## 2 Methodology  
### Deformation Hysteresis Quantification Algorithm  
In order to quantify the deformation lag time, which is primarily influenced by environmental quantities, this paper proposes a deformation lag quantification algorithm by analyzing the fluctuations and phase differences between the environmental quantities (water level, temperature, etc.) and the measured data of the effect quantity (dam deformation). This approach only considers the water level (\(H\)) and dam deformation (\(\delta\)), as this paper employs HST (hydrostatic-season-time) as the basis model. Due to the water level variation factor, the dam deformation data and water level data are similar, but there is a slight phase difference between the two.  
Therefore, the fundamental concept behind the lag quantification approach is to first employ data decomposition and reconstruction techniques to minimize the negative effects of irregular terms on the phase computation and then to compute the phase cross-entropy of both to determine the lag time. These are the precise stages in this algorithm:  
Step 1: Data pre-processing. The monitoring data were normalized to reduce the order-of-magnitude differences between different physical quantities, and the normalization equation was expressed as Equation (1). To reduce the problem of the inability to perform cross-entropy calculations due to low data density, the hot card filling method [39] was used for the monitoring data to make full use of the contextual linkage of the monitoring data for the regression interpolation of missing values, so that the monitoring data have a first-order continuous type.  
\[X_{i}^{norm}=\frac{X_{i}-X_{\text{min}}}{X_{\text{max}}-X_{\text{min}}} \tag{1}\]  
In the above equation, \(X_{i}^{norm}\) is the normalized data, \(X_{i}\) is the pre-normalized data, and \(X_{\text{max}}\) and \(X_{\text{min}}\) are the maximum and minimum values of the monitored data.  
Step 2: Breaking down and rebuilding monitoring data. VMD decomposition was used to decompose the monitoring data, and the high-frequency information obtained from the decomposition was retained and fused, reducing the detrimental effects of low-frequencyinformation on the subsequent calculation of phase cross-entropy. The specific steps of the VMD decomposition reconstruction technique are described in the literature. [40].  
Step 3: Hysteresis quantization calculation. For the reconstructed monitoring information, firstly, one set of data was transformed to the frequency of another set of data by using discrete Fourier transform (DFT) [41; 42; 43; 44; 45]. Next, the lag was calculated using the phase cross-entropy calculation method. The formula for calculating the phase cross-entropy \(CE\) is shown in Equation (2)  
\[CE=-\sum[P(i,j)\times\log(P(i,j))] \tag{2}\]  
where \(P(i,j)\) is the probability of simultaneous occurrence of the time series \(i\) and \(j\) with a phase difference of \(j\). \(CE\) denotes the phase cross-entropy between the time series \(i\) and \(j\). The phase difference between the time series \(i\) and \(j\) can be obtained by calculating the phase cross-entropy of the time series. If the phase difference is zero, they are synchronized. If the phase difference is positive, it means that the time series \(i\) is lagging behind the time series \(j\). If the phase difference is negative, then the time series is lagging behind the time series \(ji\).