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

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 are nearly constant, showing that . When this happens for , it is also true that is approximately elliptical. Of course, the exact results for 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 . There are however some significant differences among the results for , especially for and - the largest deviations from the exact curves being those for Thomsen's approximations (red curves) in both cases.

FIG7
For aligned vertical fractures and HTI symmetry:
examples of anisotropic quasi-P compressional wave speed () for
Poisson's ratio of the host medium .
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.
Figure 7. |
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FIG8
Same as Figure 7 for SH shear wave speed ().
Figure 8. |
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FIG9
Same as Figure 7 for quasi-SV shear wave speed ().
Figure 9. |
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FIG10
Same as Figure 7, for a different background medium
having Poisson's ratio
.
Figure 10. |
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FIG11
Same as Figure 8, but the value of
.
Figure 11. |
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FIG12
Same as Figure 9, but the value of
.
Figure 12. |
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Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies |

2007-09-15