Helin, T.; Lassas, M.; Oksanen, L.; Saksala, T.
(2018)
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 partial derivative(2)(t)u- Delta(g)u = chi W, where W is a random variable with the white noise statistics on R1+n, n >= 3, chi is a smooth function vanishing for negative times and outside a compact set in space, and Delta(g) is the Laplace Beltrami operator associated to a smooth non-trapping Riemannian metric tensor g on 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 chi subset of R-n, C-T(t(1), x(1), t(2), x(2)) = 1/T integral(T)(0) u(t(1) s, x(1))u(t(2) s, x(2))ds, t(1), t(2) > 0, x(1), x(2) is an element of chi, for T > 0. Supposing that chi is non-zero on chi and constant in time after t > 1, we show that in the limit T -> infinity, 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 (R-n, g) up to an isometry. (C) 2018 Elsevier Masson SAS. All rights reserved.