Direct and inverse problems of acoustic sounding in a~layered medium with discontinuous parameters

A. A. Sedipkov

Sibirskij žurnal industrialʹnoj matematiki, Tome 17 (2014) no. 1, pp. 120-134.

Résumé

We consider the direct problem of finding a solution to a one-dimensional acoustic equation with discontinuous coefficients on the whole line $y\in\mathbb R$ with boundary conditions of special kind at the interior point $y=0$. We prove that the direct problem is uniquely solvable in the corresponding function space and obtain a special presentation for its solution. Along with the direct problem, we study the inverse problem of recovering the acoustic impedance of the medium from known one-sided limits of the solution to the direct problem and its derivative at the point $y=0$. It is shown that, with the use of the obtained special representation of the direct problem, the inverse problem can be reduced to a inverse spectral problem for a Sturm–Liouville operator with discontinuous coefficients.

Détail

Keywords: direct and inverse problems, Sturm–Liouville operator, inverse spectral problem, acoustic impedance.

Bibliographie

[1] Akkuratov G. V., Dmitriev V. I., “Metod rascheta polya ustanovivshikhsya uprugikh kolebanii v sloistoi srede”, Chislennye metody v geofizike, Izd-vo MGU, M., 1979, 3–12

[2] Akkuratov G. V., Dmitriev V. I., “Metod rascheta polya ustanovivshikhsya uprugikh kolebanii v sloistoi srede”, Zhurn. vychisl. matematiki i mat. fiziki, 24:2 (1984), 272–286 | MR | Zbl

[3] Fatyanov A. G., Mikhailenko B. G., “Metod rascheta nestatsionarnykh volnovykh polei v neuprugikh sloisto-neodnorodnykh sredakh”, Dokl. RAN, 301:4 (1988), 834–839

[4] Fatyanov A. G., Nestatsionarnye seismicheskie volnovye polya v neodnorodnykh anizotropnykh sredakh s pogloscheniem energii, Preprint No 857, VTs SO RAN, Novosibirsk, 1989, 44 pp. | MR

[5] Fatyanov A. G., “Poluanaliticheskii metod resheniya pryamykh dinamicheskikh zadach v sloistykh sredakh”, Dokl. RAN, 310:2 (1990), 323–327 | MR

[6] Karchevskii A. L., “Metod chislennogo resheniya sistemy uprugosti dlya gorizontalno-sloistoi anizotropnoi sredy”, Geologiya i geofizika, 46:3 (2005), 339–351

[7] Karchevskii A. L., “Pryamaya dinamicheskaya zadacha seismiki dlya gorizontalno-sloistykh sred”, Sib. elektron. mat. izvestiya, 2 (2005), 23–61 URL: http://semr.math.nsc.ru/v2/p23-61.pdf | MR | Zbl

[8] Karchevskii A. L., “Chislennoe reshenie odnomernoi obratnoi zadachi dlya sistemy uprugosti”, Dokl. RAN, 375:2 (2000), 235–238 | Zbl

[9] Karchevskii A. L., Fatyanov A. G., “Chislennoe reshenie obratnoi zadachi dlya sistemy uprugosti s posledeistviemdlya vertikalno neodnorodnoi sredy”, Sib. zhurn. vychisl. matematiki, 4:3 (2001), 259–268 | Zbl

[10] Kurpinar E., Karchevsky A. L., “Numerical solution of the inverse problem for the elasticity system for horizontally stratified media”, Inverse Problems, 20:3 (2004), 953–976 | DOI | MR | Zbl

[11] Karchevsky A. L., “Numerical reconstruction of medium parameters of member of thin anisotropic layers”, J. Inverse Ill-Posed Probl., 12:5 (2004), 519–634 | DOI | MR

[12] Karchevskii A. L., “Analiz resheniya obratnoi dinamicheskoi zadachi seismiki dlya gorizontalno-sloistoi anizotropnoi sredy”, Geologiya i geofizika, 47:11 (2006), 1170–1184

[13] Karchevskii A. L., “Algoritm vosstanovleniya uprugikh postoyannykh anizotropnogo sloya, nakhodyaschegosya v izotropnoi gorizontalno-sloistoi srede”, Sib. elektron. mat. izvestiya, 4 (2007), 20–51 URL: http://semr.math.nsc.ru/v4/p20-51.pdf | MR | Zbl

[14] Kurpinar E., Karchevsky A. L., “Finding of the elastic parameters of a horizontal (thinly stratified) anisotropic layer”, Appl. Anal., 87:10–11 (2008), 1179–1212 | MR | Zbl

[15] Alekseev A. S., “Nekotorye obratnye zadachi teorii rasprostraneniya voln”, Izv. AN SSSR. Ser. geofiz., 1962, no. 11, 1515–1522

[16] Alekseev A. S., “Obratnye dinamicheskie zadachi seismiki”, Nekotorye metody i algoritmy interpretatsii geofizicheskikh dannykh, Nauka, M., 1967, 9–48

[17] Alekseev A. S., Belonosov V. S., “The scattering of plane waves in inhomogeneous half-space”, Appl. Math. Lett., 8:2 (1995), 13–19 | DOI | MR | Zbl

[18] Alekseev A. S., Belonosov V. S., “Direct and inverse problems associated with inclined passing of SH-waves through 1D inhomogeneous medium”, Bull. Novosibirsk Comput. Center. Ser. Numer. Anal., 1994, no. 5, 1–25 | Zbl

[19] Alekseev A. S., Belonosov V. S., “Direct and inverse problems of wave propagation through a one-dimensional inhomogeneous medium”, European J. Appl. Math., 10:1 (1994), 79–96 | DOI | MR

[20] Alekseev A. S., Megrabov A. G., “Inverse problems of plane wave scattering by 1D inhomogeneous layers”, J. Inverse Ill-Posed Probl., 15:7 (2007), 645–668 | DOI | MR | Zbl

[21] Sedipkov A. A., “Direct and inverse problems of the theory of wave propagation in an elastic inhomogeneous medium”, J. Inverse Ill-Posed Probl., 19:3 (2011), 511–523 | DOI | MR | Zbl

[22] Sedipkov A. A., “Vosstanovlenie razryvov operatora Shturma–Liuvillya s kusochno-gladkimi koeffitsientami”, Vestn. NGU, 12:1 (2012), 114–125 | Zbl

[23] Sedipkov A. A., “The inverse spectral problem for the Sturm–Liouville operator with discontinuous potential”, J. Inverse Ill-Posed Probl., 20:2 (2012), 139–167 | DOI | MR | Zbl

[24] Levitan B. M., Pochti periodicheskie funktsii, Gostekhizdat, M., 1953 | Zbl

[25] Viner N., Peli R., Preobrazovanie Fure v kompleksnoi oblasti, Nauka, M., 1964 | MR

[26] Faddeev L. D., “Obratnaya zadacha kvantovoi teorii rasseyaniya”, Uspekhi mat. nauk, 14:4 (1959), 57–119 | MR | Zbl

[27] Levitan B. M., Obratnye zadachi Shturma–Liuvillya, Nauka, M., 1984 | MR | Zbl