Mathematical models for passive imaging I: general background

Mathematical models for passive imaging I: general background. Yves Colin de Verdiere ? September 25, 2006 Abstract Passive imaging is a new technics which has been proved to be very efficient, for example in seismology: the correlation of the noisy fields be- tween different points is strongly related to the Green function of the wave propagation. The aim of this paper is to provide a mathematical context for this approach and to show, in particular, how the methods of semi- classical analysis can be be used in order to find the asymptotic behaviour of the correlations. Introduction Passive imaging is a way to solve inverse problems: it has been succesfull in seismology and acoustics [2, 3, 11, 15, 16, 20, 21, 23]. The method is as follows: let us assume that we have a medium X (a smooth manifold) and a smooth, deterministic (no randomness in it) linear wave equation in X. We hope to recover (part of) the geometry of X from the wave propagation. We assume that there is somewhere in X a source of noise f(x, t) which is a stationary random field. This source generates, by the wave propagation, a field u(x, t) = (u?(x, t))?=1,··· ,N which people do record on long time intervalls.

• dispersion relation


• relation between


• pseudo-differential equations


• high frequency limit


• wave equations


• schrodinger equation