Reconstruction of a conformally Euclidean metric from local boundary diffraction travel times

TL;DR

This paper presents an explicit method to reconstruct a conformally Euclidean metric inside a region from boundary measurements related to seismic wave scattering, accommodating conjugate points and providing coordinate conversions.

Contribution

It introduces a two-step reconstruction process for conformal metrics from boundary diffraction data, including explicit formulas and handling of conjugate points in higher dimensions.

Findings

Reconstruction of the metric in Riemannian normal coordinates.

Conversion from normal to Cartesian coordinates.

Explicit reconstruction procedure applicable in general dimensions.

Abstract

We consider a region $M$ in $\mathbb{R}^n$ with boundary $\partial M$ and a metric $g$ on $M$ conformal to the Euclidean metric. We analyze the inverse problem, originally formulated by Dix, of reconstructing $g$ from boundary measurements associated with the single scattering of seismic waves in this region. In our formulation the measurements determine the shape operator of wavefronts outside of $M$ originating at diffraction points within $M$. We develop an explicit reconstruction procedure which consists of two steps. In the first step we reconstruct the directional curvatures and the metric in what are essentially Riemmanian normal coordinates; in the second step we develop a conversion to Cartesian coordinates. We admit the presence of conjugate points. In dimension $n \geq 3$ both steps involve the solution of a system of ordinary differential equations. In dimension $n=2$ the same is true for the first step, but the second step requires the solution of a Cauchy problem for an elliptic operator which is unstable in general. The first step of the procedure applies for general metrics.