## 3 Case Studies
### Feasibility Validation of Hysteresis Factor Addition  
The hydrostatic-season-time (HST) model, a classical and well-accepted mathematical model for dam deformation prediction, predicts the deformation of dams by considering the hydrostatic pressure, seasonal factors, and time dependence of the dam. The HST model provides an alternative method to infer the dam temperature variation and correlate it with dam deformation behavior when temperature monitoring information in the dam is not sufficient. The key to the HST model is the application of the harmonic factor. The harmonic factor is a mathematical technique used to characterize the periodicity of temperature variables. By performing spectral analysis and harmonic decomposition on temperature data, different harmonic components that reflect the seasonal and periodic patterns of temperature variation can be identified. By correlating these harmonic factors with the dam deformation data, a quantitative relationship between the dam deformation and temperature variation can be established. In most cases, the predictions of the HST model coincide with the dam deformation.  
According to the HST model described, the deformation of any point of the dam (\(\delta\)) can be decomposed into a water pressure component (\(\delta_{H}\)), a temperature component (\(\delta_{T}\)), and an aging component (\(\delta_{\theta}\)). The specific equation is shown in Equation (13)  
\[\delta=\delta_{H}+\delta_{T}+\delta_{\theta}=a_{0}+\sum_{i=1}^{n}a_{i}H_{i}+\sum _{i=1}^{m}\left(b_{1i}\sin\frac{2\Pi it}{365}+b_{2i}\cos\frac{2\Pi it}{365} \right)+c_{1}\theta+c_{2}\ln\theta \tag{13}\]  
where \(a_{0}\) is the constant term, \(a_{i}\) is the regression coefficient of the water pressure component, \(b_{1i}\) and \(b_{2i}\) are the regression coefficients of the temperature component, \(c_{1}\) and \(c_{2}\) are the regression coefficients of the time-dependent component, \(n\) is the regression coefficient of the gravity dam (\(n\) is 3 depending on the dam type), \(H\) is the reservoir level, \(m\) is 2 or 3 depending on the temperature at the project site (this paper takes 2), and \(\theta\) is the cumulative number of days from the day of observation to the starting date divided by 100. In summary, the influence factor of the initial HST model in this paper is \(H,H^{2},H^{3},\sin\frac{2\Pi t}{365},\sin\frac{4\Pi t}{365},\cos\frac{2\Pi t} {365},\cos\frac{4\Pi t}{365},\theta,\ln\theta\).  
To verify the effect of adding the lag factor, the lag quantification in Section 2.1 was performed with EX1-2 as an example. The case of EX1-2 with water level decomposed via VMD is shown in Figure 5. The reconstructed case is shown in Figure 6.  
Figure 4: Time course curve of water level data.  
Figure 5: (**a**) Information of EX1-2 deformation data decomposition. (**b**) Water level data decomposition information.  
Figure 6: Time course curves of reconstructed deformation and water level data.  
under the plain HST model, although smooth, are poorly fitted to the real data and do not portray the local fluctuations of the measured values, especially in the time period when the prediction starts (1 January 2019-1 April 2019). The contrast between the two is more obvious because the input variables of the plain HST model do not contain enough information to feed back the nonlinear characteristics of the deformation, thus leading to some apparent deviation from the measured data and a local distortion. Meanwhile, the box plot of residuals in Figure 8 shows that the median and mean of the residuals are smaller and the distribution of residuals is denser for the HST model with lags considered compared with the HST model without lag quantification, which further verifies the feasibility of introducing lag factors in the HST model.