Stationary‐phase integrals in the cross correlation of ambient noise

The cross correlation of ambient signal allows seismologists to collect data even in the absence of seismic events. “Seismic interferometry” shows that the cross correlation of simultaneous recordings of a random wavefield made at two locations is formally related to the impulse response between those locations. This idea has found many applications in seismology, as a growing number of dense seismic networks become available: cross‐correlating long seismic records, the Green's function between instrument pairs is “reconstructed” and used, just like the seismic recording of an explosion, in tomography, monitoring, etc. These applications have been accompanied by theoretical investigations of the relationship between noise cross correlation and the Green's function; numerous formulations of “ambient noise” theory have emerged, each based on different hypotheses and/or analytical approaches. The purpose of this study is to present most of those approaches together, providing a comprehensive overview of the theory. Understanding the specific hypotheses behind each Green's function recipe is critical to its correct application. Hoping to guide nonspecialists who approach ambient noise theory for the first time, we treat the simplest formulation (the stationary‐phase approximation applied to smooth unbounded media) in detail. We then move on to more general treatments, illustrating that the “stationary‐phase” and “reciprocity theorem” approaches lead to the same formulae when applied to the same scenario. We show that a formal cross correlation/Green's function relationship can be found in complex, bounded media and for nonuniform source distributions. We finally provide the bases for understanding how the Green's function is reconstructed in the presence of scattering obstacles.