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Recent developments in curvelet-based seismic processing Herrmann, Felix J. 2007

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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 & multidirectional the presence of caustics, faults and pinchouts  Curvelets  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  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!  Curvelet tiling & seismic data Angular wedge  Curvelet tiling  Real data frequency bands example  Data is multiscale!  1  2  3  4  5  6  7  8  Single frequency band angular wedges  6  6th scale image  Data is multidirectional! Seismic Laboratory for Imaging and Modeling  Decomposition in angular wedges  Wavefront detection 0  -2000  Offset (m) 0  2000  Time (s)  0.5  1.0  1.5  2.0  Significant curvelet coefficient  Curvelet coefficient~0  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 Dirac  Normalized amplitude  10-1  Curvelets  Fourier  10-2  10-3 Wavelets  10-4  10-5 Total dataset True primaries True multiples  10-6 100  101  102  103  Number of coefficients  104  105  Sparsity promoting inversion  Key idea signal  y  =  A  +  n  noise  x0 curvelet representation of ideal data  ˜ = arg min x x x  1  sparsity enhancement  s.t.  Ax − y  2  ≤  data misfit  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.  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  Move-out error  Multiple prediction with erroneous move out.  Move-out error  Curvelet-based result obtained by single soft threshold given by the predicted multiples  ˜s1 = C Tλ|Cs˘2 | Cs T  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 signal  y  =  H  RC  +  n  noise  x0 curvelet representation of ideal data  Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI)   P :  look for the sparsest/most compressible, physical solution KEY POINT OF THE   RECOVERY data misfit sparsity constraint sparsity constraint         = arg ˜ H H − PC minmin x 0x 0 s.t. s.t. y −yPC x 2x≤2!≤ ! x =x˜arg ˜0 )= arg minxx Wx x s.t. Ax − y 2 ≤ (P0 )(Px 1     T  ˜   ˜H f = C x  H ˜f = ˜fC= xC ˜ x˜  * 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  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]  Shots  Receivers Shots  ∆P Receivers  Frequency  Frequency slice from data cube  Primary operator [Berkhout & Verschuur ‘96]  Maps primaries into first-order multiples. So its inverse focuses ....  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 solutionHcalculated from th inverse transform and A := R∆PC and ∆P := F block diag{∆p}F 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  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 subNyquist 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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