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Thomsen parameters^[1] are dimensionless elastic moduli which characterize transversely isotropic materials that are encountered in geophysics. In terms of the components of the elastic stiffness matrix, these parameters are expressed as

$\begin{align} \epsilon & = \frac{C_{11} - C_{33}}{ 2C_{33} } \\ \delta & = \frac{(C_{13} + C_{44})^2-(C_{33} - C_{44})^2}{ 2C_{33}(C_{33} - C_{44}) } \\ \gamma & = \frac{C_{66} - C_{44}}{ 2C_{44} } \end{align}$

where the $\mathbf{e}_3$ is the axis of symmetry. These parameters, in conjunction with the associated P wave and S wave velocities, can be used to characterize wave propagation through weakly anisotropic, layered media. It is found empirically that, for most layered rock formations the Thomson parameters are usually much less than 1.

[edit] Background

In geophysics, a common assumption is that the rock formations of the crust are locally polar anisotropic (transversely isotropic); this is the simplest case of geophysical interest. Such media support three types of elastic plane waves:

a quasi-P wave (polarization direction almost equal to propagation direction)
a quasi-S wave
a S-wave (polarized orthogonal to the quasi-S wave, to the symmetry axis, and to the direction of propagation).

Solutions to wave propagation problems in such media may be constructed from these plane waves, using Fourier synthesis. However, the equations for the angular variation of velocity are algebraically complex; the plane-wave velocities as a function of propagation angle $θ$ are ^[2]:

$\begin{align} V_{qP}(\theta) & = \frac{1}{ \sqrt{2\rho}} \left[C_{33} + C_{44} +(C_{11}-C_{33})\sin^2 \theta \right. \\ & \quad + \left\{(C_{33}-C_{44})^2 + 2[2(C_{13}+C_{44})^2-(C_{33}-C_{44})(C_{11}+C_{33}-2C_{44}) ] \sin^2 \theta \right. \\ & \left.\left. \qquad + [(C_{11}+C_{33}-2C_{44})^2 -4(C_{13}+C_{44})^2 ]\sin^4 \theta\right\}^{1/2}\right]^{1/2} \\ V_{qS}(\theta) & = \frac{1}{ \sqrt{2\rho}} \left[C_{33} + C_{44} +(C_{11}-C_{33})\sin^2 \theta \right. \\ & \quad - \left\{(C_{33}-C_{44})^2 + 2[2(C_{13}+C_{44})^2-(C_{33}-C_{44})(C_{11}+C_{33}-2C_{44}) ] \sin^2 \theta \right. \\ & \left.\left. \qquad + [(C_{11}+C_{33}-2C_{44})^2 -4(C_{13}+C_{44})^2 ]\sin^4 \theta\right\}^{1/2} \right]^{1/2} \\ V_{S}(\theta) & = \frac{1}{ \sqrt{\rho}} \sqrt{C_{44}\cos^2 \theta + C_{66}\sin^2 \theta } \end{align}$

where $ρ$ is density and the $C i j$ are elements of the elastic stiffness matrix. The Thomson parameters are used to simplify these expressions and make them easier to understand.

[edit] Simplified expressions for wave velocities

In geophysics the anisotropy in elastic properties is usually weak, in which case $\delta, \gamma, \epsilon \ll 1$ . When the exact expressions for the wave velocities above are linearized in these small quantities, they simplify to

$\begin{align} V_{qP}(\theta) & \approx V_{P0}(1 + \delta \sin^2 \theta \cos^2 \theta + \epsilon \sin^4 \theta) \\ V_{qS}(\theta) & \approx V_{S0}\left[1 + \left(\frac{V_{P0}}{ V_{S0}}\right)^2(\epsilon-\delta) \sin^2 \theta \cos^2 \theta\right] \\ V_{S}(\theta) & \approx V_{S0}(1 + \gamma \sin^2 \theta ) \end{align}$

where

$V_{P0}= \sqrt{C_{33}/\rho} ~;~~ V_{S0}= \sqrt{C_{44}/\rho}$

are the P and S wave velocities in the direction of the axis of symmetry ( $\mathbf{e}_3$ ) (in geophysics, this is usually, but not always, the vertical direction). Note that $δ$ may be further linearized, but this does not lead to further simplification.

The approximate expressions for the wave velocities are simple enough to be physically interpreted, and sufficiently accurate for most geophysical applications. These expressions are also useful in some contexts where the anisotropy is not weak.

[edit] See also

[edit] References

^ Thomsen, Leon (1986). "Weak Elastic Anisotropy". Geophysics 51(10): 1954–1966. doi:10.1190/1.1442051.
^ Nye, J. F. (2000). Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press.

[T86-0] Thomsen, Leon (1986). "Weak Elastic Anisotropy". Geophysics 51(10): 1954–1966. doi:10.1190/1.1442051.

[1] Nye, J. F. (2000). Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press.

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Contents

[edit] Background

[edit] Simplified expressions for wave velocities

[edit] See also

[edit] References