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 Matem. Mod., 2020, Volume 32, Number 6, Pages 111–126 (Mi mm4192)

Remote determination of parameters of powerful layers with the use of the intermediate model

A. S. Barashkov

MPEI (Moscow Power Engineering Institute)

Abstract: A model of the medium is introduced, 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). 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 is considered (the inverse problem for the Helmholtz equation). The conditions for the uniqueness of detection of layers are established. Examples of the ambiguity of the solution of the inverse problem according to information that initially seemed even redundant for a unique recovery of the environment are given. Algorithms and calculations for determining the characteristics of powerful layers are presented. Methods of interpreting information known for a finite set of frequencies are proposed. The natural assumption about the possibility of restoring the nlayer medium from information at $n+1$ frequencies is verified. It turned 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 was found that the $n$-layer medium can be restored from information known for $2n$ frequencies.

Keywords: intermediate model, two-dimensional medium, inverse problem for Helmholtz equation, linearized statement, infinite strip, uniqueness theorems, examples of multi-valuedness of solution when reconstructing medium, Fourier transform.

DOI: https://doi.org/10.20948/mm-2020-06-08

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Revised: 03.12.2019
Accepted:23.12.2019

Citation: A. S. Barashkov, “Remote determination of parameters of powerful layers with the use of the intermediate model”, Matem. Mod., 32:6 (2020), 111–126

Citation in format AMSBIB
\Bibitem{Bar20} \by A.~S.~Barashkov \paper Remote determination of parameters of powerful layers with the use of the intermediate model \jour Matem. Mod. \yr 2020 \vol 32 \issue 6 \pages 111--126 \mathnet{http://mi.mathnet.ru/mm4192} \crossref{https://doi.org/10.20948/mm-2020-06-08}