Random sampling: new insights into the reconstruction of coarsely-sampled wavefieldsGilles Hennenfent, EOS-UBC and Felix J. Herrmann, EOS-UBCSUMMARYIn this paper, we turn the interpolation problem ofcoarsely-sampled data into a denoising problem. Fromthis point of view, we illustrate the benefit of randomsampling at sub-Nyquist rate over regular sampling atthe same rate. We show that, using nonlinear sparsity-promoting optimization, coarse random sampling mayactually lead to significantly better wavefield reconstruc-tion than equivalent regularly sampled data.INTRODUCTIONDense sampling of seismic data is traditionally under-stood as evenly-distributed time and space measurementsof the reflected wavefield. Moreover, the sampling ratealong each axis must be equal to or above twice the high-est frequency/wavenumber of the continuous signal be-ing discretized (Shannon/Nyquist sampling theorem). Inpractice, however, seismic data is often randomly and/orsparsely sampled along spatial coordinates, which is gen-erally considered as a challenge since it breaks one orboth previously-stated conditions of dense sampling. Itturns out that these acquisition geometries are not nec-essarily a source of adversity to accurately reconstructdensely-sampled data when using nonlinear optimiza-tion promoting sparsity. This new insight, developed inthe information theory community, is referred to in theliterature by the terms ?compressed sensing? or ?com-pressive sampling? (see e.g. Donoho, 2006; Candes et al.,2005, and references therein).THEORYSampling below Nyquist rateDiscretizing a continuous function bandlimited to thefrequency interval [-B, B] corresponds to a multiplica-tion with a Dirac comb in the time domain and thusto a convolution with another Dirac comb in the fre-quency domain. As a consequence, the original spec-trum becomes periodic after discretization. Problemsoccur when regularly sampling below Nyquist rate, i.e.fs <2B. Replicas of the original spectrum overlap, whichcreates an indetermination in the reconstruction process.This is the well-known phenomenon of aliasing. In con-trast, random sampling at the same sub-Nyquist rate isless likely to create strong aliases but rather weak broad-band noise. Consider for example a random samplingoperator s over [0, N] where N is the size of the samplingregion. Suppose that s samples n < N points uniformlydistributed in the interval. Then, the expectation of thepower spectrum of s over [0, N] is given by (Leneman,1966; Dippe and Wold, 1985)Ebracketleftbig|?N(u)|2bracketrightbig =n2deltau0 +(1-deltau0)n, (1)where the symbol?denotes Fourier coefficients, E[?] themathematical expectation, and u is the frequency vari-able. The first term in this expression is only nonzeroat the origin and gives through convolution a scaled ver-sion of the original spectrum. The second term is onlyzero at the origin and can be assimilated to broadbandnoise.Fig. 1 summarizes these observations. Fig. 1(a) showsthe amplitude spectrum of a densely-sampled signal con-sisting of the superposition of three cosine functions.Figs. 1(b) and 1(c) show the spectra of the same signalbeing regularly- and randomly-sampled below Nyquistrate, respectively.Reconstruction by denoisingConsider the following linear forward model for the in-terpolation problemy =Rf0, (2)where yelementRn represents the data acquired at sub-Nyquistrate, f0 element RN the densely-sampled data to be recovered,and R element Rn?N the restriction operator that selects theacquired samples among the desired samples. Assumef0 has a sparse representation x0 in some transform do-main S where random sampling creates incoherent noise(e.g. Fourier domain, S := F), then interpolation to adense grid becomes a denoising problem in the S do-main (Donoho et al., 2007). This problem is solved bythe following nonlinear sparsity-promoting optimization(Candes et al., 2005; Hennenfent and Herrmann, 2006)? =argminx ||x||1 s.t. y =RSHx, (3)where the symbol?represents an estimated quantity, andH the conjugate tranpose. The interpolated result is givenby ? = SH ?. Note that, if the sampling at sub-Nyquistrate creates coherent noise in the S domain (see e.g.Fig. 1(b) when S := F), the separation between signaland noise generated by the acquisition in the S domainis much more delicate (if not impossible) just by impos-ing a sparsity constraint. We illustrate this comment bya simple 1-D experiment.We define S as the discrete cosine transform. We gener-ate a vector x0 of length N = 600 containing k nonzero(a) (b)Figure 2: Average recovery curves from sub-Nyquist rate samplings using nonlinear sparsity-promoting optimization.Average recovery curves (a) from regular sub-samplings by 2, . . . , 6 curves and (b) from random sub-samplings by thesame factors as in (a)(a)(b)(c)Figure 1: Spectra of a signal sampled above and belowNyquist rate. The signal consists of the superpositionof three cosine functions. Amplitude spectrum of thedensely-sampled signal (a), coarse regularly-sampledsignal (b), and coarse randomly-sampled signal (c).entries with random amplitudes, random signs, and ran-dom positions and construct f0 = SHx0. The observa-tions y are obtained by down-sampling f0 either regu-larly or randomly by a factor of 2, . . . , 6. Finally, wesolve Eq. 3 and compare the estimated representation ?of f0 to its true representation x0. The reconstruction er-ror is measured as the number of false detections in thediscrete cosine transform domain. The results presentedin Fig. 2 are averaged over 50 independent experiments.Figs. 2(a) and 2(b) show the recovery curves for regu-lar and random downsampling, respectively. Each curverepresents the results obtained for a given subsamplingfactor. Fig. 2(a) shows that, regardless of the subsam-pling factor and the sparsity of f0 in the discrete cosinetransform domain, the solution of Eq. 3 is corrupted bysome noise. However, Fig. 2(b) shows that there is asparsity zone for each subsampling factor such that thereconstruction is perfect, i.e. no false detection. Thesmaller the subsampling factor, the wider the perfect re-construction zone.NUMERICAL RESULTSIn the seismic context, the effect of coarse sampling inthe f-k domain is illustrated in Fig. 3. Figs. 3(a) and3(d) show densely-sampled data and the correspondingamplitude spectrum. Figs. 3(b) and 3(e) show regularlysub-sampled data and the corresponding amplitude spec-trum. Finally, Figs. 3(c) and 3(f) show randomly sub-sampled data and the corresponding amplitude spectrum.Note how random sampling creates incoherent noise acrossthe spectrum.Although Fourier does not provide the sparsest repre-sentation for seismic data, there exists successful inter-polation algorithms that solve Eq. 3 with S :=F (see e.g.Zwartjes and Hindriks, 2001; Xu et al., 2005). We usethe algorithm called curvelet reconstruction with sparsity-(a) (b) (c)(d) (e) (f)Figure 3: Seismic data and their corresponding spectrum. Densely-sampled data (a) and corresponding amplitudespectrum (d). Data regularly sampled below Nyquist rate (b) and corresponding amplitude spectrum (e) with strongaliasing beyond 25 Hz. Data randomly sampled at the same sub-Nyquist rate as (b) and corresponding amplitudespectrum (f) corrupted by broadband noise.promoting inversion (Herrmann, 2005; Hennenfent andHerrmann, 2005, 2006; Herrmann and Hennenfent, 2007)since curvelets provide a sparser representation for seis-mic data than Fourier (see e.g. Candes et al., 2006; Hen-nenfent and Herrmannn, 2006). In this case, S is de-fined as the curvelet transform (Candes et al., 2006, andreferences therein). The incoherent noise generated byrandom sampling remains incoherent in the curvelet do-main since curvelets are strictly localized in the f-k do-main. Fig. 4(a) and 4(b) show the interpolation resultsfor the data of Figs. 3(b) and 3(c), respectively. Thesignal-to-reconstruction-error ratios are 6.92 dB for reg-ular sub-sampling and 13.78 dB for random sub-sampling.For the same number of receivers, coarse random sam-pling leads to a much better reconstruction than coarseregular sampling. When a minimum velocity constraintis imposed during the reconstruction process, the sameconclusion holds although the difference is reduced.CONCLUSIONSWe proposed to look at the seismic data interpolationproblem from a denoising perspective. From this stand-point, we showed that, for the same amount of data col-lected, regular subsampling geometries generate coher-ent acquisition noise more difficult to remove than theincoherent noise created by random subsampling geome-tries. Hence, random subsampling leads to a more accu-rate reconstruction of the seismic wavefield than equiv-alent regular subsampling or any subsampling that gen-erates structured acquisition noise. We believe this newinsight may lead to new acquisition strategies. On land,for example, a regular sampling may lead to (severely)aliased ground-roll that needs to be interpolated to a finergrid in order to be removed. Our observations suggestone should randomly sample on the finer grid instead.This leads to a better interpolation and hence ground-roll removal.ACKNOWLEDGMENTSThis work was in part financially supported by NSERCDiscovery Grant 22R81254 and CRD Grant DNOISE334810-05 of F.J. Herrmann and was carried out as partof the SINBAD project with support, secured throughITF, from the following organizations: BG Group, BP,Chevron, ExxonMobil and Shell.(a) (b)(c) (d)Figure 4: Synthetic seismic data reconstruction using 2-D curvelet reconstruction with sparsity-promoting inversion.Interpolation result ? SNR = 6.9 dB ? (a) and corresponding amplitude spectrum (c) given data of Fig. 3(b). Inter-polation result ? SNR = 13.78 dB ? (b) and corresponding amplitude spectrum given data of Fig. 3(c). For the samenumber of receivers, coarse random sampling leads to a much better reconstruction than coarse regular sampling.REFERENCESCandes, E., L. Demanet, D. Donoho, and L. Ying, 2006, Fast discrete curvelet transforms: Multiscale Modeling andSimulation, 5, 861?899.Candes, E., J. Romberg, and T. Tao, 2005, Stable signal recovery from incomplete and inaccurate measurements:Communications on Pure and Applied Mathematics, 59, 1207?1223.Dippe, M. and E. Wold, 1985, Antialiasing through stochastic sampling: Presented at the SIGGRAPH?85.Donoho, D., 2006, Compressed sensing: IEEE Transactions on Information Theory, 52, 1289?1306.Donoho, D., Y. Tsaig, I. Drori, and J.-L. Starck, 2007, Sparse solution for underdetermined linear equations by stage-wise orthogonal matching pursuit. (submitted).Hennenfent, G. and F. Herrmann, 2005, Sparseness-constrained data continuation with frames: Applications to miss-ing traces and aliased signals in 2/3-D: Presented at the SEG International Exposition and 75th Annual Meeting.???, 2006, Application of stable signal recovery to seismic interpolation: Presented at the SEG International Expo-sition and 76th Annual Meeting.Hennenfent, G. and F. Herrmannn, 2006, Seismic denoising with non-uniformly sampled curvelets: Computing inScience and Engineering, 8.Herrmann, F., 2005, Robust curvelet-domain data continuation with sparseness constraints: Presented at the 67rdEAGE Annual Conference and Exhibition.Herrmann, F. and G. Hennenfent, 2007, Non-parametric seismic data recovery with curvelet frames. (submitted).Leneman, O., 1966, Random sampling of random processes: Impulse response: Information and Control, 9, 347 ?363.Xu, S., Y. Zhang, and G. Lambare, 2005, Antileakage fourier transform for seismic data regularization: Geophysics,70, V87?V95.Zwartjes, P. and C. Hindriks, 2001, Regularizing 3D data using Fourier reconstruction and sparse inversion: Presentedat the 63rd EAGE Annual Conference and Exhibition.
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Random sampling: new insights into the reconstruction of coarsely-sampled wavefields Hennenfent, Gilles; Herrmann, Felix J. 2007-03-10
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Title | Random sampling: new insights into the reconstruction of coarsely-sampled wavefields |
Creator |
Hennenfent, Gilles Herrmann, Felix J. |
Contributor | University of British Columbia. Seismic Laboratory for Imaging and Modeling |
Publisher | Society of Exploration Geophysicists |
Date Issued | 2007 |
Description | In this paper, we turn the interpolation problem of coarsely-sampled data into a denoising problem. From this point of view, we illustrate the benefit of random sampling at sub-Nyquist rate over regular sampling at the same rate. We show that, using nonlinear sparsity promoting optimization, coarse random sampling may actually lead to significantly better wavefield reconstruction than equivalent regularly sampled data. |
Extent | 485921 bytes |
Subject |
seismic denoising Nyquist random sampling interpolation Fourier |
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Conference Paper |
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Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-03-10 |
Provider | Vancouver : University of British Columbia Library |
DOI | 10.14288/1.0107409 |
URI | http://hdl.handle.net/2429/556 |
Affiliation |
Science, Faculty of Earth and Ocean Sciences, Department of |
Citation | Hennenfent, Gilles, Herrmann, Felix J. 2007. Random sampling: new insights into the reconstruction of coarsely-sampled wavefields. SEG International Exposition and 77th Annual Meeting. |
Peer Review Status | Unreviewed |
Scholarly Level | Graduate Faculty |
Copyright Holder | Herrmann, Felix J. |
Aggregated Source Repository | DSpace |
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