Three travel time inverse problems on simple Riemannian manifolds

TL;DR

This paper proves that various travel time data uniquely determine a simple Riemannian metric on a disc, providing new proofs and stability estimates using Myers-Steenrod theorem.

Contribution

It introduces new proofs confirming data determine the metric up to boundary fixing diffeomorphism and establishes Lipschitz stability estimates.

Findings

Travel time data determine the metric up to boundary fixing diffeomorphism.

New proofs based on Myers-Steenrod theorem.

Lipschitz stability estimates for travel time and difference data.

Abstract

We provide new proofs based on the Myers-Steenrod theorem to confirm that travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to the natural gauge of a boundary fixing diffeomorphism. Our method of the proof leads to a Lipschitz-type stability estimate for the first two data sets in the class of simple metrics.