Abstract

In my talk I give posing an inverse kinematic problem as problem of recovering the conformal Riemannian metric in a compact domain Ω from Riemannian distances between arbitrary points of the boundary of Ω. With physical point of view this problem consists in recovery a speed in Ω from given travel time τ (x, y) between arbitrary points x and y belonging to ∂Ω. A stability estimate of solutions of this problem is given. Then I introduce an integral geometry problem that consists in recovering a function from its integral along a family of geodesic line of the conformal Riemannian metric. A stability estimate of solutions of this problem is also given. As an application of the considered problems, an acoustic tomography problem is discussed. In this problem 3 unknown coefficients need to be found, namely, a speed of the sound c(x), an attenuation σ(x) and a density ρ(x). It is demonstrated that, under some convenient information about solutions of a forward problem, the recovering c(x) is reduced to the inverse kinematic problem and recovering σ(x) and ρ(x) is reduced to some integral geometry problems.