ON THE PROPERTIES OF A FAST ALGORITHM FOR SOLVING A THREE-DIMENSIONAL INVERSE PROBLEM OF SCALAR ACOUSTICS

Abstract

t Theinverse single-frequency problem of scalar acoustics in three-dimensional space is considered. It consists in determining the characteristics of acoustic inhomogeneities lying in a flat layer from the distribution of the complex amplitude of the acoustic field in a flat recording layer. To effectively solve it, a special numerical algorithm is used, previously proposed and justified by the authors. As its component part, the algorithm uses solutions to special one-dimensional Fredholm equations of the 1st kind, and in the case of ambiguity in the solution of the latter, their normal solutions (solutions with a minimum norm) are sought. To regularize this ill-posed inverse problem, Tikhonov regularization and the TSVD method are used. A systematic numerical study is carried out of the influence of various parameters in the data recording scheme on the accuracy of the approximate solutions obtained using the algorithm. In particular, we study the dependence of this accuracy on the position of point sources and composite sources that cause sound vibrations, its dependence on the position of the data recording layer, and on the number of planes in which the recording sensors lie. It is shown that acceptable accuracy of approximate solutions can be obtained even with two such layers. In addition, an approach to the numerical study of the ambiguity of the solution to the inverse problem under consideration is proposed. It is based on special theoretical statements given in the article, and on the same numerical algorithm for solving the inverse problem. The approach is focused on comparing several normal solutions of one-dimensional integral equations from the problem under consideration. These normal solutions are calculated with respect to various elements. An example of using this theory to numerically estimate the non-uniqueness of a solution is given.