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Seismic noise : the good the bad and the ugly Herrmann, Felix J.; Wilkinson, Dave 2007

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Seismic noise: the good the bad and the uglyFelix J. Herrmannasteriskmath and Dave Wilkinson?asteriskmathABSTRACTIn this paper, we present a nonlinear curvelet-based sparsity-promoting formula-tion for three problems related to seismic noise, namely the ?good?, correspondingto noise generated by random sampling; the ?bad?, corresponding to coherent noisefor which (inaccurate) predictions exist and the ?ugly? for which no predictionsexist. We will show that the compressive capabilities of curvelets on seismic dataand images can be used to tackle these three categories of noise-related problems.IntroductionIn this paper, we present recent developments of the application of the curvelet trans-form (see e.g. Candes et al., 2006; Hennenfent and Herrmann, 2006b) to problems thatinvolve different types of noise in seismic data. Our approach banks on two funda-mental properties of curvelets, namely the? detection of wave-fronts without requiring prior information on the dips oron the velocity model;? invariance of curvelets under the action of wave propagation.These two properties make this transform suitable for a robust formulation of prob-lems, such as seismic data regularization (Hennenfent and Herrmann, 2006a; Herr-mann and Hennenfent, 2007), primary-multiple separation (Herrmann et al., 2006a),ground-roll removal (Yarham et al., 2006) and stable migration-amplitude recovery(Herrmann et al., 2006b). All these methods exploit sparsity in the curvelet domainthat is a direct consequence of the above two properties and corresponds to a rapiddecay for the magnitude-sorted curvelet coefficients. This sparsity allows for a sepa-ration of ?noise? and ?signal? underlying all these problems (see e.g. Hennenfent andHerrmann, 2006b; Herrmann et al., 2006a).Curvelets: As can be observed from Fig. 1, curvelets are localized functions thatoscillate in one direction and that are smooth in the other directions. They areasteriskmathSeismic Laboratory for Imaging and Modeling, Department of Earth and Ocean Sciences, Uni-versity of British Columbia, 6339 Stores Road, Vancouver, V6T 1Z4, BC, Canada?Chevronasteriskmathe-mail: fherrmann@eos.ubc.ca2multiscale and multi-directional and because of their anisotropic shape (they obey theso-called parabolic scaling relationship, yielding a width proportional 2j/2 and a length proportional 2jwith j the scale), curvelets are optimal for detecting wavefronts. This explains theirhigh compression rates for seismic data and images as reported in the literature(Candes et al., 2006; Hennenfent and Herrmann, 2006c; Herrmann et al., 2006a,b).(a) (b)Figure 1: Example of a 3-D curvelet. Notice the oscillations in one direction and the smoothnessin the other two directions.Sparsity promoting inversion: High compression rates for signal representationsare a prerequisite for the robust formulation of stable signal recovery problems andother inverse problems. These compression rates allow for a nonlinear sparsity pro-moting solution. As such sparsity-promoting norm-one penalty functionals are notnew to the geosciences (see for instance the seminal work of Claerbout and Muir(1973), followed by many others), where sparsity is promoted on the model. Whatis different in the current surge of interest in sparsity-promoting inversion, known as?compressed sensing? (Candes et al., 2005; Donoho et al., 2006), is (i) the existence ofsparsity promoting transforms such as the curvelet transform; (ii) the deep theoreticalunderstanding on what the conditions are for a successful solution. This work can beseen as the application of these recent ideas to the seismic situation and involves thesolution of the following norm-one nonlinear program,Pepsilon1 :braceleftBiggtildewide = arg minx bardblxbardbl1 s.t. bardblAx -ybardbl2 <=< epsilon1tildewided = ST tildewidex (1)in 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(the symbol T denoting the transpose) compounded with other operators dependingon the problem. The above constrained optimization problem is solved to an accuracyof epsilon1 that depends on the noise level. Finally, tildewide stands for the recovered vector withthe symbol tildewide reserved for optimized quantities.3The ?good?: random sampling gives rise to incoherent noiseSampling of seismic wavefields is based on the assumption that equally-sampled datais good. Indeed, when Nyquist?s sampling theorem is met, equidistant samplingallows for a perfect reconstruction of the wavefield. Unfortunately, adequate samplingof steeply dipping events, such as ground roll, are often unfeasible in practice. Incase of regular subsampling, this leads to the well-known phenomonon of aliasing asillustrated in Fig. 2. This alliasing leads to a difficult to predict and separate coherent?noise? in the Fourier domain. Random subsampling, on the other hand, leads to anoisy spectrum for the same number of samples (see Fig. 2(d)). This is an exampleof ?good? noise that can easily be separated. Denoising in that case corresponds toseismic data regularization and boils down to solving Pepsilon1 with A := RCT , S := Cand y = Rd for the incomplete data. This formulation corresponds the formulationfor curvelet recovery by sparsity-promoting inversion (CRSI), which has successfullybeen applied to the recovery of incomplete seismic data (see e.g. Hennenfent andHerrmann, 2006a). In this formulation, R is the restriction matrix, selecting the rowsfrom the curvelet transform matrix that correspond to active traces. As opposed toother recovery methods, such as sparse Fourier recovery and plane wave destruction,curvelet-based methods have the advantage of working in situations where there areconflicting dips without stationarity assumptions. The method exploits the high-dimensional continuity of wavefronts and as Fig. 3 demonstrates, recovery resultsimprove when using the 3-D curvelet transform compared to the 2-D transform. Thiscan be explained because the 3-D curvelets capture more of the signal?s energy andhence are better able to separate the coherent seismic energy from the incoherent andhence ?good? noise related to random subsampling.(a) (b) (c) (d)Figure 2: Fourier spectra for incomplete data. (a) Regularly missing data leads to a strongly aliasedspectrum (b) as opposed to (c) data missing according an uniform distribution that gives rise tothe noisy Fourier spectrum (d). Observe that the Fourier spectrum for the random subsampleddata looks noisy while the regular undersampled data displays the well-known and harmful periodicimprint of aliasing.4Figure 3: Illustration of sliced versus volumetric interpolation.The ?bad?: coherent signal separation with (inaccurate) pre-dictionsSo far, we looked at exploiting the sparsity of curvelets in the data domain for thepurpose of recovery from incomplete data. The ability of curvelets to detect wave-fronts with conflicting dips, allows for a formulation of a coherent signal separationmethod that uses inaccurate predictions as weightings. By defining the synthesismatrix as A := bracketleftbigCT W1 CT W2bracketrightbig, x = bracketleftbigx1 x2bracketrightbigT and y = d = s1 + s2 and by set-ting the diagonal weighting matrices W1,2 in terms of predictions for two differentsignal components (e.g., primaries and multiples or reflectivity and ground roll), thesolution of Pepsilon1 separates the two signal components (see e.g. Herrmann et al., 2006a;Yarham et al., 2006) even for inaccurate predictions for which least-squares adaptivesubtraction fails (see Fig. 4). With this method ground-roll and reflectivity can alsosuccesfully be separated as can be seen from the example plotted in Fig. 5.The ?ugly?: migration amplitude recovery from noisy dataFinally, the precense of unknown sources of clutter in the image space can be amajor challenge. For instance, consider the situation where noisy data is migrated,yielding a noisy image, i.e. y = KT d with d = Km + n. In this expression, dis the noisy data, k the demigration operator and n a Gaussian noise term. Therecovery of the reflectivity m is challenging because the image y contains a coherentand nonstationary noise term, KT n consisting of migrated noise. To separate thisnoise term from the imaged reflectivity, we use the following approximate idendity5(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 resultobtained with a single curvelet-domain soft thresholding with lambda = 1.4. Notice that least-squaressubtractions fails.(a) (b)(c) (d)Figure 5: Example of curvelet-domain ground roll removal. (a) f - k filtered result. (b) theseparated ground roll. (c) Curvelet-domain separated result obtained by optimization. (d) Thepredicted ground roll. Notice that the predicted ground roll for the curvelet separation is clean anddoes not contain significant reflection events.6(Herrmann et al., 2006b)AAT r similarequal Psir (2)with r an appropriately chosen reference vector and Psi the discrete normal (demigration-migration) operator. The synthesis operator in this case is defined as A := CT G withG a diagonal weighting matrix. This identity diagonalizes the normal operator andallows for a stable recovery and denoising of the migrated image from ?noisy data? y.After solving for Pepsilon1, the reflectivity is obtained by applying the following synthesis,i.e. by setting ST := parenleftbigAT parenrightbig? with ? the pseudo inverse.It can be shown that the diagonal approximation of the normal operator servestwo purposes. Firstly, the inversion of the weighted curvelet transform corrects forthe amplitudes, as can be observed in Fig. 6(b). Secondly, the diagonal whitens thecoloring of the noise term in the image spaced, allowing for succesful denoising bysolving Pepsilon1 with epsilon1 set according the noise level. Results of this procedure on theSEG AA? dataset with a reverse-time migration operator are summarized in Fig. 6,confirm the validity of this approach. The resulting image shows a nice recovery ofthe amplitudes and removal of most the noise for data with a signal-to-noise ratio of3 dB.(a) (b)Figure 6: Image amplitude recovery for noisy data (SNR 3 dB). (a) Noisy image. (b) Image afternonlinear recovery from noisy data with Pepsilon1. The clearly visible non stationary noise in (a) is mostlyremoved during the recovery while the amplitudes are also restored.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 succesful removal of different types of clut-ter from seismic data. We showed that by compounding the curvelet transform withcertain matrices each ?denoising? problem can be cast into one and the same optimiza-tion problem. The solution of this optimization problem entails a denoising, wherethe curvelet coefficients are recovered through a promotion of sparsity. This sparsityallows for a separation of the different ?noisy? signal components. The results show7that (i) exploiting the multi-dimensional structure of seismic data with 3-D curveletsleads to a recovery scheme that is able to reconstruct fully sampled data volumes fromdata with > 80 % random traces missing; (ii) the sparsity of curvelet can be usedto separate coherent signal components given an (inaccurate) prediction and finallythat (iii) the invariance of curvelets under the demigration-migration operator canbe used to recover the seismic amplitudes and remove noise in the image spaced byinverting a diagonally weighted curvelet matrix.Acknowledgments: The authors would like to thank the students at the Seismic Laboratoryfor Imaging and Modeling (slim.eos.ubc.ca) and authors of CurveLab for making their codesavailable. 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 andEngineering Research Council of Canada Discovery Grant (22R81254) and Collaborative Researchand Development Grant DNOISE (334810-05) of Felix J. Herrmann and was carried out as part ofthe SINBAD project with support, secured through ITF (the Industry Technology Facilitator), fromthe following organizations: BG Group, BP, Chevron, ExxonMobil and Shell.REFERENCESCandes, E. J., L. Demanet, D. L. Donoho, and L. Ying, 2006, Fast discrete curvelet transforms:SIAM Multiscale Model. Simul., 5, 861?899.Candes, E. J., J. Romberg, and T. Tao, 2005, Stable signal recovery from incomplete and inaccuratemeasurements. to appear in Comm. Pure Appl. Math.Claerbout, J. and F. Muir, 1973, Robust modeling with erratic data: Geophysics, 38, 826?844.Donoho, D. L., M. Elad, and V. Temlyakov, 2006, Stable recovery of sparse overcomplete represen-tations in the presence of noise: IEEE Trans. Inform. Theory, 52, 6?18.Hennenfent, G. and F. Herrmann, 2006a, Application of stable signal recovery to seismic interpola-tion: 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.???, 2006c, Seismic denoising with non-uniformly sampled curvelets: IEEE Comp. in Sci. andEng., 8, 16?25.Herrmann, F. J., U. Boeniger, and D.-J. E. Verschuur, 2006a, Nonlinear primary-multiple separationwith directional curvelet frames: Geoph. J. Int. To appear.Herrmann, F. J. and G. Hennenfent, 2007, Non-parametric seismic data recovery with curveletframes. Submitted for publication.Herrmann, F. J., P. P. Moghaddam, and C. Stolk, 2006b, Sparsety- and continuity-promoting seismicimaging with curvelet frames. In revision.Yarham, C., U. Boeniger, and F. Herrmann, 2006, Curvelet-based ground roll removal: Presentedat the SEG International Exposition and 76th Annual Meeting. (submitted).

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