Optimized one-way wave equation operator for TTI

In a VTI medium, the phase velocity of qP- and qSV-waves in Thomsen's notation can be expressed as Tsvankin (1996):  

$$\frac{V^2(\theta)}{V^2_{P0}}=1+\varepsilon\sin^2(\theta)-\fr... ...{f}\right)^2- \frac{2(\varepsilon-\delta)\sin^2(2\theta)}{f} },$$ (1)

where $\theta$ is the phase angle of the propagating wave, and f=1-(V_S0/V_P0)². V_P0 and V_S0 are the qP- and qSV- wave velocities in the vertical direction, respectively. $\varepsilon$ and $\delta$ are anisotropy parameters defined by Thomsen (1986):

$$\varepsilon=\frac{C_{11}-C_{33}}{2C_{33}}, \delta=\frac{(C_{11}+C_{44})^2-(C_{33}-C_{44})^2}{2C_{33}(C_{33}-C_{44})},$$

where C_ij are elastic moduli. In equation (1), $V(\theta)$ is qP-wave phase-velocity when the sign in front of the square root is positive, and the qSV-wave phase velocity for a negative sign.

Rotating the symmetry axis from vertical to a tilted angle $\varphi$, we obtain the phase velocity of a tilted TI medium whose symmetry axis forms an angle $\varphi$ with the vertical direction:  

$$\frac{V^2(\theta,\varphi)}{V^2_{P0}}=1+\varepsilon\sin^2(\th... ...t)^2- \frac{2(\varepsilon-\delta)\sin^22(\theta-\varphi)}{f} }.$$ (2)

Here, in contrast to equation (1), $\varepsilon$ and $\delta$ are now defined in a direction tilted by the angle $\varphi$ from the vertical direction. V_P0 is the qP-wave velocity in the direction parallel to the symmetry axis.

The phase angle $\theta$ is related to the wavenumbers k_x and k_z by:  

$$\sin \theta=\frac{V(\theta,\varphi)k_x}{\omega},\ \ \ \ \ \ \ \cos \theta=\frac{V(\theta,\varphi)k_z}{\omega},$$ (3)

where $\omega$ is the temporal frequency. Let $S_x= k_x/\frac{\omega}{V_{P0}}$, and $S_z= k_z/\frac{\omega}{V_{P0}}$.We can obtain a dispersion relation equation from equations (2) and (3):

 

d₄S_z⁴+d₃S_z³+d₂S_z²+d₁S_z+d₀=0, (4)

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

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

Equation (4) is a quartic equation and there is no explicit expression for its solution.

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

$$R_{n,m}(S_r)=\frac{P_n(S_r)}{Q_m(S_r)},$$ (5)

where

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

and

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

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.

For an isotropic or VTI medium, S_z is an even function of S_x. We can approximate the dispersion relation with even rational functions, such as S_x², S_x⁴. For TTI media, S_z is not an symmetric function of S_x. Therefore, in addition to even rational functions, we need odd rational functions to approximate the dispersion relation, such as S_x, S_x³. The fourth order approximation for the dispersion relation of TTI media is as follows:  

$$S_z(S_x)\approx S_{z0}+\frac{a_1S_x^2+c_1S_x}{1+b_1S_x^2}+\frac{a_2S_x^2+c_2S_x}{1+b_2S_x^2},$$ (6)

where S_z0=S_z(0) and the coefficients c₁,b₁,a₁,c₂,b₂,a₂ can be estimated by least-squares methods. They are functions of the anisotropy parameters $\varepsilon$, $\delta$ and the tilting angle $\phi$. When these parameters vary laterally, the coefficients c₁,b₁,a₁,c₂,b₂,a₂ also vary laterally. It is too expensive to run least-squares estimation for each grid point during the wavefield extrapolation. They can be calculated and stored in a table before the wavefield extrapolation. During the wavefield extrapolation, given the anisotropy parameters $\varepsilon$, $\delta$, and the tilting angle $\phi$, we search for these coefficients from the table and put them into the finite-difference algorithm. Given the coefficients found from the table, the finite difference algorithm in TTI media is similar to the isotropic media.