# Simulation of Cross-Correlated Random Fields for Transversely Anisotropic Soil Slope by Copulas
## 4 Simulation of CCRF from Copula Aspect  
### Cross-Correlation of Soil Parameters  
As aforementioned, soil parameters show two types of correlations, namely, autocorrelation and cross-correlation, as demonstrated in Figure 2. The former characterizes the correlation of individual soil parameters at different spatial locations, mainly determined by the autocorrelation structure of the parameter. In this paper, exponential ACF is used to replace it. The latter maintains that soil parameters also tend to be dependent on each other at a given spatial location or neighborhoods, commonly measured by their cross-correlation coefficients. For example, \(c\) and \(\phi\), which are key parameters affecting slope stability, have been proven to be negatively correlated. This implies that \(c\) and \(\phi\)  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Copula** & _C(u,v; \(\mathbf{\theta}\))_ & _c(u,v; \(\mathbf{\theta}\))_ & \(\mathbf{\theta}\) \\ \hline \multirow{2}{*}{Gaussian} & \(\Phi_{\theta}\Big{(}\Phi^{-1}(u),\Phi^{-1}(v)\Big{)}\) & \(\Phi\Big{(}\frac{\Phi^{-1}(u_{2})-\Phi\Phi^{-1}(u_{1})}{\sqrt{1-\theta^{2}}} \Big{)}\) & [–1, 1] \\ \hline \multirow{2}{*}{Plackett} & \(\frac{S-\sqrt{S^{2}-4 uw\theta(\theta-1)}}{2(\theta-1)}\), \(S=1+(\theta-1)(u+v)\) & \(\frac{1}{2}-\frac{1+(\theta-1)u_{2}-(\theta+1)u_{2}}{2\left[[1+(\theta-1)(u_{ 1}+u_{2})]^{-2}-4u_{1}u_{2}\theta(\theta-1)\right]^{\frac{1}{2}}}\) & \((0,+\infty)\backslash\{1\}\) \\ \hline \multirow{2}{*}{Frank} & \(-\frac{1}{\theta}\ln\!\left[1+\frac{(e^{-u_{1}}-1)(e^{-u_{2}}-1)}{e^{-u}-1} \right]\) & \(\frac{e^{-u_{1}}(e^{-u_{2}}-1)}{(e^{-u}-1)+(e^{-u_{2}}-1)(e^{-u_{2}}-1)}\) & \((-\infty,+\infty)\backslash\{0\}\) \\ \hline \multirow{2}{*}{No. 16} & \(\frac{1}{2}\Big{(}S+\sqrt{S^{2}+4\theta}\Big{)}\), & \(\frac{1}{2}\Big{(}1+\frac{\theta}{u_{1}^{2}}\Big{)}\Big{[}1+S(S^{2}+4\theta)^{- \frac{1}{2}}\Big{]}\), & \multirow{2}{*}{[0, \(+\infty)\)} \\  & \(S=u_{1}+u_{2}-1-\theta\Big{(}\frac{1}{u_{1}}+\frac{1}{u_{2}}-1\Big{)}\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: CDFs and PDFs of candidate copulas.  
should appear dependently in any spatial location. Therefore, to consider the spatial variability and cross-correlation simultaneously, all these parameters should be simulated as cross-correlation random fields, as shown in Figure 2.  
Some scholars have studied such cross-correlation and claimed that the heterogeneity and cross-correlation of soil profiles severely affect the stability of geotechnical structures. [PERSON] and [PERSON] [7] decomposed correlation matrices of parameters using the Cholesky decomposition method, and then simulated the CCRF using lower triangular matrices. [PERSON] [25] assumes that all random fields simulated in the same space have the same ACF. Then, the cross-correlation coefficient was used to characterize the cross-correlation between random fields of \(c\) and \(\phi\). The influence of cross-correlation on slope reliability was properly accounted for. However, according to copula theory, the correlation coefficient cannot accurately capture the correlation structure among parameters. To solve this problem, copula theory can be introduced to quantify the cross-correlation of soil parameters. Copulas are able to measure complex interdependencies among soil parameters. This might be expanded to make up for the deficiency of the traditional cross-correlation coefficient in simulating CCRF. To this end, copulas are intended to link individual random fields, as illustrated in the next section.