# Dictionary:Thomsen anisotropic parameters

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The relationship between the stress ${\displaystyle \sigma }$ and strain ${\displaystyle \varepsilon }$ vectors for polar anisotropic (vertical transversely isotropic) media can be expressed as ${\displaystyle \sigma ={\textbf {C}}\varepsilon }$, where C is the stiffness tensor as shown in Figure H-7. With the z-axes as the symmetry axis, we have [1]

${\displaystyle {\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{xy}\\\sigma _{yz}\\\sigma _{zx}\end{bmatrix}}={\begin{bmatrix}c_{11}&c_{11}-2c_{66}&c_{13}&0&0&0\\c_{11}-2c_{66}&c_{11}&c_{13}&0&0&0\\c_{13}&c_{13}&c_{33}&0&0&0\\0&0&0&c_{44}&0&0\\0&0&0&0&c_{66}&0\\0&0&0&0&0&c_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\\varepsilon _{xy}\\\varepsilon _{yz}\\\varepsilon _{zx}\end{bmatrix}}}$

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

${\displaystyle \alpha _{0}={\sqrt {\frac {c_{33}}{\rho }}}}$

S-wave velocity parallel to the symmetry axis

${\displaystyle \beta _{0}={\sqrt {\frac {c_{44}}{\rho }}}}$

Half fractional change in the P-wave velocity

${\displaystyle \varepsilon ={\frac {c_{11}-c_{33}}{2c_{33}}}}$

Half fractional change in the S-wave velocity

${\displaystyle \gamma ={\frac {c_{66}-c_{44}}{2c_{44}}}}$

${\displaystyle \delta ={\frac {1}{2}}{\frac {(c_{13}+c_{44})^{2}-(c_{33}-c_{44})^{2}}{c_{33}(c_{33}-c_{44})}}}$

where ${\displaystyle c_{ij}}$ indicate elements in the stiffness matrix. Note that ${\displaystyle \varepsilon }$ , ${\displaystyle \gamma }$ and ${\displaystyle \delta }$ 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, ${\displaystyle \eta }$ (eta), captures the deviation of the long-offset P-wave moveout from what it would have been for an isotropic medium[2]:

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

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

${\displaystyle V_{p}(\theta )=\alpha _{0}(1+\delta \sin ^{2}{\theta }\cos ^{2}{\theta }+\varepsilon \sin ^{4}{\theta })}$
${\displaystyle V_{sv}(\theta )=\beta _{0}[1+{\frac {\alpha _{0}^{2}}{\beta _{0}^{2}}}(\varepsilon -\delta )\sin ^{2}{\theta }\cos ^{2}{\theta }]}$
${\displaystyle V_{sh}(\theta )=\beta _{0}(1+\gamma \sin ^{2}{\theta })}$

Not only are these expressions algebraically simpler than the corresponding "exact" equations, but they also have fewer degrees of freedom. For example, ${\displaystyle V_{p}}$ above contains three parameters: ${\displaystyle \alpha _{0}}$, ${\displaystyle \delta }$, and ${\displaystyle \epsilon }$, whereas the "exact" expression contains four: ${\displaystyle c_{11}/\rho }$, ${\displaystyle c_{33}/\rho }$, ${\displaystyle c_{13}/\rho }$, and ${\displaystyle c_{44}/\rho }$. This makes determination of the parameters from field data significantly easier.

Note also that if ${\displaystyle \delta }$ is actually small, then the expression for ${\displaystyle \delta }$ above simplifies to

${\displaystyle \delta \rightarrow \delta _{weak}={\frac {c_{13}-(c_{33}-2c_{44})}{c_{33}}}}$

but this does not further simplify the velocity expressions above.

See polar anisotropy (transverse isotropy).