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Recent results in curvelet-based primary-multiple separation: application to real data Wang, Deli; Saab, Rayan; Yilmaz, Ozgur; Herrmann, Felix J. 2007

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Recent results in curvelet-based primary-multiple separation: application to real dataDeli Wang, Jilin University, Rayan Saab, ECE-UBC, Ozgur Yilmaz, Math-UBC and Felix J. Herrmann, EOS-UBCSUMMARYIn this abstract, we present a nonlinear curvelet-based sparsity-promoting formulation for the primary-multiple separationproblem. We show that these coherent signal components canbe separated robustly by explicitly exploting the locality ofcurvelets in phase space (space-spatial frequency plane) andtheir ability to compress data volumes that contain wavefronts.This work is an extension of earlier results and the presentedalgorithms are shown to be stable under noise and moderatelyerroneous multiple predictions.INTRODUCTIONIn complex areas, multiple suppression techniques based onmove-out filtering fail because the assumptions on the hyper-bolic move-out in the CMP-offset domain are not met. Fur-thermore, the occurrence of shallow, high-velocity layers canlead to small move-out differences between primaries and mul-tiples which are difficult to interpret. These complications mayresult in unsatisfactory separation of primaries and multiples.In cases where move-out filtering based methods fail, ?wave-equation?-based predictive methods (Verschuur et al., 1992;Fokkema and van den Berg, 1993; Weglein et al., 1997) haveshown considerable improvements. Wave-equation methodsconsist of two main steps: the multiple-prediction and the primary-multiple separation step. The separation step is often referredto as adaptive subtraction, during which imperfections in thepredictions, such as the water bottom reflectivity (see e.g. Berry-hill and Kim, 1986; Wiggins, 1988; Lokshtanov, 1999) or sourceand receiver characteristics (Verschuur et al., 1992; Berkhoutand Verschuur, 1997; Ikelle et al., 1997), are absorbed by amatched-filtering procedure. This procedure is important be-cause predictions for surface-related multiples as well as in-ternal multiples based on two-dimensional input data (see e.g.Verschuur et al., 1992; Berkhout and Verschuur, 1997; Coatesand Weglein, 1996) are often inaccurate in situations where thesubsurface displays three-dimensional complexity. Other com-plications determining the success of multiple attenuation in-clude source-receiver directivity, ghosts and the obliquity fac-tor, unbalanced amplitudes of multiple predictions that con-sist of mixtures of different-order multiples (Verschuur andBerkhout, 1997; Chen et al., 2004), and incomplete data, e.g., dueto missing near offsets, or unequal source and receiver spacingboth of which may give rise to artifacts in the predicted multi-ples (see Verschuur, 2006).Several attempts have been made to improve multiple elimi-nation by either increasing the accuracy of the multiple pre-dictions or by devising a more robust subtraction/separationmethodology. Examples of the first approach are methods basedon model-driven time delays, as proposed by Ross (1997) andRoss et al. (1997), or methods based on data-driven time de-lays by Ikelle and Yoo (2000). Decomposition of the predictedmultiples into coherent and incoherent components is an exam-ple of the second approach (Kabir, 2003), where the incoherentsignal component is assumed to mostly contain diffracted mul-tiples. In that approach, both components are simultaneouslysubtracted from the input data. Another example is the ap-proach taken by Wang (2003) who improves the adaptive sub-traction by introducing additional local time and phase shifts.In this abstract, we showcase results that are obtained from thesecond-generation of curvelet-based primary-multiple separa-tion algorithms. First, we briefly summarize the first-generationalgorithm (see for detail Herrmann et al., 2007), followed bya brief outline of the next-generation Bayesian approach. Weshow that this approach has the advantage that it is less sen-sitive to the selection of the appropriate weights for the sep-aration. We conclude this abstract by applying the separationmethod on a real dataset. For more technical details on thisnew Bayesian-based separation algorithm refer to another con-tribution by the authors to the proceedings of this conference.Curvelet-domain primary-multiple separationIn this section, we briefly describe two curvelet-based primarymultiple separation methods. For more details on the curvelettransform and the first-generation of curvelet-based separationwe refer to (Herrmann et al., 2007, and the references therein).Forward model: Both curvelet-based primary-multiple sep-aration algorithms are based on the following forward signalmodel:s =s1 +s2 +n (1)with the subscript used the indicate primaries (1) and multiples(2). The symbols s and n denote the total recorded data and awhite Gaussian noise term, respectively. Both approaches useSRME (see e.g. Verschuur et al., 1992) predictions for the pri-maries (?1) and multiples (?2) as input. The two approachesdiffer in how the two unknown signal components are esti-mated.First-generation curvelet-based primary-multiple separa-tion: We solve the following nonlinear weighted lscript1-normoptimization problemP1w :8><>:minx bardblxbardblw,1 subject to bardbly-Axbardbl2 <=<epsilone1 =A1e1 and e2 =A2e2given: ?2 and w(y, ?2)(2)where A j, j = 1? ? ? 2 are the inverse curvelet transform matri-ces and w =[w1, w2]T are the weighting vectors with strictlypositive weights defined in terms of the predicted multiples,i.e., w1 =|C1 ?AT1 ?2| and w2 =|C2 ?AT2 ?1| with scale-dependentnormalization constants C1,2 (see Herrmann et al., 2007, fordetail). The estimates for the primaries and multiples are com-puted from the sparsity vector that solves P1w. During the op-timization, the sparsity vector is recovered by minimizing theCurvelet-based primary-multiple separationweighted lscript1 norm subject to a recovery that is within the toler-ance.This algorithm is based on the assumption that both signalcomponents are given by a sparse superposition of curvelets,i.e.s1 =A1x01 +n1 and s2 =A2x02 +n2 (3)with the vectors x01 and x02 assumed to be sparse.P1w : can be solved using an iterative soft thresholding methodwith thresholds set according to the weighting vector w. At thenth iteration, the following update is calculatedxnj =Tlambda?wj0@xnj +AAATj0@s-AAAjxnj -Xinegationslash=jAixi1A1A, (4)with the threshold set by the jth-component of the weightingvector and Tlambda (x) := sgn(x) ? max(0, |x| -|lambda|) the soft thresh-olding operator.Second-generation curvelet-based primary-multiple sepa-ration: Unfortunately, the first-generation of primary-multipleseparation is sensitive to the weighting, i.e. the emphasis of theweighing towards the primaries or multiples. Incorrect bal-ancing for the weights yielded, after several iterations of thesolver, a solution where all the signals energy could potentiallyend up in one (typically the primary) estimated component.To mediate this problem, we slightly modify the nonlinear sig-nal separation program intoP2w :8><>:e = argminx bardblx1bardbl1,lambda1?w1 +bardblx2bardbl1,lambda2?w2 +bardbl?2 -A2x2bardbl22 +etabardbl?1 +?2 -A1x1 -A2x2bardbl22e1 =A1e1 and e2 =A2e2.Again the ?1,2 represent the SRME-predicted primaries andmultiples, A1,2 the inverse discrete curvelet transforms, x =[x1, x2]T the unknown curvelet vector for the primaries (x1)and multiples (x2). The lambda1,2 and eta are control parametersdetermining the emphasis on the data misfit, the confidencein the multiple prediction, and the noise level. The estimatesfor the separated signal components is obtained by the inversecurvelet transform of the e1,2 that minimize above nonlinearprogram.This weighted nonlinear sparsity promoting program (P2www) at-tempts to find the sparsest sets of curvelet coefficients for theprimaries and multiples that fit the predicted multiples and thetotal data. This program is different from P1www, where the twosparsest sets of coefficients are sought that fit the total data.The additional lscript2-penalty term ensures that the coefficients forthe multiples fit the SRME-predicted multiples.The parameters lambda1 ands lambda2 and eta control the emphasis onthe sparsity prior versus the data misfit. The parameter eta ex-presses the confidence in the total data compared to the pre-diction for the multiples. The lambda1, lambda2 determine the level ofsparsity for the primaries and multiples, respectively.The above nonlinear optimization problem can be solved byan iterative thresholding procedure. After n iterations, the es-timates for the two seperated signal components are given byxn+11 = T ?lambda1?w1?xn1 +AAAT`?s1 +?s2 -AAAxn -AAAxn??(5)xn+12 = T ?lambda2?w2?xn2 +AAAT`?s2 -AAAxn?+ ?etaAAAT `?s1 -AAAx1??with ?1 = lambda12(1+eta) , ?2 = lambda22eta and ? = etaeta+1 .APPLICATION TO THE SAGA DATA SETThe proposed algorithm is tested on the SAGA dataset (seee.g. Verschuur et al., 1992, for a detailed description of thisdataset), consisting of 128 sources and receivers and 1024 timesamples. The 3-D discrete curvelet transform with wrappingwas used (Ying et al., 2005) using 32 angles at the finest scale.The SRME results (Fig. 2) were obtained using a spatiallyvarying matched filter. The single threshold results (Fig. 3)were obtained by running Eq. 4 with only one iteration andC1 = 1. The second generation results were obtained for 7iterations with eta =2.0, lambda1 =3.0 and lambda2 =3.0. Both curvelet-based results (Figs 3 and 4) contain less high-frequency cluttercompared to the SRME result in Fig. 2 while the solution ofP2www leads to an improved multiple removal compared to thesingle thresholded result (cf. Figs 3,4).DISCUSSION AND CONCLUSIONSBy virtue of their compression capabilities, localization, mul-tiscale, and multi-angular nature curvelets represent the idealdomain for primary-multiple separation. Not only does thecurvelet construction allow for a separation based on differ-ences in these curvelet attributes but their compression alsoallows for a sparsity promoting formulation of the primary-multiple separation problem. In addition, the compression re-duces the probability of having large entries for each signalcomponent at the same location in the curvelet vector. More-over, the curvelet?s multi-angular parameterization helps theseparation, even for conflicting dips and erroneous predictions.The reformulation of the separation problem resolved an ear-lier shortcoming by adding an additional constraint controllingthe misfit between the estimated and SRME-predicted multi-ples. The results of applying this method to real data show aclear improvement over SRME-predicted primaries and overthe earlier generation of curvelet-based primary-multiple sep-aration.ACKNOWLEDGMENTSThe authors would like to thank Eric Verschuur for his input inthe primary-multiple separation. We also would like to thankthe authors of CurveLab for making their codes available. Theexamples presented were prepared with Madagascar (rsf.sourceforge.net/), supplemented by SLIMPy operatoroverloading, developed by Sean Ross Ross. Norsk Hydro isthanked for making the field dataset available. This work wasin part financially supported by the NSERC Discovery (22R81254)and CRD Grants DNOISE (334810-05) of F.J.H. and was car-ried out as part of the SINBAD project with support, securedthrough ITF, from BG Group, BP, Chevron, ExxonMobil andShell.Curvelet-based primary-multiple separationFigure 1: Total data SAGA data. Figure 2: SRME-predicted primaries.Curvelet-based primary-multiple separationFigure 3: Estimated primaries with single thresholding. Figure 4: Estimated primaries according to P2www.Curvelet-based primary-multiple separationREFERENCESBerkhout, A. J. and D. J. Verschuur, 1997, Estimation of multiple scattering by iterative inversion, part I: theoretical considerations:Geophysics, 62, 1586?1595.Berryhill, J. R. and Y. C. Kim, 1986, Deep-water peg legs and multiples: Emulation and suppression: Geophysics, 51, 2177?2184.Chen, J., E. Baysal, and O. Yilmaz, 2004, Weighted subtraction for diffracted multiple attenuation: 74th Ann. Internat. Mtg., SEG,Expanded Abstracts, 1329?1332, Soc. Expl. Geophys., Expanded abstracts.Coates, R. T. and A. B. Weglein, 1996, Internal multiple attenuation using inverse scattering: Results from prestack 1 and 2-Dacoustic and elastic synthetics: 66th Ann. Internat. Mtg., SEG, Expanded Abstracts, 1522?1525, Soc. Expl. Geophys., Expandedabstracts.Fokkema, J. T. and P. M. van den Berg, 1993, Seismic applications of acoustic reciprocity: Elsevier.Herrmann, F. J., U. Boeniger, and D.-J. E. Verschuur, 2007, Nonlinear primary-multiple separation with directional curvelet frames:Geoph. J. Int. To appear.Ikelle, L., G. Roberts, and A. Weglein, 1997, Source signature estimation based on the removal of first-order multiples: Geophysics,62, 1904?1920.Ikelle, L. T. and S. Yoo, 2000, An analysis of 2D and 3D inverse scattering multiple attenuation: 70th Ann. Internat. Mtg., SEG,Expanded Abstracts, 1973?1976, Soc. Expl. Geophys., Expanded abstracts.Kabir, M. M. N., 2003, Weighted subtraction for diffracted multiple attenuation: 73rd Ann. Internat. Mtg., SEG, Expanded Ab-stracts, 1941?1944, Soc. Expl. Geophys., Expanded abstracts.Lokshtanov, D., 1999, Multiple suppression by data-consistent deconvolution: The Leading Edge, 18, 115?119.Ross, W. S., 1997, Multiple suppression: beyond 2-D. part I: theory: 67th Ann. Internat. Mtg., Expanded Abstracts, 1387?1390,Soc. Expl. Geophys., Expanded abstracts.Ross, W. S., Y. Yu, and F. A. Gasparotto, 1997, Multiple suppression: beyond 2-D. part II: application to subsalt multiples: 67thAnn. Internat. Mtg., Expanded Abstracts, 1391?1394, Soc. Expl. Geophys., Expanded abstracts.Verschuur, D. J., 2006, Seismic multiple removal techniques: past, present and future: EAGE publications b.v. edition.Verschuur, D. J. and A. J. Berkhout, 1997, Estimation of multiple scattering by iterative inversion, part II: practical aspects and ex-amples: Geophysics, 62, 1596?1611.Verschuur, D. J., A. J. Berkhout, and C. P. A. Wapenaar, 1992, Adaptive surface-related multiple elimination: Geophysics, 57,1166?1177.Wang, Y., 2003, Multiple subtraction using an expanded multichannel matching filter: Geophysics, 68, 346?354.Weglein, A. B., F. A. Carvalho, and P. M. Stolt, 1997, An iverse scattering series method for attenuating multiples in seismicreflection data: Geophysics, 62, 1975?1989.Wiggins, J. W., 1988, Attenuation of complex water-bottom multiples by wave-equation-based prediction and subtraction: Geo-physics, 53, 1527?1539.Ying, L., L. Demanet, and E. J. Cand? 2005, 3D discrete curvelet transform: Wavelets XI, Expanded Abstracts, 591413, SPIE.

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