MULTISCALE ANALYSIS OF WAVE PROPAGATION IN RANDOM MEDIA

Abstract

Wave propagation in random media can be studied by multiscale and stochastic analysis. We review some recent advances and their applications. In particular, in a physically relevant regime of separation of scales, wave propagation is governed by a Schrödinger-type equation driven by a Brownian field. We study the associated moment equations and describe the propagation of coherent and incoherent waves. We quantify the scintillation of the wave and the fluctuations of the Wigner distribution. These results make it possible to introduce and characterize correlation-based imaging methods. 1 Wave propagation in random media In many wave propagation scenarios the medium is not constant, but varies in a complicated fashion on a spatial scale that is small compared to the total propagation distance. This is the case for wave propagation through the turbulent atmosphere, the Earth’s crust, the ocean, and complex biological tissue for instance. If one aims to use transmitted or reflected waves for communication or imaging purposes it is important to characterize how such microstructure affects and corrupts the wave. Motivated by the situation described above we consider wave propagation through timeindependent complex media with a spatially varying index of refraction. Typically we cannot expect to know the index of refraction pointwise so we model it as a realization of a random process. When the index of refraction is a random process, the wave field is itself a random process and we are interested in how the statistics of the random medium affects the statistics of the wave field. The analysis of wave propagation in random media has a long history. It was first dealt with phenomenogical models such as the radiative transfer theory. The first mathematical papers were written in the 60’s by Keller [1964] who connected radiative transport theory and random wave equations. In the review presented at MSC2010: primary 35R60; secondary 35R30.