UBC Faculty Research and Publications

Seismic imaging and processing with curvelets Herrmann, Felix J. 2007-12-31

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Seismic imaging and processing with curveletsFelix J. Herrmannjoint work with Deli WangCombinations 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 formulations that are stable w.r.t. ? noise ? incomplete data? moderate phase rotations and amplitude errorsFinding a sparse representation for seismic data & images is complicated because of? wavefronts & reflectors are multiscale & multi-directional? the presence of caustics, faults and pinchoutsThe curvelet transformProperties 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 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!Obey parabolic scaling relationCurvelet tiling & seismic dataCurvelet tilingAngularwedge# of angles doubles every other scale doubling!Real 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 detectionCurvelets live in a wedge in the 3 D Fourier plane...Extenstion to 3-DCartesian Fourier space[courtesy Demanet ?05, Ying ?05] Curvelets are oscillatory in one direction and smooth in the others.3-D curveletsCurvelet sparsity promotionForward modelLinear model for the measurements of a function m0:? inversion of K either ill-posed or underdetermined.? seek a prior on m.Key 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.  Linear quadratic (lsqr):? model GaussianNon-linear       :? model Cauchy (sparse)Problem:? data does not contain point scatterers? not sparseOur contributionModel as superposition of little plane waves.Compound modeling operator with curvelet synthesis:Exploit parsimoniousness of curvelets on seismic data & images ...Sparsity-promoting programProblems boils down to solving forwith ? exploit sparsity in the curvelet domain as a prior? find the sparsest set of curvelet coefficients that match the data, i.e.,? invert an underdetermined systemsignal = + noisecurvelet representation of ideal dataSolverApplicationsProblems in seismic processing can be cast in to     ? stable under noise? stable under missing dataObtain a formulation that? explicitly exploits compression by curvelets? is stable w.r.t. noise? exploits the ?invariance? of curvelets under imagingApplications include? seismic data regularization? primary-multiple separation? seismic amplitude recoverySeismic data regularizationjoint work with Gilles HennenfentMotivationIrregular sub-samplingincoherent noiseNoisy because of irregular sampling ...Sparsity-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 dataOriginal data80 % missingCRSI recovery with 3-D curveletsPrimary multiple separationJoint work with Eric Verschuur, Deli Wang, Rayan Saab and Ozgur YilmazMotivationPrimary-multiple separation step is crucial? moderate prediction errors? 3-D complexity & noiseInadequate separation leads to? remnant multiple energy? deterioration primary energyIntroduce a transform-based technique? stable? insensitive to moderate shifts & phase rotationsExploit sparsity and parameterization transformed domainMove-out errorMove-out errorThe problemSparse signal model:with? augmented synthesis and sparsity vectors? index 1 <-> primary? index 2 <-> multipleThe solutionThe weighted norm-one optimization problem:Solution cont?dThe weights? during minimization signal components are driven apart? curvelet compression helps? separates on the basis of position, scale and directionSynthetic exampletotal data SRME predicted multiplesSynthetic exampleSRME predicted primaries curvelet-thresholdedSynthetic exampleSRME predicted primaries estimatedReal exampletotal dataSRME predicted multiplesReal examplecurvelet thresholdedcurvelet estimatedcurvelet estimatedprimariesSRME predictedprimariesSeismic amplitude recoveryJoint work with Chris Stolk and Peyman MoghaddamMotivationMigration generally does not correctly recover the amplitudes.Least-squares migration is computationally unfeasible.Amplitude recovery (e.g. AGC) lacks robustness w.r.t. noise.Existing diagonal amplitude-recovery methods? do not always correct for the order (1 - 2D) of the Hessian [see Symes ?07]? do not invert the scaling robustlyMoreover, these (scaling) methods assume that there? are no conflicting dips (conormal) in the model? is infinite aperture ? are infinitely-high frequencies? etc.Existing scaling methodsMethods are based on a  diagonal approximation of   .? Illumination-based normalization (Rickett ?02)? Amplitude preserved migration (Plessix & Mulder ?04)? Amplitude corrections (Guitton ?04)? Amplitude scaling (Symes ?07)We are interested in an ?Operator and image adaptive? scaling method which? estimates the action of    from a reference vector close to the actual image? assumes a smooth symbol of     in space and angle? does not require the reflectors to be conormal <=> allows for conflicting dips? stably inverts the diagonal Our approach?Forward? model:? diagonal approximation of the demigration-migration operator? costs one demigration-migration to estimate the diagonal weightingSolutionSolveExampleSEGAA? data:? ?broad-band? half-integrated wavelet [5-60 Hz]? 324 shots, 176 receivers, shot at 48 m? 5 s of dataModeling operator? Reverse-time migration with optimal check pointing (Symes ?07)? 8000 time steps? modeling 64, and migration 294 minutes on 68 CPU?sScaling requires 1 extra migration-demigrationMigrated data Amplitude-corrected & denoised migrated dataNoise-free data Noisy data(3 dB)Data from migrated imageData from amplitude-corrected & denoised migrated imageNonlinear dataThe combination of the parsimonious curvelet transform with nonlinear sparsity & continuity promoting program allowed us to...? recover seismic data from large percentages missing traces? separate primaries & multiples? recover migration amplitudesThis success is due to the curvelet?s ability to? detect wavefronts <=> multi-D geometry? differentiate w.r.t. positions, angle(s) and scale? diagonalize the demigration-migration operatorBecause of their parsimoniousness on seismic data and images, curvelets open new perspectives on seismic processing ...ConclusionsAcknowledgmentsThe authors of CurveLab (Demanet,Ying,  Donoho)William Symes for the reverse-time migration code.These results were created with Madagascar developed by Sergey 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), om the following organizations: oup, , vron,ExxonMobil and Shell.


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