## 2 Random Spatial FE Models of the Subsoil
### The Subsoil Material Parameters Described by Three-Dimensional Random Fields
A graphical example of a generated highly correlated random field (\(d=0.01\) m\({}^{-1}\)) is shown in Figure 3.  
The assessed displacements \(\delta_{RF}^{i}\) (i = 1,..., 200) of a highly correlated random field (\(d=0.01\) m\({}^{-1}\)) are shown in Figure 4, along with the convergence diagrams for both mean value \(\mu_{\delta_{RF}}\) and standard deviation \(\sigma_{\delta_{RF}}\) of model response. The figure detects two positive vertical axes--standard deviation is presented on a positive upward axis, while displacements refer to a positive downward axis, the latter reflecting the settlement direction. This assumption is valid in subsequent similar figures. Figure 3: Example of a three-dimensional random field representing [PERSON]’s modulus; the numbers on respective axes denote node ordinal numeration. Figure 2: Correlation range for different damping parameters \(d\). The data in Figure 4 are representative of all five cases of different correlation decay parameters, due to the similar mechanical response of the soil block in the case of the other four damping parameters \(d\). The MC simulation convergence criterion is adopted: convergence holds if the mean value relative error is lower than \(\Delta_{\mu}=0.01\%\) and the standard deviation relative error is lower than \(\Delta_{\sigma}=1\%\) in the course of a 20-sample computation. In all five cases, the adopted criterion was fulfilled. However, higher MC simulation precision requires computing a larger number of samples. Computation of the extreme central node settlements \(\delta_{RF}\), their mean values \(\mu_{\delta_{RF}}\), and standard deviations \(\sigma_{\delta_{RF}}\) for all considered FE models are shown in Table 1.  
Note that Figure 4 displays peaks in the settlement diagram, slightly disturbing the response parameter convergence. In a highly correlated random field (\(d=0.01\) m\({}^{-1}\)), the extreme value \(\delta_{RF}^{d=0.01}=0.1825\) m is reached (Table 1). The peaks stem from the FE models with a low Young's modulus taken as the basis for random field generation. Due to high correlation, an initially low modulus propagates to a significant number of finite elements in the block volume, thus generating a weak subsoil, prone to excessive displacements. The peaks vanish with correlation decay increase, as the propagation of the initial low modulus quickly fades, not allowing for an extensive material weakening of the FE subsoil model. In a slightly correlated random field (\(d=5.0\) m\({}^{-1}\)), the peaks are not observable, so the extreme displacement \(\delta_{RF}^{d=5.0}=0.1085\) m does not come too far from the mean value estimated: in this case, \(\mu_{RF}^{d=5.0}=0.1048\) m (Table 1). The computational results (Table 1) can be summarized as a closed-form relation between the correlation decay parameter and standard deviation variability of the analysed settlement. The logarithmic approximation \(\sigma_{\delta_{RF}}(d)=-0.00201831\ln(d)+0.00363733\) is shown in Figure 5. As the settlement mean value remains at a specified level throughout the computations of distinct sets (Table 1), it is not considered here. \begin{table}
\begin{tabular}{c c c c} \hline
**Correlation** & **Mean** & **Standard** & **Extreme** \\
**Decay Parameter** & **Value** & **Deviation** & **Settlement** \\ \(d\) **[m\({}^{-1}\)]** & \(\mu_{\delta_{\text{RF}}}\) **[m]** & \(\sigma_{\delta_{\text{RF}}}\) **[m]** & \(\delta_{RF}^{ext}\) **[m]** \\ \hline
0.01 & 0.1076 & 0.0132 & 0.1825 \\
0.1 & 0.1057 & 0.0066 & 0.1350 \\
0.2 & 0.1050 & 0.0048 & 0.1279 \\
1.0 & 0.1052 & 0.0016 & 0.1109 \\
5.0 & 0.1048 & 0.0009 & 0.1085 \\ \hline \end{tabular}
\end{table} 
Table 1: Soil block settlement parameters, 200-item simulation procedure, random field approach. Figure 4: Convergence of simulation-based settlement estimation, the random field model, \(d=0.01\) m\({}^{-1}\). Figure 5 shows that the standard deviation change of the settlement in a random field approach to FE models may be represented by appropriate damping parameters \(d\) in Equation (2).