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Multiple prediction from incomplete data with the focused curvelet transform 2008

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Multiple prediction from incomplete data with the focused curvelet transform Felix J. Herrmann joint work with Deli Wang and Gilles Hennenfent. The problem Total data 85 % traces missing The problem cont’d SRME from complete data SRME from missing data Our solution SRME from recovered data SRME from original data Motivation Data-driven (SRME) multiple prediction requires fully sampled data. The Focal transform (Berkhout & Verschuur ‘06) allows for  mapping of multiples => primaries  incorporation of prior information in the recovery Present a curvelet-based scheme for sparsity- promoting  recovery of missing data  prediction of primaries from multiples  data inverse ... The curvelet transform Properties curvelet transform:  multiscale: tiling of the FK domain into dyadic coronae  multi-directional: coronae sub- partitioned into angular wedges, # of angle doubles every other scale  anisotropic: parabolic scaling principle  Rapid decay space  Strictly localized in Fourier  Frame with moderate redundancy (8 X in 2-D and 24 X in 3-D) Transform Underlying assumption FK plane waves linear/parabolic Radon transform linear/parabolic events wavelet transform point-like events (1D singularities) curvelet transform curve-like events (2D singularities) k1 k2 angular wedge 2j 2j/2 Representations for seismic data fine scale data coarse scale data 2-D curvelets curvelets are of rapid decay in space curvelets are strictly localized in frequency x-t f-k Oscillatory in one direction and smooth in the others! Obey parabolic scaling relation length ≈ width2 Curvelets are oscillatory in one direction and smooth in the others. 3-D curvelets Curvelet sparsity promotion Sparsity-promoting program Solve for with  exploit sparsity in the curvelet domain as a prior.  find the sparsest set of curvelet coefficients that match the data.  invert an underdetermined system. signal =y + n noise restricted compounded curvelet representation of ideal data x0 A x0 P! : { x̃ = arg minx ‖x‖1 s.t. ‖Ax− y‖2 ≤ ! m̃ = CT x̃ Focused wavefield reconstruction with curvelets Focused recovery Non-data-adaptive Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI) derives from curvelet-sparsity of seismic data. Berkhout and Verschuur’s data-adaptive Focal transform derives from focusing of seismic data by the major primaries. Both approaches entail the inversion of a linear operator. Combination of the two yields  improved focusing => more sparsity  curvelet sparsity   => better focusing Primary operator Receivers Shots Shots Receivers Frequency  ∆P Frequency slice from data matrix with dominant primaries. Primary operator Primary operator Primaries to first-order multiples: First-order multiples into primaries: with the acquisition matrix “inverting” for source and receiver wavelet wavelets geometry and surface reflectivity. A = ( S†RD† ) ∆p !→m1 = (∆PA ∗t,x∆p) m1 !→∆p ≈ (∆PA ⊗t,x∆p) Curvelet-based Focal transform Solve with 3-D curvelet transform Curvelet-based processing 3 SPARSITY-PROMOTING INVERSION Our solution strategy is built on the premise that seismic data and images have a sparse representation, x0, in the curvelet domain. To exploit this property, our forward model reads y = Ax0 + n (1) with y a vector with noisy and possibly incomplete mea- surements; A the modeling matrix that includes CT ; and n, a zero-centered white Gaussian noise. Because of the redundancy of C and/or the incompleteness of the data, the matrix A can not readily be inverted. However, as long as the data, y, permits a sparse vector, x0, the ma- trix, A, can be inverted by a sparsity-promoting program (Candès et al., 2006b; Donoho, 2006) of the following type: P! : { x̃ = argminx ‖x‖1 s.t. ‖Ax− y‖2 ≤ ! f̃ = ST x̃ (2) in which ! is a noise-dependent tolerance level, ST the inverse transform and f̃ the solution calculated from the vector x̃ (the symbol ˜ denotes a vector obtained by non- linear optimization) that minimizes P!. Nonlinear programs such as P! are not new to seismic data processing and imaging. Refer, for instance, to the extensive literature on spiky deconvolution (Taylor et al., 1979) and transform-based interpolation techniques such as Fourier-based reconstruction (Sacchi and Ulrych, 1996). By virtue of curvelets’ high compression rates, the non- linear program P! can be expected to perform well when CT is included in the modeling operator. Despite its large- scale and nonlinearity, the solution of the convex problem P! can effectively be approximated with a limited (< 250) number of iterations of a threshold-based cooling method derived from work by Figueiredo and Nowak (2003) and Elad et al. (2005). Each step involves a descent projection, followed by a soft thresholding. SEISMIC DATA RECOVERY The reconstruction of seismic wavefields from regularly- sampled data with missing traces is a setting where a curvelet-based method will perform well (see e.g. Herr- mann, 2005; Hennenfent and Herrmann, 2006a, 2007). As with other transform-based methods, sparsity is used to reconstruct the wavefield by solving P!. It is also shown that the recovery performance can be increased when in- formation on the major primary arrivals is included in the modeling operator. Curvelet-based recovery The reconstruction of seismic wavefields from incomplete data corresponds to the inversion of the picking operator R. This operator models missing data by inserting zero traces at source-receiver locations where the data is miss- ing. The task of the recovery is to undo this operation by filling in the zero traces. Since seismic data is sparse in the curvelet domain, the missing data can be recovered by compounding the picking operator with the curvelet modeling operator, i.e., A := RCT . With this defini- tion for the modeling operator, solving P! corresponds to seeking the sparsest curvelet vector whose inverse curvelet transform, followed by the picking, matches the data at the nonzero traces. Applying the inverse transform (with S := C in P!) gives the interpolated data. An example of curvelet based recovery is presented in Figure 1, where a real 3-D seismic data volume is recov- ered from data with 80% traces missing (see Figure 1(b)). The missing traces are selected at random according to a discrete distribution, which favors recovery (see e.g. Hen- nenfent and Herrmann, 2007), and corresponds to an av- erage sampling interval of 125m . Comparing the ’ground truth’ in Figure 1(a) with the recovered data in Figure 1(c) shows a successful recovery in case the high-frequencies are removed (compare the time slices in Figure 1(a) and 1(c)). Aside from sparsity in the curvelet domain, no prior information was used during the recovery, which is quite remarkable. Part of the explanation lies in the curvelet’s ability to locally exploit the 3-D structure of the data and this suggests why curvelets are successful for complex datasets where other methods may fail. Focused recovery In practice, additional information on the to-be-recovered wavefield is often available. For instance, one may have access to the predominant primary arrivals or to the ve- locity model. In that case, the recently introduced focal transform (Berkhout and Verschuur, 2006), which ’decon- volves’ the data with the primaries, incorporates this addi- tional information into the recovery process. Application of this primary operator,∆P, adds a wavefield interaction with the surface, mapping primaries to first-order surface- related multiples (see e.g. Verschuur and Berkhout, 1997; Herrmann, 2007). Inversion of this operator, strips the data off one interaction with the surface, focusing pri- maries to (directional) sources, which leads to a sparser curvelet representation. By compounding the non-adaptive curvelet transform with the data-adaptive focal transform, i.e., A := R∆PCT , the recovery can be improved by solving P!. The solution of P! now entails the inversion of ∆P, yielding the spars- est set of curvelet coefficients that matches the incomplete data when ’convolved’ with the primaries. Applying the inverse curvelet transform, followed by ’convolution’ with ∆P yields the interpolation, i.e. ST :=∆PCT. Compar- ing the curvelet recovery with the focused curvelet recov- ery (Fig ?? and ??) shows an overall improvement in the recovered details. SEISMIC SIGNAL SEPARATION Predictive multiple suppression involves two steps, namely multiple prediction and the primary-multiple separation. In practice, the second step appears difficult and adap- and ∆P := FHblock diag{∆p}F with A := ∆PCT S := C y = p p = total data. Total data SRME estimate for the primaries Focused with the primaries Difference Solve Curvelet-based processing 3 SPARSITY-PROMOTING INVERSION Our solution strategy is built on the premise that seismic data and images have a sparse representation, x0, in the curvelet domain. To exploit this property, our forward model reads y = Ax0 + n (1) with y a vector with noisy and possibly incomplete mea- surements; A the modeling matrix that includes CT ; and n, a zero-centered white Gaussian noise. Because of the redundancy of C and/or the incompleteness of the data, the matrix A can not readily be inverted. However, as long as the data, y, permits a sparse vector, x0, the ma- trix, A, can be inverted by a sparsity-promoting program (Candès et al., 2006b; Donoho, 2006) of the following type: P! : { x̃ = argminx ‖x‖1 s.t. ‖Ax− y‖2 ≤ ! f̃ = ST x̃ (2) in which ! is a noise-dependent tolerance level, ST the inverse transform and f̃ the solution calculated from the vector x̃ (the symbol ˜ denotes a vector obtained by non- linear optimization) that minimizes P!. Nonlinear programs such as P! are not new to seismic data processing and imaging. Refer, for instance, to the extensive literature on spiky deconvolution (Taylor et al., 1979) and transform-based interpolation techniques such as Fourier-based reconstruction (Sacchi and Ulrych, 1996). By virtue of curvelets’ high compression rates, the non- linear program P! can be expected to perform well when CT is included in the modeling operator. Despite its large- scale and nonlinearity, the solution of the convex problem P! can effectively be approximated with a limited (< 250) number of iterations of a threshold-based cooling method derived from work by Figueiredo and Nowak (2003) and Elad et al. (2005). Each step involves a descent projection, followed by a soft thresholding. SEISMIC DATA RECOVERY The reconstruction of seismic wavefields from regularly- sampled data with missing traces is a setting where a curvelet-based method will perform well (see e.g. Herr- mann, 2005; Hennenfent and Herrmann, 2006a, 2007). As with other transform-based methods, sparsity is used to reconstruct the wavefield by solving P!. It is also shown that the recovery performance can be increased when in- formation on the major primary arrivals is included in the modeling operator. Curvelet-based recovery The reconstruction of seismic wavefields from incomplete data corresponds to the inversion of the picking operator R. This operator models missing data by inserting zero traces at source-receiver locations where the data is miss- ing. The task of the recovery is to undo this operation by filling in the zero traces. Since seismic data is sparse in the curvelet domain, the missing data can be recovered by compounding the picking operator with the curvelet modeling operator, i.e., A := RCT . With this defini- tion for the modeling operator, solving P! corresponds to seeking the sparsest curvelet vector whose inverse curvelet transform, followed by the picking, matches the data at the nonzero traces. Applying the inverse transform (with S := C in P!) gives the interpolated data. An example of curvelet based recovery is presented in Figure 1, where a real 3-D seismic data volume is recov- ered from data with 80% traces missing (see Figure 1(b)). The missing traces are selected at random according to a discrete distribution, which favors recovery (see e.g. Hen- nenfent and Herrmann, 2007), and corresponds to an av- erage sampling interval of 125m . Comparing the ’ground truth’ in Figure 1(a) with the recovered data in Figure 1(c) shows a successful recovery in case the high-frequencies are removed (compare the time slices in Figure 1(a) and 1(c)). Aside from sparsity in the curvelet domain, no prior information was used during the recovery, which is quite remarkable. Part of the explanation lies in the curvelet’s ability to locally exploit the 3-D structure of the data and this suggests why curvelets are successful for complex datasets where other methods may fail. Focused recovery In practice, additional information on the to-be-recovered wavefield is often available. For instance, one may have access to the predominant primary arrivals or to the ve- locity model. In that case, the recently introduced focal transform (Berkhout and Verschuur, 2006), which ’decon- volves’ the data with the primaries, incorporates this addi- tional information into the recovery process. Application of this primary operator,∆P, adds a wavefield interaction with the surface, mapping primaries to first-order surface- related multiples (see e.g. Verschuur and Berkhout, 1997; Herrmann, 2007). Inversion of this operator, strips the data off one interaction with the surface, focusing pri- maries to (directional) sources, which leads to a sparser curvelet representation. By compounding the non-adaptive curvelet transform with the data-adaptive focal transform, i.e., A := R∆PCT , the recovery can be improved by solving P!. The solution of P! now entails the inversion of ∆P, yielding the spars- est set of curvelet coefficients that matches the incomplete data when ’convolved’ with the primaries. Applying the inverse curvelet transform, followed by ’convolution’ with ∆P yields the interpolation, i.e. ST :=∆PCT. Compar- ing the curvelet recovery with the focused curvelet recov- ery (Fig ?? and ??) shows an overall improvement in the recovered details. SEISMIC SIGNAL SEPARATION Predictive multiple suppression involves two steps, namely multiple prediction and the primary-multiple separation. In practice, the second step appears difficult and adap- Recovery with focussing with A := R∆PCT ST := ∆PCT y = Rp R = picking operator. y = RP(:) P̂CRSI P̂fCRSI Multiple prediction with fCRSI incomplete data CRSI SRME y = RP(:) P̂ fCRSI P̂ recovered data ∆̂P M̂ SRME primary operator Wavefield reconstruction with fCRSI Original data 80 % missing Curvelet recovery Original data Focused curvelet recovery Original data Multiple prediction Reference predicted multiples Original Predicted multiples from missing data Predicted multiples from CRSI recovery Predicted multiples from focussed CRSI recovery Reference predicted multiples Original Primary prediction with fCRSI incomplete data CRSI SRME y = RP(:) P̂ fCRSI recovered data ∆̂P Curvelet-based Focal transform Solve Curvelet-based processing 3 SPARSITY-PROMOTING INVERSION Our solution strategy is built on the premise that seismic data and images have a sparse representation, x0, in the curvelet domain. To exploit this property, our forward model reads y = Ax0 + n (1) with y a vector with noisy and possibly incomplete mea- surements; A the modeling matrix that includes CT ; and n, a zero-centered white Gaussian noise. Because of the redundancy of C and/or the incompleteness of the data, the matrix A can not readily be inverted. However, as long as the data, y, permits a sparse vector, x0, the ma- trix, A, can be inverted by a sparsity-promoting program (Candès et al., 2006b; Donoho, 2006) of the following type: P! : { x̃ = argminx ‖x‖1 s.t. ‖Ax− y‖2 ≤ ! f̃ = ST x̃ (2) in which ! is a noise-dependent tolerance level, ST the inverse transform and f̃ the solution calculated from the vector x̃ (the symbol ˜ denotes a vector obtained by non- linear optimization) that minimizes P!. Nonlinear programs such as P! are not new to seismic data processing and imaging. Refer, for instance, to the extensive literature on spiky deconvolution (Taylor et al., 1979) and transform-based interpolation techniques such as Fourier-based reconstruction (Sacchi and Ulrych, 1996). By virtue of curvelets’ high compression rates, the non- linear program P! can be expected to perform well when CT is included in the modeling operator. Despite its large- scale and nonlinearity, the solution of the convex problem P! can effectively be approximated with a limited (< 250) number of iterations of a threshold-based cooling method derived from work by Figueiredo and Nowak (2003) and Elad et al. (2005). Each step involves a descent projection, followed by a soft thresholding. SEISMIC DATA RECOVERY The reconstruction of seismic wavefields from regularly- sampled data with missing traces is a setting where a curvelet-based method will perform well (see e.g. Herr- mann, 2005; Hennenfent and Herrmann, 2006a, 2007). As with other transform-based methods, sparsity is used to reconstruct the wavefield by solving P!. It is also shown that the recovery performance can be increased when in- formation on the major primary arrivals is included in the modeling operator. Curvelet-based recovery The reconstruction of seismic wavefields from incomplete data corresponds to the inversion of the picking operator R. This operator models missing data by inserting zero traces at source-receiver locations where the data is miss- ing. The task of the recovery is to undo this operation by filling in the zero traces. Since seismic data is sparse in the curvelet domain, the missing data can be recovered by compounding the picking operator with the curvelet modeling operator, i.e., A := RCT . With this defini- tion for the modeling operator, solving P! corresponds to seeking the sparsest curvelet vector whose inverse curvelet transform, followed by the picking, matches the data at the nonzero traces. Applying the inverse transform (with S := C in P!) gives the interpolated data. An example of curvelet based recovery is presented in Figure 1, where a real 3-D seismic data volume is recov- ered from data with 80% traces missing (see Figure 1(b)). The missing traces are selected at random according to a discrete distribution, which favors recovery (see e.g. Hen- nenfent and Herrmann, 2007), and corresponds to an av- erage sampling interval of 125m . Comparing the ’ground truth’ in Figure 1(a) with the recovered data in Figure 1(c) shows a successful recovery in case the high-frequencies are removed (compare the time slices in Figure 1(a) and 1(c)). Aside from sparsity in the curvelet domain, no prior information was used during the recovery, which is quite remarkable. Part of the explanation lies in the curvelet’s ability to locally exploit the 3-D structure of the data and this suggests why curvelets are successful for complex datasets where other methods may fail. Focused recovery In practice, additional information on the to-be-recovered wavefield is often available. For instance, one may have access to the predominant primary arrivals or to the ve- locity model. In that case, the recently introduced focal transform (Berkhout and Verschuur, 2006), which ’decon- volves’ the data with the primaries, incorporates this addi- tional information into the recovery process. Application of this primary operator,∆P, adds a wavefield interaction with the surface, mapping primaries to first-order surface- related multiples (see e.g. Verschuur and Berkhout, 1997; Herrmann, 2007). Inversion of this operator, strips the data off one interaction with the surface, focusing pri- maries to (directional) sources, which leads to a sparser curvelet representation. By compounding the non-adaptive curvelet transform with the data-adaptive focal transform, i.e., A := R∆PCT , the recovery can be improved by solving P!. The solution of P! now entails the inversion of ∆P, yielding the spars- est set of curvelet coefficients that matches the incomplete data when ’convolved’ with the primaries. Applying the inverse curvelet transform, followed by ’convolution’ with ∆P yields the interpolation, i.e. ST :=∆PCT. Compar- ing the curvelet recovery with the focused curvelet recov- ery (Fig ?? and ??) shows an overall improvement in the recovered details. SEISMIC SIGNAL SEPARATION Predictive multiple suppression involves two steps, namely multiple prediction and the primary-multiple separation. In practice, the second step appears difficult and adap- with A := ∆PCT S := C y = P(:) P = total data f̃ = focused data. Focal transform from complete data 80 % missing Focal transform from missing data An encore ... preliminary results for the data inverse Curvelet-based seismic data inverse with is the data to be inverted y = Î ST := CT A := PCT P! : { x̃ = arg minx ‖x‖1 s.t. ‖Ax− y‖2 ≤ ! p̃−1 = ST x̃ p Curvelet-sparsity regularized data inverse computed for the whole data volume ...... Curvelet-based seismic data inverse Curvelet-based seismic data inverse Curvelet-based seismic data inverse Conclusions CRSI  recovers data by curvelet sparsity promotion  uses sparsity as a prior Focused CRSI  incorporates additional prior information  strips interaction with the surface <=> more sparsity  improves the recovery and hence predicted multiples  precursor of migration-based CRSI Results of curvelet-based computation of the data inverse are encouraging. Acknowledgments The authors of CurveLab (Demanet, Ying, Candes, Donoho) Dr.  Verschuur for his synthetic data and the estimates for the primaries. The SLIM team Sean Ross Ross, Cody Brown and Henryk Modzeleweski for developing SLIMPy: operator overloading in python These results were created with Madagascar developed by Dr. Fomel. This work was in part financially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant (22R81254) and the Collaborative Research and Development Grant DNOISE (334810-05) of F.J.H. This research was carried out as part of the SINBAD project with support, secured through ITF (the Industry Technology Facilitator), from the following organizations: BG Group, BP, Chevron, ExxonMobil and Shell.

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