## 2 Methodology  
### SFEM-Based Simulation Method for Parametric Uncertainty  
#### 2.1.1 Standard Gaussian Random Fields  
To realize the analysis of the uncertainty of geotechnical materials, scholars have incorporated random field theory into the numerical computational analysis of finite elements and established the stochastic finite element method to simulate the parameter uncertainty and correlation of geotechnical materials in the spatial range. The degree of correlation of a geotechnical parameter between two points in space is usually expressed by the autocorrelation coefficient, \(\rho\), which is a function of the distance between the two points [21]: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Project Name** & **Type** & **Height** & **Impact** \\ \hline A certain rockfill & Asphalt & & The calculation results significantly underestimate the stress and deformation of the anti-seepage structure [12]. \\  dam & concrete-core-wall & 150 m & Under the influence of aftershocks, there is a significant gap between the analysis results of dam deformation, stability, and seepage and the actual situation [15]. \\  dam & rockfill dam & 156 m & The peak acceleration error of the measuring point increased from 8.16\% to 25.17\% [17]. \\ \hline  Guan Yinyan & Clay-core rockfill dam & 71 m & The maximum vertical displacement value differs from the value calculated using the inversion parameters by more than 8.3\% [16]. \\  dam & Rockfill dam & 150 m & The displacement and stress of the dam are 7\% smaller than the actual value [18]. \\ \hline  Nuo Zhadu & Core rockfill dam & 216.5 m & The probability of stable failure of the dam slope is significantly lower than the actual value, which underestimates the risk of accidents [19]. \\ \hline  Gong Boxia & Concrete-face rockfill dam & 132 m & The vertical displacement value of the dam is 10–26.7\% smaller than the measured value [20]. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The impact of not considering uncertainty on the analysis of dam structures. Figure 1: Steps for determining displacement monitoring index for rockfill dam considering material uncertainty. \[\rho\big{[}H(x_{i},y_{i},z_{i}),H\big{(}x_{j},y_{j},z_{j}\big{)}\big{]}=\frac{ \text{COV}\big{(}H(x_{i},y_{i},z_{i}),H\big{(}x_{j},y_{j},z_{j}\big{)}\big{)}}{ \sqrt{\text{Var}\big{[}H(x_{i},y_{i},z_{i})\big{]}}\sqrt{\text{Var}\big{[}H \big{(}x_{j},y_{j},z_{j}\big{)}\big{]}}} \tag{1}\]  
where \(H(x_{i},y_{i},z_{i}),H(x_{j},y_{j},z_{j})\) are the eigenvalues of the grid parameter distributions of the random field cell with center-point coordinates \((x_{i},y_{i},z_{i}),(x_{j},y_{j},z_{j})\), respectively; COV(\(\cdot\)) is the covariance of the grid parameter distributions, and \(\text{Var}(\cdot)\) is the variance of the grid parameter distributions. In accordance with the random field model of soil advanced by [PERSON] [8], it is possible to adopt a discretization method for random fields, such as the so-called center-point method and level expansion method [21; 22]. Among these methods, the center-point method based on Cholesky decomposition has been demonstrated to exhibit high computational efficiency, as well as adaptive adaptability in comparison to competing approaches [23]. In this paper, therefore, the focus is on the use of the center-point method of Cholesky decomposition for the simulation of random fields. The random field is defined by parameters including the mean, variance, correlation, and correlation distance of the variables. It is generally assumed that the mean and variance are independent of the spatial location of the random field, and the correlation is only related to the relative distance between two points in space [24].