Geodesic
Remote determination of parameters of powerful layers with the use of the intermediate model
Matematičeskoe modelirovanie, Tome 32 (2020) no. 6, pp. 111-126.

Voir la notice de l'article provenant de la source Math-Net.Ru

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
Mots-clés : Fourier transform. 
@article{MM_2020_32_6_a7,
     author = {A. S. Barashkov},
     title = {Remote determination of parameters of powerful layers with the use of the intermediate model},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {111--126},
     publisher = {mathdoc},
     volume = {32},
     number = {6},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2020_32_6_a7/}
}
TY  - JOUR
AU  - A. S. Barashkov
TI  - Remote determination of parameters of powerful layers with the use of the intermediate model
JO  - Matematičeskoe modelirovanie
PY  - 2020
SP  - 111
EP  - 126
VL  - 32
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2020_32_6_a7/
LA  - ru
ID  - MM_2020_32_6_a7
ER  - 
%0 Journal Article
%A A. S. Barashkov
%T Remote determination of parameters of powerful layers with the use of the intermediate model
%J Matematičeskoe modelirovanie
%D 2020
%P 111-126
%V 32
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2020_32_6_a7/
%G ru
%F MM_2020_32_6_a7
A. S. Barashkov. Remote determination of parameters of powerful layers with the use of the intermediate model. Matematičeskoe modelirovanie, Tome 32 (2020) no. 6, pp. 111-126. http://geodesic.mathdoc.fr/item/MM_2020_32_6_a7/

[1] Ming Li, Chuchu Chen, Li Peijun, “Inverse random source scattering for the Helmholtz equation in inhomogeneous media”, Inverse Problems, 34 (2017), 015003 | MR

[2] P. S. Martyshko, A. L. Rublev, “O reshenii trekhmernoi obratnoi zadachi dlia uravneniia Gelmgoltsa”, Rossiiskii geofizicheskii zhurnal, 1999, no. 13-14, 98–110

[3] A. S. Barashkov, A. A. Nebera, “Cases of uniform convergence of the iteratively asymptotic method for solving multidimensional inverse problems”, Differential Equations, 51:4 (2015), 558–562 | DOI | DOI | MR | MR | Zbl

[4] A. V. Goncharsky, S. Yu. Romanov, S. A. Kharchenko, “Obratnaia zadacha akusticheskoi diagnostiki trekhmernykh sred”, Vych. metody i programmer, 7:1 (2006), 117–126

[5] Deyue Zhang, Yukun Guo, “Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation”, Inverse Problems, 31 (2015), 035007 | DOI | MR | Zbl

[6] A.S. Ayriyan, E.A. Ayryan, A.A. Egorov, I.A. Maslyanitsyn, V.D. Shigorin, “Numerical Modeling of the Static Electric Field Effect on the Director of the Nematic Liquid Crystal Director”, Mathematical Models and Computer Simulations, 10:4 (2018), 714–720 | DOI | MR

[7] A. V. Baev, “On Resolving Inverse Nonstationary Scattering Problems in a Two-Dimensional Homogeneous Layered Medium by the $\tau-p$ Radon Transform”, Mathematical Models and Computer Simulations, 10:5 (2018), 659–669 | DOI | MR | Zbl

[8] A. I. Khiryanova, S. I. Tkachenko, “Vosstanovlenie impulsa toka po napriazhennosti elektricheskogo polia, izmerennoi na vnutrennei poverkhnosti trubki”, Matem. Mod., 29:5 (2017), 3–15 | MR | Zbl

[9] E. P. Shurina, B. V. Rak, P. S. Zhigalov, “Analiz effektivnosti primeneniia PML-sloia v nizkochastotnykh prilozheniiakh (morskaia geoelektrika)”, Matem. Modelirovanie, 29:2 (2017), 33–46 | MR | Zbl

[10] M. V. Klibanov, V. G. Romanov, “Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation”, Inverse Problems, 32:2 (2016), 015005 | DOI | MR | Zbl

[11] M. N. Berdichevsky, V. I. Dmitriev, Modeli i metody magnitotelluriki, Nauchnyi Mir, M., 2009, 680 pp.

[12] A. N. Tikhonov, “Mathematical basis of the theory of electromagnetic soundings”, USSR Computational Mathematics and Mathematical Physics, 5:3 (1965), 545–548

[13] A. S. Barashkov, Small Parameter Method in Multidimensional Inverse Problems, VSP, Utrecht, The Netherlands, 1998, 139 pp. | MR | Zbl

[14] A. S. Barashkov, “Asymptotic forms of the solution of inverse problems for the Helmholtz equation”, USSR Comp. Math. and Math. Physics, 28:12 (1988), 1823–1831 | MR | Zbl

[15] V. G. Romanov, Nekotorye obratnye zadachi dlia uravnenii giperbolicheskogo tipa, Nauka, Novosibirsk, 1972, 164 pp.

[16] I. A. Shishmaryov, Vvedenie v teoriiu ellipticheskikh uravnenii, Izd-vo MGU, M., 1979, 184 pp.

[17] L. E. Elsgolts, Differentsialnye uravneniia i variatsionnoe ischislenie, Nauka, M., 1969, 424 pp.

[18] S. Bochner, Lektsii ob integralakh Fure, Gos. izd-vo fiz. mat. Lit., M., 1962, 360 pp.

[19] A. S. Barashkov, “On the Feasibility of Detecting Thin Conductive Layers from Field Measurements on the Surface of a Medium”, Comp. Math. and Math. Phys., 58:12 (2018), 2043–2052 | DOI | MR | Zbl

[20] V. S. Vladimirov, Obobshchennye funktsii v matematicheskoi fizike, Nauka, M., 1979, 320 pp.

[21] A. S. Barashkov, Matematika. Vysshee obrazovanie, AST, M., 2011, 480 pp.