Uniqueness of the partial travel time representation of a compact Riemannian manifold with strictly convex boundary

TL;DR

This paper demonstrates that a compact Riemannian manifold with a strictly convex boundary can be uniquely reconstructed from partial travel time data, with implications for seismology and geometric inverse problems.

Contribution

It introduces a method to reconstruct the manifold from partial travel time data and studies the regularity properties of the distance function on such manifolds.

Findings

Unique reconstruction from partial travel time data

Global regularity properties of the distance function established

Embedding of the manifold in a function space for reconstruction

Abstract

In this paper a compact Riemannian manifold with strictly convex boundary is reconstructed from its partial travel time data. This data assumes that an open measurement region on the boundary is given, and that for every point in the manifold, the respective distance function to the points on the measurement region is known. This geometric inverse problem has many connections to seismology, in particular to micro seismicity. The reconstruction is based on embedding the manifold in a function space. This requires the differentiation of the distance functions. Therefore this paper also studies some global regularity properties of the distance function on a compact Riemannian manifold with strictly convex boundary.