## 2 Data Processing and Analysis Procedure
### Data Processing Methods
#### 2.2.1 Data Correlation and Alignment  
To examine the temporal correlation between low-frequency signal strength and XRA (X-ray flux), a cross-correlation analysis method was applied. This approach calculates the correlation between two time series at different time lags, helping to identify the potential time window during which XRA influences signal strength changes. The formula for calculating the cross-correlation function is as follows:  
\[R_{xy}(\tau)=\frac{\sum\limits_{t=1}^{N}{[x(t)-\overline{x}][y(t+\tau)- \overline{y}]}}{\sqrt{\sum\limits_{t=1}^{N}{[x(t)-\overline{x}]^{2}\sum\limits _{t=1}^{N}{[y(t+\tau)-\overline{y}]^{2}}}}} \tag{1}\]  
where \(x(t)\) and \(y(t)\) represent the signal strength data and XRA data, respectively; \(\overline{x}\) and \(\overline{y}\) are the mean values of the corresponding datasets; and \(\tau\) denotes the lag time, with positive values indicating that XRA changes lead, and negative values indicating that signal strength changes lead.  
By calculating the cross-correlation function, the optimal lag time (i.e., the lag value with the highest correlation) between signal strength and X-ray flux (XRA) can be identified. This analysis provides critical insights into the temporal interplay between solar flares and ionospheric dynamics, illustrating how XRA modulates ionospheric reflection properties and subsequently influences the propagation characteristics of low-frequency signals. A positive lag indicates that variations in XRA precede changes in signal strength, implying that an increase in X-ray flux rapidly enhances electron density within the ionosphere. This elevation modifies reflection characteristics, leading to alterations in signal propagation paths and intensity. These findings align with established theories, which posit that solar flares elicit immediate and measurable responses in the ionosphere.  
Conversely, a negative lag suggests that changes in signal strength precede variations in XRA, potentially reflecting the influence of complex ionospheric processes not exclusively driven by XRA. Previous studies have indicated that during solar storms, high-energy particle precipitation, such as protons from solar energetic particle (SEP) events, may initiate precursor disturbances in the D-region ionosphere, altering electron density and thus impacting low-frequency signal propagation before significant XRA changes aredetected [8]. This phenomenon may be explained by the faster arrival of energetic particles compared to X-rays, leading to early ionization effects that manifest as signal strength variations [8; 10]. As XRA intensifies, these emissions further disrupt ionospheric conditions, potentially amplifying or reversing initial signal trends. Additionally, negative lags may highlight cumulative or secondary effects from localized geomagnetic activity or high-energy particle flux, which act synergistically with XRA to produce complex ionospheric responses [11; 17]. These findings underscore the importance of considering multiple contributing factors when interpreting lag dynamics between signal strength and XRA. In this study, the primary focus is on the relationship between XRA and low-frequency time-code signal strength. To minimize the influence of geomagnetic activity as a confounding factor, data from periods with significant geomagnetic disturbances (e.g., Kp > 4) were excluded during preprocessing. Therefore, any observed negative lags are more likely attributable to ionospheric changes driven by XRA and associated solar flare dynamics, rather than the effects of geomagnetic activity.  
By systematically analyzing these lag times, this study bridges observed temporal patterns with the underlying physical mechanisms governing ionospheric and signal propagation responses. This integrative approach enhances our understanding of how solar flares, through both direct and indirect pathways, shape the behavior of low-frequency signals, providing a robust framework for investigating space weather phenomena.