Exact seismic velocities for VTI and HTI media and extendedThomsen Formulas for stronger anisotropies

I explore a different type of approximation to the exactanisotropic wave velocities as a function of incidence angle invertically transversely isotropic (VTI) media. This method extends theThomsen weak anisotropy approach to stronger anisotropy withoutsignificantly affecting the simplicity of the formulas. One importantimprovement is that the peak of the quasi-SV-wave speed vsv(theta) islocated at the correct incidence angle theta= theta max, rather thanalways being at the position theta = 45o, which universally holds forThomsen's approximation although max theta = 45o is actually nevercorrect for any VTI anisotropic medium. The magnitudes of all the wavespeeds are also more closely approximated for all values of the incidenceangle. Furthermore, the value of theta max (which is needed in the newformulas) can be deduced from the same data that are typically used inthe weak anisotropy data analysis. The two examples presented are basedon systems having vertical fractures. The first set of model fractureshas their axes of symmetry randomly oriented in the horizontal plane.Such a system is then isotropic in the horizontal plane and, therefore,exhibits vertical transverse isotropic (VTI) symmetry. The second set offractures also has axes of symmetry in the horizontal plane, but it isassumed these axes are aligned so that the system exhibits horizontaltransverse isotropic (HTI) symmetry. Both types of systems are easilytreated with the new wave speed formulation.