Remote Determination of the Parameters of Powerful Layers with the Use of an Intermediate Model

Abstract

A model of the medium, which makes it possible to use information more rationally for solving inverse problems in comparison with the well-known models of a layered and quasi-layered medium is introduced. A two-dimensional medium in which the fields are described by the Helmholtz equation is studied. A linearized statement of the problem of reconstructing the parameters of the medium, the inverse problem for the Helmholtz equation, is considered. The conditions for the uniqueness of the detection of layers are established. Examples of the ambiguity of the solution of the inverse problem according to information that initially even seemed redundant for the unique recovery of the medium are given. Algorithms and calculations for determining the characteristics of powerful layers are presented. Methods of interpreting the information known for a finite set of frequencies are proposed. The natural assumption about the possibility of restoring the n-layer medium from information at n + 1 frequencies is verified. It turns out that it is not possible to determine n conductivities and 2n boundaries, i.e., n functions and 2n numbers, from n + 1 functions, even if these n + 1 functions are specified by a large number of parameters. It is found that the n-layer medium can be restored from information known for 2n frequencies.