  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 $M$.
