Equipartitioning is not sufficient for Green's function extraction

The extraction of the Earth’s Green’s function from field fluctuations is a rapidly growing area of research. The principle of Green’s function extraction is often related to the requirement of equipartitioning, which stipulates that the energy of field fluctuations is distributed evenly in some sense. We show the meaning of equipartitioning for a variety of different formulations for Green’s function retrieval. We show that equipartitioning is not a sufficient condition, and provide several examples that illustrate this point. We discuss the implications of lack of equipartitioning for various schemes for the reconstruction of the Green’s function in seismology. The theory for Green’s function extraction is usually based on a statistical theory that relies on ensemble averages. Since there is only one Earth, one usually replaces the ensemble average with a time average. We show that such a replacement only makes sense when attenuation is taken into account, and show how the theory for Green’s function extraction for oscillating systems can be extended to incorporate attenuation.