# Study of the Ionospheric Scintillation Radio Propagation Characteristics with Cosmic Observations
## 3 Experimental Results
### Goodness of Fit for Different Distributions
When normalized S4 is 0.7, the standard deviation and kurtosis difference between the \(\alpha\)-\(\mu\) distribution in 2014 and the measured normalized signal-to-noise ratio probability density curve are less than the corresponding values in 2019, while the information entropy difference is larger than the corresponding values in 2019. When the normalized S4 is 0.7 in 2014, the distribution for \(\kappa\)-\(\mu\) is close to the standard deviation of the measured normalized SNR probability density curve. The information entropy and kurtosis differences are less than the corresponding values. When the normalized S4 is 0.7, the standard deviation, information entropy difference and kurtosis difference between the \(\alpha\)-\(\mu\) distribution and the measured normalized SNR probability density curve are less than the corresponding values of the other two distributions in 2014. The fitting effect of the \(\kappa\)-\(\mu\) distribution is the least ideal, but the corresponding kurtosis value is close to that of the \(\alpha\)-\(\mu\) distribution, achieving a better fitting of the tail. When the normalized S4 is 0.7, the standard deviation and information entropy difference between the \(\alpha\)-\(\mu\) distribution and the measured normalized SNR probability density curve in 2019 are less than the corresponding values of other distributions, while the kurtosis difference is the largest of all. From the perspective of kurtosis, the Nakagami-\(m\) distribution has a better fitting performance in 2019, while from the perspective of standard deviation and information entropy, the \(\alpha\)-\(\mu\) distribution has a better fitting performance in 2019. In both 2014 and 2019, the standard deviation and information entropy of the \(\kappa\)-\(\mu\) distribution are the largest. This means that from the perspective of standard deviation and information entropy, the \(\kappa\)-\(\mu\) distribution fits the worst of the three. The kurtosis indicates that the \(\kappa\)-\(\mu\) distribution achieves a better fitting of the tail. A further investigation was conducted to divide the results into two groups: an ascending solar phase from 2012 to 2015, and a descending solar phase from 2016 to 2019. At this time, the normalized S4 ranged from 0.3 to 1.0. Figure 9 shows the difference between the three models and the measured probability density curve. Slight differences in RMSE, information entropy and kurtosis are present, which means that for a general situation with large samples, the influence of solar activity on a single model turns to be very slight. The \(\alpha\)-\(\mu\) model outperforms in all considered aspects, then followed by the Nakagami-\(m\) and \(\kappa\)-\(\mu\) distributions. The fitting confidence interval is evaluated, and Figure 10 shows the result of \(S4=0.7\). Both \(\alpha\)-\(\mu\) and Nakagami-\(m\) are between the upper and lower limits, while for \(\kappa\)-\(\mu\), a part of the probability density function curve exceeds the fitting confidence boundary. This indicates that the \(\kappa\)-\(\mu\) model cannot achieve good performance when the amplitude scintillation is strong (but not very severe), consistent with the above results. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Year**} & \multicolumn{3}{c}{**RMSE**} & \multicolumn{3}{c}{**Information Entropy**} & \multicolumn{3}{c}{**Kurtosis**} \\ \cline{2-9}  & **[PERSON]-\(m\)** & \(\alpha\)-\(\mu\) & \(\kappa\)-\(\mu\) & **[PERSON]-\(m\)** & \(\alpha\)-\(\mu\) & \(\kappa\)-\(\mu\) & **[PERSON]-\(m\)** & \(\alpha\)-\(\mu\) & \(\kappa\)-\(\mu\) \\ \hline
2014 & 0.1405 & 0.0708 & 0.2043 & 0.5060 & 0.1365 & 0.727 & 0.7146 & 0.0245 & 0.0381 \\
2019 & 0.1459 & 0.0993 & 0.2021 & 0.6609 & 0.0797 & 0.8777 & 0.1145 & 0.6141 & 0.2618 \\ \hline \end{tabular}
\end{table} 
Table 3: RMSE, information entropy and kurtosis for the three distributions in 2014 and 2019 (\(S4=0.7\)).