Numerical Solution of an Inverse Multifrequency Problem in Scalar Acoustics

Abstract

A new algorithm is proposed for solving a three-dimensional scalar inverse problem of acoustic sensing in an inhomogeneous medium with given complex wave field amplitudes measured outside the inhomogeneity region. In the case of data measured in a “plane layer,” the inverse problem is reduced via the Fourier transform to a set of one-dimensional Fredholm integral equations of the first kind. Next, the complex amplitude of the wave field is computed in the inhomogeneity region and the desired sonic velocity field is found in this region. When run on a moderate-performance personal computer (without parallelization), the algorithm takes several minutes to solve the inverse problem on rather fine three-dimensional grids. The accuracy of the algorithm is studied numerically as applied to test inverse problems at one and several frequencies simultaneously, and the stability of the algorithm with respect to data perturbations is analyzed.