Exact seismic velocities for transversely isotropic media and extended Thomsen formulas for stronger anisotropies

A different type of approximation to the exact anisotropic wave velocities as a function of incidence angle in transversely isotropic (TI) media is explored. This formulation extends Thomsen's weak anisotropy approach to stronger deviations from isotropy without significantly affecting the equations' simplicity. One easily recognized improvement is that the extreme value of the quasi-SV-wave speed v(sv)(theta) is located at the correct incidence angle theta=theta(ex) rather than always being at the position theta=45 degrees. This holds universally for Thomsen's approximation, although theta(ex)equivalent to 45 degrees actually is never correct for any TI anisotropic medium. Wave-speed magnitudes are more closely approximated for most values of the incidence angle, although there may be some exceptions depending on actual angular location of the extreme value. Furthermore, a special angle theta=theta(m) (close to theextreme point of the SV-wave speed and also needed by the new formulas) can be deduced from the same data normally used in weak anisotropy data analysis. All the main technical results are independent of the physical source of the anisotropy. Two examples illustrate the use of obtained results based on systems having vertical fractures. The first set of model fractures has axes of symmetry randomly oriented in the horizontal plane. Such a system is then isotropic in the horizontal plane and thus exhibits vertical transversely 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 the system exhibits horizontal transverse isotropic (HTI) symmetry. Both VTI and HTI systems, as well as any other TI medium (whether because of fractures, layering, or other physical causes), are easily treated with the new phase-speed formulation.