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Surface-related multiple prediction from incomplete data Herrmann, Felix J. 2007

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Surface-related multiple prediction from incomplete data 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 sparsitypromoting  recovery of the data  prediction of primaries and surface-related multiples  The curvelet transform  Representations for seismic data 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)  Properties curvelet transform:     multiscale: tiling of the FK domain into dyadic coronae multi-directional: coronae subpartitioned 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)  k2 2j/2  angular wedge  2j  fine scale data  k1  coarse scale data  2-D curvelets  curvelets are of rapid decay in space  x-t  curvelets are strictly localized in frequency  f-k  Oscillatory in one direction and smooth in the others! Obey parabolic scaling relation length ≈ width2  Curvelet sparsity promotion  Sparsity-promoting program Solve for x0 signal  y  =  A  +  n  noise  x0 restricted compounded curvelet representation of ideal data  with  P :     ˜ = arg minx x x ˜ = CT x ˜ m  1  s.t.  Ax − y  2  ≤  exploit sparsity in the curvelet domain as a prior. find the sparsest set of curvelet coefficients that match the data. invert an underdetermined system.  Focused recovery with curvelets joint work with Deli Wang (visitor from Jilin university) and Gilles Hennenfent  Focused recovery Non-data-adaptive Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI) derives from 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 Shots  Receivers Shots  ∆P Receivers  Frequency  Frequency slice from data matrix with dominant primaries.  Primary operator  Primary operator Primaries to first-order multiples:  ∆p → m = (∆PA ∗t,x ∆p) 1  First-order multiples into primaries:  m → ∆p ≈ (∆PA ⊗t,x ∆p) 1  with the acquisition matrix †  A = S RD  †  “inverting” for source and receiver wavelet wavelets geometry and surface reflectivity.  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 matrix, A, can be inverted by a sparsity-promoting program Solve with 3-D curvelet transform (Cand`es et al., 2006b; Donoho, 2006) of the following type:  Curvelet-based Focal transform P :  with A S y P  x = arg minx x f = ST x  1  s.t.  Ax − y  2  ≤  (2) in which is a noise-dependent tolerance level, ST the inverse transform and f the solution calculated from the H T and ∆Pdenotes := F ablock := ∆PC vector x (the symbol vectordiag{∆p}F obtained by nonlinearCoptimization) that minimizes P . := Nonlinear programs such as P are not new to seismic = P(:) data processing and imaging. Refer, for instance, to the = totalliterature data. on spiky deconvolution (Taylor et al., extensive 1979) and transform-based interpolation techniques such as Fourier-based reconstruction (Sacchi and Ulrych, 1996).  SRME estimate for the primaries  Difference  redundancy of C and/or the incompleteness of the data the matrix A can not readily be inverted. However, a long as the data, y, permits a sparse vector, x0 , the ma trix, A, can be inverted by a sparsity-promoting program (Cand`es et al., 2006b; Donoho, 2006) of the following type Solve  Recovery with focussing  P :  x = arg minx x T f =S x  1  s.t.  Ax − y  2  ≤  (2 inwith which is a noise-dependent tolerance level, ST th T f the solution calculated from th inverse transform and A := R∆PC vector denotes a vector obtained by non T x (the symbol T S := ∆PC linear optimization) that minimizes P . y = programs RP(:) such as P are not new to seismi Nonlinear dataRprocessing and imaging. = picking operator.Refer, for instance, to th extensive literature on spiky deconvolution (Taylor et al 1979) and transform-based interpolation techniques suc  Original data  80 % missing  Original data  operation by filling in the zero traces. Since seismic the missing data can be Curvelet recovered by compounding recovery  modeling operator, i.e., A := RCT . With this definit  P corresponds to seeking the sparsest curvelet vec  followed by the picking, matches the data at the n  transform (with S := C in P ) gives the interpolated  An example of curvelet based recovery is presente data volume is recovered from data with 80 % traces  traces are selected at random according to a discrete d  data off one interaction with the surface, focusing primaries to (directional) sources, which leads to a sparser Focused curvelet urvelet representation. recovery By compounding the non-adaptive curvelet transform with the data-adaptive focal transform, i.e., A := R∆PCT , he recovery can be improved by solving P . The solution of P now entails the inversion of ∆P, yielding the sparsst set of curvelet coefficients that matches the incomplete data when ’convolved’ with the primaries. Applying the nverse curvelet transform, followed by ’convolution’ with ∆P yields the interpolation, i.e. ST := ∆PCT . Comparng the curvelet recovery with the focused curvelet recovry (Fig ?? and ??) shows an overall improvement in the ecovered details. SEISMIC SIGNAL SEPARATION  Predictive multiple suppression involves two steps, namely multiple prediction and the primary-multiple separation. n practice, the second step appears difficult and adap-  Original data  Nonlinear primarymultiple prediction joint work with Deli Wang (visitor from Jilin university) and Eric Verschuur  Multiple prediction with fCRSI incomplete data  CRSI  SRME  fCRSI  recovered data  y = RP(:)  P ∆P M P  y = RP(:)  P CRSI  M CRSI  P fCRSI  M fCRSI  Primary prediction with fCRSI incomplete data  CRSI  y = RP(:)  P  SRME  fCRSI  recovered data  ∆P  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 matrix, A, can be inverted by a sparsity-promoting program Solve (Cand`es et al., 2006b; Donoho, 2006) of the following type:  Curvelet-based Focal transform P :  x = arg minx x f = ST x  1  s.t.  Ax − y  2  ≤  (2) inwith which is a noise-dependent tolerance level, ST the inverse transform and f the solution calculated from the T A x := vector (the ∆PC symbol denotes a vector obtained by nonlinear that minimizes P . S optimization) := C Nonlinear programs such as P are not new to seismic y = P(:) data processing and imaging. Refer, for instance, to the P =literature total data extensive on spiky deconvolution (Taylor et al., 1979)˜ f and = transform-based focused data.interpolation techniques such as Fourier-based reconstruction (Sacchi and Ulrych, 1996).  Total data  P  Estimate for the primaries  ∆P  Focused with the primaries  ˜f  Multiple prediction with dfCRSI incomplete data  CRSI  y = RP(:)  P  SRME  dfCRSI  multiple data  M  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 matrix, A, can be inverted by a sparsity-promoting program Solve (Cand`es et al., 2006b; Donoho, 2006) of the following type:  Curvelet-based deFocal transform P :  with A S y P ˜f  x = arg minx x f = ST x  1  s.t.  Ax − y  2  ≤  (2) in which is a noise-dependent tolerance level, ST the inverse transform and f the solution calculated from the T T H vector x (the vectordiag{conj(∆p)}F obtained by non:= ∆P C symbol and ∆Pdenotes := F ablock linear optimization) that minimizes P . := C Nonlinear programs such as P are not new to seismic = P(:) data processing and imaging. Refer, for instance, to the extensive on spiky deconvolution (Taylor et al., = totalliterature data 1979) and transform-based interpolation techniques such = defocussed data. as Fourier-based reconstruction (Sacchi and Ulrych, 1996).  SRME predicted multiples  P = PP  (PP)  Original data  P  (P)  Multiple estimate by dfCRSI  ˜f  SRME predicted multiples  P = PP  (PP)  Original data  P  (P)  Original data  P  SRME predicted multiples  P = PP  Multiple estimate by dfCRSI  ˜f  Conclusions Focused CRSI     improves the recovery and hence predicted multiples precursor of migration-based CRSI primary estimates have higher bandwidth (deconvolution of the source)  deFocused CRSI    improves the band width contains artifacts due to remnant multiple energy & X-terms  Curvelet-based approach improves the primarymultiple prediction.  Acknowledgments The authors of CurveLab (Demanet,Ying, Candes, Donoho) Dr. Verschuur for his synthetic data and the estimates for the primaries. 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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