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Multiple prediction from incomplete data with the focused curvelet transform Herrmann, Felix J.; Wang, Deli; Hennenfent, Gilles 2007-03-11

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Multiple prediction from incomplete data with the focused curvelet transformFelix J. Herrmannasteriskmath, EOS-UBC, Deli Wang?, Jilin University and Gilles Hennenfentasteriskmath, EOS-UBCSUMMARYIncomplete data represents a major challenge for a suc-cessful prediction and subsequent removal of multiples.In this paper, a new method will be represented thattackles this challenge in a two-step approach. Duringthe first step, the recenly developed curvelet-based re-covery by sparsity-promoting inversion (CRSI) is ap-plied to the data, followed by a prediction of the pri-maries. During the second high-resolution step, the esti-mated primaries are used to improve the frequency con-tent of the recovered data by combining the focal trans-form, defined in terms of the estimated primaries, withthe curvelet transform. This focused curvelet transformleads to an improved recovery, which can subsequentlybe used as input for a second stage of multiple predictionand primary-multiple separation.INTRODUCTIONSurface-related multiple prediction and seismic interfer-ometry are examples where weighted multi-dimensionalcross-convolutions and cross-correlations of seismic datavolumes provide information on Green?s functions thatdescribe the Earth response at the surface. For instance,surface-related multiples can approximately be predictedthrough a weighted multidimensional convolution of thedata with itself, while ?daylight imaging? techniques ex-tract the Green?s function by cross-correlation of wave-fields (see e.g. Wapenaar et al., 2006, which containsa collection of the most recent papers on this topic).Recently, new approaches have been proposed, wherethe Green?s functions are extracted through inversion ordeconvolution (See the contributions by Snieder et.al,Schuster et.al. and Berkhout and Verschuur in Wape-naar et al., 2006).Unfortunately, these multidimensional techniques are sen-sitive to missing traces (see e.g. Fig. 3(a) where the pre-dicted multiples suffer significantly from the missing data).Many different techniques have been proposed to solvethe interpolation problem. The different approaches canroughly be divided into data-dependent approaches, as-suming prior (velocity) information on the wave arrivals,and non-parametric approaches that do not make suchassumptions. Examples of parametric methods are theso-called data mappings (Bleistein et al., 2001), basedon approximate solutions of the wave equation. Thesemethods require information on the seismic velocity. Para-bolic, apex-shifted Radon or migration-like transformssuch as DMO/NMO/AMO also fall in this category. Otherexamples of data-adaptive methods are predictive, dipfiltering techniques and plane-wave destructors that re-quire a preprocessing step (see e.g. Spitz, 1999; Fomeland Guitton, 2006). Examples of non-parametric ap-proaches include transform-based sparse inversion meth-ods based on the Fourier or other transforms (Sacchi andUlrych, 1996; Elad et al., 2005; Zwartjes and Gisolf,2006; Abma and Kabir, 2006).In this work, we hold the middle between data-dependentand transform-based methods by combining the data-independent discrete curvelet transform (FDCT, Candeset al., 2006; Ying et al., 2005; Hennenfent and Herr-mann, 2006b) with the recently introduced data-adap-tive focal transform (Berkhout and Verschuur, 2006). Byvirtue of its compression on seismic data and its invari-ance under wave propagation, the curvelet transform hasproven to be an excellent domain for the formulation ofseismic processing algorithms ranging from data regu-larization (Hennenfent and Herrmann, 2006a; Herrmannand Hennenfent, 2007); primary-multiple separation (Herr-mann et al., 2007) to migration-amplitude recovery (seee.g. Herrmann et al., 2006,and contributions by the au-thors to the proceedings of this conference) and com-pressed wavefield extrapolation (Lin and Herrmann, 2007).While the non-parametric curvelet-based method recov-ers incomplete data, the physics of wave propagation isnot truely exploited. Combining the non-adaptive curvelettransform with the data-adaptive focal transform (Berkhoutand Verschuur, 2006) leads to a powerful formulationwhere data is focused by inverting the primary operator(= a multidimensional ?convolution? with an estimateof the major primaries). During this curvelet-regularizedinversion of the primary operator, deltaP, propagation pathsthat include the surface are removed, yielding a more fo-cused wavefield and hence a more compressed curveletvector.The focusing operator itself is derived from the data andcontains an estimate for the major primaries obtainedfrom e.g. a SRME-primary estimation procedure (Ver-schuur and Berkhout, 1997). In this abstract, we presenta method where the focal operator is robustly inverted bycurvelet regularization, i.e. by promoting sparsity in thecurvelet domain. The robustness in this context refers tostability under noise and more importantly under miss-ing traces, leading to an improved recovery for data withlarge percentages of traces missing. First, we briefly dis-The focused curvelet transformcuss sparsity promoting inversion, followed by curveletrecovery by sparse inversion (CRSI). Next, we combinethis method with the focal transform, leading to focusedcurvelet recovery by sparse inversion (fCRSI). The pro-posed algorithm is tested on a 3D seismic data volume.SPARSITY-PROMOTING INVERSIONTo exploit curvelets, incomplete and noisy measurementsare related to a sparse curvelet coefficient vector, x0, ac-cording toy =Ax0 +nwith y a vector with noisy and incomplete measurements;A the synthesis matrix that includes the inverse curvelettransform (CT ); and n, a zero-centered white Gaussiannoise. The matrix A is a wide rectangular matrix, so thevector x0 can not readily be calculated from the mea-surements, because there exist infinitely many vectorsthat match y.Recent work in ?compressive sampling? (Cand` et al.,2006; Donoho, 2006) has shown that rectangular matri-ces can stably be inverted by solving a nonlinear spar-sity promoting program (Elad et al., 2005). These in-versions require a fast decay for the magnitude-sortedcurvelet coefficients. Following these results, the vectorx0 can be recovered from noise-corrupted and incom-plete data. Sparsity-promoting norm-one penalty func-tionals are not new to the geosciences (see for instancethe seminal work of Claerbout and Muir (1973), fol-lowed by many others). New are (i) the curvelet trans-form that obtains near optimal theoretical and empirical(Candes et al., 2006; Hennenfent and Herrmann, 2006b)compression rates on seismic data and images and (ii)the theoretical understanding of the conditions for a suc-cessful recovery.In this work, the seismic recovery problem is solved bythe norm-one nonlinear program:Pepsilon :braceleftBiggtildewide =argminx bardblxbardbl1 s.t. bardblAx-ybardbl2 <=<epsilontildewidef =SSST tildewidexin which epsilon is a noise-dependent tolerance level. Thenonlinear program Pepsilon is general and the (curvelet-based)synthesis matrix, A, and the inverse sparsity transform,ST , are defined in accordance with the application. Thevector tildewide represents the estimated solution (denoted bythe symbol tildewide). The above nonlinear program is solvedwith a threshold-based cooling method following ideasfrom Figueiredo and Nowak (2003) and Elad et al. (2005).SEISMIC DATA RECOVERYCRSIIn our formulation, seismic data regularization involvesthe solution of Pepsilon with A :=RCT , S :=C given incom-plete data, y = Rf, with f the fully sampled data and Rthe picking matrix that selects the acquired traces fromthe total data volume. In recent years, the authors re-peatedly reported on successful curvelet-based recoveryof seismic data (see e.g. Herrmann, 2005; Hennenfentand Herrmann, 2006a, 2007). Compared to other meth-ods, such as sparse Fourier recovery (Sacchi and Ul-rych, 1996; Zwartjes and Gisolf, 2006) and plane-wavedestruction (Fomel and Guitton, 2006), curvelet-basedmethods work for data with conflicting dips. Fig. 2 con-tains an example where data is recovered from 85% tracesmissing. This figure shows that CRSI is able to recoverthe complete data volume at the expense of the high-est frequency band. This estimate for the interpolateddata is used to calculate an improved estimate for theprimaries.fCRSICombining the non-adaptive curvelet transform with thedata-adaptive focal transform (Berkhout and Verschuur,2006), leads to a powerful formulation where data is fo-cused by inverting the primary operator. During thiscurvelet-regularized inversion of the primary operator,deltaP, propagation paths that include the surface are re-moved, yielding a more focused wavefield and hencea more compressed curvelet vector. This improved fo-cusing is achieved by Pepsilon with the synthesis matrix A :=RdeltaPCT and inverse sparsity transform ST :=deltaPCT . Thesolution of Pepsilon now entails the inversion of deltaPCT , yield-ing the sparsest set of curvelet coefficients that matchesthe incomplete data when ?convolved? with the primaries.The symbol deltaP refers to applying a temporal Fouriertransform, followed by a frequency-slice-by-frequency-slice matrix multiplication by hatwider(omega), followed by aninverse temporal Fourier transform for each slice. Thisoperator is compounded with the 3-D inverse curvelettransform that brings the data from the curvelet domainback to a 3D data volume. This choice for the synthesisoperator corresponds to a curvelet-regularized formu-lation of the focal transform (Berkhout and Verschuur,2006).The focal transform corresponds to an imaging towardsthe source without applying an imaging condition. Afterapplying the focal transform, the data is focused towardsthe source, a property used by Berkhout and Verschuur(2006) who ?cut out? the aliased energy in the focal do-main, prior to applying the inverse focal transform. Inour approach, we follow a less ?ad hoc? approach byonly promoting sparsity in the domain spanned by theThe focused curvelet transformfocused curvelet transform. In this way, no assumptionsexcept for sparsity are made. This sparsity assumptionseems reasonable since curvelets are sparse on wave-fields and the focused data itself is a wavefield, wherethe primaries are mapped to a directional source and thefirst-order multiples are mapped to primaries etc. etc.Because the wavefield is stripped from one interactionwith the surface, the focused wavefield will be morefocused and hence the sparsity-promoting norm in thecurvelet domain will be more effective.Aside from the focusing argument, the improved per-formance (cf. Fig 2(a) and 2(b)) can be attributed tothe increase in mutual incoherence between the Diracmeasurement basis and the columns of deltaPCT (see alsoHerrmann and Hennenfent, 2007; Hennenfent and Herr-mann, 2007). While CRSI could only recover the datavolume with the finest scale removed, fCRSI is able torecover the full data leading to a sharper recovery, espe-cially visible for the diffracted events in the time slice.The improvements for the recovery reflect in an improve-ment for the predicted multiples as shown in Fig. 3. Inturn, the fCSRI recovered data yields an improved pre-diction for the multiples.DISCUSSION AND CONCLUSIONSThe presented methodology banks on two properties ofcurvelets: their ability to detect wavefronts (the ?wave-front set?) and their approximate invariance under wavepropagation. By compounding the curvelet transformwith the focal transform, we were able to improve therecovery from incomplete data by curvelet-based spar-sity promotion. This improved performance is due tothe additional focusing by the primaries, rendering thecurvelet-sparsity promotion during the recovery moreeffective. As with curvelet-based recovery without fo-cusing, the recovery is improved by random sampling.This imperative random sampling breaks the aliasing byturning the missing data into a removable noise-term.For further details on this important observation, refer toother contributions by the authors to the proceedings ofthis conference. Since the focal transform correspondsto an imaging of seismic data towards the source, ourresults suggest that migrated images can in principle berecovered from data with large percentages of randomtraces missing.ACKNOWLEDGMENTSThe authors would like to thank Eric Verschuur for pro-viding us with the dataset. We also would like to thankthe authors of CurveLab for making their codes avail-able. The examples presented were prepared with Mada-gascar (rsf.sourceforge.net/), supplemented bySLIMPy (slim.eos.ubc.ca/SLIMpy) operator over-loading, developed by Sean Ross Ross. This work wasin part financially supported by the NSERC Discovery(22R81254) and CRD Grants DNOISE (334810-05) ofF.J.H. and was carried out as part of the SINBAD projectwith support, secured through ITF, from BG Group, BP,Chevron, ExxonMobil and Shell.(a)(b)Figure 1: Synthetic dataset. (a) Original data. (b) Ran-domly subsampled data with 85% of the traces missing.The focused curvelet transform(a) (b)Figure 2: Curvelet-based seismic data recovery. (a) Recovery with CRSI. (b) Recovery with fCRSI. Comparisonbetween the CRSI- and fCRSI results shows a clear improvement in the frequency content of the recovered data forfCRSI.(a) (b)Figure 3: SRME-multiple prediction. (a) SRME-predicted multiples from randomly subsampled data with 85% of thetraces missing (cf. Fig. 1(b)). (b) SRME-predicted from the fCRSI recovered data (cf. Fig. 2(b)).The focused curvelet transformREFERENCESAbma, R. and N. Kabir, 2006, 3D interpolation of irregular data with a POCS algorithm: Geophysics, 71,E91?E97.Berkhout, A. J. and D. J. Verschuur, 2006, Focal transformation, an imaging concept for signal restorationand noise removal: Geophysics, 71.Bleistein, N., J. Cohen, and J. Stockwell, 2001, Mathematics of Multidimensional Seismic Imaging, Migra-tion and Inversion: Springer.Cand` E., J. Romberg, and T. Tao, 2006, Stable signal recovery from incomplete and inaccurate measure-ments: Comm. Pure Appl. Math., 59, 1207?1223.Candes, E. J., L. Demanet, D. L. Donoho, and L. Ying, 2006, Fast discrete curvelet transforms: SIAMMultiscale Model. Simul., 5, 861?899.Claerbout, J. and F. Muir, 1973, Robust modeling with erratic data: Geophysics, 38, 826?844.Donoho, D. L., 2006, Compressed sensing: IEEE Trans. Inform. Theory, 52, 1289?1306.Elad, M., J. L. Starck, P. Querre, and D. L. Donoho, 2005, Simulataneous Cartoon and Texture ImageInpainting using Morphological Component Analysis (MCA): Appl. Comput. Harmon. Anal., 19, 340?358.Figueiredo, M. and R. Nowak, 2003, An EM algorithm for wavelet-based image restoration: IEEE Trans.Image Processing, 12, 906?916.Fomel, S. and A. Guitton, 2006, Regularizing seismic inverse problems by model reparameterization usingplane-wave construction: Geophysics, 71, A43?A47.Hennenfent, G. and F. Herrmann, 2006a, Application of stable signal recovery to seismic interpolation:Presented at the SEG International Exposition and 76th Annual Meeting.???, 2007, Irregular sampling: from aliasing to noise: Presented at the EAGE 69th Conference & Exhi-bition.Hennenfent, G. and F. J. Herrmann, 2006b, Seismic denoising with non-uniformly sampled curvelets: IEEEComp. in Sci. and Eng., 8, 16?25.Herrmann, F. J., 2005, Robust curvelet-domain data continuation with sparseness constraints: Presented atthe EAGE 67th Conference & Exhibition Proceedings.Herrmann, F. J., U. Boeniger, and D.-J. E. Verschuur, 2007, Nonlinear primary-multiple separation withdirectional curvelet frames: Geoph. J. Int. To appear.Herrmann, F. J. and G. Hennenfent, 2007, Non-parametric seismic data recovery with curvelet frames.Submitted for publication.Herrmann, F. J., P. P. Moghaddam, and C. Stolk, 2006, Sparsety- and continuity-promoting seismic imagingwith curvelet frames. In revision.Lin, T. and F. J. Herrmann, 2007, Compressed wavefield extrapolation. in revision.Sacchi, M. and T. Ulrych, 1996, Estimation of the discrete fourier transform, a linear inversion approach:Geophysics, 61, 1128?1136.Spitz, S., 1999, Pattern recognition, spatial predictability, and subtraction of multiple events: The LeadingEdge, 18, 55?58.Verschuur, D. J. and A. J. Berkhout, 1997, Estimation of multiple scattering by iterative inversion,part II: practical aspects and examples: Geophysics, 62, 1596?1611.Wapenaar, C., D. Draganov, and J. Robertsson, eds., 2006, Supplement Seismic Interferometry. SEG.Ying, L., L. Demanet, and E. J. Cand? 2005, 3D discrete curvelet transform: Wavelets XI, ExpandedAbstracts, 591413, SPIE.Zwartjes, P. and A. Gisolf, 2006, Fourier reconstruction of marine-streamer data in four spatial coordinates:Geophysics, 71, V171?V186.


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