Numerical-analytical algorithms of solution to the forward and the inverse problems in seismology

A. S. Alekseev ; B. G. Mikhailenko

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 3 (2000) no. 3, pp. 191-214.

Résumé

The paper deals with numerical-analytical algorithms of solution to the forward and the inverse seismological problems based on a combination of the finite integral Fourier transforms with the finite difference methods. Such an approach allows the splitting of the 3D problems to a series of the ID problems and their parallelized solution on a multiprocessor computer.

Bibliographie

[1] Alekseev A. S., “Some inverse problems of the wave propagation theory I, II”, Izv. Akad. Nauk USSR, Ser. Geophys., 11 (1962), 1515–1531

[2] Alekseev A. S., Inverse dynamic problems of seismics, some methods and algorithms for interpretation of geophysical data, Nauka, M., 1967 (In Russian)

[3] Alekseev A. S., Avdeev A. V., Fatianov A. G., Tcheverda V. A., “Closed cycle of mathematical modeling of wave processes”, J. Mathemat. Modeling, 3:10 (1991), 367–390

[4] Alekseev A. S., Belonosov V. S., “Direct and inverse problems associated with inclined passing of SH-waves through ID inhomogeneous medium”, Bulletin of the Novosibirsk Computing Center, Ser. Numer. Analysis, 1 (1994), 1–25

[5] Alekseev A. S., Dobrinsky V. I., Neprochnov L. P., “On the practical application of the theory of inverse problems”, Dokl. Akad. Nauk USSR, 228:5 (1976), 1053–1056 | MR

[6] Alekseev A. S., Megrabov A. G., “Direct and inverse problems of scattering of plane waves in inhomogeneous layers”, Mathematical Problems of Geophysics, v. 3, Novosibirsk, 1972, 8–36

[7] Alekseev A. S., Matveyeva N. N., “Inverse problems for interpretation of seismic data”, Transactions of the International Conference on the Results of the International Geophysical Year (Moscow, 1963), 57–65

[8] Alekseev A. S., Mikhailenko B. G., “Lamb's problem for inhomogeneous half-space”, Dokl. Akad. Nauk. USSR, 214 (1974), 84–86

[9] Alekseev A. S., Mikhailenko B. G., “Solution of Lamb's problem for a vertically inhomogeneous half-space”, Izv. Akad. Nauk USSR, Fizika Zemli., 12 (1976), 11–25

[10] Alekseev A. S., Mikhailenko B. G., “Numerical modeling of seismic waves propagation in radially inhomogeneous Earth's model”, Dokl. Akad. Nauk USSR, 235 (1977), 46–49

[11] Alekseev A. S., Mikhailenko B. G., “Solution of dynamic problems of elastic wave propagation in inhomogeneous media by a combination of partial separation of variables and finite difference methods”, J. Geophys., 48 (1980), 161–172

[12] Alekseev A. S., Mikhailenko B. G., ““Non-ray” effects in the theory of seismic waves propagation”, Dokl. Akad. Nauk USSR, 267 (1982), 1079–1083

[13] Alekseev A. S., Mikhailenko B. G., Tcheverda V. A., “On numerical methods of solving forward and inverse problems in the wave propagation theory”, The Actual Problems of Numerical and Applied Mathematics, Nauka, Novosibirsk, 1983, 165–172

[14] Alekseev A. S., Tsibulchik G. M., “Inverse dynamic problems of wave diffraction in seismic monitoring (dynamic problems of tomography)”, Transactions of the Internat. Conference on Inverse Problems of Geophysics (Novosibirsk, 1996), 22–25

[15] Alekseev A. S., Tsibulchik G. M., “On the relation between inverse problems of the wave propagation theory and visualization of the sources of waves”, Dokl. Akad. Nauk USSR, 242:5 (1978), 1030–1033 | MR

[16] Tcheverda V. A., Kostin V. I., “$R$-pseudoinverse for compact operators in Hilbert spaces: existence and stability”, J. of Inverse and Ill-Posed Problems, 3:2 (1995), 131–148 | DOI | MR

[17] Fatianov A. G., “A semi-analytical method for solving direct dynamic problems in layered media”, Dokl. Akad. Nauk USSR, 310:2 (1990), 323–327 | MR

[18] Gazdag J., “Numerical convective schemes based on the accurate computation of space derivatives”, J. Comput. Phys., 13 (1973), 100–113 | DOI | Zbl

[19] Gelfand I. M., Levitan B. M., “On reconstruction of some differential equations using the spectral function”, Izv. Akad. Nauk USSR, Ser. Mathem., 15 (1951), 309–360 | MR | Zbl

[20] Hron F., Mikhailenko B. G., “Numerical modeling of nongeometrical eifects by the Alekseev–Mikhailenko method”, Bull. Seism. Soc. Am., 71:4 (1981), 1011–1029 | MR

[21] Khajdukov V. G., Kostin V. I., Tcheverda V. A., “The $r$-solution and its application in linearized waveform inversion for a layered background”, Inverse Problems of Wave Propagation, IMA Volumes in Mathematics and its Application, 90, Springer-Verlag, New York, 1996, 277–294 | MR

[22] Kostin V. I., Khajdukov V. G., Tcheverda V. A., “On $r$-solution of nonlinear equations”, Advanced Mathematics: Computations and Applications, NCC Publisher, Novosibirsk, 1995, 286–291 | MR

[23] Krein M. G., “On transition function of one-dimensional second-order boundary problem”, Dokl. Akad. Nauk USSR, 88:3 (1953), 405–408 | MR

[24] Kreis H. O., Oliger J., “Comparison of accurate methods for the integration of hyperbolic equations”, Tellus, 24 (1972), 199–215 | DOI | MR

[25] Marchenko V. A., “Some questions of the theory of linear second-order operators”, Transactions of the Moscow Mathematical Society, 1 (1952), 327–420 | MR

[26] Marquardt D. W., “An algorithm for least-squares estimation of nonlinear parameters”, J. Soc. Indust. Appl. Mathem., 11 (1963), 431–441 | DOI | MR | Zbl

[27] Martynov V. N., Mikhailenko B. G., “Calculation of complete theoretical seismograms for anisotropic media”, Proc. of 6th Inter. Math. Geoph. Sem. (Berlin, 1988), 323–348

[28] Mikhailenko B. G., “The method of solution of dynamic seismic problems for 2D inhomogeneous media models”, Dokl. Akad. Nauk USSR, 246 (1979), 47–51 | MR

[29] Mikhailenko B. G., “Synthetic seismograms for complex 3D geometries using an analytical-numerical algorithm”, Geophys. J. R. Astr. Soc., 79 (1984), 963–986 | Zbl

[30] Mikhailenko B. G., “Numerical experiment in seismic investigations”, J. Geophys., 58 (1985), 101–124

[31] Mikhailenko B. G., Seismic Fields in Complex Media, Izv. Sib. Div. USSR Acad, of Sciences, Novosibirsk, 1988

[32] Mikhailenko B. G., “Elastic wave propagation in complex 3D media”, 61st Annual Internal. Mtg. Soc. Expl. Geophys. Expanded Abstracts, 1991, 1569–1572

[33] Mikhailenko B. G., Korneev V. I., “Calculation of synthetic seismograms for complex subsurface qeometries by a combination of finite integral Fourier transforms and finite difference techniques”, J. Geophys., 54 (1984), 195–206

[34] Tsibulchik G. M., “About the solution of some inverse problems for wave equation by the method of visualization of sources”, J. Geology and Geophysics, 2 (1981), 109–119