3D shot-profile migration in ellipsoidal coordinates

Relationship to elliptically anisotropic media

A naturally arising concern is whether the dispersion relationship in equation 7 can be implemented accurately and efficiently in a wavefield extrapolation scheme. We address this question by comparing the ellipsoidal and the (Cartesian-coordinate) elliptically anisotropic media dispersion relationships. By defining an effective slowness $s_A=A s$ and rewriting equation 7 as

$$\frac{k_\alpha}{\omega s_A} = \sqrt{ 1 - B^2 \frac{k_\beta^2}{\omega^2 s_A^2} - C^2\frac{ k_\gamma^2 }{\omega^2 s_A^2}},$$ (A-9)

we observe that the ellipsoidal coordinate dispersion relationship resembles that for elliptically anisotropic media (, ). Specifically, ellipsoidal coordinates relate to the case where Thomsen parameters (, ) obey $\epsilon=\delta$

$$\left. \frac{k_{x_3}}{\omega s} \right\vert _{\epsilon=\delta} = ... ... \frac{k_{x_1}^2}{\omega^2 s^2}-(1+2\epsilon) \frac{k_{x_2}^2}{\omega^2 s^2}. }$$ (A-10)

From equation 10 we see that equation 7 is no more complex than the dispersion relationship for propagating waves in elliptically anisotropic media, which is now routinely handled with optimized finite-difference approaches (, ,,).

3D shot-profile migration in ellipsoidal coordinates