## 2 Methodology
### Determination of Displacement Monitoring Index for Rockfill Dams  
#### 2.2.1 Hybrid Monitoring Model  
The hybrid monitoring model for rockfill dam displacement can be defined as a regression model that represents the nonlinear relationship between dam displacement and environmental variables [10; 35; 36]. \[\delta(\mathrm{or}\;\delta_{x},\;\delta_{y},\;\delta_{z})=\delta_{H}+\delta_{ T}+\delta_{\theta} \tag{6}\]where \(\delta\) is the displacement of the dam body; \(\delta_{x}\), \(\delta_{y}\), and \(\delta_{z}\) represent the displacement components in the three directions of the dam body, respectively, \(\delta_{H}\) is the water pressure component, which is determined by the expression of the water level-displacement relationship obtained from structural calculations and can generally be determined via polynomial curve fitting, as shown in Equation (7), \(\delta_{T}\) is the temperature component, and \(\delta_{\theta}\) is the time-dependent component. The temperature component and the time-dependent component are calculated using the expressions in the statistical model [10], as shown in Equations (8) and (9). \[\delta_{H}=\sum_{i=0}^{m}a_{i}H^{i} \tag{7}\]  
\[\delta_{T}=\sum_{i=1}^{m_{1}}b_{i}T_{i}+\sum_{i=1}^{m_{2}}\left(b_{1i}\cos \frac{2\pi it}{365}+b_{2i}\sin\frac{2\pi it}{365}\right) \tag{8}\]  
\[\delta_{\theta}=c_{1}\left(1-e^{-c_{2}\theta}\right) \tag{9}\]  
where \(H\) is the upstream water depth, \(T_{i}\) is the average temperature of \(i\) days before the observation day, and, when \(i=0\), it means the temperature on the day of observation, \(t\) is the time (days) counted from a certain day, \(m\) is the number of polynomial terms, and it is generally taken as 3-4, \(m_{1}\) and \(m_{2}\) are the number of temperature factors and cycles, and they are generally taken as 9-10, \(\theta\) is the time-dependent factor, \(\theta=t/100\), and \(a\), \(b\), and \(c\) are coefficients to be determined. In the context of the engineering safety monitoring process, a range of factors contribute to discrepancies between the activation time of the instrument and the initial displacement values recorded for the dam. These discrepancies can be attributed to the failure and replacement of monitoring equipment, variations in natural or human-made factors, and the inherent variability associated with the measurement process. Consequently, it becomes essential to implement a correction parameter, designated as \(L\), to the water pressure component to ensure the accuracy of the data. The expression of the hybrid monitoring model is thus modified as follows [37]:  
\[\delta=L\delta_{H}+\delta_{T}+\delta_{\theta} \tag{10}\]  
In order to establish the hybrid model under discussion, it is first necessary to employ the stochastic finite element method to calculate the displacement of the dam body in question under differing water levels. This calculation is then fitted to form the water pressure component, thus taking into account the effect of uncertainties pertaining to the physical and mechanical parameters of the rockfill in question. The primary structure of the rockfill dam comprises the following components: upstream, main, and secondary rockfill, with the Duncan-Chang E-B model being utilized to describe its stress-strain relationship. Due to the variability in the material properties in different rockfill zones, the random field eigenvalues of the parameters are not identical, thus necessitating the execution of random field parameter simulations according to the material partition, and subsequently assigning the stochastic results to each element for FEM analysis. The computational demands of conducting random field simulations for all parameters in the model to establish the water pressure component via SFEM can be prohibitive, particularly given the varied impact of each parameter on rockfill deformation in the E-B model.