The imaging condition

The second step of shot-profile migration is forming a subsurface image through extraction of appropriate information from the independently extrapolated source and receiver wavefields. Claerbout's imaging principle Claerbout (1971) asserts that energy in the receiver wavefield, R, that is spatially collocated with energy in the source wavefield, S, at time t=0 originates from a reflector at that model point. Mathematically, this is accomplished through the extraction of the zero-lag of the cross-correlation of the two wavefields. In practice, this translates to a summation over frequency after the multiplication of the wavefields Claerbout (1985),  

$$I(z,x) = \sum_s I_s(z, x) = \sum_s \sum_{\omega} S_s(z, x, \omega) R_s^{*}(z, x, \omega),$$ (6)

Here, I(z,x) represents the image point as a function of horizontal distance and depth, ^* represents complex conjugation, and the subscript s refers to individual shot-profile image results.

Additional subsurface reflectivity constraints are obtained by extending (6) to include subsurface offset Rickett and Sava (2002). Offset domain common image gathers (ODCIGs) are created by multiplication of the source and receiver wavefields after a lateral shift of h:  

$$I(z, x, h) = \sum_s I_s(z,x,h) = \sum_s \sum_{\omega} S_s(z,x+h,\omega) R_s^{*}(z,x-h,\omega).$$ (7)