Reciprocity Equations of Correlation Type

Assume a source at in Figure 4.2 and receivers at and . The reciprocity theorem of correlation type (Wapenaar and Fokkema, 2006) states that the virtual Green's function is given by the reciprocity theorem of correlation type:

where for the outward point unit normal on the boundary. Here, Green's function solves the Helmholtz equation for an arbitrary velocity distribution with a constant density and I follow the notation from Schuster (2009). The integration path is only over the path as the half-circle path is neglected by the Wapenaar anti-radiation condition.

Now I want the diffractions to be reinforced so is replaced by the diffraction term defined as to give, under the far-field approximation,

$$\approx$$ (42)

where is the average wavenumber and represents the diffraction contribution in the Green's function for a point scatter.

This approximation is analogous to that used in model-based redatuming of reflection data to a new datum, except in model-based datuming is a model-based extrapolation Green's function that only accounts for direct arrivals, and $G({\bf B}\vert{\bf x})$ represents the reflection data devoid of direct waves and multiples.

According to the ray diagram shown in Figure4.1(a), the correlated trace
( denotes the temporal inverse Fourier transform) for a source at has the same traveltime $\tau _{A'B}-\tau _{A'A}$ for any source location . Such source locations are considered to be at stationary points, and similar to surface wave interferometry (Xue et al., 2009) or refraction wave interferometry (Dong et al., 2006), the summation of the correlated records over source positions tend to enhance the SNR of the virtual diffraction arrival by a factor of $\sqrt {N}$ . Here, $N$ represents the number of source positions that generate the diffractions.