## 2 Reliability Analysis of Spatially Varied Soil Slopes  
### Autocorrelation Structure of Spatially Varied Soil Properties  
The spatial variability of soil properties indicates that soil properties are heterogeneous, with a certain correlation between two points in space. In general, soil properties are far more correlated at adjacent locations than those far away from each other [20]. The SOF is a quantitative statistical index indicating the range within which the soil properties are correlated [21; 22; 23]. Therefore, a large SOF implies a uniform soil material while a small one implies a heterogeneous material for which the properties change rapidly [24]. It is worthwhile to point out that, as one of the two extremes of the SOF, when the SOF becomes infinitely large, the soil properties at all locations are completely correlated, which can be characterized by a spatial constant. On the contrary, when the SOF is nearly zero or extremely small, the soil properties present little spatial autocorrelation in space. Furthermore, it is noted that SOF is direction dependent and SOFs in different directions are often not consistent. The SOF in the horizontal direction is generally larger than that in the vertical direction because of the depositional and post-depositional processes during the geological formations. Mathematically, the correlation between soil properties at any two points is governed by the ACF, which is the function of the SOF. Both the ACF and SOF are determined from measurement data by using geostatistics with the following steps [20; 25; 26; 27; 28]: (1) Evaluate the sample or experimental ACF from data, and (2) fit the sample ACF with the best correlation function. However, owing to the sparse and limited measurement data in geotechnical engineering, it is usually difficult to accurately determine the ACF and SOF from the measurement data [29; 30]. Hence, theoretical ACFs are widely used as replacements to describe the spatial variability of soil properties in the literature [16; 23; 31]. Since reliability analysis is a kind of simulation, it is acceptable to use the theoretical ACFs in slope reliability analysis. However, it is necessary to discuss the effects of different ACFs and SOFs on slope reliability analysis to further understand the potential ranges of the \(P_{f}\) of a slope. As reported in the literature [8; 12; 18], five theoretical ACFs are commonly used for characterizing the spatial variability of soil properties, as listed in Table 1. In the table, \(\rho\) denotes the autocorrelation function, and \(\tau_{x}=\left|x_{i}-x_{j}\right|\) and \(\tau_{y}=\left|y_{i}-y_{j}\right|\) denote the horizontal and vertical lag distances between two points within the soil unit, respectively. \(\delta_{h}\) and \(\delta_{v}\) represent the horizontal and vertical SOFs, respectively. Obviously, these functions are different in their expressions, thus resulting in different autocorrelation structures for soil spatial variability. However, physical differences between these functions are not discussed herein as previous studies have investigated this in detail [8; 16], and the purpose of this study was not to simply compare the differences between these ACFs. By contrast, the major aim of the current study was to further examine the effects of the combinations of different ACFs for different soil layers on the reliability analysis of layered slopes, which has not been investigated fully before.