Equation (9) applies to downward-going waves. When we do the two-pass and four-pass migration, we apply the exploding reflector concept Claerbout (1985), so we use the equation for upward-coming waves instead of the one for downward-going waves.
When the velocity is slowly variable or independent of x and y, the conditions of full separation do apply. The dispersion relation for upward-coming waves can be written as follows:
Using the splitting method, we can separate equation (10) into a thin lens term and a diffraction term as follows:
The counterpart of equation (12) in the () domain,
is the equation we use to derive the differencing star and do migration.
For the x and y directions, we use dx,dy, and dz / 2 to derive the six-point implicit finite-differencing star; for the x' and y' diagonal directions, we use , and dz / 2, assuming dx=dy. And to minimize frequency dispersion, we use the one-sixth trick Claerbout (1985).