Acoustic waveform inversion with the Lippmann-Schwinger equation constraint

Abstract

We develop the theory for performing acoustic waveform inversion in the frequency domain in both 2D and 3D, with the Lippmann-Schwinger equation as the constraint. The Lippmann-Schwinger equation that we consider has the special structure that the background velocity depends only on depth, in which case an efficient method exists to compute the forward and adjoint actions of the integral kernel. We treat the inversion as a joint optimization problem, where both the model to be inverted and the wavefields for each source and frequency are simultaneously treated as optimization variables. Here we explore the penalty method formulation of the problem, and a two-step alternating minimization strategy to solve it is presented, where each step involves solving a linear least squares problem. We point out the similarities and differences between the computational structure of this problem and the closely related wavefield reconstruction inversion problem, where the Helmholtz equation is used as the constraint instead of the Lippmann-Schwinger equation. Finally numerical examples in 2D illustrating the inversion method is presented.