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

TL;DR

This paper proves that boundary measurements of travel time differences uniquely determine a compact Riemannian manifold with boundary, under certain conditions, providing a new proof and weaker assumptions than previous results.

Contribution

It introduces a novel proof technique and relaxes boundary conditions for the inverse problem of determining manifolds from travel time data.

Findings

Travel time difference functions determine the manifold up to isometry.

A new proof method is developed for the inverse problem.

Explicit smooth atlas construction from boundary data is achieved.

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.