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Recent results in curvelet-based primary-multiple separation: application to real data Wang, Deli 2008

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Recent results in curvelet-based primary-multiple separation: application to real data Deli Wang, Jilin University, Rayan Saab, ECE-UBC, Ozgur Yilmaz, Math-UBC and Felix J. Herrmann, EOS-UBC SUMMARY In this abstract, we present a nonlinear curvelet-based sparsity- promoting formulation for the primary-multiple separation problem. We show that these coherent signal components can be separated robustly by explicitly exploting the locality of curvelets in phase space (space-spatial frequency plane) and their ability to compress data volumes that contain wavefronts. This work is an extension of earlier results and the presented algorithms are shown to be stable under noise and moderately erroneous multiple predictions. INTRODUCTION In complex areas, multiple suppression techniques based on move-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 can lead to small move-out differences between primaries and mul- tiples which are difficult to interpret. These complications may result 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) have shown considerable improvements. Wave-equation methods consist of twomain steps: the multiple-prediction and the primary- multiple separation step. The separation step is often referred to as adaptive subtraction, during which imperfections in the predictions, such as the water bottom reflectivity (see e.g. Berry- hill and Kim, 1986;Wiggins, 1988; Lokshtanov, 1999) or source and receiver characteristics (Verschuur et al., 1992; Berkhout and Verschuur, 1997; Ikelle et al., 1997), are absorbed by a matched-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; Coates andWeglein, 1996) are often inaccurate in situations where the subsurface 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 and Berkhout, 1997; Chen et al., 2004), and incomplete data, e.g., due to missing near offsets, or unequal source and receiver spacing both 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/separation methodology. Examples of the first approach are methods based on model-driven time delays, as proposed by Ross (1997) and Ross et al. (1997), or methods based on data-driven time de- lays by Ikelle and Yoo (2000). Decomposition of the predicted multiples into coherent and incoherent components is an exam- ple of the second approach (Kabir, 2003), where the incoherent signal component is assumed to mostly contain diffracted mul- tiples. In that approach, both components are simultaneously subtracted 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 the second-generation of curvelet-based primary-multiple separa- tion algorithms. First, we briefly summarize the first-generation algorithm (see for detail Herrmann et al., 2007), followed by a brief outline of the next-generation Bayesian approach. We show 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 separation method on a real dataset. For more technical details on this new Bayesian-based separation algorithm refer to another con- tribution by the authors to the proceedings of this conference. Curvelet-domain primary-multiple separation In this section, we briefly describe two curvelet-based primary multiple separation methods. For more details on the curvelet transform and the first-generation of curvelet-based separation we 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 signal model: 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 a white Gaussian noise term, respectively. Both approaches use SRME (see e.g. Verschuur et al., 1992) predictions for the pri- maries (s̆1) and multiples (s̆2) as input. The two approaches differ in how the two unknown signal components are esti- mated. First-generation curvelet-based primary-multiple separa- tion: We solve the following nonlinear weighted `1-norm optimization problem P1w : 8><>: minx ‖x‖w,1 subject to ‖y−Ax‖2 ≤ εes1 = A1ex1 and es2 = A2ex2 given: s̆2 and w(y, s̆2) (2) where A j, j = 1 · · ·2 are the inverse curvelet transform matri- ces andw= [w1, w2]T are the weighting vectors with strictly positive weights defined in terms of the predicted multiples, i.e.,w1 = |C1 ·AT1 s̆2| andw2 = |C2 ·AT2 s̆1|with scale-dependent normalization constants C1,2 (see Herrmann et al., 2007, for detail). 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 the Curvelet-based primary-multiple separation weighted `1 norm subject to a recovery that is within the toler- ance. This algorithm is based on the assumption that both signal components 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 method with thresholds set according to the weighting vectorw. At the nth iteration, the following update is calculated xnj = Tλ ·w j 0@xnj +ATj 0@s−A jxnj −X i6= j Aixi 1A1A , (4) with the threshold set by the jth-component of the weighting vector and Tλ (x) := sgn(x) ·max(0, |x| − |λ |) the soft thresh- olding operator. Second-generation curvelet-based primary-multiple sepa- ration: Unfortunately, the first-generation of primary-multiple separation is sensitive to the weighting, i.e. the emphasis of the weighing towards the primaries or multiples. Incorrect bal- ancing for the weights yielded, after several iterations of the solver, a solution where all the signals energy could potentially end up in one (typically the primary) estimated component. To mediate this problem, we slightly modify the nonlinear sig- nal separation program into P2w : 8><>: ex = argminx ‖x1‖1,λ1·w1 +‖x2‖1,λ2·w2+ ‖s̆2−A2x2‖22+η‖s̆1+ s̆2−A1x1−A2x2‖22es1 = A1ex1 and es2 = A2ex2. Again the s̆1,2 represent the SRME-predicted primaries and multiples, A1,2 the inverse discrete curvelet transforms, x = [x1, x2]T the unknown curvelet vector for the primaries (x1) and multiples (x2). The λ1,2 and η are control parameters determining the emphasis on the data misfit, the confidence in the multiple prediction, and the noise level. The estimates for the separated signal components is obtained by the inverse curvelet transform of the ex1,2 that minimize above nonlinear program. This weighted nonlinear sparsity promoting program (P2w) at- tempts to find the sparsest sets of curvelet coefficients for the primaries and multiples that fit the predicted multiples and the total data. This program is different from P1w , where the two sparsest sets of coefficients are sought that fit the total data. The additional `2-penalty term ensures that the coefficients for the multiples fit the SRME-predicted multiples. The parameters λ1 ands λ2 and η control the emphasis on the sparsity prior versus the data misfit. The parameter η ex- presses the confidence in the total data compared to the pre- diction for the multiples. The λ1, λ2 determine the level of sparsity for the primaries and multiples, respectively. The above nonlinear optimization problem can be solved by an iterative thresholding procedure. After n iterations, the es- timates for the two seperated signal components are given by xn+11 = T λ̂1·w1 ˆ xn1+A T `s̆1+ s̆2−Axn1−Axn2´˜ (5) xn+12 = T λ̂2·w2 ˆ xn2+A T `s̆2−Axn2´+ η̂AT `s̆1−Ax1´˜ with λ̂1 = λ12(1+η) , λ̂2 = λ2 2η and η̂ = η η+1 . APPLICATION TO THE SAGA DATA SET The proposed algorithm is tested on the SAGA dataset (see e.g. Verschuur et al., 1992, for a detailed description of this dataset), consisting of 128 sources and receivers and 1024 time samples. The 3-D discrete curvelet transform with wrapping was used (Ying et al., 2005) using 32 angles at the finest scale. The SRME results (Fig. 2) were obtained using a spatially varying matched filter. The single threshold results (Fig. 3) were obtained by running Eq. 4 with only one iteration and C1 = 1. The second generation results were obtained for 7 iterations with η = 2.0, λ1 = 3.0 and λ2 = 3.0. Both curvelet- based results (Figs 3 and 4) contain less high-frequency clutter compared to the SRME result in Fig. 2 while the solution of P2w leads to an improved multiple removal compared to the single thresholded result (cf. Figs 3,4). DISCUSSION AND CONCLUSIONS By virtue of their compression capabilities, localization, mul- tiscale, and multi-angular nature curvelets represent the ideal domain for primary-multiple separation. Not only does the curvelet construction allow for a separation based on differ- ences in these curvelet attributes but their compression also allows 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 signal component at the same location in the curvelet vector. More- over, the curvelet’s multi-angular parameterization helps the separation, even for conflicting dips and erroneous predictions. The reformulation of the separation problem resolved an ear- lier shortcoming by adding an additional constraint controlling the misfit between the estimated and SRME-predicted multi- ples. The results of applying this method to real data show a clear improvement over SRME-predicted primaries and over the earlier generation of curvelet-based primary-multiple sep- aration. ACKNOWLEDGMENTS The authors would like to thank Eric Verschuur for his input in the primary-multiple separation. We also would like to thank the authors of CurveLab for making their codes available. The examples presented were prepared with Madagascar (rsf., supplemented by SLIMPy operator overloading, developed by Sean Ross Ross. Norsk Hydro is thanked for making the field dataset available. This work was in part financially supported by the NSERCDiscovery (22R81254) and CRD Grants DNOISE (334810-05) of F.J.H. and was car- ried out as part of the SINBAD project with support, secured through ITF, from BG Group, BP, Chevron, ExxonMobil and Shell. Curvelet-based primary-multiple separation Figure 1: Total data SAGA data. Figure 2: SRME-predicted primaries. Curvelet-based primary-multiple separation Figure 3: Estimated primaries with single thresholding. Figure 4: Estimated primaries according to P2w . Curvelet-based primary-multiple separation REFERENCES Berkhout, 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. 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