Simply denoise: Wavefield reconstruction via jittered undersampling

We present a new, discrete undersampling scheme designed to favor wavefield reconstruction by sparsity-promoting inversion with transform elements localized in the Fourier domain. The work is motivated by empirical observations in the seismic community, corroborated by results from compressive sampling, that indicate favorable (wavefield) reconstructions from random rather than regular undersampling. Indeed, random undersampling renders coherent aliases into harmless incoherent random noise, effectively turning the interpolation problem into a much simpler denoising problem. A practical requirement of wavefield reconstruction with localized sparsifying transforms is the control on the maximum gap size. Unfortunately, random undersampling does not provide such a control. Thus, we introduce a sampling scheme, termed jittered undersampling, that shares the benefits of random sampling and controls the maximum gap size. The contribution of jittered sub-Nyquist sampling is key in formu-lating a versatile wavefi...

[1]  Oscar A. Z. Leneman,et al.  Random Sampling of Random Processes: Impulse Processes , 1966, Inf. Control..

[2]  J. Claerbout Toward a unified theory of reflector mapping , 1971 .

[3]  A. Gersztenkorn,et al.  Robust iterative inversion for the one‐dimensional acoustic wave equation , 1986 .

[4]  S. Spitz Seismic trace interpolation in the F-X domain , 1991 .

[5]  D. J. Verschuur,et al.  Adaptive surface-related multiple elimination , 1992 .

[6]  Mark A. Z. Dippé,et al.  Stochastic sampling: theory and application , 1992 .

[7]  Quasi-random Migration of 3-D Field Data , 1995 .

[8]  J. Bee Bednar Coarse is coarse of course unless , 1996 .

[9]  Anat Canning,et al.  Regularizing 3-D data sets with DMO , 1996 .

[10]  R. Tibshirani Regression Shrinkage and Selection via the Lasso , 1996 .

[11]  Gerard T. Schuster,et al.  A quasi-Monte Carlo approach to 3-D migration: Theory , 1997 .

[12]  Anat Canning,et al.  Reducing 3-D acquisition footprint for 3-D DMO and 3-D prestack migration , 1998 .

[13]  Nizar Chemingui,et al.  Azimuth moveout for 3-D prestack imaging , 1998 .

[14]  Mauricio D. Sacchi,et al.  Interpolation and extrapolation using a high-resolution discrete Fourier transform , 1998, IEEE Trans. Signal Process..

[15]  Steven G. Johnson,et al.  FFTW: an adaptive software architecture for the FFT , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).

[16]  Bernard Fino,et al.  Multiuser detection: , 1999, Ann. des Télécommunications.

[17]  Tadeusz J. Ulrych,et al.  Radon Transform: Beyond Aliasing With Irregular Sampling , 1999 .

[18]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[19]  Yanghua Wang Seismic trace interpolation in the f‐x‐y domain , 2002 .

[20]  Robert H. Stolt,et al.  Seismic data mapping and reconstruction , 2002 .

[21]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[22]  Mauricio D. Sacchi,et al.  Latest views of the sparse Radon transform , 2003 .

[23]  F. Herrmann,et al.  Sparseness-constrained data continuation with frames: Applications to missing traces and aliased signals in 2/3-D , 2005 .

[24]  Daniel Trad,et al.  Understanding Land Data Interpolation , 2005 .

[25]  Yu Zhang,et al.  Antileakage Fourier transform for seismic data regularization , 2005 .

[26]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[27]  Felix J. Herrmann,et al.  Seismic denoising with nonuniformly sampled curvelets , 2006, Computing in Science & Engineering.

[28]  Laurent Demanet,et al.  Fast Discrete Curvelet Transforms , 2006, Multiscale Model. Simul..

[29]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[30]  Emmanuel J. Candès,et al.  Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions , 2004, Found. Comput. Math..

[31]  R. Abma,et al.  3D interpolation of irregular data with a POCS algorithm , 2006 .

[32]  David L. Donoho,et al.  Sparse Solution Of Underdetermined Linear Equations By Stagewise Orthogonal Matching Pursuit , 2006 .

[33]  Mauricio D. Sacchi,et al.  High-resolution wave-equation amplitude-variation-with-ray-parameter (AVP) imaging with sparseness constraints , 2007 .

[34]  E. Berg,et al.  In Pursuit of a Root , 2007 .

[35]  Mário A. T. Figueiredo,et al.  Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems , 2007, IEEE Journal of Selected Topics in Signal Processing.

[36]  Mauricio D. Sacchi,et al.  Fourier Reconstruction of Nonuniformly Sampled, Aliased Seismic Data , 2022 .

[37]  D. Donoho,et al.  Sparse MRI: The application of compressed sensing for rapid MR imaging , 2007, Magnetic resonance in medicine.

[38]  Felix J. Herrmann,et al.  Curvelet-based seismic data processing : A multiscale and nonlinear approach , 2008 .

[39]  Felix J. Herrmann,et al.  Non-parametric seismic data recovery with curvelet frames , 2008 .