Thomsen's weak anisotropy method (Thomsen, 1986), being an approximation designed specifically for use in velocity analysis for exploration geophysics, is clearly not exact. Approximations incorporated into the formulas become most apparent for greater angles from the vertical, especially for compressional and vertically polarized shear velocities and , respectively. Angle is measured from the -vector pointing into the earth.

For reference purposes, we include here the exact velocity formulas for P, SV, and SH seismic waves at all angles in a VTI elastic medium. These results are available in many places (Rüger, 2002; Musgrave, 2003), but were taken specifically from Berryman (1979) with some minor changes of notation. The results are:

(1) |

(2) |

(3) |

(4) |

Expressions for phase velocities in Thomsen's weak anisotropy limit can be found in many places, including Thomsen (1986, 2002) and Rüger (2002). The pertinent expressions for phase velocities in VTI media as a function of angle ,measured as before from the vertical direction, are

(5) |

(6) |

(7) |

In each case, Thomsen's approximation has included a step that removes the
square on the left-hand side of the equation, by expanding a square root of the
right hand side. This step introduces a factor of multiplying
the terms on the right hand side, and -- for example -- immediately
explains how equation (7) is obtained from (4).
The other two equations for and , *i.e.*,
(5) & (6), involve additional approximations as
well that we will not attempt to explain here.

The three resulting Thomsen (1986) seismic parameters for weak anisotropy with VTI symmetry are ,,and

(8) |

5/6/2007