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

We have developed a novel strategy for simultaneous interpolation and denoising of prestack seismic data. Most seismic surveys fail to cover all possible source-receiver combinations, leading to missing data especially in the midpoint-offset domain. This undersampling can complicate certain data processing steps such as amplitude-variation-with-offset analysis and migration. Data interpolation can mitigate the impact of missing traces. We considered the prestack data as a 5D multidimensional array or otherwise referred to as a 5D tensor. Using synthetic data sets, we first found that prestack data can be well approximated by a low-rank tensor under a recently proposed framework for tensor singular value decomposition (tSVD). Under this low-rank assumption, we proposed a complexity-penalized algorithm for the recovery of missing traces and data denoising. In this algorithm, the complexity regularization was controlled by tuning a single regularization parameter using a statistical test. We tested the performance of the proposed algorithm on synthetic and real data to show that missing data can be reliably recovered under heavy downsampling. In addition, we demonstrated that compressibility, i.e., approximation of the data by a low-rank tensor, of seismic data under tSVD depended on the velocity model complexity and shot and receiver spacing. We further found that compressibility correlated with the recovery of missing data because high compressibility implied good recovery and vice versa.National Science Foundation (U.S.). Graduate Research Fellowship (Grant DGE-0806676)National Science Foundation (U.S.). Division of Computing and Communication Foundations (Grant NSF-1319653

Optimization Methods for Inverse Problems

Optimization plays an important role in solving many inverse problems. Indeed, the task of inversion often either involves or is fully cast as a solution of an optimization problem. In this light, the mere non-linear, non-convex, and large-scale nature of many of these inversions gives rise to some very challenging optimization problems. The inverse problem community has long been developing various techniques for solving such optimization tasks. However, other, seemingly disjoint communities, such as that of machine learning, have developed, almost in parallel, interesting alternative methods which might have stayed under the radar of the inverse problem community. In this survey, we aim to change that. In doing so, we first discuss current state-of-the-art optimization methods widely used in inverse problems. We then survey recent related advances in addressing similar challenges in problems faced by the machine learning community, and discuss their potential advantages for solving inverse problems. By highlighting the similarities among the optimization challenges faced by the inverse problem and the machine learning communities, we hope that this survey can serve as a bridge in bringing together these two communities and encourage cross fertilization of ideas.Comment: 13 page