Random noise attenuation via the randomized canonical polyadic decomposition

Abstract

A B S T R A C TTensor algebra provides a robust framework for multi-dimensional seismic data processing. A low-rank tensor can represent a noise-free seismic data volume. Additive random noise will increase the rank of the tensor. Hence, tensor rank-reduction techniques can be used to filter random noise. Our filtering method adopts the Candecomp/Parafac decomposition to approximates a N-dimensional seismic data volume via the superposition of rank-one tensors. Similar to the singular value decomposition for matrices, a low-rank Candecomp/Parafac decomposition can capture the signal and exclude random noise in situations where a low-rank tensor can represent the ideal noise-free seismic volume. The alternating least squares method is adopted to compute the Candecomp/Parafac decomposition with a provided target rank. This method involves solving a series of highly over-determined linear least-squares subproblems. To improve the efficiency of the alternating least squares algorithm, we uniformly randomly sample equations of the linear least-squares subproblems to reduce the size of the problem significantly. The computational overhead is further reduced by avoiding unfolding and folding large dense tensors. We investigate the applicability of the randomized Candecomp/Parafac decomposition for incoherent noise attenuation via experiments conducted on a synthetic dataset and field data seismic volumes. We also compare the proposed algorithm (randomized Candecomp/Parafac decomposition) against multi-dimensional singular spectrum analysis and classical f − xy prediction filtering. We conclude the proposed approach can achieve slightly better denoising performance in terms of signal-to-noise ratio enhancement than traditional methods, but with a less computational cost.

Expand Dimensional of Seismic Data and Random Noise Attenuation Using Low-Rank Estimation

“…The w opt i represents the thresholding and shrinkage operator on the singular values of χ. It should be noted that the optimization problem in (7) seems to be unobservable since it depends on an unknown matrix desired to be estimated; however, the optimal solution itself is computable. No structure was assumed to find the optimal solution except the low-rank characteristic in the signal matrix.…”

“…However, it is known that the exploitable structure is present in the noise portion of the eigen-spectrum, i.e., the min(m, n) − r singular values of χ. The advantage of the proposed algorithm over the optimization problem in (7) is that the presented algorithm uses an estimate of the rank of the signal matrix as input data and will return the (re)weighted approximation, which efficiently mitigates the effect of rank overestimation. When σ 1 > σ 2 > .…”

“…For instance, the more geological complex media, the less linearity behavior in the random noise. In addition, as the nature of seismic random noise, they are generally considered as stationary series in short length windows, which imposes application of adaptive and amplitude preserving methods [7]. By revealing the importance of signal preservation by filtering seismic data, conventional noise removal techniques are known to be less effective and impressive for complete preservation and/or recovery of the seismic signal.…”

A deep learning approach to separating scattered direct waves from reflection seismic records on rugged surface: Synthetic data examples

“…Rank reduction methods assume that the signal can be represented by a low‐rank matrix, whereas the noise can increase rank. The Cadzow filtering (Burroughs & Trickett, 2009; Cadzow, 1988), multichannel singular spectrum analysis (Oropeza & Sacchi, 2011) and canonical polyadic decomposition (Gao & Sacchi, 2020) are applied to reduce the rank of noisy data and retrieve the signal with low rank. For coherent noise suppression, such as DWs, surface waves and refracted waves, F–K filtering (Yilmaz, 2001) and τ–p transform (Russell et al., 1990) utilize the difference in apparent velocities between reflections and coherent noise.…”

“…These factors lead to either the leakage of signal energy or noise residuals (Almuhaidib & Toksöz, 2016). The rank reduction methods assume that the matrix of the seismic data with low‐rank represents noise‐free data, but the measurement of low rank needs subjective decisions (Gao & Sacchi, 2020); in addition, these algorithms require expensive computation (Gemechu et al., 2018). Therefore, so far it is difficult to build an accurate model to describe signal or noise for the suppressing of SDWs.…”