Now the trick to get from horizontal fractures and VTI to aligned vertical fractures and HTI symmetry is relatively simple. We will not need to make any effort to relabel the cij's. Rather we just change the meaning of the labels. As long as we stay mentally oriented in the reference frame of the fractures themselves, we can continue to view the z-direction as the symmetry axis and the xy-plane, as the plane of the fractures. The only change we need to make arises from the fact that the surface, where we shoot our seismic survey, is now at 90o from the fracture plane, whereas for horizontal fractures the surface was at 0o from the fracture plane. This observation implies that, wherever the angle (measured in radians) appeared in our previous formulas, now we must replace it by radians. Thus, and vice versa in the formulas. This algorithm is exactly right only for those planes that are vertical and also perpendicular to the fracture plane, i.e., at azimuthal angles . For all angles, we actually need to replace by .Then, when or , there is no angular dependence since we are in the plane of the fracture.
For the dependence, taking ,is actually a handier way to proceed, because then we can reduce all the formulas to the same equivalent form as the one Thomsen had originally chosen -- if we choose to do so. It is also helpful to backup one step in the Thomsen derivation and restore squares, thereby ``unexpanding'' the square root. Certain approximations are then undone, and the final formulas we obtain will be more accurate.
If , , and are the Thomsen parameters for the VTI symmetry (horizontal fracture), then, for example,
Similar calculations for vp2 and vsv2 give
Examples of these results for small () and higher () crack densities [see Berryman and Grechka (2006) for details of the methods used to obtain all the Sayers and Kachanov crack-influence parameters from simulation data and Berryman (2007) for a full discussion of the reservoir application] are presented in Figures 1-6.