A tensor higher-order singular value decomposition for prestack seismic data noise reduction and interpolation

Abstract

A patch of prestack data depends on four spatial dimensions ([Formula: see text], [Formula: see text] midpoints and [Formula: see text], [Formula: see text] offsets) and frequency. The spatial data at one temporal frequency can be represented by a fourth-order tensor. In ideal conditions of high signal-to-noise ratio and complete sampling, one can assume that the seismic data can be approximated via a low-rank fourth-order tensor. Missing samples were recovered by reinserting data obtained by approximating the original noisy and incomplete data volume with new observations obtained via the rank-reduction process. The higher-order singular value decompostion was used to reduce the rank of the prestack seismic tensor. Synthetic data demonstrated the ability of the proposed seismic data completion algorithm to reconstruct events with curvature. The synthetic example allowed to quantify the quality of the reconstruction for different levels of noise and survey sparsity. We also provided a real data example from the Western Canadian sedimentary basin.

Nuclear norm minimization and tensor completion in exploration seismology

“…The nuclear norm of a tensor will be defined as the sum of the nuclear norms of its unfoldings. Four unfoldings exist for a fourth-order tensor and the "fold" operation consists of a re-ordering of its elements into a matricial form [13]. The operations of unfolding and folding require careful manipulation of the indices of the tensor and consist of mapping a tensor to a matrix and vice versa.…”

“…sparsity) [6,5,7], (b) Prediction filtering methods that use the predictability of the signal in the frequency-space (F-X) or time-space (T-X) domain [8,9] and (c) rank-reduction methods that utilize the low-rank nature of seismic data [10,11,12,13]. The fully sampled 4D spatial volume has a natural representation via a low-rank tensor structure [13].…”

An open-source Matlab code package for improved rank-reduction 3D seismic data denoising and reconstruction

“…In this case, the multichannel seismic data is viewed as a multi-linear array, and dimensionality reduction techniques are directly applied to the multi-linear array (Gao et al, 2015). For example, Kreimer and Sacchi (2012) adopt the high-order singular value decomposition (HOSVD) to solve the 5D seismic data reconstruction problem in the frequency-space domain.…”

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

“…All of these decompositions have been applied to the problem of 5D seismic data reconstruction with good performance on real and synthetic data. Seismic data completion using tensor rank minimization under HOSVD is proposed by Kreimer and Sacchi (2012b) and Kreimer et al (2013). Although the performance was shown to be good, the proposed algorithm requires preselection of the truncation ranks along each dimension for effective interpolation.…”

“…Some of the current approaches, such as in Trad (2009) and Kreimer and Sacchi (2012b), work the f-x domain in which the data completion is carried out separately over the 4D slices corresponding to each frequency.…”