Let M be a compact Riemannian manifold with non-empty boundary M. In this paper we consider an inverse problem for the second-order hyperbolic initial-boundary-value problem utt + but + a(x, D)u = 0 in M R+, u|MR+ = f, u|t=0 = ut|t=0 = 0. Our goal is to determine (M, g), b and a(x, D) from the knowledge of the non-stationary Dirichlet-to-Neumann map (the hyperbolic response operator) RT, with sufficiently large T 0. The response operator RT is the map , where is the normal derivative of the solution of the initial-boundary-value problem.
More specifically, we show the following.
(i) It is possible to determine Rt for any t 0 if we know RT for sufficiently large T and some geometric condition upon the geodesic behaviour on (M, g) is satisfied.
(ii) It is then possible to determine (M, g) and b uniquely and the elliptic operator a(x, D) modulo generalized gauge transformations.