Semiclassical analysis and passive imaging

Abstract

The propagation of elastic waves inside the Earth provides us with informations 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 the 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 be used in order to find the asymptotic behaviour of the correlations.

Passive Sensor Imaging Using Cross Correlations of Noisy Signals in a Scattering Medium

“…In the case of a spatially localized distribution of noise sources, directional diversity of the recorded fields can be enhanced if there is sufficient scattering in the medium. An ergodic cavity with a homogeneous interior is a good example ( Figure 2.4, left): Even with a source distribution that has very limited spatial support, the reverberations of the waves in the cavity generate interior fields with high directional diversity [15,2]. Multiple scattering of waves by random inhomogeneities can also lead to wave field equipartition if the transport mean free path is short compared to the distance from the sources to the sensors [30,21,27].…”

“…When the support of the random noise sources extends over all space and they are uncorrelated, that is, their spatial correlation is a delta function, it has been shown that the derivative of the cross correlation of the recorded signals is the symmetrized Green's function between the sensors [26]. This is also true with spatially localized noise source distributions provided the waves propagate within an ergodic cavity [15,16,2]. At the physical level this result can be obtained in both open and closed environments provided that the recorded signals are equipartitioned [22,37,25,33,23].…”

Signal-to-Noise Ratio Estimation in Passive Correlation-Based Imaging

“…[10,19], provided that the noise sources surround the region of interest. In the case of closed systems, i.e., cavities, the sources can be spatially localized provided the cavity is ergodic [1,6]. The first application of these ideas to imaging was in helioseismology [7] and, more recently, in passive seismic imaging of the surface velocity of the earth [13,15].…”

Noise Source Localization in an Attenuating Medium

“…It has been shown that the Green's function of the wave equation in an inhomogeneous medium can be estimated by cross correlating the signals emitted by ambient noise sources and recorded by a passive sensor array [7,17,18]. The idea has been used for travel time estimation and background velocity estimation in geophysical contexts, and also for passive sensor imaging of reflectors [9,10], which consists of backpropagating or migrating the cross correlation matrix of the recorded signals.…”