Berryman: Extended Thomsen formulas

I explore a different type of approximation to the exact anisotropic wave velocities as a function of incidence angle in transversely isotropic (TI) media. This formulation extends the Thomsen weak anisotropy approach to stronger deviations from isotropy without significantly affecting the simplicity of the equations. One easily recognized improvement is that the extreme value of the quasi-SV-wave speed $v_{sv}(\theta)$ is located near the correct incidence angle $\theta = \theta_{ex}$ , rather than always being at the position $\theta =$ 45

, which universally holds for Thomsen's approximation -- although $\theta_{ex} \equiv 45^o$ is actually never correct for any TI anisotropic medium. Also, the magnitudes of all the wave speeds are typically (although there may be some exceptions depending on the actual angular location of the extreme value) more closely approximated for all values of the incidence angle. Furthermore, the value of a special angle $\theta_{m}$ (which is close to the location of the extreme and also required by the new formulas) can be deduced from the same data that are normally used in the weak anisotropy data analysis. All the main technical results presented are independent of the physical source of the anisotropy. To illustrate the use of the results obtained, two examples are presented based on systems having vertical fractures. The first set of model fractures has their axes of symmetry randomly oriented in the horizontal plane. Such a system is then isotropic in the horizontal plane and, thus, exhibits vertical transverse isotropic (VTI) symmetry. The second set of fractures also has its axes of symmetry in the horizontal plane, but (it is assumed) these axes are aligned so that the system exhibits horizontal transverse isotropic (HTI) symmetry. Both types of systems, as well as any other TI medium (whether due to fractures or layering or other physical causes) are more accurately treated with the new wave speed formulation.

Abstract: