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
is located near the correct incidence angle
rather than always being at the position
, which universally holds
for Thomsen's approximation -- although
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
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.