Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies

APPENDIX A: $v_{sv}(\theta_m)$

For purposes of comparison, it is useful to know the exact value and also some related approximations to the exact value of the quasi-SV wave speed $v_{sv}(\theta)$ at the angle $\theta = \theta_m$ = -- which occurs close to (but not exactly at) the extreme value of $v_{sv}(\theta)$ over all angles (see discussion after equation 25 in the main text).

Evaluation gives the exact result

$$v_{sv}^2(\theta_m) = \frac{\sin^2\theta_m}{2\rho}(c_{11}-c_{... ...rac{c_{33}+c_{44}}{c_{33}-c_{44}} - 2\sqrt{1-\zeta_m}\right].$$ (A-1)

After substituting $\sin^2\theta_m = (c_{33}-c_{44})/(c_{11}+c_{33}-2c_{44})$, expanding the square root $\sqrt{1-\zeta_m} \simeq 1 - \zeta_m/2$, and several more steps of simplification, a useful approximate expression is

$$v_{sv}^2(\theta_m) \simeq v^2_s(0)\left[1 + \frac{\zeta_m}{2... ...44})(c_{33}-c_{44})}{c_{44}(c_{11}+c_{33}-2c_{44})}\right].$$ (A-2)

And finally, by approximating the square root of this expression and using (14), we have

$$\frac{v_{sv}(\theta_m)}{v_s(0)} \simeq 1 + \frac{\zeta_m(c_... .../v^2_s(0)\right](\epsilon-\delta) \frac{\sin^2\theta_m}{2}.$$ (A-3)

Equation A-3 can be directly compared to Thomsen's formula for $v_{sv}(\theta)$ in equation 8. The only difference is a factor of $2\cos^2\theta_m$ in the final term. This factor could be unity if $\theta_m = 45^o$, but -- since this never happens for anisotropic media -- the factor always differs from unity and can be either higher or lower than unity depending on whether $\theta_m$ is less than or greater than 45$^o$.

Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies