The relationship between the stress and strain vectors for polar anisotropic (vertical transversely isotropic) media can be expressed as , where C is the stiffness tensor as shown in Figure H-7. With the z-axes as the symmetry axis, we have 
The five independent constants, c11, c13, c33, c44, c66, for weak anisotropy have been combined into the Thomsen parameters that relate more directly to seismic data:
P-wave velocity parallel to the symmetry axis
S-wave velocity parallel to the symmetry axis
Half fractional change in the P-wave velocity
Half fractional change in the S-wave velocity
where indicate elements in the stiffness matrix. Note that , and are dimensionless, reduce to zero in the special case of isotropy, and have values smaller than 0.5, frequently much smaller. For longer offsets another parameter, (eta), captures the deviation of the long-offset P-wave moveout from what it would have been for an isotropic medium:
For weak polar anisotropy, the velocities of P- and S-waves at the angle θ with the symmetry axis are :
Not only are these expressions algebraically simpler than the corresponding "exact" equations, but they also have fewer degrees of freedom.
For example, above contains three parameters: , , and , whereas the "exact" expression contains four: , , , and . This makes determination of the parameters from field data significantly easier.
Note also that if is actually small, then the expression for above simplifies to
but this does not further simplify the velocity expressions above.
See polar anisotropy (transverse isotropy).
- ↑ Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966.
- ↑ Alkhalifah, T. and Tsvankin, I., 1995, Velocity analysis for transversely isotropic media: Geophysics, 60, 1550–1566.
- ↑ Thomsen, L., 2002, Understanding seismic anisotropy in exploration and exploitation: SEG-EAGE Distinguished Instructor Series #5: Soc. Expl. Geophys.
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Thomsen anisotropic parameters