Muir's Approximation

From this viewpoint, the deviation term of Muir's approximation can be put into the form

$$q{{v_z^{-2}v_x^{-2}\cos^2\theta\sin^2\theta}\over {v_z^{-2}\cos^2\theta+v_x^{-2}\sin^2\theta}}$$ (4)

This equation also has zero value in the vertical and horizontal directions and maximum value near 45 degrees because it contains the $\sin^2\theta\cos^2\theta$, but has a kind of weighting along the angle that causes a little movement of the maximum value according to the anisotropy factor, defined as the horizontal to vertical phase velocity ratio $v_x\over v_z$. In this case the ray velocity can be represented by

$$v^{-2}(\theta)=v_z^{-2}\cos^2\theta+v_x^{-2}\sin^2\theta+q{{... ...\cos^2\theta}\over {v_z^{-2}\sin^2\theta+v_x^{-2}\cos^2\theta}}$$ (5)

Figure 2 and Figure 3 show how well these two approximations for deviation term fit to the deviation from the best fitting ellipse in the cases of weak and moderate anisotropy, respectively.