VTI Common Azimuth with bounds

In this section, we derive an analytical form for the Common Azimuth migration operator for VTI media. As before, we start with the dispersion relation given by Alkhalifah (1998), now represented in terms of ray parameters:  

$$p_{z} = \frac{1}{v}\sqrt{\frac{1 - (1 + 2\epsilon)(p_x^2 + p_y^2)v^2}{1 - 2\eta (1 + 2\delta)(p_x^2 + p_y^2)v^2}} .$$ (18)

Using the geometrical constraints on the ray path for Common Azimuth migration (equation(11)) and after some simplifications, we get:

 

a₀p_sy⁴ + a₁p_ry⁴ + a₂p_ry² + a₃psy² = 0. (19)

where

$$a_0 = 2 \eta (1 + 2\delta)v_s^4,$$ (20)

$$a_1 = - 2 \eta (1 + 2\delta)v_r^4,$$ (21)

 

$$a_2 = v_r^2\Biggl\{1 - 2\eta (1 + 2\delta)v_r^2p_{rx}^2 - (1 + 2\epsilon)p_{rx}^2v_s^2\Biggr\},$$ (22)

 

$$a_3 = - v_s^2\Biggl\{1 - 2\eta (1 + 2\delta)v_s^2p_{sx}^2 - (1 + 2\epsilon)p_{sx}^2v_r^2\Biggr\}.$$ (23)

Note that equation (19) is a quartic equation in both k_hy and k_my. This equation might be solved numerically for k_hy for given k_my values. But we want to have a closed form analytical solution. Rewriting equation (19) in a more suitable form, we get:

$$\begin{eqnarray} v_r^2p_{ry}^2\Biggl\{1 - 2\eta (1 + 2\delta)v_r^2p_{rx}^2 - 2 \... ...}^2 - 2 \eta (1 + 2\delta)v_r^2p_{sy}^2\Biggr\} & - A_0 + A_1 = 0.\end{eqnarray}$$ (24)

where

$$A_0 = v_r^2p{ry}^2(1 + 2\epsilon)p_{sx}^2v_s^2,$$ (25)

$$A_1 = v_s^2p_{sy}^2(1 + 2\epsilon)p_{rx}^2.$$ (26)

Considering the first term in the curly bracket on the RHS of equation(24), we observe that in order to drop the fourth order term in p_ry, we need:

$$1 - 2\eta (1 + 2\delta)v_r^2p_{rx}^2 \gt\gt 2 \eta (1 + 2\delta)v_r^2p_{ry}^2.$$ (27)

This can be rewritten as:

$$1 - 2\eta (1 + 2\delta) sin^2 \theta_x \gt\gt 2 \eta (1 + 2\delta) \sin^2 \theta_y.$$ (28)

Recognizing that $\sin^2\theta$ is bounded in the interval (0,1), we can write:

$$1 \gt\gt 4 \mid\eta (1 + 2\delta)\mid.$$ (29)

Thus, we obtain a bound on $\eta$ as:

$$\mid\eta\mid << \frac{1}{4(1 + 2\delta)} \approx \frac{1}{4}.$$ (30)

A similar analysis can be carried out for the second term in curly brackets on the RHS of equation (24) for dropping the fourth order term in p_sy, which again gives the same bound on $\eta$. Thus for VTI media, when the above bound holds, we can drop the fourth order terms in equation (19) giving:  

$$\frac{p_{ry}}{p_{sy}} = \frac{\sqrt{\frac{1}{v_r^2} - \Biggl... ...x}^2 + 2\eta (1 + 2\delta)\frac{v_r^2}{v_s^2}p_{rx}^2\Biggr\}}}$$ (31)

Now using equations (14) to (17), we get:  

$$\frac{\hat{k}_{hy}}{k_{my}} = \frac{A - B}{A + B},$$ (32)

where  

$$A = \sqrt{\frac{\omega^2}{v_r^2} - \frac{1}{4}\Biggl\{(1 + 2... ... (1 + 2\delta)\frac{v_s^2}{v_r^2}(k_{mx} - k_{hx})^2\Biggr\}} ,$$ (33)

and  

$$B = \sqrt{\frac{\omega^2}{v_s^2} - \frac{1}{4}\Biggl\{(1 + 2... ...a (1 + 2\delta)\frac{v_r^2}{v_s^2}(k_{mx} + k_{hx})^2\Biggr\}}.$$ (34)

This equation gives an analytical form of the stationary path approximation for VTI media. As before, the vertical wavenumber, $\hat{k}_{zvti}$, is evaluated along this stationary path as:

$$\hat{k}_{zvti} = DSR[\omega, \bold k_m, k_{hx}, \hat{k}_{hy},z ]$$ (35)

Note that when $\eta = 0$, meaning for elliptically anisotropic media equation (34), reduces to equation (9) derived earlier for elliptically anisotropy media.