An Inverse Kinematic Problem with Internal Sources

TL;DR

This paper presents an explicit method to recover a neighborhood and the conformal factor of a domain in Euclidean space from travel time data, using a relation to the conformal Killing equation.

Contribution

It introduces a novel reconstruction procedure for inverse problems involving conformal metrics and travel time data, linking it to the conformal Killing equation.

Findings

Successful explicit reconstruction method developed

Reconstruction relies on solving a Cauchy problem for the conformal Killing equation

Applicable to bounded domains with conformally Euclidean metrics

Abstract

Given a bounded domain $M$ in $\mathbb{R}^n$ with a conformally Euclidean metric $g=\rho\,dx^2$, in this paper we consider the inverse problem of recovering a semigeodesic neighborhood of a domain $\Gamma\subset \partial M$ and the conformal factor $\rho$ in the neighborhood from the travel time data (defined below) and the Cartesian coordinates of $\Gamma$. We develop an explicit reconstruction procedure for this problem. The key ingredient is the relation between the reconstruction and a Cauchy problem of the conformal Killing equation.