Efficient least‐squares imaging with sparsity promotion and compressive sensing

Abstract

A B S T R A C TSeismic 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 equationswhose 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 shows 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 a well-log constrained structure also yields excellent results underlining the importance of sparsity promotion.

Least-squares reverse time migration with radon preconditioning

“…For reducing the computational cost, phase-encoded migration (Morton and Ober, 1998;Romero et al, 2000) was proposed that was later extended to multisource LSRTM by Dai et al (2010) and several other authors. A similar approach was proposed by Herrmann and Li (2012) as they used a combination of randomized dimensionality-reduction and divide-andconquer-techniques to decimate the LSM problem as a series of smaller sub-problems where each sub-problem involved iterating on a small randomized subset of the data. However such approaches typically introduce crosstalk noise which is only slowly attenuated by additional iterations.…”

Approximate message passing meets exploration seismology

“…1). While this idea has been used successfully in situations where data is abundant, e.g., in seismic imaging where different independent randomized subsets of fully-sampled data volumes are drawn to speed up convergence [2], this solution is unworkable in data-scarse situations, such as during the recovery from incomplete field data.…”

“…By working on smaller randomized subproblems, while exploiting structure within signals, challenges related to the socalled 'curse of dimensionality' can be addressed in the field of exploration seismology [2,3]. This curse leads to exponential growth in data-collection and processing costs as the survey area and desired resolution increase.…”

A modified, sparsity-promoting, Gauss-Newton algorithm for seismic waveform inversion

“…al. Felix J. Herrmann (2011);Herrmann and Li. (2011) successfully used this approach to make sparsity-promoting seismic imaging more efficient.…”