Application of spectral methods to inverse dynamic problem of seismicity of a stratified medium

A. A. Sedipkov

Sibirskij žurnal industrialʹnoj matematiki, Tome 21 (2018) no. 4, pp. 110-120

Résumé

Under study is the wave propagation process in the half-space $y_3\ge0$ with the Cartesian coordinates $y_1,y_2$ and $y_3$ which is filled with an elastic medium. The parameters of the medium are discontinuous and depend only on the coordinate $y_3$. The wave process is induced by an external perturbation source that generates a plane wave moving from the domain $y_3>h>0$. It is proved that the direct dynamic problem is uniquely solvable in the corresponding function space, and a special presentation is found for the solution. The problem of determination of the acoustic impedance of the medium from the wave field measurements on the surface is investigated by the spectral methods of the theory of differential equations.

Détail

Keywords: stratified elastic medium, acoustic impedance, Sturm–Liouville operator, inverse spectral problem.

Bibliographie

[1] 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

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

[3] 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), 259–269 | Zbl

[4] 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

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

[6] Romanov V. G., “Opredelenie parametrov sloistoi kusochno-postoyannoi sredy pri neizvestnoi forme impulsnogo istochnika”, Sib. mat. zhurn., 48:6 (2007), 1338–1350 | MR | Zbl

[7] Romanov V. G., “O zadache opredeleniya parametrov uprugoi sloistoi sredy i impulsnogo istochnika”, Sib. mat. zhurn., 49:5 (2008), 1157–1183 | MR

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

[9] 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

[10] 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. Numerical Analysis, 1994, no. 5, 1–25 | Zbl

[11] Sedipkov A. A., “Pryamaya i obratnaya zadachi akusticheskogo zondirovaniya v sloistoi srede s razryvnymi parametrami”, Sib. zhurn. industr. matematiki, 17:1 (2014), 120–134 | MR | Zbl

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

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

[14] 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

[15] Sedipkov A. A., “Vosstanovlenie razryvov koeffitsienta operatora Shturma–Liuvillya v impedansnoi forme”, Sib. mat. zhurn., 56:2 (2015), 455–462 | MR | Zbl

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