## 3 Results and Discussion of Data Analysis
### Correlation Coefficient Analysis and Its Relationship with Low-Frequency Signal Response
For example, for a sample with a lag of \(-174\) s (approximately \(-2.9\) min), the Spearman correlation coefficient is \(0.86\), indicating a high degree of monotonic consistency between signal strength and XRA changes just before the flare peak. The Spearman results further reinforce the rapid response of the ionosphere to XRA, especially in the lag interval (\(-5\), \(0\)). This strong monotonicity suggests that even in the presence of strong noise or nonlinear effects, signal strength remains a reliable indicator of XRA's changing trend. The Kendall Tau coefficient reaches a significant positive value in the lag interval (\(-5\), \(0\)), with a slightly lower value in the (\(0\), \(5\)) interval, suggesting that the trend of signal strength changes aligns closely with XRA. Moreover, in earlier negative lag intervals (e.g., \(-15\), \(-10\)), the trend changes in signal strength may serve as a precursor to an upcoming flare event. In summary, the combined results of the three correlation coefficients indicate that, within the key lag intervals of (\(-5\), \(0\)) and (\(0\), \(5\)), the high positive correlation between XRA and signal strength reveals a direct and rapid impact of C-class solar flares on ionospheric reflection characteristics. This correlation supports the hypothesis that solar storms cause ionospheric disturbances, with the positive correlation persisting even after the event (5 to 10 min). This suggests that the high-energy radiation from the flare has a lasting effect on the ionosphere, resulting in continuous changes in signal propagation paths and signal strength. The early negative correlation observed in the lag interval (\(-20\), \(-10\)) suggests that C-class flares may act as a warning signal for ionospheric disturbances, with signal strength being affected even before the flare reaches C-class intensity. Furthermore, from the perspective of the three correlation coefficients and their relation to the geomagnetic storm levels, as illustrated in Figure 6, the Pearson Correlation shows a \begin{table}
\begin{tabular}{c c c c} \hline
**Lag Bin (Minutes)** & **Pearson Avg** & **Spearman Avg** & **Kendall Tau Avg** \\ \hline \((-20\), \(-15\)) & \(-0.15\) & \(-0.12\) & \(-0.08\) \\ \((-15\), \(-10\)) & \(-0.20\) & \(-0.19\) & \(-0.12\) \\ \((-10\), \(-5\)) & \(0.18\) & \(0.23\) & \(0.15\) \\ \((-5,0)\) & \(0.45\) & \(0.52\) & \(0.38\) \\ \((0,5)\) & \(0.38\) & \(0.42\) & \(0.31\) \\ \((5,10)\) & \(0.20\) & \(0.19\) & \(0.14\) \\ \((10,15)\) & \(-0.05\) & \(-0.02\) & \(-0.01\) \\ \((15,20)\) & \(-0.12\) & \(-0.11\) & \(-0.06\) \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of the Distribution of the Three Correlation Coefficients Across Different Lag Intervals. Figure 5: Distribution of the Three Correlation Coefficients with Lag Time. significant increase from 0.38 for C1-C3 to 0.58 for C7-C9. This indicates that as the flare intensity increases, the linear correlation between XRA and signal strength strengthens. This suggests that more intense solar flares likely have a more pronounced direct impact on ionospheric disturbances and the propagation of low-frequency signals. The Spearman Correlation (monotonic relationship) rises from 0.33 to 0.64, signifying that during stronger flare events, even nonlinear trends exhibit greater consistency. Similarly, the Kendall Tau Correlation increases from 0.27 to 0.53, further confirming that as the flare level intensifies, the stability of the relative relationships between data points improves.