Optimized one-way wave equation operator for VTI

For isotropic media, the dispersion relation for the one-way wave equation can be represented as

$$\frac{k_z}{\omega/v}=\sqrt{1-\left(\frac{k_r}{\omega/v} \right)^2},$$ (1)

where $\omega$ is the circular frequency, v=v(x,y,z) is the velocity, k_z is the wavenumber, $k_r=\sqrt{k_x^2+k_y^2}$ is the radial wavenumber, and k_x, k_y are wavenumbers for x and y respectively. Let $S_z=\frac{k_z}{\omega/v}$, and $S_r=\frac{k_r}{\omega/v}$. The square-root function can be approximated by a series of rational functions:

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

The coefficients $\alpha_i$ and $\beta_i$ can be obtained by Taylor-series analysis or rational factorization. If we consider the second-order approximation (n=1) and $\alpha_1=\frac{1}{2}$, $\beta_1=\frac{1}{4}$,we obtain the traditional $45^\circ$ equation. The coefficients $\alpha_i$ and $\beta_i$ can also be obtained by least-squares optimization, and a more accurate finite-difference scheme like the $65^\circ$ equation can be obtained Lee and Suh (1985).

For VTI media, the true dispersion relation requires solving a quartic equation numerically Shan and Biondi (2005). With the assumption that the S-wave velocity is much smaller than the P-wave velocity, the dispersion relation for VTI media can be obtained analytically and represented as follows:  

$$\frac{k_z}{\omega/v_p}=\sqrt{\frac{1-(1+2\varepsilon)\frac{k... ..._p)^2}} {1-2(\varepsilon-\delta)\frac{k_r^2}{(\omega/v_p)^2}}},$$ (3)

where v_p=v_p(x,y,z) is the vertical velocity, and $\varepsilon=\varepsilon(x,y,z)$ and $\delta=\delta(x,y,z)$ are the 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 stiffness moduli. Let $S_z=\frac{k_z}{\omega/v_p}$ and $S_r=\frac{k_r}{\omega/v_p}$. This dispersion relation can be further simplified under the weak anisotropy assumption, and it can be approximated as

$$S_z\approx1-\frac{\alpha_1S_r^2}{1-\beta_1S_r^2},$$ (4)

where $\alpha_1=0.5(1+2\delta)$ and $\beta_1=\frac{2(\varepsilon-\delta)}{1+2\delta}+0.25(1+2\delta)$ Ristow and Ruhl (1997). The coefficients $\alpha_1$ and $\beta_1$ are obtained analytically by Taylor-series analysis.

As in the isotropic case, the coefficients $\alpha_i$ and $\beta_i$ can also be obtained by least-squares optimization. The advantage of least-squares approximation is that I do not have to derive an explicit approximated expression for the dispersion relation analytically. This is especially useful for anisotropic media. For VTI media, I can use the true dispersion relation, and no assumption of small S-wave velocity and weak anisotropy is necessary.

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.

 

kz1 Figure 1 Dispersion relation: curve A is the true dispersion relation; B is the aprroximate dispersion relation by Tayor-series analysis; C is the approximate dispersion relation by optimization.

 

err1 Figure 2 Relative dispersion error: curve D is the relative dispersion error of the approximation by Taylor-series analysis; E is the relative dispersion error of the approximation by optimization.

We can obtain the coefficients a_i and b_i by solving the following optimization problem:

$$\min\int_0^{\sin(\phi)}\left(\sqrt{\frac{1-(1+2\varepsilon)S... ...lta)S_r^2}}-\frac{\sum a_iS_r^2}{\sum b_iS_r^2}\right)^2dS_r,$$ (6)

where $\phi$ is the maximum optimization angle. This problem can be changed to  

$$\min\int_0^{\sin(\phi)}\left(\sqrt{\frac{1-(1+2\varepsilon)S... ...r^2\right)-\left(\sum_{i=0}^{n} a_iS_r^2\right)\right)^2dS_r.$$ (7)

The optimization problem (7) can be solved by a least-squares method. Given $\varepsilon$ and $\delta$, we can solve a_i and b_i from equation (7), and we can approximate k_z as follows:  

$$k_z\approx\frac{\omega}{v_p}\frac{\sum_{i=0}^{n}a_iS_r^i}{\sum_{i=0}^{m}b_iS_r^i}.$$ (8)

As Ma (1981) suggested, if m=n, equation (8) can be further split into a rational-function series as follows:  

$$k_z\approx \frac{\omega}{v_p}\left(1-\sum_{i=1}^{n}\frac{\alpha_iS_r^2}{1-\beta_i S_r^2}\right).$$ (9)

The dispersion error of approximation (9) is given by

$$\Delta k_z=\frac{\omega}{v_p}\left[\sqrt{\frac{1-(1+2\vareps... ..._{i=1}^{n}\frac{\alpha_iS_r^2}{1-\beta_i S_r^2}\right)\right].$$ (10)

The relative dispersion error is defined by $\Delta k_z/k_z$.

For the second-order approximation (m=1,n=1), Figure shows the true and approximated dispersion relation, given $\varepsilon=0.4$ and $\delta=0.2$. In Figure , curve A is the true dispersion relation curve. B is the approximated dispersion suggested by Ristow and Ruhl (1997), in which $\alpha_1=0.700000$ and $\beta_1=0.635714$. C is the approximated dispersion relation by the least-squares optimization, in which $\alpha_1=0.664820$ and $\beta_1=0.948380$. The dispersion relation by optimization (C) approximates the true dispersion relation better than the approximation using Taylor-series analysis and the weak anisotropy assumption.

Figure shows the relative dispersion error. D is the relative dispersion error of the approximation using the Taylor-series analysis. E is the relative dispersion error of the optimized one-way wave operator. Figure shows that opimization greatly improves the dispersion relation. If we accept a one-percent dispersion error, the optimized one-way wave-equation is accurate to $60^\circ$ while the approximation using Taylor-series analysis is accurate to only $30^{\circ}$.

 

impulse
Figure 3 Impulse responses: (a) Optimized finite-difference method; (b) Finite-difference method by Tayor-series analysis; (c) Phase-shift method.