Time-reversal-based detection in random media

We consider the detection and imaging of inclusions buried in highly heterogeneous media. We assume that only the statistical properties of the heterogeneous media call be observed and that the wave energy density may be modelled by macroscopic equations. The detection and imaging capabilities hinge on ensuring that the measured data are statistically stable, which means that they depend only on the macroscopic statistical parameters of the random media and not on the microscopic statistical realization. Ill this paper, the macroscopic model is a diffusion equation. In this context, we construct statistical tests to detect inclusions based on macroscopic diffusion measurements and perform asymptotic expansions to image their location and Volume. We show that time-reversal measurements enjoy a much larger signal-to-noise ratio ill the presence of background noise than do direct wave energy measurements. This is a direct consequence of the enhanced refocusing properties that characterize time reversed waves propagating ill heterogeneous media. Finally, we present numerical simulations of acoustic waves propagating in heterogeneous two-dimensional media. The numerical simulations illustrate which factors contribute to 'noise' in the measured data and how they affect the detection and imaging capabilities.