Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

Abstract

We analyze the inverse problem, if a manifold and a Riemannian metric on it can be reconstructed from the sphere data. The sphere data consist of an open set U⊂M˜ and the pairs (t,Σ) where Σ⊂U is a smooth subset of a generalized metric sphere of radius t. This problem is an idealization of a seismic inverse problem, originally formulated by Dix [8], of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of waves. In this problem, one considers a domain M˜ with a varying and possibly anisotropic wave speed which we model as a Riemannian metric g. For our data, we assume that M˜ contains a dense set of point diffractors and that in a subset U⊂M˜, we can measure the wave fronts of the waves generated by these. The inverse problem we study is to recover the metric g in local coordinates anywhere on a set M⊂M˜ up to an isometry (i.e. we recover the isometry type of M). To do this we show that the shape operators related to wave fronts produced by the point diffractors within M˜ satisfy a certain system of differential equations which may be solved along geodesics of the metric. In this way, assuming that we know g as well as the shape operator of the wave fronts in the region U, we may recover g in certain coordinate systems (e.g. Riemannian normal coordinates centered at point diffractors). This generalizes the method of Dix to metrics which may depend on all spatial variables and be anisotropic. In particular, the novelty of this solution lies in the fact that it can be used to reconstruct the metric also in the presence of the caustics.