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\section{Theory}  Appleton's equation is given by \begin{equation}  \eta^2=\left(\frac{kc}{\omega}\right)^2=1-\frac{\omega_{pe}^2}{\left(\omega^2(1+\frac{i\nu}{\omega})-\frac{\omega_{ce}^2\sin^2\theta}{2\omega^2\left(1-\frac{\omega_{pe}^2}{\omega^2}\right)}\pm\sqrt{\omega_{ce}^2\sin^2\theta} \eta^2=\left(\frac{kc}{\omega}\right)^2=1-\frac{\omega_{pe}^2}{(\omega^2(1+\frac{i\nu}{\omega})-\frac{\omega_{ce}^2\sin^2\theta}{2\omega^2\left(1-\frac{\omega_{pe}^2}{\omega^2}\right)}\pm\sqrt{\omega_{ce}^2\sin^2\theta}  \end{equation}