Abstract
The following inverse kinematic problem of seismology is considered. In the compact domain M of dimension ν⩾,2 with the metric
, we consider the problem of constructing a new metricdu=nds according to the known formula
where ξ,ηεδM and Kξ,η is the geodesic in the metric du, connecting the points ξ, η. One proves uniqueness and one obtains a stability estimate
, where the refraction indices n1, n2 are the solutions of the inverse kinematic problem, constructed relative to the functions τ1, τ2, respectively,
is the differential form on δM×δM
where τ=τ2−τ1,
.
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Literature cited
V. G. Romanov, “On the uniqueness of the definition of an isotropic Riemannian metric inside a domain in terms of the distances between points of the boundary,” Dokl. Akad. Nauk SSSR,218, No. 2, 295–297 (1974).
R. G. Mukhometov, “The problem of the recovery of a two-dimensional Riemann metric and integral geometry,” Dokl. Akad. Nauk SSSR,232, No. 1, 32–35 (1977).
R. G. Mukhometov, “The inverse kinematic problem of seismology on the plane,” in: Mathematical Problems of Geophysics (collection of papers of the Computational Center, Siberian Branch, Academy of Sciences of the USSR), No. 6, Part 2 (1975), pp. 243–254.
L. Shvarts(Schwartz), Analysis [Russian translation], Vol. 2, Mir, Moscow (1972).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 84, pp. 3–6, 1979.
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Beil'kin, G.Y. Stability and uniqueness of the solution of the inverse kinematic problem of seismology in higher dimensions. J Math Sci 21, 251–254 (1983). https://doi.org/10.1007/BF01660580
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DOI: https://doi.org/10.1007/BF01660580