SLANT STACK TRANSFORMATION AND THE MIGRATION EQUATION

The most important two equations used in this method of processing are the slant stack transformation equation and the migration equation for common midpoint slant stacks.

We first have to transform the wavefield in the CMP gather from the (x,t) domain to the $(\tau,p)$ domain. Let d(x,t) be the wavefield and $u(\tau,p)$ be the transformed slant stack wavefieldClaerbout (1985), so that  

$$u(\tau,p) = \int d(x,\tau+px)dx$$ (1)

A lot of care needs to be taken to construct a high-quality slant stack CMP gather. Kostov 1990 lists several methods of obtaining a high quality gather. Poor quality is probably caused by very coarse offsets and missing offsets beyond the ends of the cable. Those offsets that are missing must be interpolated or extrapolated before slant stack transformation.

The double-square-root equationOttolini (1982) for the downward continuation of common midpoint slant stacks is  

$${\partial u \over \partial z } = {{-i\omega \over v} \left[\sqrt{1-(Y+pv)^{2}}+\sqrt{1-(Y-pv)^{2}}\right ]U}$$ (2)

where

$$Y = {{vk_{y} \over 2\omega}}$$

The solution of equation (2) is  

$$U(\omega,z+\Delta z,k)=U(\omega,z,k){\exp \left\{{ {-i\omega... ...1-({vk_{y} \over 2\omega}-pv)^{2}}\right ]\Delta z} \right \} }$$ (3)

We can use this equation to obtain the wave extrapolation for constant velocity. Velocity variations in depth are easily accommodated by varying the velocity v with depth z in equation (3). For lateral velocity variations, I use the PSPI Lin (1993) algorithm to implement the slant stack migration.