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

VTI Symmetry

Figures 1-6 present results for the case of vertical fractures having an isotropic distribution of normals (symmetry axes) in the horizontal plane. The resulting medium has VTI symmetry.

A first observation is that the low crack density results for $v_{sv}(\theta)$ are nearly constant, showing that $\epsilon - \delta \simeq 0$. When this happens for $v_{sv}(\theta)$, it is also true that $v_{p}(\theta)$ is approximately elliptical. Of course, the exact results for $v_{sh}(\theta)$ are always elliptical, but the Thomsen and new approximate results are only approximately elliptical.

Secondly, all three velocity models (exact, Thomsen, and the new approximation) give very similar results for all cases shown when $\nu _0 = 0.4375$. There are however some significant differences among the results for $\nu _0 = 0.00$, especially for $v_{sv}(\theta)$ and $v_p(\theta)$ - the largest deviations from the exact curves being those for Thomsen's approximations (red curves) in both cases.

FIG7
Figure 7. For aligned vertical fractures and HTI symmetry: examples of anisotropic quasi-P compressional wave speed ($v_p$) for Poisson's ratio of the host medium $\nu _0 = 0.00$. Velocity curves in black are exact for the fracture model discussed in the text. The Thomsen weak anisotropy velocity curves for the same fracture model are then overlain in red. Finally, the new curves for the extended Thomsen approximation valid for stronger anisotropies are overlain in blue. If any of these curves is not visible, it is because one or possibly two other curves are covering them.

FIG8
Figure 8. Same as Figure 7 for SH shear wave speed ($v_{sh}$).

FIG9
Figure 9. Same as Figure 7 for quasi-SV shear wave speed ($v_{sv}$).

FIG10
Figure 10. Same as Figure 7, for a different background medium having Poisson's ratio $\nu _0 = 0.4375$.

FIG11
Figure 11. Same as Figure 8, but the value of $\nu _0 = 0.4375$.

FIG12
Figure 12. Same as Figure 9, but the value of $\nu _0 = 0.4375$.

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