## 4 Impact of the Spatial Correlation of Soil Properties  
This section examines the impact of the length of spatial correlation on permanent horizontal, vertical, and total displacement. The permanent displacements are evaluated at the top of the slope, indicated as point A in Figure 1.  
More specifically, three cases are investigated as shown in Table 3. For case a, the autocorrelation lengths are \(l_{x}=20\) m and \(l_{y}=2\) m; for case b, they are \(l_{x}=40\) m and \(l_{y}=2\) m; and for case c, they are \(l_{x}=20\) m and \(l_{y}=4\) m. All analyses correspond to a sample greater than or equal to 30. Table 5 shows the data of the cases to be compared. Along with the lengths \(l_{x}\) and \(l_{y}\), the values of the ratios \(l_{x}/d_{x}\) and \(l_{y}/d_{y}\) are also given, where \(d_{x}=\) horizontal length of the failure surface and \(d_{y}=\) height of the failure surface. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Earthquake** & **Mw** & **R (km)** & **Recding** & **Component** & **PGA (g)** \\ \hline Kalamata (1986) & 6.0 & 12 & Prefecture & Hor. & 0.25 \\ \hline Lefkada (2003) & 6.4 & 10 & Lefkada & Trans. & 0.60 \\ \hline Kobe (1995) & 7.2 & 20 & Port Island & Hor. & 0.57 \\ \hline Northrdge (1994) & 6.7 & 30 & Rinaldi & Hor. 318 & 0.47 \\ \hline Friuli (1976) & 6.5 & 19 & FRiuli & Hor. & 0.35 \\ \hline \end{tabular}
\end{table} 
Table 4: Historic earthquake records used as excitations. Figure 3: Spectra of acceleration of the modified earthquake excitations and Eurocode 8 spectra for rock sites for a maximum ground acceleration (PGA) equal to 0.3 g.  
Figure 4 shows a representative surface of failure and the dimensions \(d_{x}\) and \(d_{y}\)([PERSON], 2017) [24]. When the lengths \(l_{x}\) and \(l_{y}\) tend to infinity, then the soil becomes homogeneous. Conversely, when the lengths \(l_{x}\) and \(l_{y}\) tend to zero then the soil changes its properties randomly without any correlation between two adjacent or distant points. The use of ratios \(l_{x}/d_{x}\) and \(l_{y}/d_{y}\) helps us better understand the impact of spatial variability of soil properties in relation to the dimensions of the expected failure surface. Indeed, when the ratios \(l_{i}/d_{i}\) take values greater than one or two (\(l_{i}/d_{i}\) > 1 or 2), then the material at a given point has properties with a strong correlation with those found in other points within the failure surface, and thus tends to approach homogeneity. The method of determining the development of the failure surface of slopes is based on monitoring the equivalent plastic strain increment throughout the analysis. When it increases significantly and abruptly along a surface, this suggests that there is significant movement along this surface and indicates the initiation of a sliding failure. The evolution of the horizontal and vertical displacement at point A for heterogeneous soil (see Figure 1) is given in Figure 5, with residual values at the end of shaking equal to 0.75 m, respectively. Figure 6 illustrates the probability density function of the permanent horizontal displacement \(u_{x}\) for the various values of \(l_{x}\) and \(l_{y}\) based on the Weibull distribution. It is observed that for a fixed value of \(l_{y}\) = 2 m, when \(l_{x}\) increases from 20 m to 40 m, there is a very small decrease of the dispersion, as the soil becomes relatively homogeneous. Conversely, for a constant value \(l_{x}\) = 20 m, when \(l_{y}\) increases from 2 m to 4 m, the dispersion increases more significantly (Figure 6). Since the sample is significant (30 analyses), this trend can be interpreted as follows: doubling the autocorrelation length at high values (e.g., from 20 m to 40 m), the effect is low as both of these lengths suggest a large spatial autocorrelation. On the contrary, for a doubling of the autocorrelation length in small values (e.g., from 2 m to 4 m), the effect is significant in that the spatial correlation with short distances is small but increases more intensely with the doubling of the length \(l_{y}\).