## 4 Results and Discussion
### Wavelet Transform Results
#### 4.2.2 XWT and WTC Results  
The XWT and WTC techniques exhibit strong signal coupling and resolution capabilities, enabling them to identify common power and relative phase information in the time-frequency domain and assess the significant coherence and confidence levels against the red noise background [25]. By performing XWT and WTC transformations on the two time series obtained from CWT, we obtain the cross-wavelet energy spectrum and the cross-wavelet condensation spectrum between the nonlinear deformation components of the characteristic points and the rainfall and reservoir water levels (Figures 12 and 13). In Figure 12a,c,e, we can observe that the cross-wavelet energy spectrum between precipitation and three feature points exhibit the highest energy density in a 1-year (30 \(\times\) 12 days) period and passed the 95% significance test, with significant periods continuously distributed in the entire sequence domain. The cross-wavelet arrows between precipitation and feature points P1 and P2 are directed toward the lower right (Figure 12a,c), indicating that the rainfall is ahead of the nonlinear displacements. This fact indicates that the nonlinear displacements exhibit a time lag response to the rainfall. To quantify the time lag within the interannual structure, we calculate the average phase angle and circular standard deviation from the phase relationship diagram of Figure 3 and Equations (5) and (6), and calculate the corresponding time delay days using Equation (7) (Shown in Table 3). The deformations of P1 and P2 lag behind precipitation by 68.7 \(\pm\) 12.2 and 72.4 \(\pm\) 8.8 days, respectively. This finding has significant implications for our understanding of the complex interplay between precipitation and deformation in this particular area. The cross-wavelet arrow between precipitation and P3 is unstable, with a circular standard deviation greater than the average phase angle, indicating that the two do not have a stable correlation,  
Figure 13: WTC results for different time series data: (**a**,**c**,**e**) denote the cross-wavelet energy spectra of precipitation and nonlinear deformation feature points p1, p2 and p3, respectively; (**b**,**d**,**f**) denote the cross-wavelet energy spectra of reservoir water level and nonlinear deformation feature points p1, p2 and p3, respectively. probably due to the impact of human factors on P3, located on the headscarp of the Guobu slope (Figure 6A), which require further investigation. In Figure 12b,d, we can observe that the cross-wavelet energy spectrum between reservoir water level and feature points P1 and P2 exhibit the highest energy density in a 1-year (30 \(\times\) 12 days) period, and pass the 95% significance test, with significant periods continuously distributed in the entire sequence domain. The arrows point upward to the left, indicating a negative correlation between P1 and P2 and the reservoir water level. In other words, when the reservoir water level decreases, the deformation of P1 and P2 increases. According to Table 3, the deformations of P1 and P2 lag behind the water level by 80.1 \(\pm\) 18.2 days and 19.3 \(\pm\) 8.3 days, respectively. The time lag at P2 is significantly smaller than that at P1 because P2 is located at the foot of the slope (Figure 6A) and is closer to the reservoir water level. However, Figure 12f shows no effective resonance period between the reservoir water level and P3. The WTC results reveal correlation and time lag information that is consistent with the XWT results, although there are some differences in local details. Figure 13a,b show the WTC between rainfall and nonlinear displacement P1 passes the 95% significance test in the 1-year (30 \(\times\) 12 days) period, while the WTC between reservoir level and P1 do not. This result implies that the nonlinear displacement of P1 is mainly affected by precipitation from 2016 to 2019. Figure 13c,d show that the WTC results for rainfall, reservoir water levels, and P2 pass the 95% significance test in the 1-year (30 \(\times\) 12 days) period.