Retrieving the Elastodynamic Green's Function of an Arbitrary Inhomogeneous Medium by Cross Correlation

Abstract

A correlation-type reciprocity theorem is used to show that the elastodynamic Green's function of any inhomogeneous medium (random or deterministic) can be retrieved from the cross correlation of two recordings of a wave field at different receiver locations at the free surface. Unlike in other derivations, which apply to diffuse wave fields in random media or irregular finite bodies, no assumptions are made about the diffusivity of the wave field. In a second version, it is assumed that the wave field is diffuse due to many uncorrelated sources inside the medium.

Seismic interferometry and ambient noise tomography in the British Isles

“…Hence, the resulting signal will approximate the Green's function between r 1 and r 2 , as if one of the receivers had actually been a source (Fig. 1b) (Wapenaar, 2003(Wapenaar, , 2004. We therefore refer to this Green's function as a seismogram from a ''virtual'' (imaginary) source at the location of one of the receivers (r 1 ).…”

“…Rickett and Claerbout (1999) summarised the application of noise crosscorrelation in helioseismology and thus conjectured for the Earth that ''by cross-correlating noise traces recorded at two locations on the surface, we can construct the wave-field that would be recorded at one of the locations if there was a source at the other'' (Rickett and Claerbout, 1999). The conjecture was finally proven mathematically by Wapenaar (2003Wapenaar ( , 2004, Snieder (2004) and van-Manen et al (2005) for acoustic media, by van-Manen et al (2006) and Wapenaar and Fokkema (2006) for elastic media, and was demonstrated in laboratory experiments by Lobkis and Weaver (2001), Weaver and Lobkis (2001), Derode et al (2003) and Larose et al (2005). Thereafter these methods became common practise in seismology.…”

“…The emergence of seismic interferometry theory (e.g. Wapenaar, 2003Wapenaar, , 2004Campillo and Paul, 2003;van-Manen et al, 2005van-Manen et al, , 2006van-Manen et al, , 2007Wapenaar and Fokkema, 2006;Slob et al, 2007;Curtis et al, 2009;Curtis and Halliday, 2010a,b;Wapenaar et al, 2011) allowed us to decode the information contained in the ambient noise wavefield to create a useful signal, in fact an artificial seismogram, from what used to be called noise. This new seismogram can then be used to image the subsurface of the Earth using traditional seismological tomographic or imaging methods.…”

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

“…Dealing with a diffuse field in open heterogeneous systems, Wapenaar (2004) and Weaver & Lobkis (2004) independently established similar identities between the Green's function and correlations of the diffuse field. 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).…”

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

“…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. The assumption underlying these formulations and physical experiments, is that the waves travel in lossless, but possibly strongly scattering, media.…”