Delayed-shot migration in TEC coordinates

3D Implicit finite-difference extrapolation

One obvious concern is whether the dispersion relationship in equation 16 can be implemented accurately and efficiently in a wavefield extrapolation scheme. I address this question by comparing the elliptical-cylindrical dispersion relationship to that for elliptically anisotropic media in Cartesian coordinates. By defining an effective slowness $s_A=A s$ and rewriting equation 16 as

$$\frac{k_{\xi_3}}{\omega s_A} = \sqrt{ 1 - A^2\frac{k_{\xi_1}^2}{\omega^2 s_A^2} - \frac{ k_{\xi_2}^2 }{\omega^2 s_A^2}},$$ (19)

the TEC coordinate dispersion relationship resembles that of elliptically anisotropic media (Tsvankin, 1996). More specifically, extrapolation in TEC coordinates is related to a special case where the Thomsen parameters (Thomsen, 1986) obey $\epsilon=\delta$:

$$\left. \frac{k_{x_3}}{\omega s} \right\vert _{\epsilon=\delta} = ... ...) \frac{k_{x_1}^2}{\omega^2 s^2}-(1+2\epsilon) \frac{k_{x_2}^2}{\omega^2 s^2}}.$$ (20)

From equation 20 we see that equation 16 is no more complex than the dispersion relationship for propagating waves in elliptically anisotropic media, which is now routinely handled with finite-difference approaches (Zhang et al., 2001; Baumstein and Anderson, 2003; Shan and Biondi, 2005).

A general approach to 3D implicit finite-difference propagation is to approximate the square-root by a series of rational functions (Ma, 1982)

$$S_{\xi_3} = \sqrt{1 - A^2 S_{\xi_1}^2 - S_{\xi_2}^2} \approx \sum_{j=1}^{n} \frac{ a_j S_r^2}{1-b_j S_r^2},$$ (21)

where $S_{\xi_j}= \frac{k_{\xi_j}}{\omega s_A}$ and $S_r^2= A^2S_{\xi_1}^2 + S_{\xi_2}^2$, for $j=1,2,3$, and $n$ is the order of the coefficient expansion.

An optimal set of coefficients can be found by solving an optimization problem (Shan and Biondi, 2005),

$$E(a_j,b_j) = {\rm min} \int_{0}^{{\rm sin} \phi} \left[ \sqrt{1-S_r^2} - \sum_{j=1}^{n} \frac{a_j S_r^2}{1 - b_j S_r^2} \right]^2 {\rm d}S_r,$$ (22)

where $\phi$ is the maximum optimization angle. I generated the following results using a 4th-order approximation and coefficients found in Table 1 (Lee and Suh, 1985).

Table 1: Coefficients used in 3D implicit finite-difference wavefield extrapolation.

Coeff. order $j$	Coeff. $a_j$	Coeff. $b_j$
1	0.040315157	0.873981642
2	0.457289566	0.222691983

Specifying a finite-difference extrapolator operator using the 4th-order approximation is equivalent to solving a cascade of partial differential equations (Shan and Biondi, 2005)

$$\frac{\partial}{\partial \xi_3} U_{\xi_3+\Delta\xi_3/3} = i \omega s U_{\xi_3},$$

$$\frac{\partial}{\partial \xi_3} U_{\xi_3+2\Delta\xi_3/3} = i \omega s \left[ \frac{ \frac{a_1}{\omega^2s^2} \frac{\partial^2... ...^2s_A^2} \frac{\partial^2}{\partial \xi_2^2} } \right] U_{\xi_3+\Delta\xi_3/3},$$ (23)

$$\frac{\partial}{\partial \xi_3} U_{\xi_3+\Delta\xi_3} = i \omega s \left[ \frac{ \frac{a_2 }{\omega^2s^2} \frac{\partial^... ...^2s_A^2}\frac{\partial^2}{\partial \xi_2^2} } \right] U_{\xi_3+2\Delta\xi_3/3}.$$

I solve these equations implicitly at each extrapolation step by a finite-difference splitting approach that alternatively advances the wavefield in the $\xi _1$ and $\xi _2$ directions. Splitting methods allow the direct application of the $A$ scaling factor in equation 21 by introducing the original slowness model, $\frac{s_A}{A}=s$, for the $\xi _1$ direction split.

One drawback to finite-difference splitting methods is that they commonly generate numerical anisotropy. To minimize these effects, I apply a Fourier-domain phase-correction filter $L[\cdot]$ (Li, 1991)

$$L[U] = U{\rm e}^{ i \Delta \xi_3 k_L},$$ (24)

where

$$k_L = \sqrt{ 1-\frac{k_{\xi_1}^2}{(\omega s_1^r)^2} -\frac{k_{\xi... ...}{\omega s_2^r})^2}{1-b_j ( \frac{k_{\xi_2}}{\omega s_2^r})^2} \right) \right],$$ (25)

and $s_1^r$ and $s_2^r$ are reference slownesses chosen to be the mean value of $s^A_{eff}$ and $s$ defined above, respectively. Note that while this phase-shift correction is explictly correct for v($\xi _3$) media, the Li filter in v( $\xi_3,\xi_1,\xi_2$) media is only approximate and will introduce error.

Delayed-shot migration in TEC coordinates