Generalized optical theorems for the reconstruction of Green's function of an inhomogeneous elastic medium

TL;DR

This paper develops generalized optical theorems for reconstructing elastic Green's functions in inhomogeneous media, linking scattering theory, reciprocity, and energy equipartition, with applications to point scatterers and absorption effects.

Contribution

It introduces generalized optical theorems involving the off-shell T-matrix for elastic waves, extending previous symmetry and reciprocity relations to inhomogeneous media.

Findings

Derived generalized optical theorems involving off-shell T-matrix.

Applied theory to a point scattering model with recurrent scattering loops.

Discussed the role of absorption and equipartition in Green's function reconstruction.

Abstract

This paper investigates the reconstruction of elastic Green's function from the cross-correlation of waves excited by random noise in the context of scattering theory. Using a general operator equation, -the resolvent formula-, Green's function reconstruction is established when the noise sources satisfy an equipartition condition. In an inhomogeneous medium, the operator formalism leads to generalized forms of optical theorem involving the off-shell $T$-matrix of elastic waves, which describes scattering in the near-field. The role of temporal absorption in the formulation of the theorem is discussed. Previously established symmetry and reciprocity relations involving the on-shell $T$-matrix are recovered in the usual far-field and infinitesimal absorption limits. The theory is applied to a point scattering model for elastic waves. The $T$-matrix of the point scatterer incorporating all recurrent scattering loops is obtained by a regularization procedure. The physical significance of the point scatterer is discussed. In particular this model satisfies the off-shell version of the generalized optical theorem. The link between equipartition and Green's function reconstruction in a scattering medium is discussed.