3D shot-profile migration in ellipsoidal coordinates

Extrapolation Algorithm

Using the 4th-order approximation is equivalent to solving a cascade of partial differential equations (, )

$$\frac{\partial}{\partial \alpha} U = {\rm i} \omega s U,$$

$$\frac{\partial}{\partial \alpha} U = {\rm i} \omega s \left[ \frac{ \frac{a_1B^2}{\omega^2s_A^2} \frac... ...+\frac{b_1C^2}{\omega^2s_A^2} \frac{\partial^2}{\partial \gamma^2} } \right] U,$$ (A-13)

$$\frac{\partial}{\partial \alpha} U = {\rm i} \omega s \left[ \frac{ \frac{a_2 B^2}{\omega^2s_A^2} \fra... ...1+\frac{b_2C^2}{\omega^2s_A^2}\frac{\partial^2}{\partial \gamma^2} } \right] U.$$

We solve these equations implicitly at each $\Delta\alpha$ extrapolation step by a finite-difference splitting method that alternately advances the wavefield in the $\beta$ and $\gamma$ directions. Splitting methods allow us to apply the $B$ and $C$ scaling factors directly by introducing rescaled effective slowness models: $s^B_{eff}=\frac{s_A}{B}= \frac{A}{B}s$ and $s^C_{eff}=\frac{s_A}{C}= \frac{A}{C}s$.

One drawback to splitting methods is that they generate numerical anisotropy. To minimize these effects, we apply a Fourier-domain phase-correction filter (, )

$$U = U{\rm e}^{ {\rm i} \Delta \alpha k_L},$$ (A-14)

where

$$k_L = \sqrt{ 1-\frac{k_\beta^2}{(\omega s_B^r)^2} -\frac{k_\gamma... ...a}{\omega s_A^r})^2}{1-b_i ( \frac{k_\gamma}{\omega s_A^r})^2} \right) \right],$$ (A-15)

and $s_B^r$ and $s_C^r$ are reference slownesses chosen to be the mean value of $s^B_{eff}$ and $s^C_{eff}$ defined above.

3D shot-profile migration in ellipsoidal coordinates