  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.
