Inverse problem for wave equation with sources and observations on disjoint sets

We consider an inverse problem for a hyperbolic partial differential equation on a compact Riemannian manifold. Assuming that ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are two disjoint open subsets of the boundary of the manifold we define the restricted Dirichlet-to-Neumann operator ${\mathrm{\Lambda }}_{{\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2}$. This operator corresponds the boundary measurements when we have smooth sources supported on ${\mathrm{\Gamma }}_1$ and the fields produced by these sources are observed on ${\mathrm{\Gamma }}_2$. We show that when ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are disjoint but their closures intersect at least at one point, then the restricted Dirichlet-to-Neumann operator ${\mathrm{\Lambda }}_{{\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2}$ determines the Riemannian manifold and the metric on it up to an isometry. In the Euclidian space, the result yields that an anisotropic wave speed inside a compact body is determined, up to a natural coordinate transformations, by measurements on the boundary of the body even when wave sources are kept away from receivers. Moreover, we show that if we have three arbitrary non-empty open subsets ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2$, and ${\mathrm{\Gamma }}_3$ of the boundary, then the restricted Dirichlet-to-Neumann operators ${\mathrm{\Lambda }}_{{\mathrm{\Gamma }}_j,{\mathrm{\Gamma }}_k}$ for $1\leq j<k\leq 3$ determine the Riemannian manifold to an isometry. Similar result is proven also for the finite-time boundary measurements when the hyperbolic equation satisfies an exact controllability condition.