# Simulation of Cross-Correlated Random Fields for Transversely Anisotropic Soil Slope by Copulas
## 6 Numerical Illustrations
### Example 1: Assumed \(c\)-\(\phi\) Soil Slope
#### 6.1.3 Effect of Marginal Distribution on CCRF  
In this section, different margins are employed to model PDFs of \(c\) and \(\phi\). Comparisons among CCRFs associated with different margins are performed. A similar Gaussian copula is adopted as a control. Lastly, the effect of marginal distribution on CCRF is discussed. Figures 10-12 respectively display the typical realizations of CCRFs associated with different candidate margins, namely, TN, TG and WB margins. The LN case is linked to Figure 4. For comparison, cases in which \(c\) and \(\phi\) respectively hold TN and LN margins are also considered, as shown in Figure 13. Similar to Section 6.2, spatially variable soil can be efficiently demonstrated by the proposed method. \(c\) and \(\phi\) develop smoothly in the horizontal direction and fluctuate severely in the vertical direction, reflecting their SOFs. In addition, it can also be clearly observed that a negative cross-correlation emerges between the two random fields. These observations are also successfully reproduced. It can also be seen that the margins of \(c\) and \(\phi\) play important roles in modeling their spatial distributions. The ranges of \(c\) and \(\phi\) with respect to different margins differ from each other. Comparably, the TG margins in Figure 11 generate the largest ranges of \(c\) and \(\phi\). However, the ranges yielded by TN margins in Figure 10 are almost similar to those induced by WB margins in Figure 12. Such differences would lead to different simulation results and further affect the final analysis of slope reliability. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Copulas** & \(\mathbf{\rho}\) & \(\mathbf{\tau}\) & \(\mathbf{\mu_{c}}\) & \(\mathbf{\sigma_{c}}\) & \(\mathbf{COV_{c}}\) & \(\mathbf{\mu_{\phi}}\) & \(\mathbf{\sigma_{\phi}}\) & \(\mathbf{COV_{\phi}}\) \\ \hline Gaussian & \(-0.5031\) & \(-0.3519\) & 10.0787 & 3.0053 & 0.2982 & 29.9707 & 5.9839 & 0.1997 \\ Plackett & \(-0.4932\) & \(-0.3739\) & 10.0787 & 3.0053 & 0.2982 & 29.9686 & 5.9893 & 0.1999 \\ Frank & \(-0.4462\) & \(-0.3326\) & 10.0787 & 3.0053 & 0.2982 & 29.9658 & 5.9739 & 0.1994 \\ No.16 & \(-0.4879\) & \(-0.4133\) & 10.0787 & 3.0053 & 0.2982 & 29.8598 & 6.0121 & 0.2011 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistics of CCRF variables associated with candidate copulas. Figure 11: A typical realization of CCRF associated with TG margins. (**a**) Random field of \(c\). (**b**) Random field of \(\phi\). Figure 10: A typical realization of CCRF associated with TN margins. (**a**) Random field of \(c\). (**b**) Random field of \(\phi\). Figure 9: A typical realization of independent and non-correlation random field. (**a**) Random field of \(c\). (**b**) Random field of \(\phi\). (**c**) Scatter plot of independent and non-correlation random field. Figure 14 gives the scatter plots of four CCRFs. The LN case can be linked to Figure 8a. As expected, all five candidate cases yield negative dependence structures, which are consistent with the presented correlation direction. However, both the realizations of CCRF and the scatter plots of soil parameters associated with different margins are comparably inconsistent. This indicates that different margins correspond to different CCRF models. This is mainly due to the fact that the variables of RF obeying a uniform distribution in rank space are first yielded according to the Gaussian copula, and then transformed into the original space through the corresponding margins. Different margins would inevitably trigger differences in the generation of RFs. Therefore, the performance of copula-based CCRF is simultaneously governed by margins and copulas. To further verify the proposed method, Table 5 compares the statistics of 2000 random variables generated by different margins.