# Study of the Ionospheric Scintillation Radio Propagation Characteristics with Cosmic Observations
## 3 Experimental Results
### Goodness of Fit for Different Distributions
The kurtosis difference (absolute value) between three standard distribution functions and the measured curve under different normalized amplitude scintillation indices is calculated as shown in Figure 7.  
For the Nakagami-\(m\) distribution, with the increase of the normalized S4, the difference of kurtosis decreases, indicating that the fitting performance of Nakagami-\(m\) distribution in this range is getting better. Meanwhile, for the \(\alpha\)-\(\mu\) distribution, the kurtosis difference is generally small regardless of the change in the normalized S4, and the value is mostly below 1. In the range of normalized S4 from 0.8 to 0.9, the kurtosis difference between the Nakagami-\(m\) and \(\alpha\)-\(\mu\) models intersects, and then the kurtosis of Nakagami-\(m\) becomes less than that of \(\alpha\)-\(\mu\), indicating that when \(54>0.9\), the Nakagami-\(m\) distribution function is better than the standard \(\alpha\)-\(\mu\) distribution. For the kurtosis difference curve of the \(\kappa\)-\(\mu\) distribution, when the normalized S4 is less than 0.5, it is below the kurtosis of the Nakagami-\(m\) distribution. When the normalized S4 is greater than 0.5, it exceeds the kurtosis of the Nakagami-\(m\) distribution. Figure 6: RMSE in relation to S4 for the three statistical distributions. Information entropy reflects disorder and uncertainty of specific results. In this study, information entropy is used to evaluate the disorder of the distribution curve. Figure 8 shows the information entropy difference (absolute value) between the measured probability density distribution and the theoretical distribution under different normalized S4. With the increase in normalized S4, the information entropy difference of the Nakagami-\(m\) distribution generally exhibits a downward trend, and there are still some fluctuations. In the case of weak ionospheric scintillation, the disorder between the Nakagami-\(m\) distribution and the actual signal-to-noise ratio distribution is quite different. When S4 increases to 1.0, the difference in information entropy reaches the minimum, indicating that the signal-to-noise ratio distribution is affected by intensity modulation, and the disorder difference has reached a low level. In the information entropy difference curve of the \(\alpha\)-\(\mu\) distribution, it is found that the size of the information entropy difference is generally small. The two curves also overlap when the normalized S4 is 0.9, indicating that the results of the goodness of fit evaluated by information entropy matches the results in Figure 5. For the \(\kappa\)-\(\mu\) distribution, the information entropy difference overlaps with the Nakagami-\(m\) distribution when S4 is 0.5, which is consistent with the results in Figure 5.  
To study the performance of the three standard distributions in different years in solar cycle 24, Table 3 shows the standard deviation, information entropy and kurtosis difference between the three distribution curves and the measured probability density distribution curves in 2014 and 2019 when the normalized S4 is set to 0.7. Figure 8: Information entropy in relation to S4 for the three distributions. Figure 7: Kurtosis in relation to S4 for the three statistical distributions. When normalized S4 is 0.7, the standard deviation and information entropy difference between the Nakagami-\(m\) distribution and the measured normalized SNR probability density curve in 2014 (solar maximum year) are slightly less than the corresponding values in 2019 (solar minimum year), while the kurtosis difference is larger than the corresponding values in 2019. This demonstrates that, from the perspective of standard deviation and information entropy, the \(\alpha\)-\(\mu\) distribution fits better in high solar activity years.