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Extended Thomsen formulas

A more direct comparison with Thomsen's approximations uses equations 24 and 25 to arrive at approximate formulas for $v_{sv}(\theta)$ and $v_p(\theta)$ analogous to Thomsen's. The resulting expressions, which may be called ``extended Thomsen formulas,'' are given by

\begin{displaymath}
v_p(\theta)/v_p(0) \simeq 1 + \epsilon\sin^2\theta -
(\epsil...
...ta_m\sin^2\theta\cos^2\theta}{[1 - \cos2\theta_m\cos2\theta]}
\end{displaymath} (A-1)

and
\begin{displaymath}
v_{sv}(\theta)/v_s(0) \simeq 1 + \left[v^2_p(0)/v^2_s(0)\rig...
...a_m\sin^2\theta\cos^2\theta}{[1 - \cos2\theta_m\cos2\theta]}.
\end{displaymath} (A-1)

Equations 26 and 27 are the two main approximate results of this paper. So far only two approximations have been made, and both of these came from expanding a square root in a Taylor series, and retaining only the first nontrivial term.

Comparing equations 26 and 27 to equations 6 and 7, the differences are found to lie in a factor of the form:

\begin{displaymath}
\frac{2\sin^2\theta_m}{[1 - \cos2\theta_m\cos2\theta]} \to
\...
...}{2\cos^2\theta_m} \qquad\mbox{as}\qquad \theta \to \theta_m,
\end{displaymath} (A-1)

which depends explicitly on the angle $\theta_m$ determined by $\tan^2\theta_m = (c_{33}-c_{44})/(c_{11}-c_{44})$, and also on $\theta$ itself. As indicated, the expression goes to $1/2\cos^2\theta_m$ in the limit of $\theta \to \theta_m$, which is also in agreement with the results for $v_{sv}(\theta_m)$ in Appendix A. But, since $\sin^2\theta_m = \tan^2\theta_m/(1+\tan^2\theta_m)$ and $\cos2\theta_m = (1-\tan^2\theta_m)/(1+\tan^2\theta_m)$, useful identities are
\begin{displaymath}
\sin^2\theta_m = \frac{c_{33}-c_{44}}{c_{11} + c_{33} - 2c_{44}} = 1 - \cos^2\theta_m
\end{displaymath} (A-1)

and
\begin{displaymath}
\cos2\theta_m = \frac{c_{11}-c_{33}}{c_{11} + c_{33} - 2c_{44}} = 1 - 2\sin^2\theta_m.
\end{displaymath} (A-1)

These results can therefore be used, after deducing some of the elastic constants from field data at near offsets, in order to extend the validity of the equations to greater angles and farther offsets. Inversion of such data is however beyond this paper's scope.

To make the formulas 26 and 27 look as much as possible like Thomsen's formulas -- and thereby arrive at a somewhat different understanding of equations 7 and 8, first eliminate $\theta_m$ by arbitrarily setting it equal to some value such as $\theta_m = 45^o$, in which case $2\sin^2\theta_m = 1$ and $\cos2\theta_m = 0$. Then, the $\theta$ dependence in the denominators goes away, and Thomsen's formulas 7 and 8 are recovered exactly. The particular choice $\theta_m = 45^o$ is however completely unnecessary as shall be shown, and furthermore is never valid for any anisotropic medium having $c_{11} \ne c_{33}$.


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Next: DEDUCING FROM SEISMIC DATA Up: EXTENDED APPROXIMATIONS FOR ANISOTROPIC Previous: Compact form for

2007-09-15