a point scat-terer, which takes the form of an hourglass (Figure 2). Here,

FIG. 2. The zero-offset migration response (i.e., the migrationGreen’s function) of point scatterers buried at different posi-tions. The dashed contour lines correspond to the amplitude ofthe migration image and fall off as the inverse of distance fromthe scatterer’s location, which is at the center of the hourglassimages. The shape of the hourglass is nearly invariant for lat-eral shifts far away from the edge of the recording array, butnot for depth shifts.

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a side of the hourglass is perpendicular to the line that con-nects the point scatterer with an end of the array. The hour-glass becomes more tilted as the scatterer nears the end of thearray, and the amplitude decreases as the inverse of the hor-izontal distance from the scatterer’s position (i.e., the centerof the hourglass). Thus, the poststack migration response of astring of buried scatterers, such as an interface, is computedby a superposition of hourglasses, each one shifted and tiltedaccording to the location of the interface’s point scatterer. Ifthe recording aperture is extremely wide, then the hourglassshape mostly stays the same as it is horizontally shifted, i.e.,it is horizontally shift invariant under these conditions. How-ever, the shape of the hourglass changes if the point scattereris lowered in depth so that G(r | ro) is not shift invariant in thez-coordinate, as shown in Figure 2.

Thus, the migration deconvolution operator [LT L]−1 in equa-tion (4) compresses each hourglass artifact in the migrationsection to approximate a fat point image, where the effectivewidth of the fat point is roughly proportional to the wavelength.

How a migration Green’s function is constructed

The migration Green’s function embodied in LT L can becomputed in several ways. One way is to use the analyticalformula derived by Schuster and Hu (2000). However, this isonly valid for a homogeneous medium under far-field condi-tions and does not account for the bending of rays. A moreaccurate method is to use ray tracing to generate G(r | ro) fora point scatterer. For example, to compute the value of themigration Green’s function kernel G(r | ro) in equation (1), useray tracing in the background velocity medium to generate syn-thetic reflected data for a scatterer at ro. Then migrate thesetraces to get the migrated image at r. To expedite the computa-tions, we assume that G(r | ro) is shift invariant in the horizontalcoordinates.

In summary, we (1) assume reference coordinates (xref, yref),(2) compute the migration Green’s function G(r | xref, yref, z)for all z and r, and (3) use this result by setting G(r | ro) ∼=G(x − xo, y − yo, z | xref, yref, zo) in equation (A-4). For mediawith strong velocity variations in the horizontal coordinates,we divide the model into subregions where the lateral velocityvariation is not too strong and compute separate migrationGreen’s functions for each subregion (see Appendix B).

SYNTHETIC AND FIELD DATA EXAMPLES

Several synthetic data sets and three field data sets are usedto test the effectiveness of the deconvolution algorithm. Thesynthetic poststack data are computed for three types of mod-els: point scatterer, meandering stream, and overthrust. Thefield data are from a 2-D marine survey in the North Sea, a3-D poststack reverse-time migration data set from the Gulf ofMexico, and a 3-D prestack Kirchhoff migration data set fromthe North Sea.

Table 1. Parameters for phase A data associated with SEG/EAGE 3-D overthrust model. Here the model is discretized into aNx ×× Nz grid, with grid intervals denoted by (dx ×× dz), a time sampling interval dt, a source spacing of ds; the number of traces isequal to Ns; and the source and receiver depths are denoted by sp and gp, respectively.

Ns Nx Nz ds (m) dx (m) dz (m) dt (ms) Wavelet Central frequency (Hz) sp (m) gp (m)

309 621 187 50 25 25 8 Ricker 15 −25 −25

Point scatterer test

This model tests the validity of the shift-invariance assump-tion in the migration Green’s function, which assumes lateralhomogeneity in the velocity model and the trace distribution.Fortunately, there are usually hundreds of traces involved inconstructing an image at any point, which lets us use the shift-invariance approximation.

This statement is validated by a 3-D synthetic numericalexperiment. Figure 3a,c shows the Kirchhoff migration im-ages of a three-point scatterer model constructed by migratingan 11 × 11 grid of traces and a 51 × 51 grid of traces, respec-tively. We deconvolved these migration images, and the resultsare shown in Figure 3b,d. The 11 × 11 trace image after mi-gration deconvolution shows strong noise, while the 51 × 51trace result shows better spatial resolution and weaker arti-facts than the original image. This result suggests that the hor-izontal shift-invariance approximation might be suitable if asufficiently large number of traces were used to construct theimage.

Meandering-stream model

The meandering-stream model is shown in Figure 4. At eachdepth level is a meandering-stream channel with a large re-flectivity contrast. The synthetic poststack migration imagesare generated by equation (1) using an analytical migrationGreen’s function (Schuster and Hu, 2000) in Figure 5. Here, thesource wavelet is a Ricker wavelet, and its central frequencyis 50 Hz. In this figure the migration amplitude is distorted bythe recording footprint artifacts, so it is difficult to delineatethe meandering-stream channel from the flat layer image. Ap-plying a migration deconvolution filter to the migrated sectionresults in Figure 6, where the meandering-stream channel isclearly delineated. The migration filter consisted of only threelayers, a 100-Hz Ricker wavelet was used in the deconvolution,and the reference position (see Appendix A) is at the middle ofthe model. The 100-Hz deconvolution wavelet was used insteadof a 50-Hz wavelet to minimize edge effects from the sides ofthe narrow velocity model. This makes sense because the ef-fective lateral extent of a migration Green’s function (Schusterand Hu, 2000) narrows for higher source frequencies and socan decrease deconvolution edge effects in the interior of themodel.

Overthrust model

The SEG/EAGE 3-D overthrust velocity model (Aminzadehet al., 1997) and the corresponding zero-offset data gener-ated by a finite-difference modeling method are shown inFigure 7; the associated recording parameters are given inTable 1.

The first step in calculating the deconvolution operator isto compute the migration Green’s function by tracing rays

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through a smoothed version of the 2-D velocity model. Thereference position is taken to be in the middle of the model.We only considered the migration influence within about a half-wavelength (i.e., 200 m) above and below an image point at ro

and therefore used fewer than 10 layers (with a depth intervalof 20 m) to calculate the migration deconvolution filter. The kx

spectrum of the Green’s function (Figure 8, top) varies rapidly,so a seven-point median filter followed by a seven-point aver-aging filter was used to smooth the spectrum G̃(kx , z | xref, zo)(Figure 8, bottom) to ensure a stable algorithm. In the process-ing, a 80-Hz Ricker wavelet was used to calculate the migrationGreen’s function.

Applying Kirchhoff migration (ignoring the obliquity term)to the zero-offset data yields the poststack migration sectionshown at the top of Figure 9. Here, all of the traces are usedto construct any point in the image, and this introduces no-ticeable noise in the migrated image. Applying a migrationdeconvolution filter to this migration section yields the resultshown at the bottom of Figure 9. Compared to the originalmigration section, the footprint artifacts are significantly re-duced, the amplitude of the reflectivity is partially recovered,and there is slightly better spatial resolution. Importantly, thenear-field artifacts of the migration Green’s function are signif-icantly deconvolved. However, the smile artifacts seen on the

FIG. 3. Comparison of the migration image and the deconvolved migration image of a three-point scatterer model, where themedium is homogeneous, the velocity is 4000 m/s, the model grid is 51 × 51 × 51, the grid interval is 20 m, and the point scatterersare located at (26, 26, 36), (11, 11, 36), and (26, 41, 36) in gridpoint numbers. (a) The 3-D poststack Kirchhoff migration imageat the level of the point-scatterer location, where the data consist of 11 × 11 traces with a 20-Hz Ricker wavelet. The traces areevenly distributed at the top of the model, and the receiver interval is five gridpoints. (b) Deconvolved image of (a), where thereference position of the migration Green’s function is chosen at the center of the model. Since the migration aperture is limitedand the two point scatterers are not at the center of the model, the migration Green’s function lateral shift invariance assumptionis inappropriate for the small number of traces used to construct these images. (c) Same as (a) except 51 × 51 traces are used in themigration and the receiver interval is one gridpoint. (d) Deconvolved image of (c). Compared with the original migration imagein (c), the deconvolved image is noticeably improved in resolution and reduced artifacts. These results indicate the suitability oflateral shift invariance when more traces are involved in image construction.

right side of the image are not deconvolved because theyare possibly (1) far-field effects of the migration Green’sfunction unaccounted for in the deconvolution operatorand (2) edge effects unaccounted for by a deconvolution

FIG. 4. Three-layer meandering-stream model where the back-ground velocity is 3000 m/s. The reflectivity values of the layerinterfaces and stream channels are 0.3 and 0.45, respectively.

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operator with a reference position in the middle of themodel.

The seismic wavelength increases with depth, so the lengthof the deconvolution filter should also increase with depth. Inthe above example, a five-layer migration deconvolution filterwas used above a depth of 1250 m, and a seven-layer migrationdeconvolution filter was used below 1250 m. Such a filter iscalled a 5 + 7 layer filter.

It would seem that more layers should suppress more noise,but a greater number of layers might introduce new artifactsbecause of the possible breakdown of the shift-invariance as-sumption. To test this possibility, an image deconvolved witha nine-layer filter is shown in Figure 10. Compared to the5 + 7 layer filter deconvolution result in Figure 9, some arti-facts are introduced in Figure 10, as shown in the boxed area at700 m. These artifacts take the form of a faint white horizontallayer not seen in the other images. So choosing the number oflayers in the deconvolution filter will likely involve a trade-offbetween eliminating some migration artifacts and introduct-ing new noise. In our experience, five to seven layers appear towork well, but those numbers may not be optimal.

FIG. 5. Standard migration images of meandering-stream chan-nel; the stream channels are barely visible.

FIG. 6. Deconvolved migration image of the stream channels;both layers and stream channels are well resolved.

To test the ability of migration deconvolution to suppressthe artifacts caused by spatial aliasing, we constructed a sub-sampled data set consisting of every other trace in the originalzero-offset data set. The half-sampled data set was migratedto give the severely aliased image in Figure 11. This image wasdeconvolved with a 5 + 7 variable-layer filter to give the resultshown at the bottom of Figure 11. Here, there are fewer alias-ing artifacts and better amplitude recovery of the reflectivitysignal than in the original Kirchhoff migration above it.

This migration deconvolution algorithm is computationallyinexpensive for the examples tested. For our examples, only40 CPU seconds were needed to accomplish the migration de-convolution for the seven-layer filter using a Pentium II 450-MHz PC workstation. In comparison, the poststack migrationrequired 5.8 CPU minutes, which included calculating the trav-eltime table.

North Sea field data

The migration deconvolution algorithm is applied to a post-stack time-migration image computed from a 2-D North Seadata set (courtesy of Robert Keys, Mobil). Here, a time-migration deconvolution filter is constructed in the same wayas a depth-migration deconvolution filter, except the depth

FIG. 7. (Top) Velocity model for the Phase A data set in theoverthrust model. (Bottom) Zero-offset Phase A data set forthe overthrust model (SEG/EAGE 3-D modeling Series No. 1).

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coordinate is replaced by a time coordinate. Conventionalprocessing is applied to the data, and the NMO velocitymodel and the poststack time-migration section are shown inFigures 12 and 13, respectively, where all traces are used toconstruct the image at any point. In these images, the numberof common depth points (CDPs) is 2142, the image gridpointspacing is dx = 12.5 m, dt = 4 ms, and the number of time sam-ples for each trace is 1500.

A 9 + 11 variable-layer time-migration deconvolution filterwas applied to the time-migration section. The result is shownat the bottom of Figure 13, where a nine-layer filter was ap-plied above 1 s and an 11-layer filter was applied below 1 s. Toaccount for the lateral velocity variations, the whole image isdivided into three vertical strips, and a different reference ve-locity profile was assumed at the center of each strip (indicatedby the arrows in Figure 12).

Compared to the original migration image at the top ofFigure 13, the deconvolved migration image at the bottom

FIG. 8. (Top) The kx amplitude spectrum of the migrationGreen’s function at a depth of 3000 m. The rapid variationsin this spectrum cause an instability problem in the process-ing. (Bottom) After filtering, the envelope of the spectrum issmoother and enhances the stability in inverting the Green’sfunction.

FIG. 9. (Top) Kirchhoff migration image of the zero-offset datashown in Figure 7, where all of the traces were used to constructthe image at any point. (Bottom) Image after applying migra-tion deconvolution filter. The filter consisted of five layers forthe top half of the model and seven layers for the bottom half.Note the decreased noise and the sharper reflectivity in theboxed area.

FIG. 10. Same as bottom image in Figure 9 except a nine-layerdeconvolution filter is used. With the increase of layers, noiseis reduced but some artifacts are introduced (see boxedarea).

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FIG. 11. (Top) Kirchhoff migration image of the half-sampledzero-offset data. The footprint noise from spatial aliasing isquite pronounced. (Bottom) Result of applying migration de-convolution to the migrated section where noise is decreasedin the boxed area. The filter used five layers for the top half ofthe model and seven layers for the bottom half.

FIG. 12. The NMO velocity model of the marine data set. ThisNMO velocity is used in time migration and migration decon-volution. The arrows at the top indicate the reference posi-tion for the three different Green’s functions used in migrationdeconvolution.

of Figure 13 is noticeably improved in resolution, migrationartifact attenuation, and reflectivity amplitude recovery. Toshow the difference more clearly, enlargements of the boxedareas in Figure 13 is presented in Figure 14. Here, the migra-tion deconvolution image reveals more geological details andis judged to have the best overall quality.

To compare the migration deconvolution image with theconventional wavelet deconvolution results, an SU spiking de-convolution with a band-pass filter and a whitening filter areapplied to the original time-migration image. The zoom-viewimages are shown in Figure 14. Compared with the spiking-deconvolution image and whitening filter results, migrationdeconvolution noticeably improves resolution and amplituderecovery.

3-D poststack reverse time migration of a Gulf of Mexicodata set

In this test, a poststack reverse time-migration (RTM) imagefrom the Gulf of Mexico was chosen to test the effectiveness of

FIG. 13. (Top) A Kirchhoff poststack time-migration image ofthis data set. (Bottom) The time-migration deconvolution re-sult for a 9 + 11 variable-layer filter. The deconvolved imageshows noticeable improvements in resolution, migration ar-tifact reduction, noise attenuation, and reflection amplitudeenhancement.

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FIG. 14. (Top four figures) Comparison of migration deconvolution images with conventional wavelet deconvolution results. Here,migration deconvolution is applied to the time-migration image. It is clear that migration deconvolution results in higher spatialresolution, reveals more geological details, and has better overall quality. (Bottom four figures) Same as top four figures except fora different area of the data set.

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migration deconvolution. In the migration deconvolution al-gorithm, the migration Green’s function was computed usingKirchhoff depth migration. In Kirchhoff depth migration, onlythe first arrival times are used for the seismic imaging condi-tion. In contrast, reverse-time migration uses the transmittedacoustic wavefield for seismic imaging so that the migrationGreen’s function may differ from that generated by Kirchhoffmigration. This test addresses the stability of applying migra-tion deconvolution to migration images constructed by wave-equation extrapolation.

The in-line section of the 3-D RTM image is shown as theleft half of Figure 15. In this data set, there is some vertical

FIG. 15. (Left) In-line section of the poststack RTM image,where the image is contaminated by some vertical noise andthe reflectivity below the saltbody is poorly defined. (Right)Migration deconvolution image of the in-line (X, 16) sectionon the left. Compared with the original RTM image, the decon-volved migration image is noticeably improved in resolution,noise attenuation, and subsalt imaging.

FIG. 16. Comparison between horizontal slices of RTM and migration deconvolution images. RTM images are at depths of 1800(upper left) and 3400 m (bottom left). To the right are the corresponding migration deconvlution images.

noise in the migrated image, and the reflectivity below saltis not well defined. The poststack RTM data were used asinput to migration deconvolution; an in-line section of the3-D migration deconvolution result is shown on the rightside of Figure 15. Compared with the original RTM data,the in-line migration deconvolution image is noticeably im-proved in resolution and noise attenuation, especially belowthe salt body. The salt boundary is more clearly delineated,and the reflectivity beneath the salt is more recognizable thanbefore.

Comparison of the horizontal RTM and migration decon-volution images at depths of 1800 and 3400 m are shown inFigure 16. Clearly, the migration deconvolution results showbetter resolution than the RTM images.

Overall, our results suggest that migration deconvolutionimproves the quality of the RTM image. These results show thatmigration deconvolution can be applied successfully to RTMsections despite the fact that the migration Green’s functionis implemented using a Kirchhoff migration in the migrationdeconvolution algorithm. Kirchhoff migration only uses thewavefield information that corresponds to minimum arrivaltimes, and RTM uses the entire wavefield. So it is reasonableto expect that the migration deconvolution algorithm would besuitable even if the data were migrated by different migrationmethods.

3-D prestack Kirchhoff migration of a North Sea data set

In this final test, migration deconvolution was applied to aprestack Kirchhoff migration image of 3-D North Sea data.

Considering the similarity between the poststack andprestack migration Green’s function (Schuster and Hu, 2000),a poststack migration deconvolution filter is applied to theprestack Kirchhoff migration image. The migrated imagesand the deconvolved migration images for in-line sections ofthe 3-D data set are shown in Figure 17. Compared with the

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original migrated image, there is a noticeable improvement inthe resolution of the migration deconvolution result. Althoughthe resolution is improved by migration deconvolution, mostof the noise in the Kirchhoff migrated image still remainsin the migration deconvolution result. The possible reasonsinclude (1) differences between the poststack migrationGreen’s function and the prestack migration Green’s function,(2) incomplete traveltime information, (3) incorrect velocitymodel, and (4) far-field migration effects. Although thepoststack migration Green’s function is different from theprestack migration Green’s function, it can still be used topartly filter the prestack migrated sections.

Figures 18 and 19 compare the Kirchhoff migration (top)and the migration deconvolution (middle) horizontal sections

FIG. 17. (Top) The prestack Kirchhoff migration image of theNorth Sea data for the in-line (X, 32) section. The image con-tains some migration noise, and the resolution is poor adjacentto the salt flank. (Middle) The migration deconvolution resultof the top image. (Bottom) The in-line section (X, 32) of theoriginal migration image after harmonizer deconvolution FXYdeconvolution plus AGC processing. Compared with the origi-nal prestack Kirchhoff migrated image (top), the deconvolvedresult (middle) is noticeably improved in resolution and am-plitude balancing. However, most of the migration noise in themigrated image remains. Comparing the bottom image withthe original migrated image (top), it is clear that harmonizerdeconvolution plus FXY deconvolution improves the verticalresolution and attenuates migration noise. In the vertical sec-tion, the appearance of the AGC plus harmonizer deconvolu-tion plus FXY deconvolution result is as good as the migrationdeconvolution result (middle); however, the top boundary ofthe saltbody is smeared.

at 1990 and 2990 m depths, respectively. The Kirchhoff migra-tion section appears as an image out of focus compared to themore focused migration deconvolution image. Thus, the spatialresolution of the Kirchhoff migrated images is improved by mi-gration deconvolution, and the deconvolved migration imagereveals more subsurface details. This result is consistent with

FIG. 18. Comparison of horizontal slices of prestack Kirchhoffmigration, migration deconvolution, and harmonizer plus FXYdeconvolution at 1990 m depth.

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the enhanced focusing seen in the point-scatterer example ofFigure 1.

Comparison of migration deconvolution with conventionaldeconvolution

Filtering is sometimes applied to migrated data to improveimage quality. For example, conventional wavelet deconvolu-

FIG. 19. Comparison of horizontal slices of prestack Kirchhoffmigration, migration deconvolution, and harmonizer plus FXYdeconvolution at 2990 m depth.

tion is sometimes used as a postmigration processing step to im-prove the vertical resolution of migrated images. Here, we com-pare the migration deconvolution result with the result afterprocessing the raw migrated image with a sample-varying pre-dictive deconvolution plus FXY deconvolution (Chase, 1992).Figure 17 (bottom) shows the sample-varying predictive decon-volution plus FXY deconvolution results for the in-line sectionof the prestack Kirchhoff migration image. Compared with themigration deconvolution result (middle) as well as the raw mi-grated data image (top) in Figure 17, sample-varying predictivedeconvolution (sometimes known as harmonizer deconvolu-tion) improves the vertical resolution in the migration imagesand partly attenuates artifacts in the migrated data. In the pro-cessed profile, the image quality is similar to the migration de-convolution result except the upper salt boundary is smearedby the sample-varying deconvolution.

The horizontal section of the sample-varying predictive de-convolution results at 1990 and 2990 m depth are shown atthe bottom of Figures 18 and 19. Compared with the raw datasection and the migration deconvolution result at the samedepth, the sample-varying predictive deconvolution results donot show improvements in the horizontal spatial resolution.The image quality of the sample-varying predictive deconvo-lution in a horizontal section is noticeably worse than the mi-gration deconvolution result.

In the sample-varying deconvolution plus FXY deconvolu-tion example, automatic gain control (AGC) is applied beforedeconvolution. Although AGC is able to balance amplitudes,it leads to another problem in the migrated image: amplitudedistortion. In migration deconvolution, the amplitudes in themigrated image are, in theory, properly corrected by the in-verse of the migration Green’s function, and the energy is re-distributed according to the migration Green’s function. Soafter migration deconvolution, the amplitude in the migratedimage is partly corrected and balanced from the top to thebottom of the image.

CONCLUSIONS AND DISCUSSION

The migration section is a blurred image of the geologic re-flectivity distribution, where the blurring kernel is the PSF,also known as the migration Green’s function. For a coarselysampled and narrow distribution of sources and receivers, themigration Green’s function introduces noticeable artifacts intothe migration image. Furthermore, LT is not the exact inverseto the forward-modeling operator, so the migration image isonly a rough approximation to the reflectivity image. To miti-gate these blurring effects in migration, we presented a methodthat partially deconvolves the effect of the poststack migrationGreen’s function from the migration image. To improve thecomputation efficiency we invoked two approximations: (1)the migration Green’s function is horizontally shift invariantand (2) the migration Green’s function is localized about theimage point to within a depth interval of about one wavelength.Deconvolution results with synthetic and field data suggest thatmigration deconvolution can be both efficient and effective ineliminating some migration artifacts. The deconvolved imagesshowed partial elimination of noise, improved spatial resolu-tion, and enhanced reflectivity amplitudes.

Although our localized migration deconvolution operatorattenuates the near-field migration artifacts, it cannot remove

950 Hu et al.

all far-field migration artifacts such as steeply dipping parts ofsmiles. Removal of such noise should begin with the carefuldesign of the migration operator, where the reflection energyshould only be smeared along a restricted portion of the mi-gration surface. This restricted region is identified from thetraveltime and incidence angle of the reflection energy (Sunand Schuster, 2000).

In the context of optimization theory, our deconvolution fil-ter is the inverse to a moderately wide band of the matrix [LT L],where [LT L] is the matrix representation of the PSF integraloperator in equation (1). Thus, this inverse should be moreeffective than the preconditioning filter, which removes geo-metrical spreading effects by multiplying the ith cell of themigration image by 1/[LT L]i i . However, it should not be as ef-fective, in principle, as the least-squares inverse [LT L]−1 usedin least-squares migration (Nemeth et al., 1999). In practice,the local nature of our migration deconvolution filter suggestsless sensitivity to an incorrect migration velocity and introducesless spurious noise than least-squares migration.

Some improvements to the design of migration deconvolu-tion filters might include the following:

1) speeding up the calculation of the migration Green’sfunction by the use of analytical formulas (Schuster andHu, 2000);

2) replacing the ad hoc or trial- and-error estimation of fil-ter parameters (e.g., truncation width, filter length, num-ber of effective layers, frequency bandwidth of sourcewavelet) by formulas based on fundamental principles(an incorrect choice of parameters will lead to subopti-mal or noisy images);

3) efficiently inverting the entire 1-D depth integral with-out the need for truncating this integral about the zoneof influence; and

4) designing an efficient filter that takes into account a lat-erally variant velocity model.

Poststack migration deconvolution cannot reduce noise frompoor spatial sampling in the prestack domain. In this case,prestack migration deconvolution (a straightforward extensionof poststack migration deconvolution) should be used (Hu andSchuster, 2000). If the velocity model is quite complicated, thenbetter results could be achieved by designing different migra-tion deconvolution operators for different reference positionsin the velocity model. The final migration image will be a com-posite of the different deconvolved images patched together.

Finally, if the migration LT and modeling L operators are de-signed to take into account other types of deterministic noise,such as multiples, then in principle such noise could also bedeconvolved from the migration image. Thus, migration de-convolution is able to lessen the dependence of image qualityon seismic recording geometry, which is very important in seis-mic interpretation and analysis.

ACKNOWLEDGMENTS

We greatly appreciate the valuable comments and sugges-tions from reviewer Gary F. Margrave, an anonymous reviewer,and associate editor P. Cary. We thank Bob Heaton for thesample-varying predictive and FXY deconvolution processingof the North Sea migration image. The authors sincerely thankGary Hoover, Phillips Petroleum Company, and other UTAM(http://utam.gg.utah.edu) sponsors for financial support of thisresearch, including Amoco, Arco, Amerada Hess, Baker-Atlas,Conoco, Chevron, Exxon, Marathon, Mobil, Noranda, INCO,Japan National Oil Co.. and RC2. We would like to extendour appreciation to Veritas Marine Surveys for allowing us toshow the Gulf of Mexico data set. We would also like to thankthe following companies for allowing us to present the NorthSea data results: Agip (UK) Ltd., BG EP Ltd., Centrica plc.,Conoco UK Ltd., Fina Exploration Ltd., and Phillips (UK) Ltd.

REFERENCES

Aminzadeh, F., Brac, J., and Kunz, T., 1997, 3-D salt and over-thrust models, SEG/EAGE 3-D Modeling Series No.1: Soc. Expl.Geophys.

Chase, M. K., 1992, Random noise reduction by FXY prediction filter-ing: Expl. Geophys., 23, 51–56.

Claerbout, J., 1992, Earth soundings analysis: Processing vs. inversion:Blackwell Scientific Publications, Inc.

Hanisch, R., White, R., and Gilliland, R., 1997, Deconvolution ofHubble telescope images and spectra, in Jansson, P., ed., Decon-volution of images and spectra, 2nd ed.: Academic Press, 310–360.

Hu, J., and Schuster, G. T., 2000, Prestack migration deconvolution:70th Ann. Internat. Mtg., Soc. Expl. Geophys., 984–987.

Jansson, P., Ed., 1997, Deconvolution of images and spectra, 2nd ed.:Academic Press.

Morse, P., and Feshbach, H., 1953, Methods of theoretical physics:McGraw-Hill Book Co.

Nemeth, T., Wu, C., and Schuster, G. T., 1999, Least-squares migrationof incomplete reflection data: Geophysics, 64, 208–221.

Schuster, G. T., and Hu, J., 2000, Migration Green’s function: Contin-uous recording geometry: Geophysics, 65, 167–175.

Sun, H., and Schuster, G. T., 2000, Wavepath migration vs. Kirchhoffmigration: 3-D prestack examples: 70th Ann. Internat. Mtg., Soc.Expl. Geophys., Expanded Abstracts, 977–980.

Yilmaz, O., 1987, Seismic data processing: Soc. Expl. Geophys.

APPENDIX A

MIGRATION GREEN’S FUNCTION AND ITS INVERSE

The solution to the acoustic wave equation can be approx-imated as a weighted summation of Green’s functions. Forexample, poststack data d(rg | rs, ω) in the frequency domainare sometimes approximated by the exploding reflector model(Yilmaz, 1987):

d(rg | rs, ω)

=∫

Model−Spaceg(rg | ro, ω)g(ro | rs, ω)m(ro)S(ω) dro,

(A-1)

where g(rg | ro, ω) and g(ro | rs, ω) describe the one-way prop-agation of acoustic waves from the source rs and the receiverrg to the spatial point ro, respectively; m(ro) is the reflectivitydistribution; and S(ω) is the source wavelet. Without loss ofgenerality, we conveniently neglect other weighting terms tosimplify the explanation.

In the following discussion, the Green’s function for theforward modeling operator is assumed to be that for theHelmholtz equation, but the extension of this Green’s func-tion notation to the space–time wave equation is straight-forward.

Poststack Migration Deconvolution 951

Migration of seismic data can be carried out by applying theadjoint (Claerbout, 1992) of the modeling operator to the data.In the case of the Helmholtz equation, the adjoint Green’s func-tion is the complex conjugate of the forward-modeling Green’sfunction (Morse and Feshbach, 1953) and yields the migrationimage denoted by m(r)′, i.e.,

m(r)′ =∫ ∞

−∞

[ ∫Data−Space

S∗(ω)g∗(rg | r, ω)

g∗(r | rs, ω) d(rg | rs, ω) drg drs

]dω,

=∫

Model−SpaceG(r | ro)m(ro) dro, (A-2)

where

G(r | ro) =∫ ∞

−∞

[ ∫Data−Space

S(ω)S∗(ω)g∗(rg | r, ω)

g∗(r | rs, ω)g(rg | ro, ω)g(ro | rs, ω) drg drs

]dω.

Comparing equation (1) to equation (A-2), we see that G(r | ro)is the PSF for diffraction-stack migration. Physically, G(r | ro)describes the migrated section at r for a point scatterer locatedat ro and is appropriately named the migration Green’s func-tion (Schuster and Hu, 2000).

Computing G(r | ro) and its inverse is computationally pro-hibitive, so a simplifying approximation is invoked. If the mi-gration Green’s function is shift invariant in the horizontal di-rection, i.e., velocity is only a function of the z-coordinate andthe recording aperture is extremely wide (see Figure 2), thenequation (A-2) reduces to a 2-D spatial convolution in the hor-izontal coordinates and a 1-D integration in depth:

m(r)′=∫ zmax

zmin

∫(xo,yo)ε�

G(x − xo, y − yo, z | xref, yref, zo)

× m(xo, yo, zo) dxo dyo dzo, (A-3)

where � denotes the horizontal coordinates in model spaceand (xref, yref) is the reference position of the migration Green’sfunction, as discussed in the text.

Applying a 2-D Fourier transform in x and y to equation (A-3) yields

m̃(kx , ky, z)′

=∫ zmax

zmin

G̃(kx , ky, z | xref, yref, zo)m̃(kx , ky, zo) dzo,

(A-4)

where the tilde denotes the transformed function and therecording aperture in x and y is assumed to be laterallyinvariant.

The migration Green’s function is strongest near the point-scatterer location at ro and decreases inversely with horizontaldistance from ro (Schuster and Hu, 2000). Thus, we make a lo-calization approximation and assume G̃(kx , ky, z | xref, yref, zo)is negligible beyond some critical depth interval D above or

below ro, i.e.,

G̃(kx , ky, z | xref, yref, zi ) = 0; if z, zi < zo − D

or z, zi > zo + D, (A-5)

where z increases downward. This localization in depth ap-proximation is somewhat justified by recognizing that the am-plitudes of the hourglasses in Figure 2 fall off inversely withthe horizontal distance from their center.

Substituting equation (A-5) into (A-4), and assuming 2N +1thin layers of thickness δz centered about zo, equation (A-4)can be approximated by the discrete sum

m̃(kx , ky, z)′

≈

N∑i=−N

G̃(kx , ky, z | xref, yref, zi )m̃(kx , ky, zi )zoδz;

if zo − D ≤ z ≤ D + zo

0 otherwise

,

where zi = zo + i · δz and m̃(kx , ky, zi )zo is the deconvolvedimage for the 2N + 1 layers centered at the depth zo. Herewe implicitly assume that the migration image m̃(kx , ky, z)′ atz ε (zi ; i = −N , . . . , N) is primarily influenced by the reflectivitydistribution between the depth levels zo + N · δz and zo − N ·δz.

The above 2N+1 equations can be regularized and solved forthe 2N + 1 reflectivity values m̃(kx , ky, zi )zo at each wavenum-ber. Then m̃(kx , ky, zi )zo can be inverse transformed to give thedeconvolved migration image m(x, y, zi )zo . This procedure isrepeated for all values of zo sampled at depth intervals of δz.The final composite migration image is given by

m(x, y, zi ) =∑

zo

m(x, y, zi )zo . (A-6)

Physically, the assumption of shift invariance in the horizontalcoordinates implies that the medium is laterally homogeneousand the recording aperture is laterally invariant. If the record-ing aperture was laterally variant in width, then the hourglassof the migration Green’s function would tilt (i.e., become shiftvariant) as it approached the edge of the array (see Figure 2).Although these appear to be unrealistic conditions, our numer-ical results in the text suggest their usefulness.

Practical implementation

In practice, several procedures are used to ensure efficiencyand robustness in solving for m(x, y, zi )zo . Calculating the mi-gration Green’s function can be done, as described in the text.In an effort to stabilize the calculation of the deconvolutionfilter, the velocity field was sometimes smoothed to ensure amore regular distribution of rays.

To minimize computation costs, five to seven layers (i.e.,depth samples) of migration influence about ro was assumedso a small linear system of regularized equations is solved. Thetotal thickness of the layers (i.e., the region of influence) isabout a wavelength or so. The implication of assuming a local-ized region of influence is that migration artifacts near ro arereduced but that smiling artifacts far from ro are untouched.Thus, this filter deconvolves the near-scatterer artifacts, whichare the strongest (Schuster and Hu, 2000). The far-field migra-tion artifacts can be reduced by limiting the migration surface

952 Hu et al.

to a small region around the reflector as dictated by the inci-dence angle of the reflection arrivals.

To reduce noise in the filtered image, it is convenient tosmooth G̃(kx , ky, z | xref, yref, zi ) by a five- or seven-point me-dian filter followed by a five- or seven-point averaging filter.This is equivalent to truncating the migration Green’s func-tion G(r | ro) in the horizontal direction, i.e., the hourglasses inFigure 2 become truncated in both the horizontal and depthdirections.

The best results are obtained by replacing G̃(r | ro) in equa-tion (A-4) by |G̃(r | ro)|. For a layered medium and xo at thecenter of the model, G(x, z | xo, zo) is a symmetric, real func-tion in x . Therefore G̃(k, z | xo, zo) is real and symmetric in k,

so that we can replace G̃(k, z | xo, zo) in equation (A-4) by itsmagnitude. Consequently, xo at the center of the model is ourpreferred choice for the reference position. In this case, themigration deconvolution filter is the inverse to the magnitudespectrum of G(r | ro).

An important parameter is the wavelet spectrum in calculat-ing the migration Green’s function. In the synthetic and fielddata examples, we selected the wavelet’s frequency band inthe deconvolution operator to be as wide as that in the data.In the wavenumber domain, the bandwidth of the deconvo-lution operator should be as wide as the effective bandwidthof the migration image, which is determined by inspecting thewavenumber spectrum of the migration image.

APPENDIX B

SUBDIVISION OF THE MIGRATION IMAGE FOR MIGRATION DECONVOLUTION

To account for lateral variations in velocity, the migrationimage volume can be subdivided into several parts and a mi-gration deconvolution filter can be computed for each part.Mathematically, the volume of integration in equation (1) canbe subdivided as

m(r)′| �1 =∫

�1

G(r | ro)m(ro) dro +∫

�2

G(r | ro)m(ro) dro

+ · · · +∫

�n

G(r | ro)m(ro) dro

≈∫

�1

G(r | ro)m(ro) dro, (B-1)

where the total model volume � = ∧ni=1 �i is divided into ver-

tical slabs �i with slight overlap. For each �i slab, we solveequation (B-1) for m(r)|�i . Note the migration Green’s func-tion G(r | ro) [see equation (A-2)] involves all of the traces alongthe entire recording plane, even though r and ro are restrictedto �i for the ith slab.