Acoustic inverse scattering via Helmholtz operator factorization and optimization

We present a joint acoustic/seismic inverse scattering and finite-frequency (reflection) tomography program, formulated as a coupled set of optimization problems, in terms of inhomogeneous Helmholtz equations. We use a higher order finite difference scheme for these Helmholtz equations to guarantee sufficient accuracy. We adapt a structured approximate direct solver for the relevant systems of algebraic equations, which addresses storage requirements through compression, to yield a complexity for computing the gradients or images in the optimization problems that consists of two parts, viz., the cost for all the matrix factorizations which is roughly O ( rN ) (for example O ( rN log N ) when d = 2) times the number of frequencies, and the cost for all solutions by substitution which is roughly O ( N ) (for example O ( N log ( r log N ) ) when d = 2) times the number of frequencies times the number of sources (events), where N = nd if n is the number of grid samples in any direction, and r is a parameter depending on the preset accuracy and the problem at hand. With this complexity, the multi-frequency approach to inverse scattering and finite-frequency tomography becomes computationally feasible with large data sets, in dimensions d = 2 and 3.