Primary estimation with sparsity-promoting bi-convex optimization

This thesis establishes a novel inversion methodology for the surface-related primaries from a given recorded seismic wavefield, called the Robust Estimation of Primaries by Sparse Inversion (Robust EPSI, or REPSI). Surface-related multiples are a major source of coherent noise in seismic data, and inferring fine geological structures from active-source seismic recordings typically first necessitates its removal or mitigation. For this task, current practice calls for data-driven approaches which produce only approximate multiple models that must be non-linearly subtracted from the data, often distorting weak primary events in the process. A recently proposed method called Estimation of Primaries by Sparse Inversion (EPSI) avoids this adaptive subtraction by directly inverting for a discrete representation of the underlying multiple-free subsurface impulse response as a set of band-limited spikes. However, in its original form, the EPSI algorithm exhibits a few notable shortcomings that impede adoption. Although it was shown that the correct impulse response can be obtained through a sparsest solution criteria, the current EPSI algorithm is not designed to take advantage of this finding, but instead approximates a sparse solution in an ad-hoc manner that requires practitioners to decide on a multitude of inversion parameters. The Robust EPSI method introduced in this thesis reformulates the original EPSI problem as a formal bi-convex optimization problem that makes obtaining the sparsest solution an explicit goal, while also reliably admit satisfactory solutions using contemporary self-tuning gradient methods commonly seen in large-scale machine learning communities. I show that the Robust EPSI algorithm is able to operate successfully on a variety of datasets with minimal user input, while also producing a more accurate model of the subsurface impulse response when compared to the original algorithm. Furthermore, this thesis makes several contributions that improves the capability and practicality of EPSI: a novel scattering-based multiple prediction model that allows Robust EPSI to deal with wider near-offset receiver gaps than previously demonstrated for EPSI, as well as a multigrid-inspired continuation strategy that significantly reduces the computation time needed to solve EPSI-type problems. These additions are enabled by and built upon the formalism of the Robust EPSI as developed in this thesis.