Numerical solution of the inverse problem for a~system of elasticity with the aftereffect for a~vertically inhomogeneous medium

A. L. Karchevsky ; A. G. Fatianov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 3, pp. 259-268.

Résumé

In this paper, the results of numerical solution of a 1-D inverse problem for a system of elasticity with the aftereffect for a vertically inhomogeneous medium are presented. The Boltzmann model is chosen to take into account the aftereffect. The purpose of the inverse problem is to reconstruct the longitudinal and transverse velocities by using the known longitudinal and transverse displacements at the surface. A series of model calculations is carried out, and the results of reconstructions of a velocity model of the medium typical for Western Siberia are presented.

Bibliographie

[1] N. N. Puzyrev (red.), Poperechnye i obmennye volny v seismorazvedke, Nedra, M., 1967

[2] Nefedkina T. N., “Vydelenie obmennykh otrazhennykh PS-voln po sisteme OGT s nesimmetrichnymi vyborkami”, Geologiya i Geofizika, 1980, no. 3, 113–122

[3] Nefedkina T. N., Kondrateva G. P., Oleinik L. V., “Tsifrovaya obrabotka obmennykh otrazhennykh voln”, Geologiya i Geofizika, 1980, no. 4, 67–77

[4] Puzyrev N. N., Trigubov A. V., Brodov L. Yu. i dr., Seismicheskaya razvedka metodom poperechnykh i obmennykh voln, Nedra, M., 1985

[5] N. N. Puzyrev (red.), Mnogovolnovye seismicheskie issledovaniya, Nauka, M., 1987

[6] Karchevsky A. L., “Numerical solution of the inverse problem for the system of elasticity for vertically inhomogeneous medium”, Bulletin of Novosibirsk Computing Center, Series Mathematical Modeling in Geophysics, 5 (1999), 135–142

[7] Alekseev A. S., Dobrinskii V. I., “Nekotorye voprosy prakticheskogo ispolzovaniya obratnykh dinamicheskikh zadach seismiki”, Matematicheskie problemy geofiziki, ch. 2, v. 6, VTs SO AN SSSR, Novosibirsk, 1975, 7–53

[8] Shemyakin E. I., Dinamicheskie zadachi teorii uprugosti i plastichnosti, NGU, Novosibirsk, 1968

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

[10] Fatyanov A. G., Nestatsionarnye seismicheskie volnovye polya v neodnorodnykh anizotropnykh sredakh s pogloscheniem energii, Preprint / AN SSSR. Sib. otd-nie. VTs; 857, Novosibirsk, 1989 | MR

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

[12] Avdeev A. V., Goruynov E. V., “Inverse problem of acoustics: determination of source wavelet and velocity”, J. of Inverse and Ill-posed Problems, 4:6 (1996), 475–482 | DOI | MR | Zbl

[13] Avdeev A. V., Goruynov E. V., Soboleva O. N., Priimenko V. I., “Numerical solution of some direct and inverse problems of electromagnetoelasticity”, J. of Inverse and Ill-posed Problems, 7:5 (1999), 453–462 | DOI | MR

[14] Avdeev A. V., Lavrentiev-jr. M. M., Priimenko V. I., Inverse Problems and Some Applications, ICMMG Publ., Novosibirsk, 1999

[15] Alekseev A. S., Avdeev A. V., Fatyanov A. G., Cheverda V. A., “Zamknutyi tsikl matematicheskogo modelirovaniya volnovykh protsessov v vertikalno-neodnorodnykh sredakh (pryamye i obratnye zadachi)”, Matematicheskoe modelirovanie, 3:10 (1991), 80–94 | MR | Zbl

[16] Alekseev A. S., Avdeev A. V., Fatianov A. G., Cheverda V. A., “Wave processes in verticallyinhomogeneous media: a new strategy for a velocity inversion”, Inverse Problems, 9:3 (1993), 367–390 | DOI | MR | Zbl

[17] Bamberger A., Chavent G., Lailly P., “Une application de la tlieori du control a un probleme inverse de seismique”, Ann. Geophys., 33:1–2 (1977), 183–200

[18] Bamberger A., Chavent G., Hemon Ch., Lailly P., “Inversion of normal incidence seismograms”, Geophysics, 47 (1982), 757–770 | DOI | Zbl

[19] Cao D., Boydoun W. B., Singh S. C., Tarantola A., “A Simultaneous Inversion for Background Velocity and Impedance Maps”, Geophysics, 55 (1990), 458–469 | DOI

[20] Cook D. A., Schneider W. A., “Generalized linear inversion of reflection seismic data”, Geophysics, 48 (1983), 665–676 | DOI

[21] Santosa F., Symes W. W., An Analysis of Least-squares Velocity Inversion., Geophysical Monograph Series, 4, Society of Exploration Geophysicists, 1989

[22] Symes W. W., “A Differential resemblance algorithm for the inverse problem of reflection seismology”, Computers Math. Applic., 22:4–5 (1991), 147–178 | DOI | MR | Zbl

[23] Hildebrand S. I., McMechan G. A., “1-D seismic inversion of dual wavefield data: Part I, Nonuniqueness and stability”, Geophysics, 59:5 (1994), 782–788 | DOI

[24] Hildebrand S. I., McMechan G. A., “1-D seismic inversion of dual wavefield data: Part II, A Gulf of Mexico example”, Geophysics, 59:5 (1994), 789–800 | DOI