## 2 Methodology
### Multi-Dimensional Joint Distribution Model
#### 2.1.3 The Construction Method of a Multi-Dimensional Joint Distribution Model for Parameters  
In the reliability analysis of rockfill dams, many studies have used a multi-dimensional normal distribution of random variables to establish the joint probability distribution function [31] or applied the Nataf transform to convert non-normal variables into independent variables. However, geotechnical parameters do not always follow a multi-dimensional normal distribution. Moreover, using a normal marginal distribution to describe the multi-dimensional probability distribution of random variables may underestimate the failure probability [32]. Thus, it is essential to develop a multi-dimensional joint probability distribution model to address the non-normal distribution of rockfill dam material parameters. In the statistical analysis of rockfill dam material parameters, constructing the joint probability distribution function is challenging when material parameter distributions are non-normal. Copula theory offers a solution by allowing the establishment of a joint distribution function for non-normal distributions. Using Copula functions, a mathematical model can be developed that incorporates both the marginal distribution functions of the random parameters and the correlation functions between them. Assume that \(X=\left\{X_{1},\,X_{2}...,\,X_{n}\right\}\) is an \(n\)-dimensional random vector and \(F_{i}(x_{i})\) is the marginal distribution function of random variables; then, the joint distribution of random variables can be expressed by the Copula function as  
\[F_{X}(x)=C(F_{1}(x_{1}),F_{2}(x_{2}),\ldots,F_{n}(x_{n});\theta) \tag{3}\]  
where \(X=\left\{X_{1},X_{2},\ldots,X_{n}\right\}\) represents the random vectors; \(F_{X}(x)\) is the joint distribution function of random vectors; and \(\theta\) is the correlation coefficient matrix of the Copula function. Copula theory can be used to independently implement model construction including a marginal distribution of random parameters and correlation between parameters, thus avoiding the thorny problem of coupling modeling. There are many types of Copula functions that represent different correlation relationships between parameters. The Gaussian Copula, \(t\) Copula, Frank Copula, No. 16 Copula, and Clayton Copula can be used to express the symmetrical correlation between the random parameters. The Clayton Copula, Gumbel Copula, and Plackett Copula can be applied to represent the positive correlation between random parameters. Moreover, the \(t\) Copula, Clayton Copula, Clayton Copula, Gumbel Copula, and No. 16 Copula can represent the tail correlation between random parameters. The multi-dimensional Gaussian Copula function and ellipse Copula function can describe the positive and negative correlations between parameters, and the correlation coefficient ranges from \(-1\) to \(1\). These two Copula functions are widely used in ascertaining the uncertainty distribution of a multi-dimensional parameter field. So, these two Copulafunctions were used to construct the multi-dimensional joint probability density and distribution function of material parameters or constitutive model parameters of high core rockfill dams in this study.