Seismic Theory 8: Migration Resolution AnalysisTuesday a.m., Nov. 4

S T 8 . 1

Migration Operator Compression by Wavelet Transform: Beamlet MigratorRu-Shan Wu*, Fusheng Yang, Zhenli Wang and Ling Zhang, Institute of Tectonics, University ofCalifornia, Santa Crux, CA

Summary

We study one-way wave propagation in the frequency-wavelet domain (frequency-beamlet domain) for dif-ferent wavelet bases, and compare their performancesfor migration/imaging. We show that the Kirchhoffoperator in space domain is a dense matrix, whilethe compressed beamlet-propagator matrix which isthe wavelet decomposition of the Kirchhoff opera-tor (or other one-way wave propagators), is a highlysparse matrix. For sharp and short bases, such asthe Daubechies 4 (D4), both the ‘interscale and in-trascale coupling are strong. However, the interscalecoupling is relatively weak for smooth bases, such ashigher-order Daubechies wavelets, Coiflets, and splinewavelets. The images obtained by the compressedbeamlet operator are almost identical to the imagesfrom a full-aperture Kirchhoff migration. Comparedwith the limited aperture Kirchhoff migration, beam-let migration can obtain much wider effective aper-tures, hense higer resolution and image quality, withsimilar computational efficiency. The compression ra-tio of the migrator ranges from a few times to afew hundred times, depending on the frequency, steplength and the wavelet basis. Combining with the data(wavefield) compression, even greater efficiency can beexpected.

Introduction

Seismic data compression has achieved remarkablesuccess. However, at present data compression is fo-cused on data transportation and storage problems,and is assumed to be transparent to processing. Littlehas been done towards directly processing compresseddata in the wavelet domain and, furthermore, com-pressing the processing operators themselves. Nev-ertheless some progress has been reported in the ap-plication of wavelet transform to wave propagationand migration (Le Bras and Mellman, 1994; Dess-ing and Wapenaar,1995; Mosher and Foster, 1995;Wu and McMechan, 1995; Mosher et al., 1996; Wangand Pann, 1996). Our approach is to to study bothdata compression and migration operator compres-sion using wavelet transform. This approach requiresmuch dedicated theoretical study on wavelet transformand wave propagation. Operator compression mayhave quite different requirements on the properties ofwavelet bases. Seismic data compression is intended

to find efficient ways of representing data. Yet DavidMarr reminded us,“how information is presented cangreatly affect how easy it is to do different things withit”. Not all of the good compression algorithms suitseismic imaging. As Yves Meyer pointed out, “Analgorithm that is optimal for compression can be dis-astrous for analysis” (Meyer, 1993). We need to findthe best algorithm not only for data compression butalso for compression of migration operators. In thiswork we study one-way wave propagation operators inthe wavelet domain, and their compression in differentwavelet bases.

Beamlet Decomposition andMigrators

Compressiono f

As an example, we study the one-way wave propaga-tion and imaging using Kirchhoff integral (or Rayleighintegral for flat surfaces):

= -2 (1)

S

where m is the extrapolated wave field for forwardproblem, and the image function for imaging problem,s is the surface on which the wave field u (pressure)is known, n is the normal of the surface, and g is theGreen’s function. If we replace the Green’s functionin above formulas by its conjugate (changing forwardpropagation into backpropagation), the surface inte-grals become migration operators.

For the sake of simplicity, we use the Kirchhoff mi-grator in homogeneous background (Rayleigh back-propagation integral) as an example. Figure la andlb show the matrix representations of a full-aperturemigrator for different frequencies (5.9 Hz and 25 Hz).This matrix will migrate a wave field 128 points longwith sampling interval = 25 m to a depth of z = 25m. In the figures only real parts of the complex op-erators are shown.We see that the Kirchhoff op-erator in space domain is a dense matrix, i.e.theoff-diagonal coefficients fall off slowly when movingaway from the diagonal.The second and third rowsof Figure 1 (c-f) show the matrix representations ofthe same operator in wavelet domain with differentbases, Daubechies 4 (D4) and Coiflet 5 (C5), respec-tively. These figures demonstrate the different behav-iors of migrators in wavelet domain of different waveletbases. A wavelet-decomposed migrator represents the

1646

Migrator compression by wavelet transform

backpropagation of beams of different angles in dif-ferent scales. Therefore we name it as “BEAMLETMIGRATOR”. We see that for sharp and short bases,such as D4, both the interscale coupling (the couplingbetween large and small angle (scale) beamlets) andintrascale (the coupling between beamlets in the samescale) are strong. However, the interscale coupling isrelatively weak for smooth bases, such as higher-orderDaubechies wavelets, Coiflets, and spline wavelets. InFigure 1e and 1f is shown the case of C5. Note thatthe beamlet matrices are highly sparse matrices com-pared with the original Kirchhoff matrix. This meansthat great efficiency can be achieved with beamlet mi-gration. The bottom two rows in Figure 1 (g-j) showthe compressed beamlet migrators (D4 and C5) witha cut level of one percent of the maximum coefficient.The compression ratio of the migrator ranges from afew times to a few hundred times, depending on thefrequency, step length and the wavelet basis.

Imaging with Compressed Beamlet Migrators

The operator compression in the wavelet domain isvery different from more traditional techniques for re-ducing computations for Kirchhoff migration, such asaperture limitation. Aperture limitation can result indegradation of image quality. On the other hand, thecompression in wavelet domain will retain the effectiveaperture untouched. This is due to the efficient way ofrepresenting the migration operator in wavelet domainand the multi-scale nature of the wavelet transform.In order to see the effectiveness and the aperture pre-served nature of the beamlet migrator compression, inFigure 2 and 3 we show some examples of beamlet mi-gration for simple objects: point scatterers. Figure 2bis the synthetic seismic section of a point scatterer (anexploding reflector), and Figure 2a shows the migratedimage by a full aperture Kirchhoff migration, which isserved as a reference image to compare with. Figure 2cand 2d show the migrated image by a compressed D4beamlet migrator and its residual image with respectto the reference image in figure 2a, respectively. Theresidual image has been amplified 10 times in strength.Figure 2e and 2f are parallel to Figure 2c and 2d butusing a compressed C5 beamlet migrator. We see thatthe images using the compressed beamlet migratorsare almost identical to that using the uncompressedmigrator. The averaged compression ratios over fre-quency is 14:1 for the D4 beamlet migrator, and 17:1for the C5 beamlet migrator. In comparison, Figure2g and 2h give the migrated image and the residualimage, respectively, for a 33 point aperture Kirchhoffmigrator which requires a similar amount of computa-tion time as the compressed D4 beamlet migrator. Wesee that the image quality is much deteriorated due tothe limited aperture problem.

Figure 1: Comparison of migration operator in dif-ferent domains. The left panel is for f = 5.9 Hz. and theright pane,, f = 25 Hz. Only real parts of the complex operatorare plotted. (a)-(b): in space domain; (c)-(d): in D4 waveletdomain; (e)-(f): in C5 wavelet domain; (g)-(h): compressedoperator in D4 wavelet domain. (i)-(j) compressed operatorin C5 wavelet domain. The cut off level is 0.01 of maximumcoefficient.

1647

Migrator compression by wavelet transform

Figure 2: Performance comparison of compressed migrators in wavelet domain and migratorsin space domain. The synthetic zero-offset seismic section for a point scatterer is shown in (b) with 128geophones, = 25m, = 50 Hz, and a Ricker source wavelet. (a): Migrated image by a full apertureKirchhoff migration; (c), (e) and (g) are the migrated images by the compressed D4 beamlet migrator (compressionratio = 14), compressed C5 beamlet migrator (compression ratio = 17) and a 33 point width Kirchhoff migrator,respectively. On the right, (d), (f) and (h) are the corresponding residual images (amplified by a factor of 10).

1646

Migrator compression by wavelet transform

As another example, Figure 3a gives the syntheticsex-offset seismic section of three point scatterers andFigure 3b shows the migrated images by the com-pressed D4 beamlet migrator. We see the high res-olution and good image quality have been retained af-ter the operator compression in wavelet domain. Incontrast, the images of the three points migrated bythe 33 point Kirchhoff migrator become fuzzy and arenot separatable due to the interference between thepoorly-resolved and distorted individual image fieldsof these points.

Conclusions

We have shown that the compressed b-let matrix isa highly sparse matrix. The beamlet migrator repre-sents the backpropagation of beams of different anglesin different scales. For sharp and short bases, suchas D4, both the interscale coupling (the coupling be-tween large and small angle (scale) beamlets) and in-trascale coupling (the coupling between beamlets inthe same scale) are strong. However, the interscalecoupling is relatively weak for smooth bases, such ashigher-order Daubechies wavelets, Coiflets, and splinewavelets. The images obtained by the compressedbeamlet operator is almost identical to the images by afull-aperture Kirchhoff migration. Compared with thelimited aperture Kirchhoff migration, beamlet migra-tion can obtain much wider effective apertures, hensehiger resolution and image quality, with similar com-putational efficiency.

Acknowledgements

The support from the WTOPI (Wavelet TransformOn Propagation and Imaging for seismic exploration)Consortium at UCSC and the facility support from theW. M. Keck Foundation are acknowledged. Contribu-tion number 325 of the Institute of Tectonics, Univer-sity of California, Santa Cruz.

References

Daubechies, I., 1992, Ten Lectures on Wavelets, SIAM.Dessing, F.J. and Wapenaar, C.P.A., 1995 Efficient mi-

gration with one-way operators in the wavelt transformdomain, Expanded Abstracts of the Technical Program,SEG, 65th Annual Meetign, 1240-1243

Le Bras, R. and Mellman, G., 1994 Wavelet Transformsand Downward Continuation, Wavelets and Their Ap-plication, Kluwer Academic Publishers. 291-296

Meyer, Y., 1993, Wavelets: Algorothortism and ApplicationsS I A M .

mic data compressed by matching pursuits decomposi-tion, Expanded Abstracts of the Technical Program, SEG66th Annual Meeting, 1642-1645

Wu, Y. and McMechan, G. 1995. Wavefield Extrapolationin the wavelet domain with application to poststack m-gration, Expanded Abstracts of the Technical ProgramsSEG 65th Annual Meeting, 1236-1239

Mosher, C. and Foster, D., 1995, Frequency domain repre-sentations of avelet transforms, “Mathematical meth-ods in geophysical imaging III", (ed. Hassanzadeh, S.),SPIE-The International Society for Optical Engineering,196210.

(b)

Figure 3: Synthetic zero-offset seismic section for threepoint scatterers; (b) Migrated images by the compressedD4 beamlet migrator = = with a compression ratio of 14:1; (c) Migrated images bythe 33 point Kirchhoff migrator (same parameters).

Mosher, C.C., Foster, D.J. and R.S. Wu, 1996, Phase shiftmigration with wave packel algorithms, "MathematicalTextmethods in geophysical imaging IV", (ed. Hassanzadeh,S.), Proc. SPIE., 2822, 2-16.

Wang, Bin and Pann, K., 1996, Kirchhoff migration of seis-

1 6 4 9