The downward continued wavefield in Equation (6) can be rewritten as
When computed using the operator in Equation (15), the scattered wavefield, Equation (14), exhibits a nonlinear relation to the slowness perturbation. A straightforward linearization is to approximate for the constant background slowness. Biondi and Sava (1999) implement the prestack version of the scattering operator using a 4th order Taylor series expansion, under the constant velocity assumption ( and s=so) for all the terms in the sum:
Figure 3 Comparison of various approximation for the linear scattering operator. The solid lines correspond to scattering operators computed using the spatially variable slowness function (s), and the dashed lines correspond to the scattering operator computed using only the background slowness (so). The accuracy of the operator improves when using the variable slowness function. The horizontal axis is the ratio , and the vertical axis is the relative error of the linear scattering operator with respect to the true one, in logarithmic scale.
Figure shows a comparison of various approximation for the linear scattering operator, Equation (15). As expected, the relative error of the backprojection operator increases with increasing ratio . Also, Muir's continuous fraction expansion gives a significantly better approximation to the scattering operator for the same order of the expansion. The solid horizontal line corresponds to relative error of the approximate to the true scattering operator.
After we apply the imaging condition to the downward continued scattered wavefield Equation (13), we can formulate the relationship between the image () and slowness () perturbations as
Next section presents a simple example, in which I simulate and compute using a prestack backprojection operator derived from Equation (16).