To the numerical solution of the inverse multi-frequency scalar acoustics problem

Abstract

A new algorithm is proposed for solving the three-dimensional scalar inverse problem of acoustic sounding in an inhomogeneous medium. The data for the algorithm are the complex amplitudes of the wave field measured outside the inhomogeneity region. For the data recording scheme in a flat layer, the inverse problem is reduced using the Fourier transform to solving a set of one-dimensional Fredholm integral equations of the first kind. Solving these equations by the use of regularizing methods, we calculate the complex amplitude of the wave field in the inhomogeneity region and then we find the desired field of sound velocities in this domain. The proposed algorithm allows us to solve the inverse problem on a personal computer of average performance (without parallelization) for sufficiently fine three-dimensional grids in a few minutes. We demonstrate the results of solving model inverse problems at one frequency and several frequencies simultaneously along with a study of the accuracy of the proposed algorithm. We also investigate the issues of numerical stability of the algorithm with respect to data perturbations.