Identification of Green’s Functions Singularities by Cross Correlation of Noisy Signals

In this paper we consider the problem of estimating the singular support of the Green’s function of the wave equation in a bounded region by cross correlating noisy signals. A collection of sources with unknown spatial distribution emit stationary random signals into the medium, which are recorded at two observation points. We show that the cross correlation of these signals has enough information to identify the singular component of the Green’s function, which provides an estimate of the travel time between the two observation points. As in the recent work of Y. Colin de Verdière [math-ph/0610043], we use semiclassical arguments to approximate the wave dynamics by classical dynamics. We also use the ergodicity of the ray dynamics to obtain estimates of the travel times even when the noisy sources have limited spatial support in the region. We show furthermore that this approach is statistically stable when the averaging time is long enough, and that the accuracy of the travel time estimation is directly related to the regularity of the spatial correlation function of the sources. PACS numbers: 86A15,81Q20,37A25,60G60 Submitted to: Inverse Problems Identification of Green’s Functions Singularities 2