Geodesic
Algorithm for reconstruction of the elastic constants of an anisotropic layer lying in an isotropic horizontally stratified medium
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 20-51.

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

In the work the algorithm for numerical solving of the inverse problem for reconstruction of the elastic constants of an anisotropic (thinly stratified) layer is presented. In force of the basic restriction the algorithm allow us to reconstruct only part of the elastic constants. Conditions of symmetry for the trace of the direct problem solution on the surface was used together with usual form of the additional information (value of the direct problem solution on the surface). The numerical solution of the inverse problem was suggested to find in the frequency domain. The general statement of the inverse problem was decomposed to the series of the standard statements of inverse problems to reconstruct sequentially the part of the elastic constants. 
@article{SEMR_2007_4_a2,
     author = {A. L. Karchevsky},
     title = {Algorithm for reconstruction of the elastic constants of an anisotropic layer lying in an isotropic horizontally stratified medium},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {20--51},
     publisher = {mathdoc},
     volume = {4},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2007_4_a2/}
}
TY  - JOUR
AU  - A. L. Karchevsky
TI  - Algorithm for reconstruction of the elastic constants of an anisotropic layer lying in an isotropic horizontally stratified medium
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2007
SP  - 20
EP  - 51
VL  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2007_4_a2/
LA  - ru
ID  - SEMR_2007_4_a2
ER  - 
%0 Journal Article
%A A. L. Karchevsky
%T Algorithm for reconstruction of the elastic constants of an anisotropic layer lying in an isotropic horizontally stratified medium
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2007
%P 20-51
%V 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2007_4_a2/
%G ru
%F SEMR_2007_4_a2
A. L. Karchevsky. Algorithm for reconstruction of the elastic constants of an anisotropic layer lying in an isotropic horizontally stratified medium. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 20-51. http://geodesic.mathdoc.fr/item/SEMR_2007_4_a2/

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

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

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

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

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

[6] V. M. Pavlov, “A convienent technique for calculating synthetic seismograms in layered half-space”, Proceedings of the International Conference “Problems of Geocosmos” (St. Peteburg, 03–08 June 2002), 320–323

[7] A. L. Karchevsky, “A numerical solution to a system of elasticity equations for layered anisotropic media”, Russian Geology and Geophysics, 46:3 (2005), 339–351

[8] A. L. Karchevskii, “Pryamaya dinamicheskaya zadacha seismiki dlya gorizontalno-sloistykh sred”, Sibirskie Elektronnye Matematicheskie Izvestiya, 2 (2005), 23–61 http://www.emis.ams.org/journals/SEMR/V2/v2p23-61.pdf | MR | Zbl

[9] A. L. Karchevsky, “Several remarks on numerical solution of the one-dimensional coefficient inverse problem”, Journal of Inverse and Ill-Posed Problems, 10 (2002), 361–384 | MR

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

[11] A. L. Karchevsky, “The analytical formulas for the gradient of the residual functional for the coefficient inverse problem for the elasticity system”, Journal of Inverse and Ill-Posed Problems, 11 (2003), 619–629 | DOI | MR | Zbl

[12] E. Kurpinar, A. L. Karchevsky, “Optimization inversion of seismic data from layered media: an algorithm for gradient”, Russian Geology and Geophysics, 46:4 (2005), 439–447

[13] A. L. Karchevsky, “Numerical reconstruction of medium parameters of member of thin anisotropic layers”, Journal of Inverse and Ill-Posed Problems, 12 (2004), 519–634 | DOI | MR

[14] A. L. Karchevskii, “Analiz resheniya obratnoi dinamicheskoi zadachi seismiki dlya gorizontalno-sloistoi anizotropnoi sredy”, Geologiya i Geofizika, 2006 (to appear)

[15] A. L. Karchevsky, “Numerical solution of the inverse problem for the system of elasticity for verticaly inhomogeneous medium”, Bulletin of the Novosibirsk Computing Center, series Mathematical Modeling in Geophysics, 5 (1999), 63–69 | Zbl

[16] A. L. Karchevskii, “Chislennoe reshenie odnomernoi obratnoi zadachi dlya sistemy uprugosti”, Doklady RAN, 375 (2000), 235–238

[17] A. L. Karchevskii, A. G. Fatyanov, “Chislennoe reshenie obratnoi zadachi dlya sistemy uprugosti s posledeistviem dlya vertikalno neodnorodnoi sredy”, Sibirskii zhurnal vychislitelnoi matematiki, 4 (2001), 259–269

[18] G. I. Petrashen, B. M. Kashtan, A. A. Kovtun, I. V. Mukhina, “Metod konturnykh integralov v sluchae transversalno-izotropnykh sred s osyu simmetrii, normalnoi granitsam razdela”, Voprosy dinamicheskoi teorii rasprostraneniya seismicheskikh voln, 24, ed. G. I. Petrashen, Nauka, Leningrad, 1984, 4–91 | MR

[19] A. Bakulin, V. Grechka, I. Tsvankin, “Estimation of fracture parameters from reflection seismic data”, Part I, II, III, Geophysics, 65 (2000), 1788–1830 | DOI

[20] S. G. Lekhnitskii, Teoriya uprugosti anizotropnogo tela, Gos. izd. tekhniko-teoreticheskoi literatury, Moskva, Leningrad, 1950 | MR