# A Method for Evaluating Systematic Risk in Dams with Random Field Theory
## 1 Introduction
[25] performed a stochastic analysis of seepage using three types of random fields, and the application showed that the spatial variability of hydraulic conductivity induces differences using different random fields. However, it should be noted that there are very few studies on dam risk analysis considering the spatial variability of the mechanical parameters of rock masses and concrete. Therefore, this study intends to propose a system risk analysis model and method considering the spatial variability of dams and rock foundations, which lacks investigation and needs to be urgently solved. The purpose of this paper is to study the key technologies and methods of systematic risk analysis used for dams a complex foundations, considering the spatial variability of the mechanical parameters of rock masses and concrete. The main contents of the current research are arranged as follows: (a) studying the simulation method of the spatial variability of rock masses and concrete based on random field theory and the midpoint method; (b) constructing composite response surface equations (RSEs) for strength and stability failure performance function considering the spatial variability of parameters and putting forward a calculation method and technical process for systematic failure risk using the Monte Carlo method; and c) by taking a concrete gravity dam-rock foundation system as an example, the applicability and rationality of the proposed method in this paper are analyzed and discussed. A Finite Element Simulation Method for Gravity Dams Considering the Spatial Variability of Parameters  
For geotechnical engineering projects like the complex foundations of gravity dams, the influences of faults, joints, and fissures in the foundation cannot be ignored. In this study, a random field theory-based spatial variability simulation method is introduced to describe the uncertainties of parameters in gravity dams that can be achieved with the midpoint method. As we know, the parameters of dam bodies and foundation materials mostly follow a normal or log-normal distribution [26], so the normal and log-normalcorrelation random field simulation methods are put forward, and they are applied to a concrete gravity dam with a complex foundation. According to the midpoint method, the two-dimensional dam-rock foundation system model needs to be discretized into n elements (V\({}_{1}\), V\({}_{2}\),..., V\({}_{i}\),..., V\(n\)), and their center point is defined as X\({}_{i}\) (\(i\) = 1, 2,..., \(n\)). In this study, the spatial variability of mechanical parameters in the rock and concrete foundation is determined using an autocorrelation function, with an exponential autocorrelation function being adopted [27; 28; 29]:  
\[\rho_{i,j}=\exp\left(-(\frac{\left|x_{i}-x_{j}\right|}{L_{x}}+\frac{\left|y_{i }-y_{j}\right|}{L_{y}})\right) \tag{1}\]  
where \(\rho_{i,j}\) stands for the parameters' autocorrelation coefficient in two arbitrary elements: \(i\), \(j\); \(x_{i}\) and \(y_{i}\) stand for the \(i\) element's x- and y-coordinates; \(x_{j}\) and \(y_{j}\) stand for the j element's x- and y-coordinates; and \(L_{x}\) and \(L_{y}\) stand for the autocorrelation distances in the x- and y-directions. Firstly, according to the coordinate information, \(X_{i}\) (\(i\) = 1, 2,..., \(n\)) in the dam-rock foundation system and the exponential autocorrelation function (Equation (1)), the autocorrelation coefficient matrix \(\rho_{n\times n}\) can be expressed:  
\[\rho_{n\times n}=\left[ \begin{array}{cccc}\rho_{11}&\rho_{12}&\cdots&\rho_{ 1n}\\ \rho_{21}&\rho_{22}&\cdots&\rho_{2n}\\ \cdots&\cdots&\cdots\\ \rho_{n1}&\rho_{n2}&\cdots&\rho_{nn}\end{array} \right] \tag{2}\]  
Secondly, the autocorrelation coefficient matrix, \(\rho_{n\times n}\), can be decomposed based on the Cholesky decomposition method [30]:  
\[\rho_{n\times n}=\textbf{B}_{n\times n}^{T}\cdot\textbf{B}_{n\times n} \tag{3}\]  
where \(\textbf{B}_{n\times n}\) is an upper triangular matrix.