Seismic data interpolation using sparsity constrained inversion

Abstract

Missing data reconstruction is an ongoing challenge in seismic processing for incomplete and irregular acquisition. The problem of missing data negatively affects several important processing steps such as migration. While many methods have been developed to address this problem, most of the recent research on the subject focuses on transform domain approaches due to their low computational cost and independence of medium property estimation, such as velocity and density. Because of the underdetermined nature of the interpolation problem, transform domain techniques use sparsity constraints in order to obtain better solutions than the conventional least squares approach. Motivated by the compressive sensing framework, which proves that sparse signals can be recovered from a highly incomplete set of measurements, two approaches for sparsity constrained inversion are commonly used: the synthesis and analysis approaches. Although similarly posed, they present differences when the transform domain used in the inversion is redundant. Due to the high dimensionality and aliasing that occurs in the interpolation problem, such approaches are not able to handle large gaps between traces and display artifacts in the reconstruction. Weighted L1 minimization algorithms have been proposed as a way to mitigate such problems. In this paper, we show the problems associated with the synthesis and analysis approaches when applied to seismic interpolation and propose an approach based on a reweighted L1 algorithm that estimates weights recursively, i.e., from previous weighted steps. Experiments, carried out on 2D and 3D data using different undersampling schemes, show that the proposed approach is able to improve estimation inside large gaps, decrease artifacts and obtain better SNR when compared with conventional analysis and synthesis sparsity constrained inversion.