Abstract. We consider an active source embedded in a randomly layered medium. We study the cross-correlation functions of the signals recorded at a series of points located at the surface. We show that this information can be processed to locate the source inside the medium. The analysis is based on a separation of scales technique and limit theorems for random differential equations. The statistical stability of the imaging method is proved. The analogy with the time-reversal of waves is enlightened, but the main difference is also put forward: we propose a passive way of imaging an unknown medium without the use of any active device. We finally extend these ideas for the location of a scatterer illuminated by a controlled source located at the surface or by a set of unknown sources generating random noise. Key words. Acoustic waves, random media, asymptotic theory. AMS subject classifications. 76B15, 35Q99, 60F05. 1. Introduction. Imaging in complex media is a difficult but exciting problem, with many potential applications [30] . We consider the problem of the source localization when the source is embedded in a randomly layered half-space. We study the statistical properties of the signals recorded at the surface. We focus our attention on the cross-correlation function between the signals recorded at two observation points. From the exact statistical description of this function, we propose a way to recover the precise location of the source from a data set consisting of the signals recorded at five (or more) observation points. We then extend this technique to the case of a point scatterer embedded in the random half-space and illuminated by a known or unknown source, or by a set of known or unknown sources. The waves recorded at the observation points have scattered from the random heterogeneities of the medium. We shall first consider configurations where a coherent front pulse can be detected. According to the O'Doherty Anstey theory [24] , this front pulse is spread out and shifted due to multiple scattering. In particular a random time shift makes it difficult to recover precise information about travel times. We shall also consider configurations where the signal arriving at the surface is so noisy that no coherent or ballistic front can be detected. Information about an individual scatterer could be understood to be lost in such a case. However time-reversal experiments [15, 16] and the explanations by Clouet and Fouque [11] extended by Papanicolaou et al. of the self-averaging property of time-reversal for broadband signals [5, 3] have shown that multiple scattering is not always a nuisance. The utility of time-reversal for imaging has been pointed out recently [6, 17] . Unfortunately it is not always possible to perform real time-reversal experiments, in geophysics for instance, as it requires very specific active devices. In this paper we shall see that cross-correlation functions of recorded noisy signals are useful to image a random medium. The first mathematical guess is that a cross-correlation function is in some sense close to achieve a time reversal experiment [35, 26] , as it involves the product of two Green's functions, one of them being complex-conjugated. We should mention that the idea to exploit the cross-correlation of noisy signals to retrieve information about ballistic wave motion has first been proposed in seismology
In REF a similar cross-correlation procedure was proposed for passive source localization which removes random components due to multiple scattering in randomly layered media.
Imaging in Randomly Layered Media by Cross-Correlating Noisy Signals
Multiscale Model. Simul.
Multiscale Model. Simul.
Mathematics Computer Science