Optimized finite-difference for TTI media

The dispersion relation of TTI media can be characterized by a quartic equation as follows:

 

d₄S_z⁴+d₃S_z³+d₂S_z²+d₁S_z+d₀=0, (1)

where the coefficients d₀,d₁,d₂,d₃, and d₄ are defined as follows:

$$\begin{array} {l} d_0=(2+2\varepsilon\cos^2\varphi-f)S_x^2- ... ...phi-\frac{f}{2}(\varepsilon-\delta)\sin^22\varphi, \end{array}$$

where $\varepsilon$ and $\delta$ are Thomsen anisotropy parameters Thomsen (1986) and $\varphi$ is the tilting angle of the media. Theoretically, equation 1 can be solved analytically, but there is no explicit analytical expression for its solution. The solid line in Figure shows how the dispersion relation looks, given the anisotropy parameters $\varepsilon=0.4$, $\delta=0.2$ and the tilting angle $\varphi=30^\circ$.Note that S_z is not a symmetric function of S_x. And S_z has two branches when S_x>0.8. One of them represents the up-going waves and the other one represents the down going waves. Therefore, in a TTI medium waves may overturn even though it is homogeneous.

Conventional implicit finite-difference methods are designed by truncating the Taylor series of the dispersion relation. The dispersion relation for TTI media is so complex that it is difficult to derive an analytical Taylor series used for an implicit finite-difference scheme.

Generally, the Padé approximation suggests that if the function $S_z(S_x)\in C^{n+m}$, then S_z(S_x) can be approximated by a rational function R_n,m(S_x):

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

where

$$P_n(S_x)=\sum_{i=0}^{n}a_iS_x^i$$

and

$$Q_m(x)=\sum_{i=0}^{m}b_iS_x^i$$

are polynomials of degree n and m, respectively. The coefficients a_i and b_i can be obtained either analytically by Taylor-series analysis or numerically by least-squares fitting.

S_z is an even function of S_x for isotropic and VTI media. In contrast, S_z is not an symmetric function of S_x for TTI media. It's well known that an general function can be decomposed into an even function and an odd function. We approximate the even part of the dispersion with the even rational functions, such as S_x², S_x⁴ and approximate the odd part with odd rational functions, such as S_x,S_x³.

 

dispersion
Figure 1 Comparison of the true and approximate dispersion relations for a TTI medium with $\varepsilon=0.4$, $\delta=0.2$ and $\varphi=30^\circ$: the solid line is the true dispersion relation for TTI media; the dashed line is the approximate dispersion relation for finite-difference scheme. The dispersion relation for the finite-difference scheme is very close to the true one for negative S_x. When the phase-angle is close to $90^\circ$ or more than $90^\circ$ for the positive S_x, the dispersion for the finite-difference scheme diverge from the true one.

Considering the stability of the finite-difference scheme, I approximate the dispersion relation of TTI media with rational functions as follows:  

$$S_z(S_x)\approx S_{z0}+\frac{a_1S_x^2+c_1S_x}{1+b_1S_x^2}+\frac{a_2S_x^2+c_2S_x}{1+b_2S_x^2},$$ (3)

where S_z0=S_z(0) and the coefficients a₁,b₁,c₁,a₂,b₂,c₂ are estimated by least-squares optimization. They are functions of the anisotropy parameters $\varepsilon$, $\delta$ and the tilting angle $\varphi$. Figure compares the true dispersion relation with the approximate dispersion relation. The solid line is the true dispersion relation (equation 1) and the dashed line is the approximate dispersion relation for the finite-difference scheme ( equation 3). The dispersion relation for the finite-difference scheme is very close to the true one for the negative S_x. When the phase-angle is close to $90^\circ$ or more than $90^\circ$ for positive S_x, the dispersion for the finite-difference scheme diverges from the true one. Figure shows the relative dispersion error defined as follows:

$$E(S_x)=\frac{S_z^{fd}(S_x)-S_z^{true}(S_x)}{S_z^{true}(S_x)},$$ (4)

where S_z^true(S_x) is the value of S_z calculated from equation 1 and S_z^fd(S_x) is the value of S_z from equation 3 using the coefficients from the least-squares estimation.

 

err
Figure 2 Relative dispersion relation error of finite-difference approximation for a TTI medium with $\varepsilon=0.4$, $\delta=0.2$ and $\varphi=30^\circ$.

For a laterally varying medium, the anisotropy parameters vary laterally. As a consequence, the coefficients for the finite-difference scheme vary laterally. It is too expensive to estimate these coefficients for each discrete grid during the wavefield extrapolation. After estimating the minimum and maximum value of the anisotropy parameters and the tilting angle, I compute the coefficients for the anisotropy parameters and the tilting angle in these ranges and store them in a table before the migration. During the wavefield extrapolation, given the anisotropy parameters $\varepsilon$, $\delta$, and the tilting angle $\varphi$, I search the coefficients in the table and put them into the finite-difference scheme. Given the coefficients found from the table, the finite difference algorithm in TTI media is the same as the isotropic media. The table of coefficients is small, and the computation cost for table-searching is trivial compared to that of solving the finite-difference equation. Therefore, the cost of the optimized implicit finite-difference for TTI media is similar to that of the conventional finite-difference methods for isotropic media.

 

model
Figure 3 The velocity model and anisotropy parameters: (a) the tilting angle of the TTI medium; (b) the velocity paralleling the symmetry axis; (c) the anisotropy parameter $\varepsilon$; (d) the anisotropy parameter $\delta$.