## 3 Results and Discussion of Data Analysis
### Correlation Analysis Results  
Figure 4 presents the statistical histogram of the time lag corresponding to the maximum correlation between the signal strength data and XRA data, based on a sample space of 152 cases. To evaluate the significance of the lag distribution, descriptive statistical metrics were calculated, including the mean lag time (\(-147.90\) s), standard deviation (316.64 s), and skewness (0.50), indicating a slightly right-skewed distribution. Additionally, a Kolmogorov-Smirnov (K-S) test was conducted, comparing the observed lag distribution with a random noise baseline generated using a Gaussian distribution (mean = 0, standard deviation = 5). The choice of 5 s as the standard deviation is supported by studies such as [PERSON] et al. (2005) and [PERSON] et al. (2011) [9; 10] which identified this range as representative of typical background noise variations in low-frequency signal propagation. The test results (K-S statistic = 0.72, \(p\)-value = \(7.15\times 10^{-76}\), \(p\) < 0.01) confirm that the lag distribution is significantly different from random noise, supporting the hypothesis that it is driven by ionospheric dynamics rather than stochastic fluctuations.  
From the figure, it is evident that the most probable time lag occurs within the interval of \(-5\) to 0 min, where the signal strength changes precede the C-class XRA changes. This finding aligns with the ionospheric pre-disturbance phenomenon reported in previous studies. For example, [PERSON] et al. (2017) pointed out that in weaker solar flare events, background disturbances in the ionosphere can cause fluctuations in signal strength several minutes before the onset of the flare event [7]. This may be due to early signals from high-energy particle flows or solar radiation fluctuations, which, although not directly observed, have already begun to influence the propagation path of low-frequency signals.  
The next most probable time lag occurs between 0 and 5 min, indicating that the signal strength was further disturbed after the XRA event began. This is consistent with the findings of [PERSON] et al. (2005), who observed that C-class solar flare events significantly increased the electron density in the ionospheric D-layer within minutes, thereby altering the propagation path of low-frequency signals [9].  
A relatively earlier negative time lag (from \(-10\) to \(-5\) min) accounted for 21.48% of occurrences, suggesting that significant changes in signal strength might have already been triggered by precursor disturbances before the C-class XRA event. These disturbances are likely associated with the instability of the ionosphere in the polar regions or the early stages of high-energy solar particle activity. Such early responses were also noted in the study by [PERSON] et al. (2001), which indicated that even low-level solar flare events could affect the propagation of low-frequency time-code signals in the polar regions due to precursor disturbances [8].  
Figure 4: Histogram of Lag Time (Minutes).  
The significance of time lags longer than 5 to 10 min, or beyond, shows a notable decline. This phenomenon is consistent with the findings of [PERSON] et al. (2011), who noted that the impact of C-class events on the ionosphere is typically short-lived, usually recovering within a few minutes ([PERSON] et al., 2011).