# Simulation of Cross-Correlated Random Fields for Transversely Anisotropic Soil Slope by Copulas
## 4 Simulation of CCRF from Copula Aspect
### Transformation of CCRF by Copulas  
As discussed in the previous section, a dependence structure exists among soil parameters. Random fields corresponding to different soil parameters are therefore correlated with each other. However, the cross-correlation coefficient matrix cannot accurately represent the cross-correlation between random fields. The key issue to be solved in this section is how to connect the independent standard normal random fields \(\mathbf{H}^{\textit{IAG}}\) with a specific correlation structure or a copula function. In essence, simulating CCRF is still a construction process of the joint distribution of multiple random variables, which obey the specified dependence structure. Therefore, \(\mathbf{H}^{\textit{IAG}}\) can be first converted into an independent standard uniform random field \(\mathbf{H}^{\textit{CAU}}\). Then, multivariate CCRFs can be constructed according to the predefined copula.  
The general framework for generating a cross-correlation standard uniform random field \(\mathbf{H}^{\textit{CAU}}\) based on copula function is as follows.  
The proposed copula-based CCRF is advantageous in that it can efficiently capture the interdependency of soil parameters. If required, \((\mathbf{H}_{1}^{\textit{CAU}},\mathbf{H}_{2}^{\textit{CAU}},\cdots,\mathbf{H}_{l}^{\textit {CAU}})\) can readily be converted into expected non-Gaussian random fields by giving the mean, standard deviation and marginal probability distribution of soil parameters.  
It is worth noting that dependent structures characterized by different copulas are severely different. This implies that the CCRFs simulated by different copulas also differ from each other. Care must be taken to ensure the selected copula matches the target interdependency of soil parameters. Such matching is commonly achieved through goodness-of-fit tests. Namely, Algorithm 1 should be modified to cater to real spatial correlation based on specific copulas. Fortunately, several efforts have been made by us and other researchers.  
Figure 2: Schematic diagram of correlations among soil parameters.  
Gaussian, Plackett, Frank and No. 16 copulas, as shown in Table 1, have been proven to be able to match the requirements. To this end, the four candidates are also employed to exploit transversely anisotropic CCRF in the subsequent sections.  
(1) Define the correlation structure between multivariate correlation standard uniform random fields as \(C_{i}=C\left(\mathbf{H}_{1}{}^{CAU},\mathbf{H}_{2}{}^{CAU},\cdots,\mathbf{H}_{i}{}^{CAU}\right)\), where \(i=2,3,\cdots,n\);  
(2) Extract \(\mathbf{H}_{1}{}^{AAU}\) from the standard uniform distribution U(0, 1), let \(\mathbf{H}_{1}{}^{CAU}=\mathbf{H}_{1}{}^{IAU}\);  
(3) Extract \(\mathbf{H}_{2}{}^{CAU}\) from \(C_{2}\left(\mathbf{H}_{2}{}^{CAU}\right)\left|\mathbf{H}_{1}{}^{CAU}\right\rangle\);  
(4) Similarly, extract \(\mathbf{H}_{n}{}^{CAU}\) from \(C_{n}\left(\mathbf{H}_{n}{}^{CAU}\right|\mathbf{H}_{1}{}^{CAU},\mathbf{H}_{2}{}^{CAU}, \cdots,\mathbf{H}_{n-1}{}^{CAU}\right)\).  
**Algorithm 1** Simulation algorithm of copula-based standard uniform CCRF