Green's function representations for seismic interferometry

Abstract

The term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations. The first version of this principle was derived by Claerbout ͑1968͒, who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response. For an arbitrary 3D inhomogeneous lossless medium it follows from Rayleigh's reciprocity theorem and the principle of time-reversal invariance that the acoustic Green's function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points. The integral is along sources on an arbitrarily shaped surface enclosing these points. No assumptions are made with respect to the diffusivity of the wavefield. The Rayleigh-Betti reciprocity theorem leads to a similar representation of the elastodynamic Green's function. When a part of the enclosing surface is the earth's free surface, the integral needs only to be evaluated over the remaining part of the closed surface. In practice, not all sources are equally important: The main contributions to the reconstructed Green's function come from sources at stationary points. When the sources emit transient signals, a shaping filter can be applied to correct for the differences in source wavelets. When the sources are uncorrelated noise sources, the representation simplifies to a direct crosscorrelation of wavefield observations at two points, similar as in methods that retrieve Green's functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface.

Seismic interferometry for sparse data: SVD-enhanced Green's function estimation

“…We follow the derivation of SI in [10]. Let x i for 1 ≤ i ≤ N be the location of sources, ω the angular frequency, andĜ(x A , x i , ω) andĜ(x B , x i , ω) the Fourier transforms of the signal for a source located at x i and recorded at receivers A and B, respectively.…”

Application of Surface-Wave Methods for Seismic Site Characterization

“…Recent theoretical studies have shown that the cross-correlation of diffuse wavefields can provide an estimate of the Green's functions between receivers Lobkis 2001, 2004;Snieder 2004;Wapenaar 2004;Wapenaar and Fokkema 2006). Using coda waves of seismic events (Campillo and Paul 2003) and long seismic noise sequences (Shapiro and Campillo 2004), it was confirmed that it is possible to estimate the Rayleigh wave component of Green's functions between two stations by the cross-correlation of simultaneous recordings, a method now generally referred to as seismic interferometry.…”

A Comparison of Strategies for Seismic Interferometry

“…Only closed systems have, in general, normal modes. Alternative derivations, also valid for open systems of acoustic waves, have been developed using time-reversal invariance (Derode et al 2003a, b;Bakulin and Calvert 2004), representation theorems (Wapenaar et al 2005;Wapenaar and Fokkema 2006), and the summation of random incident plane waves (Weaver and Lobkis 2005). The derivation based on representation theorems has been extended to elastic waves (Wapenaar 2004).…”