Recent developments in curvelet-based seismic processing Felix J. Herrmann Seismic Laboratory for Imaging and Modeling slim.eos.ubc.ca EAGE, London, June 11 Combinations of parsimonious signal representations with nonlinear sparsity promoting programs hold the key to the next-generation of seismic data processing algorithms ... Since they allow for a formulation that is stable w.r.t. noise & incomplete data do not require prior information on the velocity or locations & dips of the events Seismic data and images are complicated because wavefronts & reflectors are multiscale & multi- directional the presence of caustics, faults and pinchouts Curvelets 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 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! Curvelet tiling & seismic data Curvelet tiling Angular wedge Real data frequency bands example 1 2 43 8 65 7 Data is multiscale! Seismic Laboratory for Imaging and Modeling Decomposition in angular wedges 6th scale image Single frequency band angular wedges 6 Data is multidirectional! 00.5 1.0 1.5 2.0 Ti m e (s ) -2000 0 2000 Offset (m) 0 0.5 1.0 1.5 2.0 T im e ( s ) -2000 0 2000 Offset (m) Significant curvelet coefficient Curvelet coefficient~0 Wavefront detection curvelet coefficient is determined by the dot product of the curvelet function with the data Nonlinear approximation Nonlinear approximation Nonlinear approximation Nonlinear approximation Nonlinear approximation Nonlinear approximation rates 100 Number of coefficients N or m al iz ed a m pl itu de Total dataset True primaries True multiples 10-1 10-2 10-3 10-4 10-5 10-6 100 101 102 103 104 105 Curvelets Wavelets Dirac Fourier Sparsity promoting inversion Key idea x̃ = arg min x ‖x‖1 s.t. ‖Ax− y‖2 ≤ ! data misfitsparsity enhancement When a traveler reaches a fork in the road, the l1 -norm tells him to take either one way or the other, but the l2 -norm instructs him to head off into the bushes. John F. Claerbout and Francis Muir, 1973 New field “compressive sampling”: D. Donoho, E. Candes et. al., M. Elad etc. Preceded by others in geophysics: M. Sacchi & T. Ulrych and co-workers etc. signal =y + n noise curvelet representation of ideal data x0 A Applications Sparsity promotion can be used to recovery from incomplete data: “Curvelet reconstruction with sparsity promoting inversion: successes & challenges and “Irregular sampling: from aliasing to noise” migration amplitude recovery: “Just diagonalize: a curvelet-based approach to seismic amplitude recovery ground-roll removal: “Curvelet applications in surface wave removal” multiple prediction: “Surface related multiple prediction from incomplete data” seismic processing: “Seismic imaging and processing with curvelets” Primary-multiple separation Joint work with Eric Verschuur, Deli Wang, Rayan Saab and Ozgur Yilmaz Multiple prediction with erroneous move out. Move-out error Move-out error Curvelet-based result obtained by single soft threshold given by the predicted multiples s̃1 = CTTλ|Cs̆2| ( Cs ) Approach Bayesian formulation of the primary-multiple separation problem promotes sparsity on estimated primaries & multiples minimizes misfit between total data and sum of estimated primaries and multiples exploits decorrelation in the curvelet domain new: minimizes misfit between estimated and (SRME) predicted multiples Separation formulated in terms of a sparsity promoting program robust under moderate timing and phase errors noise Synthetic example total data SRME predicted multiples Synthetic example SRME predicted primaries curvelet-thresholded Synthetic example SRME predicted primaries estimated Curvelet-based recovery joint work with Gilles Hennenfent Sparsity-promoting inversion* Reformulation of the problem Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI) look for the sparsest/most compressible, physical solution KEY POINT OF THE RECOVERY * inspired by Stable Signal Recovery (SSR) theory by E. Candès, J. Romberg, T. Tao, Compressed sensing by D. Donoho & Fourier Reconstruction with Sparse Inversion (FRSI) by P. Zwartjes signal =y + n noise curvelet representation of ideal data x0 RCH (P0) x̃= arg sparsity constraint︷ ︸︸ ︷ min x ‖x‖0 s.t. ‖y−PC Hx‖2 ≤ ! f̃= CH x̃ (P0) x̃= arg sparsity constraint︷ ︸︸ ︷ min x ‖x‖0 s.t. data misfit︷ ︸︸ ︷ ‖y−PCHx‖2 ≤ ! f̃= CH x̃ P! : { x̃ = arg minx ‖Wx‖1 s.t. ‖Ax− y‖2 ≤ ! f̃ = CT x̃ Focused recovery with curvelets joint work with Deli Wang (visitor from Jilin university) and Gilles Hennenfent Motivation Can the recovery be extended to “migration-like” operators? How can we incorporate prior information on the wavefield, e.g. information on major primaries from SRME? How can we compress extrapolation operator? Compound primary operator with inverse curvelet transform. Primary operator [Berkhout & Verschuur ‘96] Frequency slice from data cube Receivers Shots Shots Receivers Frequency ∆P Primary operator [Berkhout & Verschuur ‘96] Maps primaries into first-order multiples. So its inverse focuses .... 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 wit A := R∆PCT ST := ∆PCT y = RP(:) R = picking operator. and ∆P := FHblock diag{∆p}F Original data 80 % missing Original data Curvelet recovery 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. Herrmann, 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 information 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 missing. 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 definition 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 recovered 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. Hennenfent and Herrmann, 2007), and corresponds to an average sampling interval of 125m . Comparing the ’ground truth’ in Figure 1(a) with the recovered data in Figure 1(c) 5 Focused curvelet recovery 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- Original data Conclusions Curvelets represent a versatile transform that brings robustness w.r.t. moderate shifts and phase rotations to primary multiple separation allows for the nonlinear recovery for severely sub- Nyquist data leads to an improved recovery when compounded with “migration like” operators Opens tentative perspectives towards a new sampling theory for seismic data that includes migration operators ... Acknowledgments The authors of CurveLab (Demanet, Ying, Candes, Donoho) Eric Verschuur for providing us with the synthetic and real data examples. 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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Recent developments in curvelet-based seismic processing Herrmann, Felix J. 2007
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Title | Recent developments in curvelet-based seismic processing |
Creator |
Herrmann, Felix J. |
Contributor | University of British Columbia. Seismic Laboratory for Imaging and Modeling |
Publisher | European Association of Geoscientists & Engineers |
Date Issued | 2007 |
Extent | 6961609 bytes |
Subject |
nonlinear sparsity curvelet transform tiling decomposition curvelet cooefficient nonlinear approximation sparsity move-out error SRME sub-Nyquist |
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Presentation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-03-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | All rights reserved |
DOI | 10.14288/1.0102590 |
URI | http://hdl.handle.net/2429/581 |
Affiliation |
Science, Faculty of Earth and Ocean Sciences, Department of |
Citation | Herrmann, Felix J. 2007. Recent developments in curvelet-based seismic processing. Workshop presented at 69th EAGE Conference & Exhibition. |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty |
Copyright Holder | Herrmann, Felix J. |
AggregatedSourceRepository | DSpace |
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