Dictionary:Thomsen anisotropic parameters

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

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

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

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

S-wave velocity parallel to the symmetry axis

$\beta _{0}={\sqrt {\frac {c_{66}}{\rho }}}$

Half fractional change in the P-wave velocity

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

Half fractional change in the S-wave velocity

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

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

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

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

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

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

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

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

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

but this does not further simplify the velocity expressions above.

See polar anisotropy (transverse isotropy).

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