UBC Faculty Research and Publications

Seismic imaging and processing with curvelets Herrmann, Felix J.; Hennenfent, Gilles; Moghaddam, Peyman P. 2007-03-10

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ABSTRACTIn this paper, we present a nonlinear curvelet-based sparsity-promoting formulation forthree problems in seismic processing and imaging namely, seismic data regularizationfrom data with large percentages of traces missing; seismic amplitude recovery for sub-salt images obtained by reverse-time migration and primary-multiple separation, givenan inaccurate multiple prediction. We argue why these nonlinear formulations are ben-eficial.EAGE 69th Conference & Exhibition ? London, UK, 11 - 14 June 2007In this paper, we report recent developments on the application of the curvelet trans-form (see e.g. Candes et al., 2005a; Hennenfent and Herrmann, 2006b) to seismic pro-cessing and imaging. Our approach banks on two fundamental properties of curvelets,namely the? detection of wave-fronts without requiring prior information on the dips or onthe velocity model;? invariance under the action of wave propagation.These two properties make this transform suitable for a robust formulation of problemssuch as seismic data regularization (Hennenfent and Herrmann, 2006a; Herrmann andHennenfent, 2007), migration-amplitude recovery (Herrmann et al., 2006b) and pri-mary multiple separation (Herrmann et al., 2006a). All these methods exploit sparsityin the curvelet domain that is a direct consequence of the above two properties andcorresponds to a rapid decay for the magnitude-sorted curvelet coefficients.Curvelets: As can be observed from Fig. 1, curvelets are localized functions that os-cillate in one direction and that are smooth in the other directions. They are multiscaleand multi-directional and because of their anisotropic shape (they obey the so-calledparabolic scaling relationship, yielding a width proportional 2j/2 and a length proportional 2j with j thescale), curvelets are optimal for detecting wavefronts. This explains their high com-pression rates for seismic data and images as reported in the literature.(a) (b)Figure 1: Example of a 3-D curvelet. Notice the oscillations in one direction and the smoothness in theother two directions.Sparsity promoting inversion: High compression rates for signal representations area prerequisite for the robust formulation of stable signal recovery problems and otherinverse problems. These compression rates allow for a nonlinear sparsity promotingsolution. As such sparsity-promoting norm-one penalty functionals are not new to thegeosciences (see for instance the seminal work of Claerbout and Muir (1973), followedby many others), where sparsity is promoted on the model. What is different in the cur-rent surge of interest in sparsity-promoting inversion, known as ?compressed sensing?(Candes et al., 2005b; Donoho et al., 2006), is (i) the existence of sparsity promotingtransforms such as the curvelet transform; (ii) the deep theoretical understanding onwhat the conditions are for a successful solution. This work can be seen as the ap-plication of these recent ideas to the seismic situation and involves the solution of thefollowing norm-one nonlinear program,Pepsilon1 :braceleftBiggtildewide = arg minx bardblxbardbl1 s.t. bardblAx-ybardbl2 <=< epsilon1tildewided = ST tildewidex (1)EAGE 69th Conference & Exhibition ? London, UK, 11 - 14 June 2007in which y is the (incomplete) data, A the synthesis matrix and ST the inverse sparsitytransform. Both these matrices consist of the inverse curvelet transform matrix, ST (thesymbol T denoting the transpose) compounded with other operators depending on theproblem. The above constrained optimization problem is solved to an accuracy of epsilon1 thatdepends on the noise level. Finally, tildewide stands for the recovered vector with the symbol tildewidereserved for optimized quantities.CRSI: An important topic in seismic processing is the seismic regularization prob-lem, where attempts are made to recover fully-sampled seismic data volumes from in-complete data, i.e., data with large percentages (> 80 %) of traces missing. By choosingA := RCT , S := Cand y = Rdfor the incomplete data, one arrives at the formulationfor curvelet recovery by sparsity-promoting inversion (CRSI), which has successfullybeen applied to the recovery of incomplete seismic data (see e.g. Hennenfent and Herr-mann, 2006a). In this formulation, R is the restriction matrix, selecting the rows fromthe curvelet transform matrix that correspond to active traces. As opposed to other re-covery methods, such as sparse Fourier recovery and plane wave destruction, curvelet-based methods have the advantage of working in situations where there are conflictingdips without stationarity assumptions. The method exploits the high-dimensional conti-nuity of wavefronts and as Fig. 2 demonstrates, recovery results improve when using the3-D curvelet transform compared to the 2-D transform. For application of this methodto real data, refer to other contribution by the authors to the proceedings of this meeting.Figure 2: Illustration of sliced versus volumetric interpolation.Migration amplitude recovery: Because of the ?alleged? invariance of curvelets un-der wave propagation, there has been a substantial interest in deriving migration op-erators in the curvelet domain (Douma and de Hoop, 2006; Chauris, 2006). In theseapproaches, one comes to benefit when strict sparsity is preserved. Strict sparsity is asignificant stronger assumption than the preservation of high decay rates for the sortedcoefficients. Curvelets are discrete and hence move around on grids and this makes it achallenge to define fast migration operators. Curvelets, however, prove to be very usefulfor solving the seismic amplitude recovery problem, during which curvelets are beingimaged. On theoretical grounds (Herrmann et al., 2006b), one can expect the followingidentity to approximately holdAAT r similarequal Psir (2)with r an appropriately chosen discrete reference vector and Psithe discrete normal op-erator, formed by compounding the discrete scattering and its transpose, the migrationEAGE 69th Conference & Exhibition ? London, UK, 11 - 14 June 2007operator. The synthesis operator in this case is defined as A := CT G with G a diagonalweighting matrix. This identity diagonalizes the normal operator and allows for a sta-ble recovery of the migration amplitudes by setting y = KT d, with KT the migrationoperator and d the seismic data, and ST := parenleftbigAT parenrightbig? with ? the pseudo inverse. Resultsof this procedure on the SEG AA? dataset with a reverse-time migration operator, aresummarized in Fig. 3. The resulting image shows a nice recovery of the amplitudes.For more details on this method and on a method to remove remaining artifacts, refer toanother contributions by the authors to the proceedings of this meeting.(a) (b)Figure 3: Reverse-time migration on the SEG AA? salt model. (a) Conventional migrated imagey = KT d. (b) Seismic amplitude recovery through Pepsilon1. Notice the improved amplitudes. Some mi-nor artifacts remain.Primary-multiple separation: So far, we looked at exploiting the sparsity of curveletsin the data and image domains for the purpose of recovery. The ability of curvelets todetect wavefronts with conflicting dips, allows for a formulation of a coherent signalseparation method that uses inaccurate predictions as weightings. By defining the syn-thesis matrix as A := bracketleftbigCT W1 CT W2bracketrightbig, x = bracketleftbigx1 x2bracketrightbigT and y = d and by setting thediagonal weighting matrices W1,2 in terms of predictions for the primaries and multi-ples, the solution of Pepsilon1 separates primaries and multiples (Herrmann et al., 2006a) evenfor inaccurate predictions for which least-squares subtractions fails (see Fig. 4).DiscussionThe methodology presented in this paper banks on two favorable properties of curvelets,namely their ability to detect wavefronts and their approximate invariance under wavepropagation. These properties allow for a formulation of seismic processing and imag-ing problems that promote sparsity through the nonlinear optimization problem Pepsilon1. Weshowed that by compounding the curvelet transform with certain matrices each prob-lem can be cast into one and the same optimization problem. The results show that(i) exploiting the multi-dimensional structure of seismic data with 3-D curvelets leadsto a recovery scheme that is able to reconstruct fully sampled data volumes from datawith > 80 % traces missing; (ii) the invariance of curvelets under the demigration-migration operator can be used to recover the seismic amplitudes by inverting a diago-nally weighted curvelet matrix. This latter approach is an extension of recent work bySymes (2006). Finally, we also showed that (iii) the sparsity of curvelet can be used toseparate coherent signal components given an (inaccurate) prediction. The successfulapplication to this wide spectrum of problems opens an enticing perspective of extend-ing this framework to multiple prediction and the compression of (migration) operators.EAGE 69th Conference & Exhibition ? London, UK, 11 - 14 June 2007(a) (b)(c) (d)Figure 4: Example of primary-multiple separation through Pepsilon1 for predicted multiples with moveouterrors. (a) the total data with primaries and multiples. (b) the true multiples used for the prediction. (c)the result obtained with least-squares adaptive subtraction with localized windows. (d) the result obtainedwith a single curvelet-domain soft thresholding with lambda = 1.4. Notice that least-squares subtractions fails.Acknowledgments: The authors would like to thank the authors of CurveLab for making theircodes available. We also would like to thank Dr. William Symes for making his reverse-time migrationcode available to us. This work was in part financially supported by the Natural Sciences and EngineeringResearch Council of Canada Discovery Grant (22R81254) and Collaborative Research and DevelopmentGrant DNOISE (334810-05) of Felix J. Herrmann and was carried out as part of the SINBAD project withsupport, secured through ITF (the Industry Technology Facilitator), from the following organizations: BGGroup, BP, Chevron, ExxonMobil and Shell.REFERENCESCandes, E., L. Demanet, D. Donoho, and L. Ying, 2005a, Fast discrete curvelet transforms: SIAMMultiscale Model. Simul.Candes, E., J. Romberg, and T. Tao, 2005b, Stable signal recovery from incomplete and inaccurate mea-surements. to appear in Comm. Pure Appl. Math.Chauris, H., 2006, Seismic imaging in the curvelet domain and its implications for the curvelet design:Presented at the 76th Ann. Internat. Mtg., SEG, Soc. Expl. Geophys., Expanded abstracts.Claerbout, J. and F. Muir, 1973, Robust modeling with erratic data: Geophysics, 38, 826?844.Donoho, D., M. Elad, and V. Temlyakov, 2006, Stable recovery of sparse overcomplete representationsin the presence of noise: IEEE Trans. Inform. Theory.Douma, H. and M. de Hoop, 2006, Leading-order seismic imaging using curvelets: Presented at the 76thAnn. Internat. Mtg., SEG, Soc. Expl. Geophys., Expanded abstracts.Hennenfent, G. and F. Herrmann, 2006a, Application of stable signal recovery to seismic interpolation:Presented at the SEG International Exposition and 76th Annual Meeting.Hennenfent, G. and F. J. Herrmann, 2006b, Seismic denoising with non-uniformly sampled curvelets:IEEE Comp. in Sci. and Eng., 8, 16?25.Herrmann, F. J., U. Boeniger, and D.-J. E. Verschuur, 2006a, 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, 2006b, Sparsety- and continuity-promoting seismicimaging with curvelet frames. In revision.Symes, W. W., 2006, Optimal scaling for reverse time migration: Technical Report TR 06-19, Departmentof Computational and Applied Mathematics, Rice University, Houston, Texas, USA.EAGE 69th Conference & Exhibition ? London, UK, 11 - 14 June 2007


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