Extracting the Green’s function from the correlation of coda waves: A derivation based on stationary phase

TLDR

Cross‑correlating multiply scattered waves recorded at two receivers yields the Green’s function, enabling passive imaging that does not require a source at either location and is traditionally explained by equipartitioning of uncorrelated normal modes. This work proposes an alternative derivation of passive imaging for the ballistic wave that does not rely on normal‑mode assumptions. The derivation applies to scalar waves in three dimensions and elastic surface waves, relying on destructive interference from scatterers distant from the receiver line and constructive interference from secondary sources near the line. It demonstrates that the global equipartitioning requirement can be replaced by a local condition of average isotropic propagation of scattered waves near the receivers.

Abstract

The Green's function of waves that propagate between two receivers can be found by cross-correlating multiply scattered waves recorded at these receivers. This technique obviates the need for a source at one of these locations, and is therefore called "passive imaging." This principle has been explained by assuming that the normal modes of the system are uncorrelated and that all carry the same amount of energy (equipartitioning). Here I present an alternative derivation of passive imaging of the ballistic wave that is not based on normal modes. The derivation is valid for scalar waves in three dimensions, and for elastic surface waves. Passive imaging of the ballistic wave is based on the destructive interference of waves radiated from scatterers away from the receiver line, and the constructive interference of waves radiated from secondary sources near the receiver line. The derivation presented here shows that the global requirement of the equipartitioning of normal modes can be relaxed to the local requirement that the scattered waves propagate on average isotropically near the receivers.