UBC Faculty Research and Publications

Recent developments in curvelet-based seismic processing Herrmann, Felix J. 2007-03-14

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Recent developments in curvelet-based seismic processingFelix J. HerrmannSeismic Laboratory for Imaging and Modeling slim.eos.ubc.caEAGE, London, June 11Combinations 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 eventsSeismic data and images are complicated because? wavefronts & reflectors are multiscale & multi-directional? the presence of caustics, faults and pinchoutsCurveletsProperties 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 redundancyTransform Underlying assumptionFK plane waveslinear/parabolic Radon transform linear/parabolic eventswavelet transform point-like events (1D singularities)curvelet transform curve-like events (2D singularities)Representations for seismic datafine scale data coarse scale data 2-D curveletscurvelets are of rapid decay in spacecurvelets are strictly localized in frequencyx-t f-kOscillatory in one direction and smooth in the others!Curvelet tiling & seismic dataCurvelet tilingAngularwedgeReal data frequency bands             example 1 24867Data is multiscale!Seismic Laboratory for Imaging and ModelingDecomposition in  angular wedges6th scale imageSingle frequency band      angular wedges6Data is multidirectional!Significantcurvelet coefficient Curveletcoefficient~0Wavefront detectioncurvelet coefficient is determinedby the dot product of the curveletfunction with the dataNonlinear approximationNonlinear approximationNonlinear approximationNonlinear approximationNonlinear approximationNonlinear approximation ratesSparsity promoting inversionKey ideadata misfitenhancementWhen 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 = + noisecurvelet representation of ideal dataApplicationsSparsity 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 separationJoint work with Eric Verschuur, Deli Wang, Rayan Saab and Ozgur YilmazMultiple prediction with erroneous move out.Move-out errorMove-out errorCurvelet-based result obtained by single soft threshold given by the predicted multiplesApproachBayesian 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 multiplesSeparation formulated in terms of a sparsity promoting program robust under? moderate timing and phase errors? noiseSynthetic exampletotal data SRME predicted multiplesSynthetic exampleSRME predicted primaries curvelet-thresholdedSynthetic exampleSRME predicted primaries estimatedCurvelet-based recoveryjoint work with Gilles HennenfentSparsity-promoting inversion*Reformulation of the problemCurvelet 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. Cand?s, J. Romberg, T. Tao, Compressed sensing by D. Donoho & Fourier Reconstruction with Sparse Inversion (FRSI) by P. Zwartjessignal = + noisecurvelet representation of ideal dataFocused recovery with curveletsjoint work with Deli Wang (visitor from Jilin university) and Gilles HennenfentMotivationCan 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 cubeReceiversShotsShotsReceiversFrequency Primary operator[Berkhout & Verschuur ?96]Maps primaries into first-order multiples. So its inverse focuses ....SolveRecovery with focussingOriginal data80 % missingOriginal dataCurvelet recoveryFocused curvelet recoveryOriginal dataConclusionsCurvelets 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? operatorsOpens tentative perspectives towards a new sampling theory? for seismic data? that includes migration operators ...AcknowledgmentsThe authors of CurveLab (Demanet,Ying,  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), om the following organizations: oup, , vron,Shell.


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