THE DIFFERENTIAL EQUATION AND THE DIFFERENCING STAR

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:

 

$$k_z = {1 \over 2} {\left [-2{\omega \over v} + \frac{k_x^2}... ...{v({{-k_x+k_y} \over \sqrt{2}})^2} \over {2 \omega}}} \right ]}$$ (10)

Using the splitting method, we can separate equation (10) into a thin lens term and a diffraction term as follows:

$$\frac{\partial U}{\partial z}=-i{\omega \over v}U$$ (11)

 

$$\frac{\partial U}{\partial z}= {1 \over 2}{\left [\frac{ik_... ...}{2{ \omega \over v} -{{vk_y'^2} \over {2 \omega}}} \right ]}U$$ (12)

The counterpart of equation (12) in the ($\omega,x,y,z$) domain,

$$\frac{\partial U}{(\partial{z \over 2})}= {\left [\frac{-i \... ...frac{v \frac {\partial^2}{\partial y'^2}}{2 \omega}} \right ]}U$$ (13)

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 $\sqrt{2}dx, \sqrt{2}dy$, and dz / 2, assuming dx=dy. And to minimize frequency dispersion, we use the one-sixth trick Claerbout (1985).