Inverse problem for the wave equation with a white noise source

TL;DR

This paper investigates an inverse problem for the stochastic wave equation with a white noise source, demonstrating that a single measurement can determine the scattering relation of the Riemannian manifold, and under certain conditions, the metric itself.

Contribution

It proves that measurements from a single realization of the white noise source can recover the scattering relation and, in specific cases, the metric of the manifold.

Findings

Measurement determines the scattering relation with probability one.

In simple, conformally Euclidean manifolds, the metric can be uniquely recovered.

Single realization of the stochastic source suffices for inverse problem solutions.

Abstract

We consider a smooth Riemannian metric tensor $g$ on $\R^n$ and study the stochastic wave equation for the Laplace-Beltrami operator $\p_t^2 u - \Delta_g u = F$. Here, $F=F(t,x,\omega)$ is a random source that has white noise distribution supported on the boundary of some smooth compact domain $M \subset \R^n$. We study the following formally posed inverse problem with only one measurement. Suppose that $g$ is known only outside of a compact subset of $M^{int}$ and that a solution $u(t,x,\omega_0)$ is produced by a single realization of the source $F(t,x,\omega_0)$. We ask what information regarding $g$ can be recovered by measuring $u(t,x,\omega_0)$ on $\R_+ \times \p M$? We prove that such measurement together with the realization of the source determine the scattering relation of the Riemannian manifold $(M, g)$ with probability one. That is, for all geodesics passing through $M$, the travel times together with the entering and exit points and directions are determined. In particular, if $(M,g)$ is a simple Riemannian manifold and $g$ is conformally Euclidian in $M$, the measurement determines the metric $g$ in $M$.