## 3 Case Study
### Project Specification
#### 3.2.2 Distribution Types of the Duncan-Chang Model Parameters
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Parameters**} & \multicolumn{6}{c}{**AIC**} \\ \cline{2-6}  & \multicolumn{1}{c}{\begin{tabular}{c} **Truncated Normal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **Lognormal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **Extremal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\
\begin{tabular}{c} **Weibull** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{
\begin{tabular}{c} **Optimal Marginal Distribution** \\ \end{tabular} } \\ \hline \(\varphi_{0}/^{\circ}\) & 388.22 & / & / & 490.91 & Truncated normal distribution \\ \(\Delta\varphi/^{\circ}\) & 478.74 & / & 595.76 & 457.02 & Weibull distribution \\ \(R_{f}\) & \(-185.30\) & \(-187.36\) & / & / & \(\mathrm{Lognormal}\) distribution \\ \(K\) & 1125.10 & 1126.40 & / & 1105.90 & Weibull distribution \\ \(K_{b}\) & 1032.50 & / & / & 1008.60 & Weibull distribution \\ \(m\) & \(-123.15\) & \(-100.23\) & / & \(-159.68\) & Weibull distribution \\ \(n\) & / & / & / & / & \(\mathrm{Lognormal}\) distribution \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The optimal marginal distribution of the Duncan–Chang model in the rockfill. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Parameters**} & \multicolumn{6}{c}{**K-S Test**} & \multicolumn{6}{c}{**A-D Test**} \\ \cline{2-6}  & \multicolumn{1}{c}{ \begin{tabular}{c} **Truncated Normal** \\ **Distribution** \\ \end{tabular}  } & \multicolumn{1}{c}{ \begin{tabular}{c} **Lognormal** \\ **Distribution** \\ \end{tabular}  } & \multicolumn{1}{c}{ \begin{tabular}{c} **Extremal** \\ **Distribution** \\ \end{tabular}  } & \multicolumn{1}{c}{ \begin{tabular}{c} **Reibull** \\ **Distribution** \\ \end{tabular}  } & \multicolumn{1}{c}{ \begin{tabular}{c} **Truncated** \\ **Normal** \\ **Distribution** \\ \end{tabular}  } & \multicolumn{1}{c}{ \begin{tabular}{c} **Lognormal** \\ **Distribution** \\ \end{tabular}  } & \multicolumn{1}{c}{ \begin{tabular}{c} **Extremal** \\ **Distribution** \\ \end{tabular}  } & \multicolumn{1}{c}{ \begin{tabular}{c} **Weibull** \\ **Distribution** \\ \end{tabular}  } & \multicolumn{1}{c}{\ \begin{tabular}{c} **Reibull** \\ **Distribution** \\ \end{tabular}  } & \multicolumn{1}{c}{ \begin{tabular}{c} **Reibull** \\ **Distribution** \\ \end{tabular}  } \\ \hline \(c/kPa\) & ✗ & ✗ & ✓ & ✓ & ✗ & ✗ & ✓ \\ \(\varphi/^{\circ}\) & ✓ & ✗ & ✓ & ✗ & ✗ & ✓ \\ \(R_{f}\) & ✓ & ✓ & ✗ & ✗ & ✓ & ✓ & ✗ \\ \(K_{b}\) & ✓ & ✓ & ✗ & ✓ & ✓ & ✗ & ✗ \\ \(K_{b}\) & ✓ & ✓ & ✗ & ✓ & ✓ & ✗ & ✗ \\ \(m\) & ✓ & ✓ & ✓ & ✓ & ✓ & ✗Based on the above results, this study employed the bootstrap sampling method to generate 10,000 groups of parameter subsamples for the Duncan-Chang E-B model to verify whether the distribution features of statistical data were reasonable by analyzing the variability of statistical eigenvalues of the subsamples. The results of our efforts at generating subsamples for each parameter and testing their marginal distribution are presented in Tables 10-12. As can be seen in these tables, the marginal distribution test results for the bootstrap subsample of the high core rockfill dam material are consistent with the statistical sample test results, thereby verifying the correctness of the marginal distribution test results for the material parameters. Figures 5-7 show the marginal distribution probability density curves and histograms of the Duncan-Chang E-B model parameters of samples for the core wall material, rockfill, and dam foundation overburden, respectively. Here, the marginal distribution probability density curves closely match the probability distribution of the sample parameters.