Semi classical analysis and passive imaging

Niveau: Supérieur, Doctorat, Bac+8Semi-classical analysis and passive imaging Yves Colin de Verdiere ? March 16, 2009 Abstract The propagation of elastic waves inside the earth provides us information's about the geological structure of the earth's interior. Since the beginning of seismology, people are using waves created by earthquakes or by artificial explosions. They record the waves as functions of the time using seismome- ters located at different stations on the earth's surface. Even without any earthquake or explosion, a weak signal is still recorded which has no evident structure: it is a ”noise”. How to use these noises? This is the goal of the method of ”passive imaging”. The main observation is the following one: the time correlation of the noisy fields, computed from the fields recorded at the points A and B, is ”close” to the Green's function G(?, A,B) 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 cor- relations. Introduction The seismologists want to recover the physical parameters of the earth's interior from records (called seismogram's), at the earth's boundary, of the elastic waves propagating inside the earth. From the mathematical point of view, it is an example of a so-called ”inverse problem”: the most famous inverse problems are the Calderon problem (recovering the conductance of a domain from boundary measurements) and the Kachas been used main result surface waves problem has wave equations makes quite