# Study of the Ionospheric Scintillation Radio Propagation Characteristics with Cosmic Observations
## 2 Data and Methods
### Ionospheric Scintillation Intensity  
The amplitude scintillation index is defined as [40]  
\[S4=\sqrt{\frac{\left\langle I^{2}\right\rangle-\left\langle I\right\rangle^{2} }{\left\langle I\right\rangle^{2}}} \tag{1}\]  
where S4 indicates the amplitude scintillation intensity. \(I=R^{2}\) represents the strength of the received signal. \(R\) is the signal amplitude. \(\left\langle\cdot\right\rangle\) indicates the time average. Amplitude scintillation occurs at a higher probability at low latitudes, while it seldom occurs at high latitudes. The phase scintillation index is given as  
\[\sigma_{\phi}=\sqrt{\phi^{2}-\phi^{2}} \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**S4** & **Ionospheric Scintillation Intensity** \\ \hline \(<\)0.3 & Very weak \\
0.3–0.5 & Weak \\
0.5–0.7 & Moderate \\
0.7–1 & Strong \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ionospheric scintillation intensity levels. where \(\phi\) is the tracked signal phase after detrending. Severe phase scintillation mainly occurs at high latitudes and polar regions. There are different distributions to describe the scintillation signal propagations, while the most traditional distribution is the Nakagami-\(m\) distribution, given as [41]  
\[f_{R}(r)=\frac{2m^{m}r^{2m-1}}{\Gamma(m)\Omega^{m}}e^{-mr^{2}/\Omega} \tag{3}\]  
where \(r\) is the unit vector of the received signal amplitude vector \(R\). \(\Gamma(\cdot)\) indicates the Gama function. \(m=1/\left(S_{4}\right)^{2}\) represents the attenuation. \(\Omega=E\left\{r^{2}\right\}\) is the average power of received signals. After the normalization, Equation (3) is rewritten as  
\[f_{R}(r)=\frac{2m^{m}r^{2m-1}}{\Gamma(m)}e^{-mr^{2}} \tag{4}\]  
The \(\alpha\)-\(\mu\) distribution is considered to be a rewritten form of the generalized Gama distribution. It is used to describe the distribution of the GPS scintillation signals at low latitudes in America, where amplitude scintillation is the most prominent [25]. This model assumes the signal as a composed of clusters of multipath waves propagating in a nonhomogeneous environment. This distribution has two degrees of freedom and is thus more suitable for describing the amplitude distribution under scintillation. The \(\alpha\)-\(\mu\) distribution is given by [42]  
\[f(r)=\frac{\alpha r^{\alpha m-1}}{\xi^{\alpha m/2}\Gamma(\mu)}exp\left(-\frac{ r^{\alpha}}{\xi^{\alpha/2}}\right) \tag{5}\]  
where \(\xi\) is calculated by \(\xi=\frac{\Gamma(\mu)}{\Gamma(\mu+2/\alpha)}\). The \(\alpha\), \(\mu\) parameters are estimated by [42]  
\[\frac{E^{2}\left(R^{\beta}\right)}{E\left(R^{2\beta}\right)-E^{2}\left(R^{\beta }\right)}=\frac{\Gamma(\mu)\Gamma(\mu+4/\alpha)-\Gamma^{2}(\mu+2/\alpha)}{ \Gamma^{2}(\mu+2/\alpha)} \tag{6}\]  
where \(\beta\) is a parameter to be determined. As there are two unknowns, assuming \(\beta=3\) and \(\beta=4\), Equation (6) can be rewritten as two equations, and \(\alpha\) and \(\mu\) can be calculated. It should be noted that the same \(S_{4}\) can correspond to different \(\alpha\) parameters. A larger value of \(\alpha\) indicates that the signal is more attenuated. When \(\alpha=2\) and \(\mu=m\), the \(\alpha\)-\(\mu\) distribution equals the Nakagami-\(m\) distribution. Thus, Nakagami-\(m\) is a special case of \(\alpha\)-\(\mu\) distributions. The \(\alpha\)-\(\mu\) distribution is a kind of envelope distribution that uses two parameters, \(\alpha\) and \(\mu\), to establish a generalized fading model considering multipath effects. It assumes that the signals propagating in a nonhomogeneous environment have similar delay times, with delay-time spreads of different clusters being relatively large. The scattered signals diverge over certain distances, but all of the scattered signals have the same impact on the phase, with the same power. Then, the envelope can be abstracted as a nonlinear function of the sum of multipath scatters. The parameter \(\alpha\) indicates the extent of nonlinearity, while the parameter \(\mu\) is related to the number of multipath scatters.