## 3 Case Study
### Project Specification
#### 3.2.3 The Multi-Dimensional Joint Distribution Model for the Duncan-Chang Model Parameters  
Correlation Analysis of Duncan-Chang Model Parameters  
Many geotechnical tests have shown that the mechanical parameters of geotechnical materials are not mutually independent but exhibit a certain degree of correlation. Based on the parameter test data, the Pearson correlation coefficients (PCCs) between the Duncan-Chang E-B model parameters are given in Tables 13-15. The results show that the correlation coefficients of \(c\)-\(\varphi\) and \(K\)-\(K_{b}\) in the core wall, as well as those of \(\varphi_{0}\)-\(\Delta\varphi\) and \(K\)-\(K_{b}\) in the rockfill, all range between 0.6 and 0.8, indicating a strong correlation. Additionally, the correlation coefficient of \(K\)-\(K_{b}\) in the overburden is 0.87, indicating an extremely strong correlation. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Parameters** & \(c\)**/kPa** & \(\varphi\)\({}^{\text{\tiny{O}}}\) & \(R_{f}\) & \(K\) & \(K_{b}\) & \(m\) & \(n\) \\ \hline \(c/kPa\) & 1.00 & \(-\)0.68 & 0.31 & 0.15 & 0.17 & \(-\)0.31 & 0.06 \\ \(\varphi\)\({}^{\text{\tiny{O}}}\) & & 1.00 & 0.12 & 0.34 & 0.22 & \(-\)0.35 & 0.28 \\ \(R_{f}\) & & & 1.00 & 0.53 & 0.07 & 0.41 & \(-\)0.08 \\ \(K\) & & & & 1.00 & 0.63 & 0.41 & 0.02 \\ \(K_{b}\) & & & & & 1.00 & 0.11 & 0.18 \\ \(m\) & & & & & & 1.00 & 0.40 \\ \(n\) & & & & & & & 1.00 \\ \hline \end{tabular}
\end{table} 
Table 13: The correlation coefficient among the parameters in the Duncan–Chang E-B model in the core wall. Figure 7: Marginal distribution probability density curve of the Duncan–Chang model parameters in the overburden. In total, 10,000 groups of samples were generated using the bootstrap sampling method. Aside from the strong correlations for \(c\)-\(\varphi\) and \(K\)-\(K_{b}\) in the core wall, the bootstrap sampling analysis results indicated that 57.87% of the samples for \(m\)-\(n\) had strong correlations, which meant that the strong correlation for \(m\)-\(n\) would be neglected with traditional small-sample processing methods. Similarly, there was also a strong correlation for \(m\)-\(n\) in the overburden. However, the strong correlation for \(m\)-\(n\) in the rockfill was only supported by 20% of the sample data. Therefore, parameters that can be considered to have a strong correlation include \(c\)-\(\varphi\), \(K\)-\(K_{b}\), and \(m\)-\(n\) in the core wall, \(\varphi_{0}\)-\(\Delta\varphi\) and \(K\)-\(K_{b}\) in the rockfill, and \(K\)-\(K_{b}\) and \(m\)- \(n\) in the overburden. Construction of the Multi-Dimensional Joint Distribution Model for the Duncan-Chang Model Parameters  
Based on Equations (4)-(8) and the marginal distribution types of Duncan-Chang model parameters, the corresponding multi-dimensional joint probability density functions and distribution functions could be constructed using the Gaussian Copula and \(t\) Copula as follows:  
\[\left\{ \begin{array}{l}D(c,\varphi,R_{f},K,K_{b},m,n;\theta)=|\theta|^{-1/2} \exp[-\frac{1}{2}\zeta^{\prime}(\theta^{-1}-\mathbf{I})\zeta]\\ C(c,\varphi,R_{f},K,K_{b},m,n;\theta)=\Phi_{n}(\Phi^{-1}(c),\Phi^{-1}(\varphi),\dots,\Phi^{-1}(n);\theta)\end{array} \right. \tag{14}\]  
\[\left\{ \begin{array}{l}D(c,\varphi,R_{f},K,K_{b},m,n;\theta)=|\theta|^{-1/2} \frac{\Gamma(\frac{\
u+n}{2})(\Gamma(\frac{\
u}{2}))^{n}}{(\Gamma(\frac{\
u+1 }{2}))^{n}}\frac{(1+\frac{1}{\
u}\zeta^{\prime}\theta^{-1}\zeta)\frac{\
u+n} {2}}{\prod\limits_{i=1}^{n}{(1+\frac{\zeta_{i}^{2}}{\
u})}\frac{\
u+1}{2}}\\ C(c,\varphi,R_{f},K,K_{b},m,n;\theta)=T_{n}\big{(}T^{-1}(c),T^{-1}(\varphi), \dots,T^{-1}(n),\theta,\
u)\end{array} \right. \tag{15}\]  
where \(c\), \(\varphi\), \(R_{f}\), \(K\), \(K_{b}\), \(m\), and \(n\) are the Duncan-Chang model parameters in the high core rockfill dams. To verify the applicability of the two functions, 10,000 sample sets were analyzed, and the test results for the multi-dimensional joint probability model of Duncan-Chang model parameters are presented in Table 16. As shown in Table 16, the \(t\) Copula function performs