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

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.

Retrieving the Green's Function From Cross Correlation in a Bianisotropic Medium

“…Not all source locations are equally important and practical configurations with open boundaries may result in accurate results, because the main contributions come from sources located at stationary points [9]. The presence of weak losses has no effects on the phase of the extracted Green's function, while some amplitude errors but no spurious events occur [17].…”

“…where the contrast function ∆M is the same as in Equation (9). No assumptions about the material matrixM has been made, other than that its adjoint exists.…”

“…Draganov et al [7] retrieved seismic subsurface reflections from noise records. Schuster et al [8] and Snieder [9] showed with stationary phase theory how not all sources on the boundary play an equally important role, but only those that are stationary for the resulting event. Wapenaar [10] showed that the elastodynamic response of an arbitrary heterogeneous elastic half space can be retrieved from contributions of sources in the subsurface below a pressure free surface.…”

Statistical properties of the seismic noise field: influence of soil heterogeneities

“…Using correlational forms of the reciprocity theorems, Wapenaar & Fokemma (2006) showed how inter-receiver Green's functions could be constructed by crosscorrelating recorded fields whereas other studies have also revealed the limitations of that method (Snieder 2004;Roux et al 2005;Gouédard et al 2008). For obtaining a diffuse field Halliday & Curtis (2008) analyse the requirements in terms of source distribution for the retrieval of scattered surface waves.…”

“…Whereas the randomness of the coda results from multiple scattering by randomly distributed scatterers, the fact that seismic noise is assumed to be random is mostly because of the distribution of sources at the surface of the Earth (e.g. Derode et al 2003;van Tiggelen 2003;Malcolm et al 2004;Snieder 2004). …”