Kinematics in iterated correlations of a passive acoustic experiment

Introduction

It has long been known that correlations of seismic background noise recorded at two stations can yield the Green's function between the two stations (Aki, 1957; Lobkis and Weaver, 2001; Claerbout, 1968; Wapenaar, 2004), hereafter referred to as the estimated Green's function (EGF). A variety of proofs exist for this relation, including many based upon diffusivity of the wave fields (Weaver and Lobkis, 2001; Roux et al., 2005; Sánchez-Sesma et al., 2006; Sánchez-Sesma and Campillo, 2006), stationary-phase analysis (Schuster et al., 2004; Snieder et al., 2006; Snieder, 2004), and propagation invariants and reciprocity theorems (van Manen et al., 2005; Weaver and Lobkis, 2004; Wapenaar and Fokkema, 2006; Claerbout, 1976; Wapenaar, 2004). In general, these proofs require energy equipartitioning in the background seismic field; i.e., the energy flow must be equal in all directions. It is generally assumed that energy equipartitioning should be obtained after averaging over sources that excite the background field (Snieder et al., 2007). If the background noise field does not satisfy this condition, we expect the field correlations to recover imperfect EGFs (Malcolm et al., 2004; Paul et al., 2005).

Recently it has been argued that multiple scattering by random inhomogeneities can excite a secondary wave field that satisfies the assumption of equipartitioning, even if the primary wave field does not (Stehly et al., 2008). It is also known that correlation of coda waves can yield the Green's function (Malcolm et al., 2004; Snieder, 2004; Paul et al., 2005; de Ridder, 2008). Stehly et al. (2008) describes a way to use the coda waves of background noise to improve the quality of EGFs (Stehly et al., 2008). Garnier and Papanicolaou (2009) give a proof for this procedure to enhance Green's function estimation in random media based upon stationary-phase analysis of the four leading terms in the higher order correlation.

This paper discusses the problems associated with Green's function retrieval in directional wave fields. Then we proceed to briefly repeat the stationary-phase analysis of Garnier and Papanicolaou (2009) in the case of a wave field excited by one source in a homogeneous medium with the addition of one scatterer. The kinematics of the four leading terms are investigated using correlation gathers of auxiliary station position and source positions. Our examples show the basic procedure for reconstructing a Green's function by iterated correlations and provides a physical understanding of the elementary requirements for the positions of sources, random inhomogeneities, and auxiliary stations. This study has implications for seismic exploration using ambient seismic noise for different acquisition geometries, as in a network of stations only on the surface recording the ambient field above a reservoir, or a borehole survey with stations both down-hole and on the surface.

Kinematics in iterated correlations of a passive acoustic experiment