**1. Introduction**

Shallow landslides are slope instabilities of a mass of soil and/or debris, which could involve the most superficial colluvial layers till around 2.0 m from ground level. Although they involve small volumes (101–105 m3) of soil, they can be densely distributed across small catchments [1] and can affect slopes close to urbanized areas, provoking significant damages to cultivations and infrastructures, and sometimes causethe loss of human lives [2].

Rainfall is generally the main triggering factor [3]. Rainfall features leading to shallow landslides and the consequent temporal probability of occurrence at regional scale are generally estimated by means of rainfall thresholds, defined for different geological, geomorphological, and environmental

settings [4]. These thresholds represent the main tool to estimate the daily or hourly level of hazard across a territory prone to shallow landslides or to implement earlywarning systems [5], representing the lower bound of rainfall conditions that caused the triggering of shallow landslides [3,6]. These thresholds are expressed as curves which separate the rainfall conditions leading to shallow slope failures from the ones where stability is maintained, sometimes with associated different probabilities of occurrence with uncertainties related to the possible incompleteness of the input data required to define the same thresholds [5,7].

The most widespread type of rainfall thresholds is the empirical one. These thresholds are reconstructed through the statistical analysis of empirical distributions of rainfall conditions that presumably resulted in the triggering of shallow landslides in a particular testsite [8]. The comparison between a multi-temporal inventory of shallow-landslides events and rainfall parameters measured in several points of the study area (e.g., in correspondence of raingauges) during the same analyzed time span is required in order to estimate these types of thresholds. Several authors proposed different methods for the estimation of empirical rainfall thresholds in different contexts all over the world [4,8–20]. In all cases, two different rainfall parameters were considered to build up boundary thresholds, namely cumulated event rainfall vs. rainfall duration or mean rainfall intensity vs. rainfall duration.

The use of only easily measurable rainfall data and the reconstruction based on the analysis of real past events, whether or not they triggered shallow landslides, makes empirical thresholds a reliable tool to estimate temporal probability of occurrence of shallow landslides at a large scale (catchment, regional, and national) [4,5]. Instead, these are sometimes limited in their effectiveness for different reasons. First, the shape of the thresholds is affected by the following: (i) the availability and quality of rainfall and of landslide information across the analyzed study area [21,22]; and (ii) the correct definition of the real rainfall features responsible for slope failures during a particular triggering event, generally linked to leakage of precise information about the moment of shallow landslides occurrence during a particular event [4,5]. Moreover, these types of thresholds do not take into account the unsaturated/saturated flow processes and the hydromechanical conditions of soils at the beginning of a particular rainfall event. The mechanical processes, which lead to shallow-slope failures, are in fact related to rainwater flows and water accumulation in the subsurface that provoke the increase in pore-water pressure and the consequent decrease of soil shear strength [22–26].

To overcome these limitations, rainfall thresholds can be estimated by means of a physicallybased model that can provide the assessment of the link between the rainfall features, the soil hydromechanical conditions before a rainfall event, and the shear strength response of the soils during the rainwater infiltration. In this case, the deterministic model estimates the response of the typical geological–geomorphological frame prone to shallow landsliding toward a particular rainfall event, defined by those parameters that are generally involved also for the reconstruction of an empirical threshold (cumulated event rainfall vs. rainfall duration; mean rainfall intensity vs. rainfall duration). This response is represented by the trend in time of the slope safety factor (Fs), during the modeled event. Triggering conditions are then represented by the rainfall patterns, which provoke the decrease of Fs below 1 (unstable conditions). Instead, if Fs stay higher than 1, shallow failures are not modeled (stable conditions). Some attempts were proposed to build up reliable physicallybased thresholds in some areas prone to shallow landslides, such as in Italian alpine catchments [26], catchments of the Central Italian Apennines [27], hilly catchments of Southern Italy [20,28–31], western hilly and mountainous settings of United States [32], and Chinese areas susceptible to shallow landsliding [33,34].

The main limitations of physicallybased thresholds are related to the most important disadvantages of the deterministic methods [35]: (i) requiring a significant amount of geotechnical, mechanical, and hydrological parameters for model simulation; and (ii) reconstructing the boundary conditions which represent, in the best way, the real soil and slope behaviors. Integration of meteorological measurements (e.g., rainfall) and hydrological soil parameters (e.g., pore-water pressure and water content) could help in obtaining a better insight into the quantitative effects of antecedent soil conditions on the triggering mechanism of shallow landslides. Thus, field monitoring allows us to improve the calibration of the physicallybased models used to reconstruct rainfall thresholds [23,32–34].

This paper aims to reconstruct and compare empirically and physicallybased rainfall thresholds for the occurrence of shallow landslides in a susceptible area of the Northern Italian Apennines (Figure 1). The main objectives of this work can be summarized as follows: (i) assessing empirical thresholds through the analysis of time series of rainfall data and of shallow-landslide inventories for the identification of the triggering and non-triggering events; (ii) calibrating a physicallybased model by the comparison between monitored and simulated soil hydrological parameters in correspondence of a test-site slope, which can be assumed to be representative of the typical geological, geomorphological, and environmental settings prone to shallow landsliding in the study area; (iii) assessing physicallybased thresholds through the application of the calibrated deterministic model in correspondence with the representative testsite for different rainfall events; (iv) comparing the two typologies of estimated thresholds and verifying their predictive capabilities through different inventories of occurred shallow landslides not used for the threshold reconstruction. Considered rainfall events corresponded to the ones that occurred in the 2000–2018 period and to other synthetic rainfalls characterized by strong average intensities and limited durations, which are not typical of the current climate of Oltrepò Pavese. Instead, their probability of occurrence may increase in the future due to the effects of climate change, which could cause an increase in very intense and short-duration rainfalls in Northern Italy, where the study area is located [36,37].
