Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map

In this paper we consider the stability of the inverse problem of determining a function q(x) in a wave equation in a bounded smooth domain in Rn from boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet-to-Neumann map associated to the solutions to the wave equation. We prove in the case of n⩾2 that q(x) is uniquely determined by the range restricted to a subboundary of the Dirichlet-to-Neumann map whose stability is a type of double logarithm.