There are three Thomsen parameters: ,, and . Parameter is essentially the fractional difference between the SH-wave velocities in the horizontal and vertical directions for a VTI medium. Similarly, parameter is essentially the difference between the P-wave velocities in the horizontal and vertical directions. Parameter is more difficult to interpret, but contributes in an essential way both to near vertical P-wave speed variations, and also to the angular dependence of the SV-wave speed. There are a great many steps that go into Thomsen parameter calculations since the crack density effects are most conveniently expressed in terms of the compliance matrix while the Thomsen parameters are usually defined instead in terms of the stiffness matrix (inverse of the compliance matrix). I will not show my work here, but merely quote the final result for the case of randomly oriented vertical fractures considered in the previous subsection.
For present purposes, I just want to show in a quick way how this method works, so I will concentrate on the easiest two parameters which are and . For these two parameters, I have the following results:
Figure 2 Computed values of the Thomsen parameters ,, , for four distinct EMT models: noninteracting (black), CPA (red), DS (blue) and the Budiansky-O'Connell self consistent (green). The parameter is not seen separately here because for this choice of crack microstructure (randomly oriented vertical cracks) to the order to which we are working, for small crack densities.
Examples of Thomsen's parameters for various choices of EMT are displayed in Figure 2. The results illustrate how estimates of and obtained from four different isotropic estimation schemes [noninteracting, DS (Zimmerman, 1991), CPA (Berryman, 1980), and nonsymmetric self-consistent scheme of Budiansky and O'Connell (1976), and O'Connell and Budiansky (1977)] can then be used to predict what values Thomsen's parameters should take in field data.
Some judgment is required then as to the most appropriate EMT to use, and prior work shows that some knowledge of microstructure can serve as a very useful guide when making this choice (Berge et al., 1993a; 1995).