## 1 Introduction  
The Loess Plateau in China covers approximately 440,000 km\({}^{2}\), which makes up 4.6% of China's land area [1]. Due to the population growth and the acceleration of urbanization, agricultural lands expanded significantly during the past several decades in the Loess Plateau. To improve the irrigation of these agricultural lands, many water resources and hydropower facilities were constructed by the government. However, these irrigation facilities may have adverse effects on the geological environment and cause geological disasters,such as landslides, debris flow, ground subsidence, and soil erosion [2; 3; 4; 5]. Among these geological disasters, retrogressive landslides are the most common and destructive. For instance, more than 120 retrogressive landslides induced by agricultural irrigation have been recorded on the Heifangtai Platform since 1984, resulting in more than 70 casualties and significant economic loss [6; 7; 8]. Recently, geological researchers investigated several typical retrogressive landslides caused by agricultural irrigation due to their severe geological and social consequences [9; 10; 11; 12; 13; 14]. Retrogressive landslides are defined as that the stable area of an initial landslide is subject to continuously subsequent failures until a stable state of the slope is reached [15; 16; 17]. The triggering factors for retrogressive landslides include rainfall [18; 19; 20], groundwater rise [21; 22], and irrigation [4; 23], all of which play critical roles in destabilizing the slope. Thus far, the studies of retrogressive landslides were mainly focused on slopes in clayey soils [21; 24; 25; 26; 27; 28; 29; 30]. Based on [PERSON] et al. [31], retrogressive landslides in clayey soils are usually caused by cracks, subsidence, and disruption of a slide body over a developing liquefied zone. [PERSON] et al. [32] quantified the volume, profile, and time series of a retrogressive landslide through terrestrial laser scanning. [PERSON] et al. [33] analyzed the characteristics of a retrogressive landslide using physical model tests and proposed a simplified numerical approach to predict the retrogressive landslide triggered by rainfall events. [PERSON] et al. [30] used physical plume tests to investigate the inclinations of back scarps in retrogressive landslides induced by irrigation. Based on the two-surface failure mechanism, a new theoretical method was developed to predict these inclinations. In addition, centrifuge tests and numerical simulations were also performed to determine the failure mechanisms of retrogressive landslides. [PERSON] [34], [PERSON] [35], and [PERSON] et al. [36] used the limit equilibrium method and numerical simulation to analyze the strength reduction of soil during retrogressive landslides. [PERSON] et al. [4] and [PERSON] et al. [37] conducted studies where they compared monitoring data with the predictions from existing empirical equations to evaluate the occurrence probability and runout distances of retrogressive landslides in terrains with quick clays. These studies have provided invaluable insights into the failure mechanisms of landslides in clayey soils but did not address the unique challenges posed by loess soils. In loess regions, retrogressive landslides exhibit unique characteristics due to the high moisture sensitivity of loess. Loess, an aeolian sediment, is highly susceptible to moisture changes, which can lead to structural failure [38]. In recent years, the issue of retrogressive landslides has become more severe in the Loess Plateau due to the increasing prevalence of irrigation practices. Previous studies have primarily focused on the effects of groundwater fluctuations on loess slopes. For instance, [PERSON] et al. [3] pointed out that an increase in groundwater levels leads to soil saturation, weakening its stability and triggering landslides. However, although [PERSON] et al.