We prove that knowing the lengths of geodesics joining points of the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

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Abstract. Given a bounded domain M in R n with a conformally Euclidean metric g = ρ dx 2 , in this paper we consider the inverse problem of recovering a semigeodesic neighborhood of a domain Γ ⊂ ∂M and the conformal factor ρ in the neighborhood from the travel time data (defined below) and the Cartesian coordinates of Γ. We develop an explicit reconstruction procedure for this problem. The key ingredient is the relation between the reconstruction and a Cauchy problem of the conformal Killing equation.

In this paper we develop numerical algorithm for solving inverse problem for the wave equation using Boundary Control method. The results of numerical experiments are represented.

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