Three-dimensional inverse problem for inhomogeneous transversely isotropic media

Summary

The basic formula used in the presented paper gives the relation between the P wave travel-time perturbation and the perturbation of an inhomogeneous transversely isotropic medium, expressed by four perturbations of elastic parameters and by two angles of orientation of the axis of symmetry of transverse isotropy in space. The travel time perturbation is computed along the ray in the unperturbed inhomogeneous isotropic medium. Four elastic parameters and two angles are parametrized in the model under study and a system of equations for many rays is constructed. The equations are linear in the sought elastic parameters and nonlinear in the sought angles, and the iterative Levenberg-Marquardt algorithm is thus used to solve them. The theoretical 3-D inverse problem was solved in the presented numerical example. The data, simulating teleseismic data, were computed in the direct problem and then inverted. The results indicate the applicability and limitation of the presented algorithm in real problems.

Резюме

Основнaя формулa, uсnользовaннaя в nре¶rt;лaгрaемоŭ рaбоmе, ¶rt;aеm оmношенuе меж¶rt;у uзмененuем временu nробегa волны Р u uзмененuем нео¶rt;норо¶rt;ноŭ nоnеречно uзоmроnноŭ сре¶rt;ы, вырaженноŭ чеmырьмя уnругuмu naрaмеmрaмu u ¶rt;вумя углaмu орuенmaцuu осu сuммеmрuu nоnеречноŭ uзоmроnuu в nросmрaнсmве. Измененuе временu nробегa вычuсляеmся в¶rt;оль лучa в нaчaльноŭ нео¶rt;норо¶rt;ноŭ uзоmроnноŭ сре¶rt;е. Уnругuе naрaмеmры u ¶rt;вa углa naрaмеmрuзuровaны в мо¶rt;елu u nосmроенa сuсmемa урaвненuŭ ¶rt;ля несколькuх лучеŭ. Урaвненuя лuнеŭны nо неuзвесmным уnругuм naрaмеmрaм u нелuнеŭны nо неuзвесmным углaм орuенmaцuu осеŭ сuммеmрuu, nоэmому uсnо льзуеmся umерamuвныŭ aлгорumм Левенбергa-Мaрквaр¶rt;ma ¶rt;ля uх решенuя. В рaбоmе nрuбе¶rt;ен мо¶rt;ельныŭ mрехмерныŭ чuсленныŭ nрuмер. Для nолученuя mелесеŭсмuческuх ¶rt;aнных снaчaлa решенa nрямaя зa¶rt;aчa u зamем ¶rt;aнные обрaщены. Резульmamы nокaзывaюm огрaнuченuя u возможносmu nрuмененuя nрuве¶rt;енноголгорumмa в реaльных случaях.