Parabolic dictionary learning for seismic wavefield reconstruction across the streamers

Abstract

Dictionary learning (DL) methods are effective tools to automatically find a sparse representation of a data set. They train a set of basis vectors on the data to capture the morphology of the redundant signals. The basis vectors are called atoms, and the set is referred to as the dictionary. This dictionary can be used to represent the data in a sparse manner with a linear combination of a few of its atoms. In conventional DL, the atoms are unstructured and are only numerically defined over a grid that has the same sampling as the data. Consequently, the atoms are unknown away from this sampling grid, and a sparse representation of the data in the dictionary domain is not sufficient information to interpolate the data. To overcome this limitation, we have developed a DL method called parabolic DL, in which each learned atom is constrained to represent an elementary waveform that has a constant amplitude along a parabolic traveltime moveout. The parabolic structure is consistent with the physics inherent to the seismic wavefield and can be used to easily interpolate or extrapolate the atoms. Hence, we have developed a parabolic DL-based process to interpolate and regularize seismic data. Briefly, it consists of learning a parabolic dictionary from the data, finding a sparse representation of the data in the dictionary domain, interpolating the dictionary atoms over the desired grid, and, finally, taking the sparse representation of the data in the interpolated dictionary domain. We examine three characteristics of this method, i.e., the parabolic structure, the sparsity promotion, and the adaptation to the data, and we conclude that they strengthen robustness to noise and to aliasing and that they increase the accuracy of the interpolation. For both synthetic and field data sets, we have successful seismic wavefield reconstructions across the streamers for typical 3D acquisition geometries.