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

SUMMARY AND CONCLUSIONS

The main technical result of the paper is equation 22, showing directly how $\zeta(\theta)$ is related to $\theta_m$ and $\zeta_m$. The most significant applications of this result are summarized in equations 26 and 27. These formulas generalize (i.e., extend the validity of) Thomsen's weak anisotropy approach to wider ranges of angles, and stronger anisotropies. These formulas have the clear advantage that they require no more data analysis than Thomsen's formulas for weak anisotropy, but they give more accurate predictions of the wave speeds for longer offsets. In particular, the new formulas are designed to give the peak (or possibly the trough -- if the difference $\epsilon - \delta$ happens to be negative) of the quasi-SV-wave in the right location, (i.e., the correct angle $\theta = \theta_m$ with the vertical), even though the magnitude of these velocities may still be a bit off. For quasi-SV waves, this error made in the velocity magnitude is always less than that made using the original Thomsen formulas. For quasi-P waves the results are somewhat mixed because the errors introduced by poor approximations to $\zeta(\theta)$ can have either sign, positive or negative, depending on what the actual value of $\theta_m$ happens to be. This fact shows that Thomsen's approximation will sometimes give better results at smaller $\theta$ than the new formulas, but other times they will be worse. This fluctuation in the behavior for quasi-P waves is observed in the examples contained in the Figures. Thus, the new approximation does have the advantage of consistency.

Furthermore, the only new parameter needed for implementation is the angle $\theta_m$ itself; however, the value $\theta_m$ is also determined directly from the same data required to compute all the Thomsen parameters (for example, see TABLE 1). A final advantage that is especially helpful for the practical use of the method is that the corrections needed for all the NMO velocities do not change, and so are exactly the same as for Thomsen's formulation.

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