# Lipschitz Stability of Travel Time Data

## TL;DR

This paper shows that reconstructing certain spaces from travel time data is stable under small changes in the data.

## Contribution

The paper introduces a proof of Lipschitz stability for reconstructing length spaces from travel time data.

## Key findings

- Reconstruction from travel time data is Lipschitz stable under specific conditions.
- The result applies to non-simple Riemannian manifolds and Euclidean domains with complex topology.
- Metric trees are among the spaces covered by the stability proof.

## Abstract

We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen closed subset. The case of a Riemannian manifold with boundary with the boundary as the measurement set appears is a classical geometric inverse problem arising from Gel’fand’s inverse boundary spectral problem. Examples of spaces satisfying our assumptions include some non-simple Riemannian manifolds, Euclidean domains with non-trivial topology, and metric trees.