Inverse problem of Travel time difference functions on compact Riemannian manifold with boundary

Abstract

We show that the travel time difference functions, measured on the boundary, determine a compact Riemannian manifold with smooth boundary up to Riemannian isometry, if boundary satisfies a certain visibility condition. This corresponds with the inverse microseismicity problem. The novelty of our paper is a new type of a proof and a weaker assumption for the boundary than it has been presented in the literature before. We also construct an explicit smooth atlas from the travel time difference functions.