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

Thomsen's weak anisotropy formulation (Thomsen, 1986), being a collection of approximations designed specifically for use in velocity analysis for exploration geophysics, is clearly not exact. Approximations incorporated into the formulas become most apparent for angles greater than about 15 from the vertical, especially for compressional and vertically polarized shear wave velocities and , respectively. For VTI media, angle is measured from the -vector pointing directly into the earth.

For reference purposes, I include here the exact velocity formulas for;
quasi-P, quasi-SV, and SH seismic waves at all angles in a VTI elastic medium.
These results are available in many places
(Postma, 1955; Musgrave, 1959, 2003; Rüger, 2002; Thomsen, 2002),
but were taken directly from Berryman (1979)
with only some minor changes of notation; specifically, the ,,,,, notation for stiffnesses
has been translated to the Voigt stiffness notation wherein , ,
, , , and . The results are:

where

and, finally,

I have purposely written equations 1 and 2 in this way to emphasize the fact that and are closely related since they are actually the two solutions of a quadratic equation having the form:

Any approximations made to one of these two wave speeds should therefore always be reflected in the other for this reason. In particular, any approximation to the square root in should be made consistently for both and .

For VTI symmetry, the stiffness matrix is defined for
by

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 previously mentioned from the vertical direction, are

and

In our present context, , and , where , , and are two stiffnesses of the cracked medium and the mass density of the isotropic host elastic medium. [For the specific physical examples that follow involving models of fractured reservoirs, I assume that the cracks contain insufficient volume to affect the overall mass density significantly.] The three Thomsen (1986) seismic parameters appearing in equations 7-9 for weak anisotropy with VTI symmetry are , , and

Parameter is a measure of the shear wave anisotropy and birefringence. Parameter is a measure of the quasi-P wave anisotropy. Parameter controls the complexity of the shape of the wave fronts for quasi-P and quasi-SV waves;

All three of these parameters , , can play important
roles in the velocities given by equations 7-9
when the anisotropy is large, as would be the case in fractured reservoirs when the
crack densities are high enough. If crack densities are very low, then the SV shear wave will
actually have no dependence on angle of wave propagation. Note that the
so-called anellipticity parameter (Dellinger *et al.*, 1993; Fomel, 2004; Tsvankin, 2005, p. 253),
, vanishes when
--
which (as will be shown) does happen to a very good approximation for low crack densities.
Then, the results are anisotropic but have the special (elliptical) shape to the wave front
mentioned previously.

For each of these phase velocities, the derivation of Thomsen's approximation has
included a step that removes the square on the left-hand side of equations
1, 2, and 4 --
obtained 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 8
is obtained from equation 4. The other two equations for and
, *i.e.*,
equations 7 and 8,
involve additional approximations. More of the details about the nature of these
approximations are elucidated by first obtaining an alternative approximate formulation.

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

2007-09-15