Diffuse fields in dynamic elasticity

In this article the problem of Green function retrieval from correlations is approached from a theoretical point of view and for this purpose an integral identity is considered: a representation theorem of the correlation type for an inhomogeneous region embedded in a homogeneous space. The full homogeneous case is studied with the theorem and it is concluded that, in the resulting field, the energy is equipartitioned. In infinite space this means that the ratio of P and S energy densities stabilizes to a constant value. That equipartition is reached in the classical sense is also demonstrated. Thus, in infinite space the energy densities associated with the possible degrees of freedom tend to share in equal parts the available energy.

The representation theorem permits the verification of the well known result that by averaging correlations of motions from diffuse, equipartitioned fields within an inhomogeneous, anisotropic, elastic medium it is possible to retrieve its Green function. As a result of this it is shown that the average autocorrelation of the diffuse displacement field at a point is proportional to the imaginary part of the Green function at the source precisely at this point. As a consequence, the energy density of the diffuse field is proportional to the trace of the imaginary part of the Green tensor at the source. Thus, the analytical form of the Green function permits the establishment, in and around an inhomogeneous region, of the theoretical energy density of a diffuse field.

In both homogeneous and inhomogeneous cases (i.e. localized elastic inclusions or cavities) the equipartition of the background illumination (the so called incident field in scattering theory) is a necessary and sufficient condition to retrieve the exact Green function from correlations. Local effects lead to energy ratios that fluctuate in space and frequency. The boundary of a half-space produces in its interior fluctuations of energy densities that are local effects of the diffuse field. These results may be useful to assess the diffuse nature of seismic ground motion from a limited set of observation points and to detect the presence of a target by its signature in the distribution of diffuse energy.