Travel time estimation by cross correlation of noisy signals

¹ Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris VII, 2 Place Jussieu, 75251 Paris Cedex 5
² Mathematics Department, Stanford University, Stanford, CA 94305

Abstract

In this review paper we consider the problem of estimating the singular support of the Green's function of the wave equation by using passive sensors. We assume that noise sources emit stationary random signals into the medium which are recorded by sensors. We show that the cross correlation of the recorded signals has enough information to identify the singular component of the Green's function, which provides an estimate of the travel time between the sensors. We consider different situations, such as when the support of the noise distribution extends over all space or is spatially limited, the medium is open or bounded, homogeneous or inhomogeneous, dissipative or not. We discuss the limitations of the cross correlation technique and identify configurations under which travel time estimation is possible. We show that iterated cross correlations using auxiliary sensors can be efficient for travel time estimation when the support of the noise source distribution is spatially limited.