Stationary phase analysis of ambient noise cross-correlations: Focusing on non-ballistic arrivals

Stacked cross-correlation functions have become ubiquitous in the ambient seismic imaging and monitoring community as approximations to the Green's function between two receivers. While theoretical understanding of this approximation to the ballistic arrivals is well established, the equivalent analysis for the non-ballistic arrivals is alarmingly inadequate compared to the exponential growth of its applications. To provide a fundamental understanding of the cross-correlation functions beyond the ballistic arrivals, we derive analytical stationary phase solutions for ambient noise cross-correlations with a focus on non-ballistic arrivals. We establish the mathematical and corresponding physical conditions that drastically differentiate the non-ballistic arrivals in the stacked cross-correlation and the actual Green's functions. In ambient noise environments, the coda waves due to random medium scatterings of an impulsive source cannot be distinguished from the cross-talk artifacts due to overlapping random noise sources. Therefore, changes in the non-ballistic arrivals cannot be uniquely attributed to changes in the medium or changes in the noise source environment without additional constraints. The theoretical results demand that interpreting large-elapse-time arrivals in the stacked cross-correlation functions as coda waves for deterministic information about the propagation medium should be conducted only after the source influence is sufficiently ruled out. Once the source influence is eliminated, the stationary phase solutions for scattering waves provide a solid basis for extracting reliable scattering information from the noise correlation functions for higher-resolution imaging and monitoring.