# Dictionary:delta

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${\displaystyle \delta ^{*}}$ is one of the Thomsen anisotropic parameters (q.v.):

${\displaystyle \delta ^{*}={\frac {1}{2(c_{33})^{2}}}\left[2(c_{13}+c_{44})^{2}-(c_{33}-c_{44})(c_{11}+c_{33}-2c_{44})\right]}$

where cij indicate elements in the elastic constants matrix [1]. Another Thomsen anisotropic parameter is ${\displaystyle \varepsilon }$, and with weak anisotropy, ${\displaystyle \delta }$, which is independent of ${\displaystyle \varepsilon }$, is generally used instead of ${\displaystyle \delta ^{*}}$ it is the most critical factor for transverse isotropy[2]:

${\displaystyle \delta ={\frac {1}{2}}\left[{\frac {\varepsilon +\delta ^{*}}{1-{\frac {\beta ^{2}}{\alpha ^{2}}}}}\right]={\frac {1}{2}}{\frac {(c_{13}+c_{44})^{2}-(c_{33}-c_{44})^{2}}{c_{33}(c_{33}-c_{44})}}}$

Several seismic expressions involve ${\displaystyle \delta }$, such as the short-offset moveout correction to the vertical velocity,

${\displaystyle V_{NMO}=\alpha _{\parallel }(1+\delta )}$

For long offsets, another anisotropy parameter, ${\displaystyle \eta }$ (eta) captures the deviation of long-offset P-wave moveout from what it would have been for isotropicity [3]:

${\displaystyle \eta ={\frac {\varepsilon -\delta }{1+2\delta }}}$

## References

1. Thomsen, Leon (1986). "Weak elastic anisotropy". GEOPHYSICS 51 (10): 1954–1966. doi:10.1190/1.1442051. PDF version.
2. Thomsen, Leon (2002). Understanding Seismic Anisotropy in Exploration and Exploitation. Society of Exploration Geophysicists. doi:10.1190/1.9781560801986.
3. Alkhalifah, Tariq; Tsvankin, Ilya (1995). "Velocity analysis for transversely isotropic media". GEOPHYSICS 60 (5): 1550–1566. doi:10.1190/1.1443888.PDF version.