View source for Dictionary:Thomsen anisotropic parameters

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{{#category_index:T|Thomsen anisotropic parameters (tom&#x2019; s&#x2202;n)}}
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The relationship between the stress <math>\sigma</math> and strain <math>\varepsilon</math> vectors for polar anisotropic (vertical transversely isotropic) media can be expressed as <math>\sigma=\textbf{C}\varepsilon</math>, where <b>C</b> is the stiffness tensor as shown in Figure [[Special:MyLanguage/Dictionary:Fig_H-7|H-7]]. With the ''z''-axes as the symmetry axis, we have <ref> Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966. </ref>

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<center>
<math>
\begin{bmatrix}
\sigma_{xx} \\
\sigma_{yy} \\
\sigma_{zz} \\
\sigma_{xy} \\
\sigma_{yz} \\
\sigma_{zx} 
\end{bmatrix}
=
\begin{bmatrix}
c_{11}            & c_{11} - 2 c_{66} & c_{13} & 0      & 0      & 0  \\
c_{11} - 2 c_{66} & c_{11}            & c_{13} & 0      & 0      & 0  \\
c_{13}            & c_{13}            & c_{33} & 0      & 0      & 0  \\
0                 & 0                 & 0      & c_{44} & 0      & 0  \\
0                 & 0                 & 0      & 0      & c_{66} & 0  \\
0                 & 0                 & 0      & 0      & 0      & c_{66} 
\end{bmatrix}

<!--T:3-->
\begin{bmatrix}
\varepsilon_{xx} \\
\varepsilon_{yy} \\
\varepsilon_{zz} \\
\varepsilon_{xy} \\
\varepsilon_{yz} \\
\varepsilon_{zx} 
\end{bmatrix}

<!--T:4-->
</math>
</center>

<!--T:5-->
The five independent constants, c<sub>11</sub>, c<sub>13</sub>, c<sub>33</sub>, c<sub>44</sub>, c<sub>66</sub>, for weak anisotropy have been combined into the Thomsen parameters that relate more directly to seismic data: 

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P-wave velocity parallel to the symmetry axis

<!--T:7-->
<center><math>
\alpha_0 = \sqrt{\frac{c_{33}}{\rho}}
</math></center>

<!--T:8-->
S-wave velocity parallel to the symmetry axis

<!--T:9-->
<center><math>
\beta_0 = \sqrt{\frac{c_{44}}{\rho}}
</math></center>

<!--T:10-->
Half fractional change in the P-wave velocity

<!--T:11-->
<center><math>
\varepsilon = \frac{c_{11}-c_{33}}{2 c_{33}} </math></center>

<!--T:12-->
Half fractional change in the S-wave velocity

<!--T:13-->
<center><math>
\gamma = \frac{c_{66}-c_{44}}{2 c_{44}} </math></center>


<!--T:14-->
<center><math>
\delta = \frac {1} {2} \frac{(c_{13}+c_{44})^2-(c_{33}-c_{44})^2}{c_{33}(c_{33}-c_{44})}
</math></center>


<!--T:15-->
where <math>c_{ij}</math> indicate elements in the stiffness matrix. Note that <math>\varepsilon</math> , <math>\gamma</math> and <math>\delta</math> 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, <math>\eta</math> (eta), captures the deviation of the long-offset P-wave moveout from what it would have been for an isotropic medium<ref>Alkhalifah, T. and Tsvankin, I., 1995, Velocity analysis for transversely isotropic media: Geophysics, 60, 1550–1566.</ref>: 

<!--T:16-->
<center><math>
\eta=\frac{\varepsilon-\delta}{1+2\delta}
</math></center>

<!--T:17-->
For weak polar anisotropy, the velocities of P- and S-waves at the angle &#x03B8; with the symmetry axis are <ref> Thomsen, L., 2002, Understanding seismic anisotropy in exploration and exploitation: SEG-EAGE Distinguished Instructor Series #5: Soc. Expl. Geophys.  </ref>:

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<center><math>
V_p(\theta)=\alpha_0 (1+\delta\sin^2{\theta}\cos^2{\theta}+\varepsilon\sin^4{\theta})
</math></center>

<!--T:19-->
<center><math>
V_{sv}(\theta)=
\beta_{0} 
[1 + \frac{\alpha_0^2}{\beta_{0}^2}  (\varepsilon-\delta) \sin^2{\theta}\cos^2{\theta}]
</math></center>

<!--T:20-->
<center><math>
V_{sh}(\theta)=
\beta_{0} (1+\gamma\sin^2{\theta})
</math></center>

Not only are these expressions algebraically simpler than the corresponding "exact" equations, but they also have fewer degrees of freedom. 
For example, <math>V_p</math> above contains three parameters: <math>\alpha_0</math>, <math>\delta</math>, and <math>\epsilon</math>, whereas the [[Seismic_anisotropy|"exact" expression]] contains four: <math>c_{11}/\rho</math>, <math>c_{33}/\rho</math>, <math>c_{13}/\rho</math>, and <math>c_{44}/\rho</math>. This makes determination of the parameters from field data significantly easier.

Note also that if <math>\delta</math> is actually small, then the expression for <math>\delta</math> above simplifies to 
<center><math>
\delta\rightarrow \delta _{weak}=\frac{c_{13}-(c_{33}-2c_{44})}{c_{33}}
</math></center>

but this does not further simplify the velocity expressions above.

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See [[Special:MyLanguage/Dictionary:polar_anisotropy|''polar anisotropy'']] (transverse isotropy).

== References == <!--T:22-->

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==External links== <!--T:23-->

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