Inverse problem for the wave equation with a white noise source

Abstract

We consider a smooth Riemannian metric tensor $g$ on ${\mathbb{R}}^n$ and study the stochastic wave equation for the Laplace-Beltrami operator $\p_t^2 u - Δ_g u = F$. Here, $F=F(t,x,\omega )$ is a random source that has white noise distribution supported on the boundary of some smooth compact domain $M\subset {\mathbb{R}}^n$. 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 $M$.

25 pages

References in corpus (2)

• Sparse Deterministic Approximation of Bayesian Inverse Problems
• Level sets of the stochastic wave equation driven by a symmetric Lévy noise

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