UBC Faculty Research and Publications

Surface related multiple prediction from incomplete data Herrmann, Felix J. 2007

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ABSTRACTIncomplete data, unknown source-receiver signatures and free-surface reflectivity rep-resent challenges for a successful prediction and subsequent removal of multiples. Inthis paper, a new method will be represented that tackles these challenges by combiningwhat we know about wavefield (de-)focussing, by weighted convolutions/correlations,and recently developed curvelet-based recovery by sparsity-promoting inversion (CRSI).With this combination, we are able to leverage recent insights from wave physics to-wards a nonlinear formulation for the multiple-prediction problem that works for in-complete data and without detailed knowledge on the surface effects.EAGE 69th Conference & Exhibition ? London, UK, 11 - 14 June 2007Surface-related multiple prediction and seismic interferometry are examples whereweighted multi-dimensional cross-convolutions and cross-correlations of seismic datavolumes provide information on Green?s functions that describe the Earth response atthe surface. For instance, surface-related multiples can approximately be predictedthrough a weighted multidimensional convolution of the data with itself, while ?daylightimaging? techniques extract the Green?s function by cross-correlation of wavefields (seee.g. Wapenaar et al., 2006, which contains a collection of the most recent papers on thistopic). Recently, new approaches have been proposed, where the Green?s functions areextracted through inversion or deconvolution (See the contributions of Snieder et.al,Schuster et.al. and Berkhout and Verschuur in Wapenaar et al., 2006). We follow asimilar approach, where we are interested in finding an alternative formulation for thefollowing two operations:? wavefield defocusing, where the wavefield is convolved with the ?primary? wave-field. This convolution maps the primaries into first-order multiples and first-order multiples into second-order multiples etc., i.e., we have the mappingp(m)(x, t) mapstoarrowright p(m+1)(x, t) = parenleftbigGbracketleftbig?0, Abracketrightbigpmparenrightbig(x, t) (1)with Gbracketleftbig?0, psibracketrightbig? := A? asteriskmathx,t ?0 asteriskmathx,t ?, the symbol asteriskmathx,t denoting multi-dimensionalcross-convolution and A? the weighting;? wavefield focusing, where the wavefield is correlated with the ?primary? wave-field. This correlation maps the primaries into first order multiples and first-ordermultiples into second-order multiples etc., i.e., we have the mappingp(m+1)(x, t) mapstoarrowright p(m)(x, t) = parenleftbigFbracketleftbig?0, Bbracketrightbigpm+1parenrightbig(x, t) (2)with Pbracketleftbig?0, Bbracketrightbig? := ?0 circlemultiplyx,t B? asteriskmathx,t ?, the symbol circlemultiplyx,t denoting multi-dimensionalcross-correlation and B? another weighting.In these expressions, pm refers to the (m)th-order multiple in the data (p0 represents theprimary wavefield) and ?0 represents an estimate for the ?primaries? that is assumed tobe given (we used ? to indicate that in practice we only have approximate knowledgeof the primaries since the sole purpose of this work is to estimate these primaries). Gand F are the defocusing and focussing operators that map the mth-order component tothe (m+ 1)th-order component and back. The defocusing operator consists of a multi-dimensional weighted cross-convolution between the ?primaries?, ?0, and the wavefield,followed by a deconvolution by A? that contains the surface reflectivity and the pseudoinverse (denoted by the ?) of the source and receiver directivity and time signatures.After cross-convolution, the wavefield has a tendency to spread out, i.e., we added a?travel path?, hence the name defocusing. The wavefield focusing operator, on the otherhand, constitutes a weighted cross-correlation of the wavefield with the ?primaries?,removing a ?travel path?. This weighting by B? is defined by the damped pseudo inverseof the ?autocorrelation? of the ?primary? operator.In this paper, we seek an alternative formulation, where (i) no information is re-quired on A; (ii) that is stable w.r.t. incomplete data and (iii) where focusing is accom-plished by sparsity-promoting inversion (replacing B?). Fig. 1 illustrates the effect ofmissing data on the prediction of multiples. To accomplish these goals, we combinethe focusing property and the sparsity of curvelets. After discretization (lower- andupper-case bold symbols refer to discretized vectors and matrices), define the followingEAGE 69th Conference & Exhibition ? London, UK, 11 - 14 June 2007(a) (b)Figure 1: SRME multiple prediction (b) from incomplete data (a) with 80 % of the traces missing.Notice the artifacts due to the missing data leading to a deterioration of the multiple prediction.incomplete data representationy = Ax with A := RPCT (3)with R the restriction operator; P := tildewide0asteriskmathx,t the ?primary? operator; CT the trans-pose (inverse) of the curvelet transform (see e.g. Hennenfent and Herrmann, 2006b, andthe references therein), x the curvelet coefficient vector and y = Rd the incompletedata. This signal representation differs from standard-CRSI (Hennenfent and Herr-mann, 2006a; Herrmann and Hennenfent, 2007) by including the ?primary? operator P.CRSI with focusing: By inverting the defocusing, seismic data is focused, bootstrap-ping the sparsity obtained by the curvelet transform. Using this property, the recoveryfrom incomplete data can be written as followsF :bracelefttpbraceexbraceleftmidbraceexbraceleftbttildewide = arg minx bardblxbardbl1 subject to Ax = ytildewided(m-1) = CT tildewidextildewided = PCT tildewidex.(4)In words, the solution of F involves finding the sparsest set of curvelet coefficients thatmatches the incomplete data when convolved with the primaries. The data, d, in thiscase includes primaries and multiples (see Fig. 1). As such, the estimated coefficientsrepresent an estimate for the focused data since they are converted back into data bythe ?primary? operator during the optimization. Eq. 4 corresponds to a curvelet-sparsityregularized inverse of Berkhout?s focusing matrix and of the convolution operator ininterferometric imaging (Vasconcelos and Snieder, 2006). The symbol d(m-1) refers tofocused data with primaries mapped to the focal point and mth-order multiples mappedto (m - 1)th-order. The result of the sparse recovery from the incomplete data usingstandard-CRSI (Herrmann and Hennenfent, 2007) and CRSI + focusing are summa-rized in Fig 2. Expectedly, the curvelet transform compounded with the primary opera-tor improves the recovery.Defocusing with CRSI: After successful recovery of the incomplete data, multi-ples can be predicted using the nonlinear mapping defined in Eq. 1. This mappingthrough multi-dimensional convolution, however, has the disadvantage that an estimateis needed for A. By defining A := PasteriskmathCT with Pasteriskmath = tildewide0circlemultiplyx,t the adjoint of the primaryEAGE 69th Conference & Exhibition ? London, UK, 11 - 14 June 2007(a) (b)Figure 2: Comparison between CSRI (a) and CSRI + focusing (b) for data with 80 % of the tracesmissing. Notice the significant uplift from compounding the inverse curvelet transform with the focusing?primary? operator.operator, multiples can be predicted by solvingG :braceleftBiggtildewide = arg minx bardblxbardbl1 subject to Ax = ytildewided(m+1) = CT tildewidex (5)with y = tildewide (estimated above). In words, the solution of G corresponds to finding thesparsest set of curvelet coefficients that matches the data when cross-correlated with theprimaries. As such the estimated coefficients represent an estimate for the multiples,since this estimate for the multiples is converted to the primaries after applying thecorrelation during the optimization. More precisely, this formulation corresponds to asparse inversion of the operator that maps multiples to primaries. The advantage of thisformulation is that it does not require information on Aas can be observed from Fig. 3.Examples: Without loss of generality, we considered the acoustice reflection responseof a 1-D medium consisting of three layers and a free surface. For this type of medium,the multi-dimensional cross-convolutions and correlations become simple convolutionsand correlations that are diagonal in the f - k domain. In practice, the primaries arenot known and the data itself is used instead, as part of an iterative procedure. In thiscase, spurious non-physical events may occur an observation reported in the literature(Snieder et.al. in Wapenaar et al., 2006).DiscussionThe methodology presented in this paper banks on two complementary aspects of wavephenomena, namely, (i) the focusing and defocusing by multidimensional cross-con-volutions/correlations, reflecting certain physical relations, and (ii) the existence of amultiscale and multi-directional curvelet transform that sparsely represents high-frequen-cy solutions of wave equations. Pairing these two aspects leads to a new formulationfor the prediction of multiples from incomplete data, without knowledge on the sur-face effects. The focusing is found to improve the recovery because the data becomessparser in the curvelet domain after focusing and this explains the improvement overcurvelet-only CRSI. We also observed that F corresponds to the focal transform andinterferometric imaging by deconvolution formalisms, opening the interesting new per-spective of adding more robustness. The prediction for the multiples after the recoveryalso benefited from the sparsity promotion. Again the sparsity of curvelets, that isEAGE 69th Conference & Exhibition ? London, UK, 11 - 14 June 2007(a) (b)(c) (d)Figure 3: Comparison between convolution-based multiple prediction (a) and sparsity-based multipleprediction (b). Aside from the acausal artifact, the sparsity-promoting multiple prediction according toEq.5 improves the frequency content and makes it closer to the spectrum of the true multiples. Trace-wise comparisons in (c-d) between the true (including internal) mutiples (blue), the multiples predictedwith conventional (green) and sparsity promoted predicted multiples (red) confirm this observation. Thedifference in the spectrum are partially due to the fact that we only predicted the surface-related multiples.related to the invariance of curvelets under wave propagation, leads to an improved pre-diction. This improvement can be understood because the method inverts the adjoint ofthe primary operator that contains the surface effects. The improved predictions will inturn improve curvelet-based primary-multiple separation (Herrmann et al., 2006). Inter-ferometric prediction of ground roll will be discussed elsewhere in these proceedings.Acknowledgments: The authors would like to thank the authors of CurveLab for making theircodes available. 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.REFERENCESHennenfent, 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, 2006, 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.Vasconcelos, I. and R. Snieder, 2006, Interferometric imaging by deconvolution: theory and numericalexamples: Presented at the SEG International Exposition and 76th Annual Meeting.Wapenaar, C., D. Draganov, and J. Robertsson, eds. 2006, Supplement Seismic Interferometry. SEG.EAGE 69th Conference & Exhibition ? London, UK, 11 - 14 June 2007

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