Generalized optical theorems for the reconstruction of Green's function of an inhomogeneous elastic medium illuminated by random noise: derivations and application to a point scattering model.
Résumé
Imagining an arbitrary heterogeneity embedded in an isotropic and homogeneous elastic medium, we study the reconstruction of Green's function from the cross-correlation function of waves excited by randomly oriented body forces of which the distribution is stationary in time and homogeneous in space. A short derivation of the retrieval of Green's function of a homogeneous slightly dissipative elastic medium is presented. Using a general operator setting, we generalize this result and derive in a concise way the reconstruction of Green's function of an arbitrary inhomogeneous medium. In a scattering medium, we show that the reconstruction process is intimately related to generalized forms of the optical theorem which involve the ``off-shell'' $T$-matrix of elastic waves. As an application of the theory, we introduce a simple scattering model for elastic waves, composed of a single point-like heterogeneity. The $T$-matrix of the point scatterer incorporating all recurrent scattering loops is obtained by a regularization procedure. We discuss the physical significance of the point scattering model and show in particular that it satisfies the generalized optical theorem. The link between the equipartition state and the excitation of the system by random forces with white-noise statistics is established.
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