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Semiclassical analysis and passive imaging

The propagation of elastic waves inside the Earth provides us with information about the geological structure of the Earth's interior. Since the beginning of seismology, people have been using waves created by earthquakes or by artificial explosions. They record the waves as functions of time using seismometers 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 Green's function G(τ, 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 semiclassical analysis can be used in order to find the asymptotic behaviour of the correlations.