The relationship between cross correlations and Green's functions in ambient noise interferometry with Bayesian constraints

Cross correlation of ambient seismic noise has begun to provide detailed images of the Earth that are not possible to attain using earthquake-based or active-source imaging techniques. However, the theoretical justification for the ambient noise correlation or seismic interferometry approach typically relies on an equipartition or diffuse field assumption that is not expected to be satisfied on Earth. Here we demonstrate that a Bayesian inference approach to calculating the expected cross correlation of seismic signals leads to an improved understanding of its relationship with the Green's function. The Bayesian derivation is found to exactly replicate the equipartition result under a global energy constraint, and is able to accommodate any number of other constraints that typically cannot be utilized in other approaches. With these stronger constraints, the cross correlation is found to deviate more and more strongly from the 'expected' Green's function, but nonetheless a specific prediction can be made regarding the expected cross correlation which can be used as a more reliable point of comparison with observations than the Green's function. The approach therefore provides a path forward for how to use seismic cross correlation data under more and more realistic conditions and improvements in knowledge, and should eventually help improve the accuracy of resulting images of the Earth.