Several numerical approximations have been published to describe quasi-SV (qSIO, quasi-P (qP) and Stoneley wave speeds in transversely isotropic media with a vertical axis of symmetry (TIV). The significance of the approximations can be summarized as follows:
General mathematical expressions for calculating the wave speeds are quite complicated as their relevance to various TI parameters is not straightforward. This significantly complicates the processing and interpretation of seismic and acoustic data in anisotropic media.
Approximations in terms of anisotropic parameters c, 6, and 7" provide a better understanding of the dependence of wave signatures on anisotropy parameters.
A significant amount of work using approximations has been made to simplify the general mathematical expressions. Most approximations, however, assume that the medium is weakly anisotropic (i.e., anisotropy is less than 10-~20 %) and the phase angle0 (i.e., angle between wavefront normal and the TI symmetry axis) is small. Nevertheless, in many shale formations anisotropy is often higher than 20%. In deviated and horizontal acoustic logging practices, wave phase angles vary anywhere from 0 ° to 90 °. In these situations, the existing approximations become inaccurate. Using the physical constraints for the elastic stiffness of TIV media, this study derives accurate approximations of qSV, qP, and Stoneley wave speeds in arbitrarily anisotropic formations for the entire range of phase angles. The derived approximations, compared to the exact theory, give quite accurate results. Further, because of their simplicity, the approximations are efficient and easy to use, especially in the estimation of anisotropic parameters using inversion methods. As an example, a strongly anisotropic rock (Mesaverde clayshale, 5501, as in Thmosen, 1986), with c = 0. ~ 4, 6=0.730, and 7,=0.575, is used to demonstrate the new approximation results. The largest error for the approximated qSV wave speed, which occurs around a 45 ° well deviation, is less than 4%. For the same formation, a 30% error is obtained by using the classical approximation for weakly anisotropic media. Stoneley waves in a deviated well penetrating a TIV formation are sensitive to the TI parameters (Tang, 2003). An analytical solution of Stoneley-wave speed as a function of well deviation has been developed. Using the approximations of qSV and qP wave speeds, a quasi-linear approximation of the Stoneley speed for a general anisotropic formation is obtained. For the same formation, the largest error between the approximate and exact results occurs at 90 ° and is less than 2%. The newly developed approximations can be used in the linear inversion of anisotropic parameters. The mathematical expressions for the approximations of qSV and qP are similar to those of weak anisotropy except for the presence of a correction coefficient, which is a function of elastic stiffness, Cll, c33, Can, and phase angle, O.
