Phase correction filter

In the 3D case, as in the isotropic migration, the dispersion relation is split into x and y components as follows:

$$\frac{\partial}{\partial z}P=i\frac{\omega}{v_p}\left [ \fra... ...frac{v_p^2}{\omega^2}\frac{\partial^2}{\partial y^2}}\right ]P.$$ (13)

This two-way splitting causes numerical anisotropy, which can be remedied by a phase-correction filter Li (1991) in the Fourier domain as follows:

$$P=Pe^{i\Delta zk_L},$$ (14)

where

$$k_L=\sqrt{\frac{1-(1+2\varepsilon_r)\frac{k_r^2}{(\omega/v_p... ...{v_p^r}k_y)^2}{1-\beta_1^r(\frac{\omega}{v_p^r}k_y)^2}\right ],$$ (15)

where v_p^r is the reference vertical velocity, $\varepsilon_r$ and $\delta_r$ are the reference anisotropy parameters, and $\alpha_1^r$and $\beta_1^r$ are the optimized finite-difference coefficients corresponding to the anisotropy parameters $\varepsilon_r$ and $\delta_r$.