# Study of the Ionospheric Scintillation Radio Propagation Characteristics with Cosmic Observations
## 3 Experimental Results
### Goodness of Fit for Different Distributions  
In this part, residuals of the estimation deviation are analyzed. Figure 6 shows the standard deviation of the normalized amplitude scintillation index for three distributions under different intensities. This result indicates that a better fitting affect is achieved the smaller the values of the standard deviation. The difference between the Nakagami-\(m\) distribution curve and measured probability density curve decreases with increasing normalized S4, indicating that the fitting performance of the Nakagami-\(m\) distribution curve improves with increasing normalized S4. The results are different from those obtained from ground-based observations. From the ground-based results, when normalized \(S4\leq 0.7\), the standard deviation between the Nakagami-\(m\) distribution function curve and the measured probability density curve decreases with increasing S4, showing that the fitting performance of the Nakagami-\(m\) distribution function is better with increasing S4 under weak and medium scintillation [29]. When the normalized S4 is greater than 0.7, the root mean square error of the Nakagami-\(m\) distribution has an upward trend in the ground-based results. This reflects the difference between the analysis results of radio occultation data and ground data. The standard deviation of the \(\alpha\)-\(\mu\) distribution is generally low, with more slight fluctuation, indicating that the fitting effect of the \(\alpha\)-\(\mu\) distribution is relatively better as a whole. \begin{table}
\begin{tabular}{c c c} \hline
**S4** & \(\alpha\) & \(\mu\) \\ \hline
0.3 & 5.3467 & 0.239 \\
0.4 & 4.2748 & 0.3481 \\
0.5 & 3.788 & 0.3846 \\
0.6 & 3.1258 & 0.514 \\
0.7 & 2.5877 & 0.8942 \\
0.8 & 2.3442 & 1.0059 \\
0.9 & 1.9489 & 1.3512 \\
1.0 & 1.4625 & 2.1049 \\ \hline \end{tabular}
\end{table} 
Table 2: Calculated values of parameters \(\alpha\) and \(\mu\) under different scintillation intensities. Figure 5: Probability density curves of normalized SNR in relation to S4 obtained by analyzing ten-year data from 2010 to 2019. When the normalized S4 is equal to 0.9, the standard deviation of the two analytical distribution curves basically overlaps. This is because when the normalized S4 is equal to 0.9, the value of \(\alpha\) approaches 2, and the two probability density distribution curves are basically equal, so the standard deviation is also similar. When the normalized S4 is greater than 0.9, the standard deviation of the Nakagami-\(m\) distribution is less than that of the \(\alpha\)-\(\mu\) distribution, indicating that the fitting performance of the Nakagami-\(m\) distribution exceeds that of the \(\alpha\)-\(\mu\) distribution when the ionospheric scintillation intensity is extremely severe. The results of the \(\kappa\)-\(\mu\) distribution demonstrate a similar variation trend of the standard deviation as that of the Nakagami-\(m\) distribution, and intersection occurs when the normalized S4 is 0.5. It also shows that the fitting performance of the \(\kappa\)-\(\mu\) distribution improves with increasing normalized S4, but the fitting performance is not as good as that of the Nakagami-\(m\) distribution when the scintillation intensity becomes extremely severe. The kurtosis and information entropy analysis methods are used to analyze the fitting performance and evaluate the actual fitted probability density curve. Kurtosis reflects the nonuniformity of the curve tail distribution. The greater the kurtosis value, the wider the tail. When studying the statistical distribution of SNR under a certain S4, it is observed that the probability density distribution does not fit the normal distribution. In some actual curves, the two ends of the normalized amplitude do not decrease monotonically, but there are abnormal peaks in some places.