# Simulation of Cross-Correlated Random Fields for Transversely Anisotropic Soil Slope by Copulas
## 7 Conclusions  
Aiming at simulating the spatial variability of cross-correlated soil parameters, this paper proposes a method of simulating cross-correlated random fields for transversely \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Margins** & **Copula** & \(\mu_{c}\) & **COV\({}_{c}\)** & \(\mu_{\phi}\) & **COV\({}_{\phi}\)** & \(\rho_{c,\phi}\) & \(\tau_{c,\phi}\) \\ \hline (LN, LN) & Gaussian & 14.1232 & 0.7055 & 23.6476 & 0.2466 & \(-\)0.6255 & \(-\)0.4945 \\ (LN, LN) & Plackett & 14.1232 & 0.7055 & 23.4610 & 0.2513 & \(-\)0.5815 & \(-\)0.4755 \\ (LN, LN) & Frank & 14.1232 & 0.7055 & 23.5756 & 0.2527 & \(-\)0.5777 & \(-\)0.4855 \\ (LN, LN) & No.16 & 14.1232 & 0.7055 & 23.4873 & 0.2622 & \(-\)0.4617 & \(-\)0.3938 \\ (TN, TN) & Gaussian & 16.0378 & 0.5499 & 23.5136 & 0.2704 & \(-\)0.7000 & \(-\)0.4945 \\ (TG, TG) & Gaussian & 14.9426 & 0.6617 & 23.7249 & 0.2364 & \(-\)0.6355 & \(-\)0.4945 \\ (WB, WB) & Gaussian & 14.2407 & 0.7232 & 23.4831 & 0.2770 & \(-\)0.7033 & \(-\)0.4945 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Statistics of CCRF variables associated with candidate margins and copulas. Figure 26: A typical realization of CCRF for soil parameters of the Papillion River Basin slope. (**a**) Random field of \(c\). (**b**) Random field of \(\phi\). (**c**) Scatter plot of CCRF variables and measured data. anisotropic soil slopes based on copulas. Four algorithms of CCRF obeying different dependent structures are established. The differences among CCRFs associated with different copulas are comparatively demonstrated. The effects of the cross-correlation and spatial variability of geotechnical parameters on the proposed copula-based CCRF are deeply discussed. Lastly, the feasibility and efficiency of the proposed method are verified by two examples. Some main conclusions can be drawn, as follows. The distribution of random variables derived from independent random fields entirely depends on marginal distribution, which cannot account for the interdependency of soil parameters. On this account, independent random fields cannot reflect the real distribution of soil profiles. Subsequent structural stability analysis is bound to cause large deviation and uncertainty. The copula-based CCRF accounts for the autocorrelation and cross-correlation of soil profiles simultaneously. It can be interpolated within each element to represent spatially variability. With the aid of the proposed method, the statistical characteristics of soil parameters can be recreated to agree with the observations. Soil profiles can be reflected with a relatively high accuracy. The cross-correlation coefficient and copula affect the characterization of the dependent structure of CCRFs. A unique dependent structure cannot be determined from the marginal distribution and cross-correlation coefficient of soil parameters. Ignoring the interdependency of soil parameters would lead to significant bias, which might further expand as more negligence occurs. Therefore, it is not reasonable to construct the CCRF only using cross-correlation coefficient, which is often the case in reality. The performance of the copula-based CCRF is simultaneously governed by the dependent structures and marginal distributions of random variables. Different copulas correspond to different CCRF models and interdependencies. Uncertain margins of soil parameters also affect the realizations of random fields. Before implementing the algorithm, the copula and margins of soil parameters should be precisely determined. In addition, the CCRFs constructed by different copulas are very different. More algorithms derived from other copulas should be established to cater to various complicated dependent structures. Meanwhile, the proposed method in this paper is non-intrusive. It can therefore be employed in conjunction with the random finite element method. On this basis, further advanced and efficient slope reliability methods should be investigated in the future to reduce the computational burden caused by more and more complex algorithms.