# Simulation of Cross-Correlated Random Fields for Transversely Anisotropic Soil Slope by Copulas
## 6 Numerical Illustrations
### Example 2: Chicago Congress Street Cut Slope
#### 6.2.2 Simulation of Multi-Layer CCRF  
According to Table 7, seven typical realizations of multi-layer CCRFs associated with different simulation strategies are derived, as shown in Figures 17-23. It should be pointed out that all realizations are carried out based on the same seeds in rank space. It is obvious that, except for strategy 4, the rest successfully generate multi-layer CCRFs with the aid of copulas. Negative correlations can be clearly observed from their element colors, illustrating the feasibility of the proposed method in modeling spatially variable multi-layer soils. Moreover, strategies 1-4 yield the same RFs of \(c\), which is due to the presence of the same seeds in each realization in rank space. However, the corresponding RFs of \(\phi\) are considerably different from each other. Similar to the one-layer slope in Section 6, this is attributed to the fact that different copulas characterize different dependent structures, and therefore reproduce different CCRFs. From Figures 17 and 21, Figures 22 and 23, it can be seen that uncertain margins also affect the simulation of multi-layer CCRFs. Under similar dependent structures of geotechnical parameters, CCRFs transformed from standard space by different marginal PDFs show some differences. Therefore, before the probabilistic assessment of the slope system, deriving sufficient on-site observations of strata parameters is essential. \begin{table}
\begin{tabular}{l l l l l l l l} \hline \multirow{2}{*}{**Soil Layer**} & \multirow{2}{*}{\(\mathbf{\gamma}\) (kN/m3)} & \multicolumn{2}{c}{\(\mathbf{c}\) (kPa)} & \multicolumn{4}{c}{\(\mathbf{\phi}\) (\({}^{\circ}\))} \\ \cline{3-8}  & & **Margin** & \(\mathbf{\mu_{c}}\) & \(\mathbf{\mathrm{COV}_{c}}\) & **Margin** & \(\mathbf{\mu_{\phi}}\) & \(\mathbf{\mathrm{COV}_{\phi}}\) \\ \hline Sand layer 1 & 21.0 & - & 0 & - & - & - & - \\ Clay layer 2 & 19.5 & LN & 55 & 0.37 & LN & 5 & 0.2 \\ Clay layer 3 & 19.5 & LN & 43 & 0.19 & LN & 7 & 0.21 \\ Clay layer 4 & 20.0 & LN & 56 & 0.20 & LN & 15 & 0.24 \\ \hline \end{tabular}
\end{table} 
Table 6: Statistics of soil parameters of the Chicago Congress Street cut slope. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Strategies** & **Margins** & **Copulas** \\ \hline Strategy 1 & (LN, LN) & ([PERSON], [PERSON], [PERSON]) \\ Strategy 2 & (LN, LN) & ([PERSON], Gaussian, No. 16) \\ Strategy 3 & (LN, LN) & (Independent, Plackett, No. 16) \\ Strategy 4 & (LN, LN) & (Independent, Independent, Independent) \\ Strategy 5 & (TN, TN) & ([PERSON], [PERSON], [PERSON]) \\ Strategy 6 & (TG, TG) & ([PERSON], [PERSON], [PERSON]) \\ Strategy 7 & (WB, WB) & ([PERSON], [PERSON], [PERSON]) \\ \hline \end{tabular}
\end{table} 
Table 7: Simulation strategies of multi-layer CCRFs. Figure 19: A typical realization of transversely anisotropic CCRF yielded by strategy 3. (a) Random field of \(c\). (b) Random field of \(\phi\). Figure 17: A typical realization of transversely anisotropic CCRF yielded by strategy 1. (a) Random field of \(c\). (b) Random field of \(\phi\). Figure 20: A typical realization of transversely anisotropic CCRF yielded by strategy 4. (a) Random field of \(c\). (b) Random field of \(\phi\). Table 8 gives the statistics of multi-layer CCRFs associated with seven strategies. The corresponding average errors of the statistics with respect to three clay layers are shown in Figure 24. It can be seen that the average comprehensive errors of the seven cases are 4.62%, 4.96%, 10.72%, 23.87%, 6.47%, 5.96% and 7.14%, respectively. The statistics deduced by copula-based CCRF are basically consistent with the real values shown in Table 6. This demonstrates the accuracy of the proposed method in modeling multi-layer CCRFs for spatially variable soils. Notably, strategies 3 and 4 show a relatively high bias. This can be explained by the huge errors in the estimated \(\rho_{c,\phi}\) values, as shown in Figure 24. In Strategy 3, the soil parameters of clay layer 1 are assumed to be independent variables, and yield 37.48% bias.