Difference between revisions of "Dictionary:Thomsen anisotropic parameters"

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{{#category_index:T|Thomsen anisotropic parameters (tom’ s∂n)}}
 
{{#category_index:T|Thomsen anisotropic parameters (tom’ s∂n)}}
 
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The relationship between the stress <math>\sigma</math> and strain <math>\varepsilon</math> vectors for polar anisotropic (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>
 
The relationship between the stress <math>\sigma</math> and strain <math>\varepsilon</math> vectors for polar anisotropic (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>
 
<center>
 
<math>
 
<math>
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\end{bmatrix}
 
\end{bmatrix}
  
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\begin{bmatrix}
 
\begin{bmatrix}
 
\varepsilon_{xx} \\
 
\varepsilon_{xx} \\
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\end{bmatrix}
 
\end{bmatrix}
  
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</math>
 
</math>
 
</center>
 
</center>
  
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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:  
 
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
 
P-wave velocity parallel to the symmetry axis
  
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<center><math>
 
<center><math>
 
\alpha_0 = \sqrt{\frac{c_{33}}{\rho}}
 
\alpha_0 = \sqrt{\frac{c_{33}}{\rho}}
 
</math></center>
 
</math></center>
  
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S-wave velocity parallel to the symmetry axis
 
S-wave velocity parallel to the symmetry axis
  
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<center><math>
 
<center><math>
 
\beta_0 = \sqrt{\frac{c_{44}}{\rho}}
 
\beta_0 = \sqrt{\frac{c_{44}}{\rho}}
 
</math></center>
 
</math></center>
  
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Half fractional change in the P-wave velocity
 
Half fractional change in the P-wave velocity
  
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<center><math>
 
<center><math>
 
\varepsilon = \frac{c_{11}-c_{33}}{2 c_{33}} </math></center>
 
\varepsilon = \frac{c_{11}-c_{33}}{2 c_{33}} </math></center>
  
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Half fractional change in the S-wave velocity
 
Half fractional change in the S-wave velocity
  
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<center><math>
 
<center><math>
 
\gamma = \frac{c_{66}-c_{44}}{2 c_{44}} </math></center>
 
\gamma = \frac{c_{66}-c_{44}}{2 c_{44}} </math></center>
  
  
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<center><math>
 
<center><math>
 
\delta = \frac {1} {2} \frac{(c_{13}+c_{44})^2-(c_{33}-c_{44})^2}{c_{33}(c_{33}-c_{44})}
 
\delta = \frac {1} {2} \frac{(c_{13}+c_{44})^2-(c_{33}-c_{44})^2}{c_{33}(c_{33}-c_{44})}
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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 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>:  
 
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 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>:  
  
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<center><math>
 
<center><math>
 
\eta=\frac{\varepsilon-\delta}{1+2\delta}
 
\eta=\frac{\varepsilon-\delta}{1+2\delta}
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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>:
 
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>
 
<center><math>
 
V_p(\theta)=\alpha_0 (1+\delta\sin^2{\theta}\cos^2{\theta}+\varepsilon\sin^4{\theta})
 
V_p(\theta)=\alpha_0 (1+\delta\sin^2{\theta}\cos^2{\theta}+\varepsilon\sin^4{\theta})
 
</math></center>
 
</math></center>
  
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<center><math>
 
<center><math>
 
V_{sv}(\theta)=
 
V_{sv}(\theta)=
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</math></center>
 
</math></center>
  
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<center><math>
 
<center><math>
 
V_{sh}(\theta)=
 
V_{sh}(\theta)=
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</math></center>
 
</math></center>
  
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See [[Special:MyLanguage/Dictionary:polar_anisotropy|''polar anisotropy'']] (transverse isotropy).
 
See [[Special:MyLanguage/Dictionary:polar_anisotropy|''polar anisotropy'']] (transverse isotropy).
  
  
  
== References ==
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== References == <!--T:22-->
  
 
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==External links==
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==External links== <!--T:23-->
  
 
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Revision as of 01:49, 14 April 2019

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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, 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 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

  1. Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966.
  2. Alkhalifah, T. and Tsvankin, I., 1995, Velocity analysis for transversely isotropic media: Geophysics, 60, 1550–1566.
  3. Thomsen, L., 2002, Understanding seismic anisotropy in exploration and exploitation: SEG-EAGE Distinguished Instructor Series #5: Soc. Expl. Geophys.


External links

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Thomsen anisotropic parameters
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