Angle gathers in wave-equation imaging for transversely isotropic media

Acknowledgments

We are grateful to KACST, KAUST, and the Bureau of Economic Geology, University of Texas at Austin for their support. We thank Gilles Lambare, Jeff Shragge and Jun Cao for their critical and helpful review of the paper.

$a$	$$\begin{array}{c} \frac{1}{2} \left(k_h^2-k_m^2\right)^4 \left... ... \eta +1) k_m^2-4 \omega ^2\right)^2\right)\right]^2\end{array}$$
$b$	$$\begin{array}{c} \left(k_h^2-k_m^2\right)^4 \left(-\left(v^2 ... ... v^2 (4 \eta +1) \omega^2\right) -8 \omega ^4\right)\end{array}$$
$c$	$$\begin{array}{c} \frac{1}{2} \left(k_h^2-k_m^2\right)^4 \left... ... \eta +1) k_m^2-4 \omega ^2\right)^2\right)\right]^2\end{array}$$

Table 1. Exact analytical equations for the coefficients of equation (4).

$a$	$$\begin{array}{c} -\left(k_h^2-k_m^2\right){}^4 \left(v k_h-v ... ...k_m^2-4 \omega ^2\right){}^2-k_m^4 v_z^4\right){}^2 \end{array}$$
$b$	$$\begin{array}{c} 2 \left(k_h^2-k_m^2\right){}^4 \left(v k_h-v... ...^2 k_m^2-4 \omega ^2\right){}^2+k_m^4 v_z^4\right) \end{array}$$
$c$	$$\begin{array}{c} -\left(k_h^2-k_m^2\right){}^4 \left(v k_h-v ... ...k_m^2-4 \omega ^2\right){}^2-k_m^4 v_z^4\right){}^2 \end{array}$$

Table 2. Exact analytical equations for the coefficients of equation (4) in the case of elliptic anisotropy ($\eta =0$ ).

$a$	$$\begin{array}{c} -\left(k_h^2-k_m^2\right){}^4 \left(v k_h-v ... ...^2 \left(k_h^2+k_m^2\right)-2 \omega ^2\right){}^2 \end{array}$$
$b$	$$\begin{array}{c} \left(k_h^2-k_m^2\right){}^4 \left(v k_h-v k... ... k_m^4-4 \omega ^2 k_m^2\right)+8 \omega ^4\right) \end{array}$$
$c$	$$\begin{array}{c} -\left(k_h^2-k_m^2\right){}^4 \left(v k_h-v ... ...^2 \left(k_h^2+k_m^2\right)-2 \omega ^2\right){}^2 \end{array}$$

Table 3. Exact analytical equations for the coefficients of equation (4) in the case of isotropy ($\eta =0$ , $v_z=v$ ).

Angle gathers in wave-equation imaging for transversely isotropic media