## 3 Numerical Model with Random Fields of Material Properties  
An example of the numerical model discretization of a slope with inclination 2:1 (H:V) using the FLAC program is given in Figure 1. This model is used to examine the effects of the variation of soil properties, the characteristic autocorrelation lengths \(l_{x}\) and \(l_{y}\), and the intensity and frequency content of the seismic excitation. Two additional numerical models with different slope inclinations are examined later in the paper. Figure 1: Discretization of geometry A having slope inclination equal to 2:1 (HV). Table 1 gives the mean value \(\mu\) and standard variation \(\sigma\) of the cohesion \(c\), friction angle \(\varphi\), Young's elastic modulus \(E\), and density \(\rho\). The dilation angle \(\psi\) and Poisson's ratio \(v\) are also given, but without considering any variation. The cross-correlation coefficients \(\rho_{ij}\) of the properties \(i\) and \(j\), based on published experimental data, are given in Table 2. The \(\rho_{ij}\) values for which no experimental data were found have been considered as equal to zero ([PERSON], 2000, [PERSON], 2013) [20; 21]. It should be noted that random fields following the Gauss distribution are defined by their mean value \(\mu\), the standard variation \(\sigma\), and the autocorrelation function \(\rho(x,y)\). The most commonly used expression for the autocorrelation function between two points \((x,y)\) and \((x^{\prime},y^{\prime})\) in a two-dimensional anisotropic space is given by  
\[\rho(x,y)=Exp[-\frac{|x-x^{\prime}|}{l_{x}}-\frac{|y-y^{\prime}|}{l_{y}}] \tag{1}\]  
where \(l_{x}\) and \(l_{y}\) are the characteristic autocorrelation lengths in the horizontal and vertical directions, respectively. These lengths define the scale of fluctuation of the random field. Higher values of autocorrelation lengths indicate a stronger statistical correlation between two points. Table 3 shows the autocorrelation lengths \(l_{y}\), \(l_{x}\) in the vertical and horizontal directions, respectively, used in this study ([PERSON] et al., 1999) [22; 23]. \begin{table}
\begin{tabular}{c c c} \hline
**Property** & **Mean Value \(\mu\)** & **Coefficient of Variation \(\sigma\)/\(\mu\)** \\ \hline Cohesion \(c\), kPa & 30 & 0.3 \\ \hline Friction angle \(\varphi^{\circ}\), degrees & 20 & 0.2 \\ \hline Dilation angle \(\psi^{\circ}\), degrees & \(0^{\circ}\) & 0 \\ \hline Density \(\rho\), kg/m\({}^{3}\) & 2000 & 0.1 \\ \hline Modulus \(E\), MPa & 60 & 0.2 \\ \hline Poisson’s ratio \(v\) & 0.3 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Mean value and standard deviation of soil properties. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{**Correlation Coefficient \(\rho_{ij}\)**} \\ \hline
**Property** & \(c\) & \(\varphi\) & \(\rho\) & \(E\) & \(\
u\) \\ \hline \(c\) & 1 & \(-0.5\) & 0.5 & 0.2 & 0 \\ \hline \(\varphi\) & \(-0.5\) & 1 & 0.5 & 0.2 & 0 \\ \hline \(\rho\) & 0.5 & 0.5 & 1 & 0 & 0 \\ \hline \(E\) & 0.2 & 0.2 & 0 & 1 & 0 \\ \hline \(v\) & 0 & 0 & 0 & 0 & 1 \\ \hline \end{tabular}
\end{table} 
Table 2: Correlation coefficients of soil properties. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Autocorrelation Lengths** & \multicolumn{4}{c}{**Cases**} \\ \cline{2-5}  & a & **b** & **c** \\ \hline \(l_{x}\), m & 20 & 40 & 20 \\ \hline \(l_{y}\), m & 2 & 2 & 4 \\ \hline \end{tabular}
\end{table} 
Table 3: Autocorrelation lengths. To illustrate the development of a numerical model of the slope using random fields of properties, the geometry in Figure 1 is considered. The soil properties are shown in Table 1 and the cross-correlation coefficients are shown in Table 2. The characteristic autocorrelation lengths used are \(l_{y}\) = 2 m in the vertical direction and \(l_{x}\) = 20 m in the horizontal direction. The generation of the random field of the material properties using the LAS method assumes a Gaussian distribution.