Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

TL;DR

This paper presents a method to determine the local isometry type of a Riemannian manifold, modeling seismic wave speeds, from boundary measurements of single scattered wave fronts, even in the presence of caustics.

Contribution

It generalizes Dix's geophysical method to anisotropic, variable metrics and introduces a differential equation system for shape operators to recover the metric.

Findings

Reconstruction of the metric up to isometry using shape operators.

Method applicable to anisotropic and variable wave speeds.

Effective even with wave caustics present.

Abstract

We analyze the inverse problem, originally formulated by Dix in geophysics, of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of seismic waves. We consider a domain $\tilde M$ with a varying and possibly anisotropic wave speed which we model as a Riemannian metric $g$. For our data, we assume that $\tilde M$ contains a dense set of point scatterers and that in a subset $U\subset \tilde M$, modeling a region containing measurement devices, we can measure the wave fronts of the single scattered waves diffracted from the point scatterers. The inverse problem we study is to recover the metric $g$ in local coordinates anywhere on a set $M \subset \tilde M$ up to an isometry (i.e. we recover the isometry type of $M$). To do this we show that the shape operators related to wave fronts produced by the point scatterers within $\tilde M$ satisfy a certain system of differential equations which may be solved along geodesics of the metric. In this way, assuming we know $g$ as well as the shape operator of the wave fronts in the region $U$, we may recover $g$ in certain coordinate systems (e.g. Riemannian normal coordinates centered at point scatterers). This generalizes the well-known geophysical method of Dix to metrics which may depend on all spatial variables and be anisotropic. In particular, the novelty of this solution lies in the fact that it can be used to reconstruct the metric also in the presence of the caustics.