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Random sampling: new insights into the reconstruction of coarsely-sampled wavefields Hennenfent, Gilles 2008

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Random sampling: new insights into the reconstruction of coarsely-sampled wavefields Gilles Hennenfent, EOS-UBC and Felix J. Herrmann, EOS-UBC SUMMARY 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 reconstruc- tion than equivalent regularly sampled data. INTRODUCTION Dense sampling of seismic data is traditionally under- stood as evenly-distributed time and space measurements of the reflected wavefield. Moreover, the sampling rate along 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). In practice, however, seismic data is often randomly and/or sparsely sampled along spatial coordinates, which is gen- erally considered as a challenge since it breaks one or both previously-stated conditions of dense sampling. It turns out that these acquisition geometries are not nec- essarily a source of adversity to accurately reconstruct densely-sampled data when using nonlinear optimiza- tion promoting sparsity. This new insight, developed in the information theory community, is referred to in the literature by the terms “compressed sensing” or “com- pressive sampling” (see e.g. Donoho, 2006; Candes et al., 2005, and references therein). THEORY Sampling below Nyquist rate Discretizing a continuous function bandlimited to the frequency interval [−B,B] corresponds to a multiplica- tion with a Dirac comb in the time domain and thus to a convolution with another Dirac comb in the fre- quency domain. As a consequence, the original spec- trum becomes periodic after discretization. Problems occur when regularly sampling below Nyquist rate, i.e. fs < 2B. Replicas of the original spectrum overlap, which creates 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 is less likely to create strong aliases but rather weak broad- band noise. Consider for example a random sampling operator s over [0,N] where N is the size of the sampling region. Suppose that s samples n < N points uniformly distributed in the interval. Then, the expectation of the power spectrum of s over [0,N] is given by (Leneman, 1966; Dippe and Wold, 1985) E [|ŝN(u)|2]= n2δu0 +(1−δu0)n, (1) where the symbolˆdenotes Fourier coefficients, E [·] the mathematical expectation, and u is the frequency vari- able. The first term in this expression is only nonzero at the origin and gives through convolution a scaled ver- sion of the original spectrum. The second term is only zero at the origin and can be assimilated to broadband noise. Fig. 1 summarizes these observations. Fig. 1(a) shows the 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 signal being regularly- and randomly-sampled below Nyquist rate, respectively. Reconstruction by denoising Consider the following linear forward model for the in- terpolation problem y = Rf0, (2) where y∈Rn represents the data acquired at sub-Nyquist rate, f0 ∈ RN the densely-sampled data to be recovered, and R ∈ Rn×N the restriction operator that selects the acquired samples among the desired samples. Assume f0 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 a dense grid becomes a denoising problem in the S do- main (Donoho et al., 2007). This problem is solved by the following nonlinear sparsity-promoting optimization (Candes et al., 2005; Hennenfent and Herrmann, 2006) x̃ = argmin x ||x||1 s.t. y = RSHx, (3) where the symbol˜represents an estimated quantity, and H the conjugate tranpose. The interpolated result is given by ˜f = SH x̃. Note that, if the sampling at sub-Nyquist rate creates coherent noise in the S domain (see e.g. Fig. 1(b) when S := F), the separation between signal and noise generated by the acquisition in the S domain is much more delicate (if not impossible) just by impos- ing a sparsity constraint. We illustrate this comment by a 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 the same factors as in (a) (a) (b) (c) Figure 1: Spectra of a signal sampled above and below Nyquist rate. The signal consists of the superposition of three cosine functions. Amplitude spectrum of the densely-sampled signal (a), coarse regularly-sampled signal (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, we solve Eq. 3 and compare the estimated representation x̃ of f0 to its true representation x0. The reconstruction er- ror is measured as the number of false detections in the discrete cosine transform domain. The results presented in 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 curve represents the results obtained for a given subsampling factor. Fig. 2(a) shows that, regardless of the subsam- pling factor and the sparsity of f0 in the discrete cosine transform domain, the solution of Eq. 3 is corrupted by some noise. However, Fig. 2(b) shows that there is a sparsity zone for each subsampling factor such that the reconstruction is perfect, i.e. no false detection. The smaller the subsampling factor, the wider the perfect re- construction zone. NUMERICAL RESULTS In the seismic context, the effect of coarse sampling in the f -k domain is illustrated in Fig. 3. Figs. 3(a) and 3(d) show densely-sampled data and the corresponding amplitude spectrum. Figs. 3(b) and 3(e) show regularly sub-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 across the 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 use the 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 amplitude spectrum (d). Data regularly sampled below Nyquist rate (b) and corresponding amplitude spectrum (e) with strong aliasing beyond 25 Hz. Data randomly sampled at the same sub-Nyquist rate as (b) and corresponding amplitude spectrum (f) corrupted by broadband noise. promoting inversion (Herrmann, 2005; Hennenfent and Herrmann, 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, and references therein). The incoherent noise generated by random 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 results for the data of Figs. 3(b) and 3(c), respectively. The signal-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 coarse regular sampling. When a minimum velocity constraint is imposed during the reconstruction process, the same conclusion holds although the difference is reduced. CONCLUSIONS We proposed to look at the seismic data interpolation problem 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 the incoherent 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 new insight 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 finer grid in order to be removed. Our observations suggest one should randomly sample on the finer grid instead. This leads to a better interpolation and hence ground- roll removal. ACKNOWLEDGMENTS This work was in part financially supported by NSERC Discovery Grant 22R81254 and CRD Grant DNOISE 334810-05 of F.J. Herrmann and was carried out as part of the SINBAD project with support, secured through ITF, 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 same number of receivers, coarse random sampling leads to a much better reconstruction than coarse regular sampling. REFERENCES Candes, E., L. Demanet, D. Donoho, and L. Ying, 2006, Fast discrete curvelet transforms: Multiscale Modeling and Simulation, 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 in Science and Engineering, 8. Herrmann, F., 2005, Robust curvelet-domain data continuation with sparseness constraints: Presented at the 67rd EAGE 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: Presented at the 63rd EAGE Annual Conference and Exhibition.


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