# Dictionary:Thomsen anisotropic parameters

The relationship between the stress and strain vectors for polar anisotropic (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 ^{[1]}

The five independent constants, c_{11}, c_{13}, c_{33}, c_{44}, c_{66}, 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 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^{[2]}:

For weak polar anisotropy, the velocities of P- and S-waves at the angle θ with the symmetry axis are ^{[3]}:

See *polar anisotropy* (transverse isotropy).

## References

- ↑ 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.