An Inverse Kinematic Problem with Internal Sources

Abstract

Given a bounded domain $M$ in $\mathbb{R}^n$ with a conformally Euclidean metric $g=ρ\,dx^2$, in this paper we consider the inverse problem of recovering a semigeodesic neighborhood of a domain $Γ\subset \partial M$ and the conformal factor $ρ$ in the neighborhood from the travel time data (defined below) and the Cartesian coordinates of $Γ$. 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.

7 pages