Wave-equation migration velocity analysis for VTI media using optimized implicit finite difference

Optimized implicit finite difference for VTI media

Assuming the S-wave velocity is much slower than the P-wave velocity, we can approximate the dispersion relationship for VTI media as follows (Shan, 2009):

$$S_z = \sqrt{\frac{1-(1+2\delta)S_r^2}{1-2\eta(2\delta+1)S_r^2}},$$ (1)

where $S_z = \frac{k_z}{w/v_v}$ , $S_r = \frac{k_r}{w/v_v}$ and $k_r=\sqrt{k_x^2+k_y^2}$ . Anisotropic parameter $\delta$ relates the vertical P-wave velocity $v_v$ with the NMO velocity $v_n$ , while the anellipticity parameter $\eta$ relates the horizontal velocity $v_h$ with the NMO velocity $v_n$ . Shan (2009) suggests that the exact dispersion relationship 1 can be approximated by a rational function $R_{n,m}(S_r)$ :

$$R_{n,m}(S_r)=\frac{P_n(S_r)}{Q_m(S_r)},$$ (2)

where

$$P_n(S_r)=\sum_{i=0}^{n} a_i S_r^i$$ (3)

and

$$Q_m(S_r)=\sum_{i=0}^{m} b_i S_r^i.$$ (4)

Moreover, when the polynomials in equations 3 and 4 are of the same degree, namely $m=n$ , dispersion relationship 2 can be further split as follows:

$$S_z = 1 - \sum_{i=1}^{n}\frac{\alpha_i S_r^2}{1-\beta_i S_r^2}.$$ (5)

The coefficients $\alpha_i$ and $\beta_i$ can be obtained by solving the least-square problem below:

$$\min \sum_{S_r} \left ( \sqrt{\frac{1-(1+2\delta)S_r^2}{1-2\eta(2... ...t(1 - \sum_{i=1}^{n}\frac{\alpha_i S_r^2}{1-\beta_i S_r^2} \right ) \right )^2.$$ (6)

For second-order coefficients the exact and approximated dispersion curves are shown in Figure 1(a), given $\eta=0.14$ and $\delta=0.2$ . Curve A is the exact dispersion relation from Equation 1. Curve B is obtained from a previous estimation by Ristow and Ruhl (1997), and curve C is obtained using the optimized coefficients. Apparently, the dispersion relation using the optimized coefficients is a better approximation compared with the previous method which uses Taylor expansion and assumes weak anisotropy. The relative errors between these two approximated curves and the exact dispersion curve are plotted in Figure 1(b). Within a tolerance of 1% relative error in the dispersion relation, the optimized dispersion is accurate up to $60^\circ$ , while the Taylor approximation is only accurate up to $30^\circ$ .

kz1,err1
Figure 1. (a) Dispersion relation curves: A, exact dispersion relation curve from equation 1; B, approximated dispersion curve from weak anisotropy and Taylor expansion; C, approximated dispersion curve from optimization. (b) Relative dispersion error: D, relative error between B and A; E, relative error between C and A.

The tables for coefficients $\alpha$ and $\beta$ for $\eta$ ranging from 0 to $0.15$ and $\delta$ ranging from $-0.004$ to $0.2$ are shown in Figure 2. In general, parameter $\alpha$ is more sensitive to the change in $\delta$ than to the change in $\eta$ . Parameter $\beta$ has similar sensitivities to both $\eta$ and $\delta$ .

coef
Figure 2. (a) Table for $\alpha$ and (b) table for $\beta$ at discrete $\eta$ and $\delta$ locations.

Wave-equation migration velocity analysis for VTI media using optimized implicit finite difference