Elastodynamic 2D Green function retrieval from cross-correlation: Canonical inclusion problem

TL;DR: In this paper, the authors study the problem of retrieval of 2D elastodynamic Green tensors in an infinite space containing a cylinder inclusion, and show that the Fourier transform of azimuthal average of cross-correlation of motion between two points is proportional to the imaginary part of the corresponding exact Green tensor.

Abstract: [1] It has been demonstrated experimentally and theoretically that cross-correlation at two points subjected to diffuse wavefields leads to the emergence of the Green function. Theoretical derivations imply several ideas of great generality but distinct interpretations exist. One assumes diffuse wavefields generated by multiple scattering. In other there is the assumption of multiple independent, uncorrelated sources giving rise to the phenomenon. Here we study the canonical problem of the retrieval of 2D elastodynamic Green function in an infinite space containing a cylinder inclusion. We illuminate isotropically the space with plane waves. We assume the spectra for both P and SV uniform and such that the energy ratio ES/EP = (α/β)2, which is predicted by equipartition theory in 2D. We then show that the Fourier transform of azimuthal average of cross-correlation of motion between two points is proportional to the imaginary part of the corresponding exact Green tensor. Some implications are discussed.