Imaging in Randomly Layered Media by Cross-Correlating Noisy Signals

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.