Stability and uniqueness of the solution of the inverse seismo-kinematic problem in multidimentional case

G. Ya. Beil'kin

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 11, Tome 84 (1979), pp. 3-6

Résumé

The following kinematic problem is considered. Let $M$ be a compact $\nu$-dimensional domain with the metric $ds^2=g_{ij}dx^idx^j$. Given the function $\tau(\xi,\eta)=\int_{K_{\xi,\eta}}n\,ds$, a new metric $du=nds$ is constructed. $K_{\xi,\eta}$ is the geodesic connecting $\xi$ and $\eta$ in metric $du$; $\xi\eta\in\partial M$. The uniqueness of the solution is proved and the estimate $$ \int_M(n_2-n_1)(n_2^{\nu-1}-n_1^{\nu-1})\,dx^1\wedge\dots\wedge dx^\nu \leq\int_{\partial M\times\partial M}\Omega^{\tau_1,\tau_2} $$ is obtained. Refractive indices $n_1, n_2$ are the solutions of the inverse kinematic problem corresponding to functions $\tau_1,\tau_2$; $\Omega^{\tau_1,\tau_2}$ is the differential form on $\partial{M}\times\partial{M}$. $$ \Omega^{\tau_1,\tau_2}=-\frac{\Gamma(\nu/2)(-1)^{(\nu-1)(\nu-2)/2}} {2\pi^{\nu/2}(\nu-1)!} \sum_{\alpha+\beta=\nu-2}D_\eta\tau\wedge D_\xi\tau(D_\eta D_\xi\tau_1)^\alpha\wedge(D_\tau D_\xi\tau_2)^\beta, $$ $\tau=\tau_2-\tau_1$, $D_\xi=d\xi^i\partial/\partial\xi^i$, $D_\eta=d\eta^i\partial/\partial\eta^i$, $i=1,\dots,\nu-1$.