Table-driven implicit finite-difference migration

For the second-order approximation (m=1,n=1), equation (9) is the following cascaded partial differential equation in the space domain:

$$\begin{eqnarray} \frac{\partial}{\partial z}P&=&i\frac{\omega}{v_p}P,\ \frac{\p... ...frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})}P.\end{eqnarray}$$ (11) (12)

In isotropic migration, $\alpha_1$ and $\beta_1$ are constant. In VTI media, $\alpha_1$ and $\beta_1$ are functions of the anisotropy parameters $\varepsilon$ and $\delta$. For laterally varying media, the value of $\alpha_1$ and $\beta_1$ also vary laterally. It is too expensive to calculate $\alpha_1$ and $\beta_1$ for each grid point during the wavefield extrapolation. I calculate $\alpha_1$ and $\beta_1$ for a range of $\varepsilon$ and $\delta$ and store them in a table before the migration. I then generate maps of $\alpha_1$ and $\beta_1$ from the table. With the map of the coefficents $\alpha_1$ and $\beta_1$, the finite-difference scheme for VTI media can be performed in the same way as an isotropic migration.