Correlation based passive imaging with a white noise source

Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information is available. In this paper we study the stochastic wave equation ${\mathrm{\partial }}_t^{2}u-{\mathrm{\Delta }}_{g}u=\chi W$, where $W$ is a random variable with the white noise statistics on ${\mathbb{R}}^{1+n}$, $n\ge 3$, $\chi$ is a smooth function vanishing for negative times and outside a compact set in space, and ${\mathrm{\Delta }}_g$ is the Laplace-Beltrami operator associated to a smooth non-trapping Riemannian metric tensor $g$ on ${\mathbb{R}}^n$. The metric tensor $g$ models the medium to be imaged, and we assume that it coincides with the Euclidean metric outside a compact set. We consider the empirical correlations on an open set $\mathcal{X}\subset {\mathbb{R}}^n$, $$C_T(t_1,x_1,t_2,x_2)=\frac{1}{T}{\int }_0^{T}u(t_1+s,x_1)u(t_2+s,x_2)ds,t_1,t_2>0,x_1,x_2\in \mathcal{X},$$ for $T>0$. Supposing that $\chi$ is non-zero on $\mathcal{X}$ and constant in time after $t>1$, we show that in the limit $T\to \mathrm{\infty }$, the data $C_T$ becomes statistically stable, that is, independent of the realization of $W$. Our main result is that, with probability one, this limit determines the Riemannian manifold $({\mathbb R}^n,g)$ up to an isometry. To our knowledge, this is the first result showing that a medium can be determined in a passive imaging setting, without assuming a separation of scales.