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Identification of Green’s functions singularities by cross correlation of noisy signals. (English) Zbl 1145.35307

Summary: In this paper, we consider the problem of estimating the singular support of 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 Green’s function, which provides an estimate of the travel time between the two observation points. As in the recent work of Colin de Verdière (2006 Preprint math-ph/0610043v1), 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.

MSC:

35A08 Fundamental solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L05 Wave equation
35R60 PDEs with randomness, stochastic partial differential equations
35R30 Inverse problems for PDEs
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