Efficient least-squares imaging with sparsity promotion and compressive sensing

Seismic imaging is a linearized inversion problem relying on the minimization of a least-squares misfit functional as a function of the medium perturbation. The success of this procedure hinges on our ability to handle large systems of equations–-whose size grows exponentially with the demand for higher resolution images in more and more complicated areas–-and our ability to invert these systems given a limited amount of computational resources. To overcome this "curse of dimensionality" in problem size and computational complexity, we propose a combination of randomized dimensionality-reduction and divide-and-conquer techniques. This approach allows us to take advantage of sophisticated sparsity-promoting solvers that work on a series of smaller subproblems each involving a small randomized subset of data. These subsets correspond to artificial simultaneous-source experiments made of random superpositions of sequential-source experiments. By changing these subsets after each subproblem is solved, we are able to attain an inversion quality that is competitive while requiring fewer computational, and possibly, fewer acquisition resources. Application of this concept to a controlled series of experiments showed the validity of our approach and the relationship between its efficiency–-by reducing the number of sources and hence the number of wave-equation solves–-and the image quality. Application of our dimensionality-reduction methodology with sparsity promotion to a complicated synthetic with well-log constrained structure also yields excellent results underlining the importance of sparsity promotion.