A fast reduced-rank interpolation method for prestack seismic volumes that depend on four spatial dimensions

Abstract

Rank reduction strategies can be employed to attenuate noise and for prestack seismic data regularization. We present a fast version of Cadzow reduced-rank reconstruction method. Cadzow reconstruction is implemented by embedding 4D spatial data into a level-four block Toeplitz matrix. Rank reduction of this matrix via the Lanczos bidiagonalization algorithm is used to recover missing observations and to attenuate random noise. The computational cost of the Lanczos bidiagonalization is dominated by the cost of multiplying a level-four block Toeplitz matrix by a vector. This is efficiently implemented via the 4D fast Fourier transform. The proposed algorithm significantly decreases the computational cost of rank-reduction methods for multidimensional seismic data denoising and reconstruction. Synthetic and field prestack data examples are used to examine the effectiveness of the proposed method.

A fast SVD for multilevel block Hankel matrices with minimal memory storage

“…Alternatively, fast MBH matrix-vector multiplication can be achieved by first transforming an MBH matrix into a Multilevel Block Circulant (MBC) matrix then applying multi-dimensional FFT [6,13]. Zero entries are introduced when transforming an MBH matrix into an MBC matrix, resulting in more storage requirement.…”

“…A multidimensional seismic data array is first transformed into a two-dimensional MBH matrix, then the 1-D FFT [1] is applied to perform a fast MBH matrix-vector multiplication. In contrast to other fast MBH matrix-vector multiplication methods employing multidimensional FFTs in [6,13], our algorithm only requires 1-D FFT and minimal memory storage. Therefore, the main contribution of this article is our fast MBH matrix-vector multiplication with minimal memory storage requirement and its application to fast MBH SVD.…”

“…In 2013, Gao et al [13] developed a rank reduction denoising and reconstruction scheme that is used to reconstruct prestack data that depend on four spatial dimensions by forming a Level-4 Block Toeplitz matrix. This is often called 5D interpolation because reconstruction algorithms operate on volumes that depend on four spatial dimensions and time or frequency.…”

Dreamlet-based interpolation using POCS method

“…Methods for interpolation can be divided into three categories (Gao et al, 2013): First, methods based on mathematical transform and signal analysis Naghizadeh and Innanen, 2011;Naghizadeh and Sacchi, 2010;Trad et al, 2002;Wang et al, 2010); second, prediction filter based methods (Naghizadeh and Sacchi, 2007;Spitz, 1991); third, interpolation methods based on wave equation (Ronen, 1987). In the past several years, rank-reduction method (Gao et al, 2013;Kreimer et al, 2013;Ma, 2013) has also played an important role in seismic data interpolation.…”

5D seismic data completion and denoising using a novel class of tensor decompositions

“…Iterative thresholding and imposing sparsity on the singular values of the matrix enforces the reduced rank of the data, requiring numerous costly singular value decompositions (SVDs). Other variations enforce the low rank through less expensive matrix decomposition, completely avoiding the calculation of the SVD (Gao et al, 2011(Gao et al, , 2013Kumar et al, 2013) or applying the SVD to a randomized subset of the original matrix Sacchi, 2010, 2011). In contrast to the aforementioned methods, other methods use a priori velocity information and the wave equation to interpolate missing traces (Ronen, 1987;Stolt, 2002;Fomel, 2003;Kaplan et al, 2010), thereby incorporating actual physical information and principles governing the system.…”