# Probabilistic Analysis of Infinite Slope Stability Considering Variation in Soil Depth
## 1 Introduction  
Landslides are major natural geological hazards that annually cause human casualties and enormous property damage worldwide. Heavy rainfall is a main triggering factor for landslides [1, 2], and climate change could cause an increase in the intensity, duration, and frequency of extreme rainfall events. Numerous studies have predicted that the frequency and risk of landslides would increase because of the effects of climate change [3, 4, 5]. Physical or statistical approaches are used widely to evaluate landslide vulnerability [6], and recently, machine learning approaches have also been actively conducted [7, 8]. Each of these approaches has advantages and limitations. Physically based models are commonly used to assess slope stability by considering external physical factors, such as rainfall or earthquakes that trigger landslides, and to develop early warning systems for landslides. The main parameters in the physically based model are the shear strength parameters of soils (cohesion, \(c\), and friction angle, \(\phi\)) and soil unit weight. However, soil is naturally heterogeneous, and its properties exhibit high variability. In particular, soil propertiesare spatially correlated, and their variability is known to significantly affect soil behavior. Slope stability is a geotechnical engineering problem, the analysis results of which are dominated by uncertainty [9], and numerous studies have investigated the effects of the spatial variability of soil on slope stability analysis [10; 11; 12; 13]. In a probabilistic analysis of slope stability, cohesion and friction angle are known to have the greatest influence on failure probability. Therefore, these two parameters are commonly treated as random variables, and their large variability is associated with a higher slope failure probability. Furthermore, considering the correlation between these two parameters is crucial, and it is known that accounting for a negative correlation between them reduces the slope failure probability. The unit weight of soil is occasionally considered a random variable, but compared with other variables, its variability is lower, leading to its influence being disregarded. Another variable in constructing a physically based model is the slope angle and soil depth, which are essential factors for generating cross-sections of the slope and, generally, are treated as deterministic variables. Slope angles could easily be determined through field surveys or using a digital elevation model (DEM). However, soil depth, defined as the vertical distance from the soil surface to the bedrock, is an unknown variable in the hillslope system [14; 15]. Moreover, estimating soil depth is difficult, as it is controlled by complex interactions of several factors, including topography, parent material, vegetation, climate, and weathering processes [14; 16]. In particular, as the strata are formed continuously and mostly exhibit gradual weathering changes, defining clear boundaries for soil depth is not straightforward. Soil depth plays a crucial role in the prevention and management of landslides and is a key parameter in analysis to predict shallow landslides and estimate debris flows [17; 18; 19; 20]. Nevertheless, compared with other parameters, the importance of soil depth has often been overlooked [21]. While it is known that soil depth exhibits high spatial variation compared with other soil properties [14; 22], its variability has not been evaluated. To quantitatively account for the vertical variability of soil depth in probabilistic analysis, the probabilistic distribution and magnitude of this factor should be identified as a priority. However, currently, no known information on such aspects exists. Therefore, despite numerous studies on probabilistic slope stability analyses being conducted, soil depth has always been considered a deterministic value, and there has been no research that treats it as a random variable. This study aims to achieve two primary objectives.