Finite difference calculation of the operators

Now that we can calculate the partial derivatives of traveltimes and surface horizontal slownesses, the next step is to solve equation (4) with proper initial conditions. Let us take the residual depth migration of post-stack images as an example. Assume that calculations are done in polar coordinates. The initial conditions at the surface are

$$\hat{r} = 0 \ \ \ \ \ \hbox{and} \ \ \ \ \ \hat{\theta}=\si... ...{s(0,\theta) \over \hat{s}(0,\hat{\theta})} \sin \theta \right)$$

where $s(0,\theta)$ and $\hat{s}(0,\hat{\theta})$ are slowness at the shot-receiver position. From these initial conditions we can extrapolate the kinematic operators of residual depth migration in the radius direction as follows:

$$\left\{ \begin{array} {lll} \hat{r}_{j+1} & = & \hat{r}_j+\l... ...heta} \over \partial r}\right]_j \Delta r \\ \end{array}\right.$$ (8)

where j is the index of the grid in the radius direction. Finally, the calculated operators are mapped from polar coordinates to Cartesian coordinates.