True-amplitude shot-modeling

 Under the single-scattering (first-order Born) approximation the scattered field at r due to a shot at s is given by  

$$q_{\rm scat}(r\vert s) \sim \omega^2 \int G(r'\vert s) \; G(r\vert r') \; \Delta(r') \; dV(r'),$$ (76)

where G(r|s) is the Green's function response at r of the medium to an impulse at s, and $\Delta(r')$ represents perturbation in medium parameters.

Equation () describes the process of calculating the volume integral in equation () first by downward continuing the shot wavefield to give G(r'|s), then by upward continuing the scattered field to the surface, accumulating contributions from each depth slice on the way back up. The $\omega^2$term can be applied as a data-space filter after modeling. However, for the amplitudes to be correct in equation (), the extrapolation operators need to represent true-amplitude Green's functions.

For v(z) media, the WKBJ Green's function for a smoothly-varying background velocity field [e.g. Aki and Richards (1980); Stolt and Benson (1986)] is given by  

$$q(k_x,k_y,z_2,\omega) = q(k_x,k_y,z_1,\omega) \; \sqrt{\frac... ...(z_2)}} \; \exp \left[ i \int_{z_1}^{z_2} k_z(z') dz' \right],$$ (77)

$${\rm where} \hspace{0.15in} k_z = \frac{\omega^2}{v(z)^2} - k_x^2 - k_y^2.$$ (78)

The amplitude term in equation () can be incorporated into the shot-profile modeling procedure; however, different operators are required for upward versus downward continuation. Firstly, the true-amplitude downgoing WKBJ extrapolators should be substituted into equation (), and secondly, the upgoing WKBJ extrapolators should be substituted for the adjoint of ${\bf D}$ in equation ().