# Simulation of Cross-Correlated Random Fields for Transversely Anisotropic Soil Slope by Copulas
## 6 Numerical Illustrations
### Example 1: Assumed \(c\)-\(\phi\) Soil Slope
#### 6.1.2 Simulation of CCRF  
In this section, bivariate CCRFs of \(c\) and \(\phi\) are established according to Algorithms 1-4. Then, their performances and differences are discussed in detail. Figures 4-7 respectively display the typical realizations of CCRFs associated with four candidate copulas. In the subfigures of the slope geometry, the depth of mesh color represents the magnitude of soil parameters. Namely, the deeper the color, the larger the value of \(c\) or \(\phi\). It is evident that spatially variable soil properties can be efficiently demonstrated by the CCRF models. Soil parameters \(c\) and \(\phi\) develop smoothly in the horizontal direction, but fluctuate severely in the vertical direction. This is mainly attributed to the domination of SOFs \(\delta_{h}\) and \(\delta_{v}\). As \(\delta_{v}\) is much smaller than \(\delta_{h}\) (2 m vs. 20 m), \(c\) and \(\phi\) fluctuate more severely in the vertical direction than in the horizontal direction, further indicating the reasonable characterization of spatial variability. In addition, it can also be clearly observed from Figures 4-7 that as \(c\) develops a deeper color, the corresponding \(\phi\) in the same location develops a lighter color, and vice versa. This indicates that a negative cross-correlation exhibits between the two random fields. The above ratiocinations can be further validated using the corresponding scatter plots in Figure 8. As expected, all the four candidate copulas yield negative dependence structures, which are consistent with the presented correlation direction. The copulas seem to be sufficient to give an excellent description of the cross-correlation of soil parameters. However, the realizations of CCRF and scatter plots of soil parameters associated with different copulas are comparably inconsistent, especially for the No. 16 copula. Such discrepancies with respect to Gaussian, Plackett and Frank copulas are comparatively unobvious. These may further propagate and affect the final geotechnical structure analysis. To this end, it can be concluded that different copulas correspond to different CCRF models and interdependencies. The performance of copula-based CCRF is simultaneously governed by dependent structures and cross-correlations of random variables. Figure 4: A typical realization of Gaussian copula-based transversely anisotropic CCRF. (**a**) Random field of \(c\). (**b**) Random field of \(\phi\). Figure 5: A typical realization of Plackett copula-based transversely anisotropic CCRF. (**a**) Random field of \(c\). (**b**) Random field of \(\phi\). Figure 8: Scatter plots of soil parameters associated with different CCRFs. (**a**) Gaussian copula. (**b**) Plackett copula. (**c**) Frank copula. (**d**) No. 16 copula. Figure 6: A typical realization of Frank copula-based transversely anisotropic CCRF. (**a**) Random field of \(c\). (**b**) Random field of \(\phi\). Figure 7: A typical realization of No. 16 copula-based transversely anisotropic CCRF. (**a**) Random field of \(c\). (**b**) Random field of \(\phi\). To further verify the differences of the algorithms, Table 4 compares the average statistics of CCRFs via 1000 runs of algorithms. It can be observed that the cross-correlation coefficients of simulated \(c^{\prime}\) and \(\phi^{\prime}\) associated with four candidate copulas are \(-0.5031\), \(-0.4932\), \(-0.4462\) and \(-0.4879\), which are almost similar to the original value of \(-0.5\). Other statistics, including \(\mu\), \(\sigma\), and COV, are also consistent with the original values. Namely, the proposed method recreates soil profiles similar to observations. This implies that the proposed copula-based CCRF is capable of considering the randomness, cross-correlation and spatial variability of soil parameters simultaneously, and effectively reflecting the real state of soil profiles with a relatively high accuracy. For comparison, Figure 9 shows a typical realization of random fields without considering the cross-correlation of soil parameters.