Byun's approximation

Among the possible forms of deviation from the ellipse, one probable function is $\sin^2\theta\cos^2\theta$ which has zero value in the vertical and horizontal directions, and maximum value of 45 degrees. In this approximation, the ray velocity equation becomes

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

where q^-2 has four times the value of the deviated sloth from the besting fitting ellipse at 45 degrees and where the case q=0 corresponds to elliptically anisotropic velocity. By using trigonometric identity, equation(2) becomes Byun's approximation :

$$v^{-2}(\theta)=a_0^2+a_1^2\cos^2\theta-a_2^2\cos^4\theta$$ (3)

where

a₀²=v_x^-2

a₁²=v_z^-2-v_x^-2+q^-2

a₂²=q^-2