Efficient matrix completion for seismic data reconstruction

Abstract

Despite recent developments in improved acquisition, seismic data often remains undersampled along source and receiver coordinates, resulting in incomplete data for key applications such as migration and multiple prediction. We interpret the missing-trace interpolation problem in the context of matrix completion and outline three practical principles for using low-rank optimization techniques to recover seismic data. Specifically, we strive for recovery scenarios wherein the original signal is low rank and the subsampling scheme increases the singular values of the matrix. We employ an optimization program that restores this low rank structure to recover the full volume. Omitting one or more of these principles can lead to poor interpolation results, as we show experimentally. In light of this theory, we compensate for the high-rank behavior of data in the source-receiver domain by employing the midpoint-offset transformation for 2D data and a source-receiver permutation for 3D data to reduce the overall singular values. Simultaneously, in order to work with computationally feasible algorithms for large scale data, we use a factorization-based approach to matrix completion, which significantly speeds up the computations compared to repeated singular value decompositions without reducing the recovery quality. In the context of our theory and experiments, we also show that windowing the data too aggressively can have adverse effects on the recovery quality. To overcome this problem, we carry out our interpolations for each frequency independently while working with the entire frequency slice. The result is a computationally efficient, theoretically motivated framework for interpolating missing-trace data. Our tests on realistic two-and three-dimensional seismic data sets show that our method compares favorably, both in terms of computational speed and recovery quality, to existing curvelet-based and tensor-based techniques.

Beating Level-Set Methods for 5-D Seismic Data Interpolation: A Primal-Dual Alternating Approach

“…In this context, we can expect low-rank penalization to help. We use the rank-revealing transforms of [1], [9], [19]. To motivate the use of these transforms, we first explore the singular value decay of full and subsampled volumes in the context of 3D seismic data acquisition.…”

Relaxation algorithms for matrix completion, with applications to seismic travel-time data interpolation

“…Hence, researchers often use prior information to denoise and interpolate the data prior to inversion. Parsimonious representations [6,23] of the data in transform domains such as Fourier [24] and Curvelet [9] have been used in exploration seismology [8,13], along with low-rank representations [1,7].…”

A joint matrix minimization approach for seismic wavefield recovery

“…To avoid the expensive computations in solving the involved matrix completion optimization problems, a matrix factorization strategy was developed in 31,32 . This paper proposes a different matrix minimization approach based on l 2, q − l 2, p norm which naturally generalizes the representative vector to matrix in joint distribution sense.…”