THOMSEN'S SEISMIC WEAK ANISOTROPY METHOD

Thomsen's weak anisotropy method (Thomsen, 1986), being an approximation designed specifically for use in velocity analysis for exploration geophysics, is clearly not exact. Approximations incorporated into the formulas become most apparent for greater angles $\theta$ from the vertical, especially for compressional and vertically polarized shear velocities $v_p(\theta)$ and $v_{sv}(\theta)$, respectively. Angle $\theta$ is measured from the $\hat{z}$-vector pointing into the earth.

For reference purposes, we include here the exact velocity formulas for P, SV, and SH seismic waves at all angles in a VTI elastic medium. These results are available in many places (Rüger, 2002; Musgrave, 2003), but were taken specifically from Berryman (1979) with some minor changes of notation. The results are:  

$$v_p^2(\theta) = \frac{1}{2\rho}\left\{\left[\left(c_{11}+c_{... ...t(c_{33}+c_{44}\right)\cos^2\theta\right] + R(\theta)\right\}$$ (1)

and  

$$v_{sv}^2(\theta) = \frac{1}{2\rho}\left\{\left[\left(c_{11}+... ...(c_{33}+c_{44}\right)\cos^2\theta\right] - R(\theta)\right\},$$ (2)

where  

$$R(\theta) = \sqrt{\left[\left(c_{11}-c_{44}\right)\sin^2\the... ...t]^2 + 4\left(c_{13}+c_{44}\right)^2\sin^2\theta\cos^2\theta}$$ (3)

and, finally,  

$$v_{sh}^2(\theta) = \frac{1}{\rho}\left[c_{44} + (c_{66}-c_{44})\sin^2\theta\right].$$ (4)

Expressions for phase velocities in Thomsen's weak anisotropy limit can be found in many places, including Thomsen (1986, 2002) and Rüger (2002). The pertinent expressions for phase velocities in VTI media as a function of angle $\theta$,measured as before from the vertical direction, are  

$$v_p(\theta) \simeq v_p(0)\left(1 + \delta\sin^2\theta\cos^2\theta + \epsilon\sin^4\theta\right),$$ (5)

 

$$v_{sv}(\theta) \simeq v_s(0)\left(1 + [v^2_p(0)/v^2_s(0)](\epsilon - \delta)\sin^2\theta\cos^2\theta\right),$$ (6)

and  

$$v_{sh}(\theta) \simeq v_s(0)\left(1 + \gamma\sin^2\theta\right).$$ (7)

In our present context, $v_s(0) = \sqrt{c_{44}/\rho_0}$,and $v_p(0) = \sqrt{c_{33}/\rho_0}$, where c₃₃, c₄₄, and $\rho_0$are two stiffnesses of the cracked medium and the mass density of the isotropic host elastic medium. We assume that the cracks have insufficient volume to affect the mass density $\rho_0$ significantly.

In each case, Thomsen's approximation has included a step that removes the square on the left-hand side of the equation, by expanding a square root of the right hand side. This step introduces a factor of $\frac{1}{2}$ multiplying the $\sin^2\theta$ terms on the right hand side, and -- for example -- immediately explains how equation (7) is obtained from (4). The other two equations for $v_p(\theta)$ and $v_{sv}(\theta)$, i.e., (5) & (6), involve additional approximations as well that we will not attempt to explain here.

The three resulting Thomsen (1986) seismic parameters for weak anisotropy with VTI symmetry are $\gamma = (c_{66}-c_{44})/2c_{44}$,$\epsilon = (c_{11}-c_{33})/2c_{33}$,and  

$$\delta = \frac{(c_{13}+c_{44})^2-(c_{33}-c_{44})^2}{2c_{33}(... ...ht) \left(\frac{c_{13}+2c_{44}-c_{33}}{c_{33}-c_{44}}\right).$$ (8)

All three of these parameters can play important roles in the velocities given by (5)-(7) when the crack densities are high enough. If crack densities are very low, then the SV shear wave will actually have no dependence on angle of wave propagation. Note that the so-called anellipticity parameter ${\sc A} = \epsilon - \delta$, vanishes when $\epsilon \equiv \delta$, which we will soon see does happen for low crack densities.