User:DeltaGeo/Draft

Thomsen Parameters
In geophysics, a common assumption is that the rock formations of the crust are locally polar anisotropic; this is the simplest case of geophysical interest. It is easily established that such media support three types of elastic plane waves: one is a quasi-P wave (polarization direction almost equal to propagation direction), one a quasi-S wave, and one an S-wave (polarized orthogonal to the quasi-S wave, to the symmetry axis, and to the direction of propagation). All solutions 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 $$\theta$$ are :

$$ V_{qP}(\theta)= \frac{1}{ \sqrt{2\rho}} \sqrt{C_{33} + C_{44} +(C_{11}-C_{33})\sin^2 \theta + \sqrt{(C_{33}-C_{44})^2 + 2[2(C_{13}+C_{44})^2-(C_{33}-C_{44})(C_{11}+C_{33}-2C_{44}) ] \sin^2 \theta + [(C_{11}+C_{33}-2C_{44})^2 -4(C_{13}+C_{44})^2 ]\sin^4 \theta} } $$

$$ V_{qS}(\theta)= \frac{1}{ \sqrt{2\rho}} \sqrt{C_{33} + C_{44} +(C_{11}-C_{33})\sin^2 \theta - \sqrt{(C_{33}-C_{44})^2 + 2[2(C_{13}+C_{44})^2-(C_{33}-C_{44})(C_{11}+C_{33}-2C_{44}) ] \sin^2 \theta + [(C_{11}+C_{33}-2C_{44})^2 -4(C_{13}+C_{44})^2 ]\sin^4 \theta} } $$

$$ V_{S}(\theta)= \frac{1}{ \sqrt{\rho}}  \sqrt{C_{44}\cos^2 \theta + C_{66}\sin^2 \theta } $$

where $$\rho$$ is density and the $$C_{\alpha \beta}$$ are elements of the elasticity matrix. These expressions are too complicated to be easily understood (and too long to be easily viewed!).

In geophysics, the anisotropy is usually weak, in which case the equations for the seismic velocities should simplify. In order to define precisely the meaning of "weak anisotropy", one could employ standard perturbation theory. Alternatively, an examination of the exact solutions reveals the following combinations :

$$ \epsilon = \frac{C_{11} - C_{33}}{ 2C_{33} } $$

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

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

These nondimensional measures of anisotropy are collectively known as Thomsen parameters. It is found empirically that, for most bulk rock formations (treated as polar anisotropic), they are usually <<1. When the exact solutions above are linearized in these small quantities, they simplify substantially:

$$ V_{qP}(\theta)\approx V_{P0}(1 + \delta sin^2 \theta cos^2 \theta + \epsilon sin^4 \theta) $$

$$ V_{qS}(\theta)\approx V_{S0}(1 + (\frac{V_{P0}}{ V_{S0}})^2(\epsilon-\delta) sin^2 \theta cos^2 \theta) $$

$$ V_{S}(\theta)\approx V_{S0}(1 + \gamma sin^2 \theta ) $$

where

$$ V_{P0}= \sqrt{C_{33}/\rho} $$

$$ V_{S0}= \sqrt{C_{44}/\rho} $$

are the P and S velocities in the direction of the symmetry (3) axis (in geophysics, this is usually, but not always, the vertical direction). (It is clear that $$\delta$$ may be further linearized, but this does not lead to further simplifcation, and the current definition proves useful in some contexts where the anisotropy is not weak.)

The approximate expressions above are sufficiently simple that one may easily understand their meaning, and are sufficiently accurate for most geophysical applications.