Sensitivity Analysis of a Physically Based Model to Assess Rainfall-Triggered Shallow Landslides

Rainfall-triggered shallow landslides are one of the most frequent geomorphological processes in tropical mountainous terrains, and they deserve special attention due to their potential negative consequences in society. Such processes involve several variables including those regarding precipitation, terrain morphology, and hydraulic and shear strength parameters of saturated and unsaturated soils. A spatially distributed and physically-based model for transient rainfall infiltration and grid-based regional slope stability analysis (TRIGRS), was implemented. It was used to analyze the influence of the mentioned variables on the triggering of shallow landslides by rainfall. A robust sensitivity analysis was carried out to quantify the influence of parameter variation on the change of the factor of safety (Δfs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta f_\textrm{s}$$\end{document}) for shallow landslides, including a one-parameter-at-a-time analysis of a unique cell representing the simplest model space. Four combinations of saturated/unsaturated models and finite/infinite infiltration models were analyzed. The interaction among the hydraulic parameters was analyzed through small-multiple plots to observe their influence on Δfs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta f_\textrm{s}$$\end{document}. It was found that Δfs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta f_\textrm{s}$$\end{document} is most sensitive to variation in the maximum depth at which the landslide can be triggered, the slope angle, the cohesion, and the friction angle. In addition, it was found that the parameters of the soil–water characteristic curve for unsaturated soils have little influence on Δfs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta f_\textrm{s}$$\end{document}. Finally, the interactions among the remaining parameters determine the impact of each one of these on Δfs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta f_\textrm{s}$$\end{document}. Thus, the results showed that when the baseline set of parameters is changed, the influence of each parameter is modified.


Introduction
Landslides result when masses of soil move down slope along a surface at the slide base (Cruden 1991;Cruden and Varnes 1996). These processes have caused numerous victims and economic losses worldwide (Assis et al. 2019;Gómez et al. 2020;Garcia-Delgado et al. 2022), leading to an increase of interest in understanding their triggering mechanisms, modeling them, and predicting possible scenarios in order to reduce negative impacts and improve landuse planning.
Slide-type mass movements (landslides) are classified as translational or rotational based on their kinematic mechanism. Translational landslides tend to be shallower and much longer than rotational failures and the mass moves along a planar surface (Burns 2013;Melo et al. 2022). Among landslides detonated by rainfall (Wieczorek 1996;Iverson 2000; Aristizábal and Sánchez 2020), translational landslides are triggered by high-intensity events, while rotational deep-seated landslides are usually caused by low-intensity and long-duration rainfall events (Sidle and Ochiai 2006a;Rahimi et al. 2011). A wide range of topographic gradients and other morphometric terrain attributes (i.e., predisposition factors), as well as frequent rainfall events with variable intensity and duration (i.e., triggering factor), contribute to landslides (Melo et al. 2022).
The mobilized mass in a landslide usually affects the uppermost soil horizons of the weathering profiles (i.e., in-situ soils) or hillside deposits (i.e., sedimentary soils) accumulated on pre-existing geomaterials (Sidle and Ochiai 2006b;Aristizábal et al. 2017;Borrelli et al. 2018). Geotechnical characterization of both types of soil takes additional effort, since they may be unsaturated. Traditionally, the shear strength of unsaturated soils depends on the strength parameters of saturated soil mechanics and the soil suction, often estimated through the soil-water characteristic curve as a function of the water content (Fredlund et al. 2011;Meza Ochoa 2012).
During a rainfall event, part of the water runs off on the terrain surface, and the other part infiltrates into the soil, changing pore pressures that change the conditions of the effective stresses and, therefore, the shear strength. As a result, landslides may be triggered either by a decrease in suction when unsaturated, or by an increase in positive pore pressure due to the water table rise (Sidle and Bogaard 2016;Conte et al. 2022). Infiltration and runoff are both determined by the hydraulic properties of the soil and rainfall intensity, which means that more variables are incorporated into the problem.
Therefore, accurately assessing landslides requires the implementation of numerical models capable of incorporating the topographic effect of the terrain surface and the effect of rainwater infiltration on the variation of the shear strength of unsaturated soils. It means that geomorphological, geotechnical, hydraulic, and meteorological variables must be taken into account in such physically-based models.
TRIGRS is an acronym for Transient Rainfall Infiltration and Grid-based Regional Slope-stability analysis. TRIGRS is a physically-based model developed and improved by Baum et al. (2002Baum et al. ( , 2008, and Alvioli and Baum (2016). It incorporates the variables associated with topography (slope angle), precipitation events (rain intensity and duration), unsaturated shear strength parameters, and hydraulic parameters of the soil to evaluate shallow landslides. The assessment is performed spatially by calculating the factor of safety ( f s ) of terrain prone to shallow landslides driven by rainfall, which is one of the most frequent triggering factors in tropical mountainous terrains such as the Colombian Andes. TRIGRS has been proven to be effective and applicable in tropical mountainous terrains due to its successful prediction rates in registered historical cases, where the mechanisms are compatible with those evaluated by the TRIGRS model (e.g., Marin et al. 2021).
The assessment of landslides is attributed to numerous parameters (Huat et al. 2006), and they need to be considered simultaneously in models to evaluate the occurrence of this type of process (Assis et al. 2019). Reducing the associated uncertainty in the input parameters may noticeably decrease the uncertainty measured in the model response (Mahmoudi et al. 2019). Sensitivity analysis (SA) evaluates the effects of parameters on the model response. Then, detecting the most sensitive model inputs is desirable to focus resources for parameters' characterization on those parameters in order to reduce the overall model uncertainty. Even though TRIGRS has been implemented in numerous studies to analyze rainfall-triggered shallow landslides, some limitations remain in the studies of sensitivity analyses of its parameters (e.g., Bordoni et al. 2015;Gioia et al. 2016;Zieher et al. 2017;Marin and Velásquez 2020;Osorio-Ríos et al. 2022). First, in previous works on one-parameter-at-a-time (OAT) SA, authors have not considered all the parameters of the model. Second, both OAT and regional analyses for shallow landslides usually compare the change in f s ( Δf s ) referring to a unique initial set of parameters, however, the magnitude of Δf s might be dependent on the initial set of parameters. Third, there is a lack of outcomes regarding multivariable sensitivity analyses in which the interaction among physically related parameters is taken into account.
This research aims to perform a robust sensitivity analysis of TRIGRS's parameters for assessing rainfall-triggered shallow landslides. Initially, through a single cell OAT analysis, then, by analyzing the effect of changing the baseline set of parameters. Multivariable sensitivity analysis via small multiple charts will also be addressed in order to understand the influence of each variable on the magnitude of Δf s under different reference sets of hydraulic and rainfall parameters. As a result, TRIGRS users will have a valuable tool, as they may concentrate effort and optimize resources when characterizing parameters while reducing model uncertainties.
This paper is outlined as follows. First, a brief analysis of previous works on sensitivity analysis using TRIGRS is presented in Sect. 2. Then, Sect. 3 shows the main numerical aspects of the model and the methodology. The obtained outcomes are presented and analyzed in Sect. 4. Finally, the conclusions of this research and a description of possible future works derived from the results are pointed out in Sect. 5.

Previous Works on Sensitivity Analysis Using TRIGRS
This section reviews previous studies on sensitivity analyses of TRIGRS input parameters for analyzing rainfall-triggered shallow landslides (RTSL) stability. The main findings are discussed next. Zieher et al. (2017) is probably one of the most significant works on sensitivity analysis of TRIGRS variables to evaluate RTSL. As a result, an in-depth review of this study is presented. First, they performed a local OAT-SA, varying TRIGRS parameters in addition to parameters related to vegetation effects on stability, which were not originally included in TRIGRS. They calculated the change of the factor of safety ( Δf s , Eq. 2) for different values of: • Terrain-derived parameters: slope angle ( ) and soil depth ( d LZ ).
• Hydraulic and rainfall parameters: Saturated hydraulic conductivity ( K s ), specific storage ( S s ), precipitation intensity ( I nZ ), and initial water table depth ( d).
From the OAT analysis, they found that an increase in ′ , c ′ or c r can affect Δf s positively, while an increase in , d LZ or s t affect Δf s negatively. OAT variation may have a non-linear effect on Δf s for example, and d LZ effect varies for different values. Overall, Δf s is more sensitive to geotechnical parameters than to vegetation-related ones.
K s , S s , and d have different effects on Δf s depending on I nZ . In general, reducing K s leads to an increase of Δf s , while increasing S s has the opposite effect as diffusivity ( D 0 ) decreases and makes infiltration rate quicker. I nZ showed a similar effect on Δf s as K s but is less sensitive.
Second, ′ , c ′ , K s , and S s , identified as sensitive parameters, were systematically sampled and regionally calibrated for a landslide-triggering rainfall event in the Laternser valley, Vorarlberg (Austria) in 2005. A model ensemble was made up of the best 25 behavioral model runs of the tested parameters. These models were selected by maximizing the true positive rate (TPR) and the true negative rate (TNR), i.e., minimizing the distance to perfect classification. The same model ensemble was later validated using a second rainfall event in 1999. The prediction rates of the observed shallow landslides were 73.0% and 91.5% for both rainfall events, respectively.
Although Zieher et al. (2017) performed a considerable number of tests to identify the effect of TRIGRS parameters on slope stability, additional concerns remain to be researched. For instance, one of those aspects is the influence of shear strength parameters of unsaturated soils on the RTSL assessment. Those parameters depend on the soil suction that may be calculated through the SWCC (Fredlund et al. 2011;Meza Ochoa 2012). TRIGRS features for unsaturated material allow the input of the SWCC parameters such as the saturated ( sat ) and residual ( res ) water content, as well as the fitting parameter ( ) for the Gardner (1958) model. As Zieher et al. (2017) did not include those parameters in the analyses, there is still a question on how sensitive Δf s is to the variation in SWCC parameters. Zieher et al. (2017) looked at precipitation, but only at its intensity variation, and it is unclear what the effect of increasing or reducing rainfall duration for the same intensity is. The hydraulic parameter of the initial (steady or background) infiltration rate ( I ZLT ) was set to zero and its variation was not taken into account for OAT analysis. However, Baum et al. (2010) stated that model results are very sensitive to the steady seepage initial condition.
It is important to highlight that Δf s is calculated as the difference in f s divided by a baseline value of f s ( f s, 0 ) obtained by evaluating an initial set of parameters. Yet, none of the described behaviors of sensitivity have been compared to a different set of parameters to check whether they are affected or not. Additionally, it was pointed out by the authors that OAT analysis did not consider interactions between parameters. Research on these previous aspects needs to be pursued.
Other studies include Bordoni et al. (2015)'s work, in which an unsaturated TRIGRS model was also implemented, initially on a sample slope and then on a regional area in Oltrepò Pavese (northern Italy). They performed an OAT analysis for s , ′ , and c ′ . Hydrological parameters ( res , sat , , and K s ) were varied simultaneously because of the connection among them. Bordoni et al. (2015) sensitivity analyses showed that the variation in ′ and c ′ had more effect in the calculation of the TPR and the false positive rate (FPR) than the effect of changing s and the hydrological properties ( sat , res , , and K s ). The mean values of the soil properties showed the best results in terms of the ratio between TPR and FPR compared to the results of evaluating the mean values plus/minus one standard deviation.
Both Zieher et al. (2017) and Bordoni et al. (2015) only considered an unsaturated model with finite depth to an impermeable basal boundary. It is acknowledged that the author's selected model was the more suitable one for their regional study zone. The finite depth is preferred only when there is a strong contrast in permeability at shallow depth, for example in hillsides that have a colluvial mantle only a few meters thick (Baum et al. 2010). However, TRI-GRS offers the possibility of choosing another model with infinite depth to an impermeable basal boundary, either for saturated or unsaturated conditions. The infinite depth is suitable for relatively homogeneous hillsides.
In contrast to Zieher et al. (2017) and Bordoni et al. (2015), Gioia et al. (2016) tested two TRIGRS models, a saturated initial condition with an impermeable basal boundary at hypothetical infinite depth (SAT-INF) and an unsaturated initial condition with an impermeable basal boundary at 1.0 m (UNS-FIN). For each model and for three soil types, K s , D 0 , and d were varied to evaluate their influence on the transient pressure head for two rainfall-triggered landslides events in 1996 and 1998 that took place in the eastward section of the Esino river catchment, Marche region (Central Italy). Although Gioia et al. (2016)'s work was not focused on Δf s sensitivity, they used both SAT-INF and UNS-FIN TRIGRS models and the time response of the pressure head ( (Z, t) ) varying parameters as a validation method for a set of parameters calibrated in their study. Such implementations encourage taking advantage of TRIGRS features not having been fully explored in sensitivity analysis for RTSL. Marin and Velásquez (2020) used TRIGRS to analyze the influence of the hydraulic properties on the stability results and the rainfall thresholds for the occurrence of shallow landslides.
The study was carried out in a tropical mountainous basin in Envigado (Colombia) and the model consisted of an unsaturated condition with an impermeable basal boundary at a finite depth. They analyzed the transient variation of (Z, t) and f s in cells with different values of , I nZ , and K s . Then, they extended the analysis to study the effect of the hydraulic properties on the rainfall thresholds. Marin and Velásquez (2020) identified that the infiltration in the grid cells reached a steady state in which the maximum pressure head did not vary despite the mean rainfall intensity simulated. However, the time when the steady condition is reached mainly depends on I nZ , Z max , d , , and K s . More recently, Osorio-Ríos et al. (2022) calibrated and evaluated a TRIGRS model to study a regional area with two geological surface units located in San Antonio de Prado in Medellín (Colombia). Then, they performed a sensitivity analysis of the calibrated geotechnical parameters, d , and rainfall recurrence intervals ( T r ) on stability. It was found that rainfall duration plays a significant role in most cases, and is more prevalent when the geotechnical parameters are reduced by one standard deviation and when the groundwater table is one meter below the sliding surface. Osorio-Ríos et al. (2022) combined three variations of the calibrated geotechnical parameters, d , and T r for a total of 27 cases, which is an interesting first approach to interactions among parameters. Nevertheless, as for the previously cited studies, the variation of some inputs for the TRIGRS model was left out, and the influence of the variation in those inputs on the regional landslide assessment remains unknown.
Finally, Velásquez and Marin (2022) used TRI-GRS with an unsaturated infiltration model to evaluate the influence of soil mechanical properties ( c ′ , ′ , and s ) on the definition of rainfall intensity and duration thresholds. The parametric analysis was applied in four watersheds located in the Aburrá Valley (Colombia). They found that those mechanical parameters affect the threshold significantly, as small changes in them can increase or decrease the stability of the study areas. An increase in c ′ and ′ shifts the thresholds to lower values, while an increase in s displaces them to higher ones. Despite their promising findings, it is important to highlight that the variation range of two of the three evaluated parameters ( ′ and s ) reaches magnitudes that may be considered unrealistic under typical geotechnical characterizations.
The previous works on sensitivity analysis using TRIGRS have been limited to analyzing a few parameters, do not consider all the models available at TRI-GRS, and ignore interaction among variables. This research will address these shortcomings through more robust sensitivity analyses than those presented so far for a better understanding of the parameter influence on RTSL modeling. The approach and methods used in this study will be explained in the next section.

Approach and Methods
The previous section showed that evaluating RTSL demands numerical models that incorporate terrainderived, geotechnical, hydraulic, and rainfall parameters. TRIGRS was chosen for this study, not only because it includes the mentioned parameters, but because it also incorporates the transient nature of infiltration and rainfall events characterized by fluctuations in intensity over time and its effects on unsaturated soil behavior. The following subsections explain the selected model and the scheme for the sensitivity analysis of the model outcomes when its parameters are modified.

The TRIGRS Model
TRIGRS is an open-source program in FORTRAN for physically-based modeling of the timing and spatial distribution of RTSL. Although the program runs over a spatial domain, the slope stability calculations are derived from a one-dimensional domain represented by cells of a grid-based space. Those calculations are obtained using analytical solutions for partial differential equations of models for either saturated or unsaturated soils (Alvioli and Baum 2016). The program uses the infinite-slope stability method for translational planar landslides. The transient infiltration is assumed vertical and runoff to neighboring downslope cells is instantaneous where precipitation exceeds the infiltrability. A complete development of the mathematical model of infiltration, subsurface flow, runoff, flow routing, and their effects on slope stability can be found at Baum et al. (2010).

Slope Stability Model
For assessing the stability of a slope prone to shallow landslides, TRIGRS uses the infinite-slope method by Taylor (1948). It represents a landslide as a continuous and homogeneous mass sliding over a plane Fig. 1 Depiction of general slope model. Adapted from Baum et al. (2008) parallel to the slope surface. The mass has a considerable extent, and any vertical slice, as depicted in Fig. 1, may be taken as representative of the entire sliding mass.
Neither deformations nor interactions between adjacent cells are taken into account in the model. The stability is evaluated by calculating f s as the ratio of the strength to the driving forces at the slip surface. In TRIGRS, f s is calculated at different vertical depths, Z n ∈ 0, Z max , using Eq. 1 and the minimum value is assigned to the cell: where ′ is the effective angle of friction, c ′ is the effective cohesion, w is the water's unit weight, and s is the soil's unit weight. is the slope angle measured from a horizontal reference line.
The groundwater pressure head is represented by as a function of depth (Z), and time (t). depends on the saturation status, the infiltration condition, and several hydraulic parameters. These parameters include the initial water table depth ( d ), saturated hydraulic conductivity ( K s ), diffusivity ( D 0 ), precipitation intensity ( I nZ ), and initial (steady or background) infiltration rate ( I ZLT ). In this study, f s is calculated at Z max and at the end of the rainfall, which means that = (Z max , t n+1 ).
The initial model condition might be saturated (SAT) or unsaturated (UNS). If unsaturated (Fig. 1), which is the most generalized condition, the soil is divided into two zones, a saturated one below the water table, and an unsaturated zone above the capillary fringe that overlies the water table. The soil-water characteristic curve (SWCC) that relates the matric suction and saturation degree of soils is approximated through Gardner (1958) model. Parameters such as the saturated ( sat ) and residual ( res ) water content, as well as the fitting parameter ( ) are then incorporated. Evaluation of f s also relies on the saturation condition as the matric suction, (Z, t) w in Eq. 1, is multiplied by , the Bishop (1959)'s effective stress parameter for the unsaturated zone.
Moreover, TRIGRS also incorporates two infiltration models depending on the depth at which an impermeable boundary is located. Referring to Fig. 1, if K s above and below the basal boundary are approximately equal, then it is an infinite-depth impermeable boundary model (INF). However, if there is a strong hydraulic conductivity contrast, it is more appropriate to use a finite-depth impermeable boundary model (FIN).
To sum up, TRIGRS provides four possible models for infiltration scenarios (finite or infinite) and for initial saturation conditions (saturated or unsaturated). These can be combined as SAT-FIN, SAT-INF, UNS-FIN, and UNS-INF. Each one of them has its particular numerical solution for determining the water table rise as a consequence of infiltration and the calculation of (Z, t) above and below the water table. The detailed mathematical development is explained by Baum et al. (2008Baum et al. ( , 2010.

Sensitivity Analysis
Sensitivity analyses are used to assess the effects of the model parameters on the model response. This statement has implications in terms of the model's overall uncertainty. Reducing the associated uncertainty in some parameters may noticeably decrease the uncertainty measures in the model response (Mahmoudi et al. 2019). Hence, identifying the most sensitive parameters is desirable to focus resources for parameter characterization in order to reduce the overall model uncertainty.
In this study, an OAT-SA on a local one-cell spatial model is performed. The procedure consists of varying each parameter individually and registering the variation effect on f s against a baseline value, f s, 0 calculated with an initial set of parameters P n=0 = {⋯} . In all cases, variation of individual parameters must be within a physically reasonable range, otherwise, it will lead to unrealistic slope stability scenarios. The variation effect is calculated as the change in f s , Δf s , through Eq. 2: where f s, n, i is the factor of safety obtained from the variation of the ith parameter of the nth set of parameters.
The OAT-SA procedure is illustrated in Fig. 2  OAT-SA is proposed to be conducted over a new P n by modifying the most sensitive parameters detected in a previous run ( P mod ). Alternatively, the four different models provided by TRIGRS can also be analyzed in order to detect how initial saturation condition and infiltration models affect f s .

Model Parameters
Parameters of the baseline set, P 0 , are presented in Table 1. The 14 parameters were grouped into three categories. The first group was named miscellaneous parameters and includes , Z max , d , and the rainfall duration (t). The second group contains the shear strength parameters of the soil ( c ′ , ′ ) and s . The third group includes hydraulic parameters of the soil ( K s , D 0 , and I ZLT ) along with sat , res , and that are strength parameters as well, and the rainfall intensity, I nZ . P 0 corresponds to the set calibrated by Osorio-Ríos et al. (2022) for Altavista Stock's residual soil, which is one of the two geological units mapped over the study area by Consorcio Microzonificación (2006). The f s value evaluated at P 0 for the saturated models is 1.41 and for the unsaturated models is 1.92. For this set of parameters, there is no difference between the finite and infinite depth models, as will be shown in Table 3 when comparing them with another set of parameters. Figure 3 shows the influence on f s of the individual variation of each of the 14 parameters in P 0 for a single-cell spatial model. As explained in Sect. 3.3, the influence of parameter variation is expressed in terms of a percentage of change in f s regarding f s, 0 calculated with P 0 as shown in Eq. 2. The OAT-SA showed that some parameters have a positive influence on f s , i.e., f s increases when increasing the parameter, while other parameters cause f s to decrease (negative influence). The most negatively influential parameter is closely followed by Z max as shown in Fig. 3a. They can reduce f s by 15% for a 30% increase in the parameter, also, they can increase f s by 20% for a reduction of 20% in the parameter. These results are expected as steeper planes are more prone to instabilities. As Z max increases, f s decreases because Z is in the   Fig. 3 Results from the OAT-SA using the initial set of parameters P 0 denominator of the second term of Eq. 1, and because there is an increase in (Z, t) at higher depths. However, any systematic increase in Z max must be carefully analyzed, as such depths may exceed the limitations of the physical model for shallow landslides and other failure mechanisms should be considered instead. Δf s is also negatively sensitive to s and its influence increases with unsaturated models as it may be seen with the higher slopes in Fig. 3b. The most positively influential parameters on Δf s are the shear strength parameters, c ′ and ′ as shown in Fig. 3b. f s increases by 10-15% for a 20% increase in c ′ and increases by 5-10% for a 10% increase in ′ . For saturated conditions, c ′ has more effect on f s than ′ ; but, for unsaturated models, ′ and c ′ have basically the same effect on f s , as the effect of c ′ decreases and the effect of ′ increases. The c ′ parameter is identified as the only parameter that produces a linear effect on f s , as it is a linear term in the calculation of f s (see Eq. 1) and (Z, t) does not depend on it. Overall, in the UNS-FIN model, the Δf s response when varying , Z max , and the shear strength parameters is consistent with the behavior observed by Zieher et al. (2017). sat , res , and have no effect on f s in the saturated finite and infinite models and little effect in the unsaturated models. In the latter, a decrease of 30% and 65% of sat and respectively, can reduce f s by up to 5% in the UNS-FIN model; while no change is seen for an increase of these values. Saturated infiltration models do not take into account the soil water retention curve but only saturated volumetric, which is why sat , res , and have no effect on Δf s . On the other hand, the little effect of the same parameters in the unsaturated infiltration models is a circumstantial behavior of the evaluated case. Among other variables, infiltration in unsaturated porous media depends on the initial value of d , and, then, on the initial value of matric suction; as a consequence, a different relationship between d and Z max can modify the observed effect. Additionally, the response of Δf s is observed at the end of the rain, when, probably, the maximum value of (Z, t) and saturated conditions have been reached; then, the effect of changing sat , res , and on Δf s can be stronger at earlier stages of the rainfall event.

One-Parameter-at-a-Time Sensitivity Analysis
Changes in d and t have different effects depending on the model as shown in Fig. 3a. For saturated models, where (Z, t) is evaluated as the superposition of the steady component (initial d , I ZLT , and K s ) and the transient infiltration, there is no effect on f s , because the maximum physical value of (Z, t) imposed by Baum et al. (2008Baum et al. ( , 2010 following Iverson (2000), has been reached at the end of the rainfall regarding the initial position of the water table or the rainfall time. However, for rainfall time reduced by more than 60%, f s increases for the SAT-INF model, which means that the rainfall occurred so quickly, that (Z, t) did not reach its maximum physical allowed value during the infiltration process. A similar behavior is noticed for both unsaturated models, but the required reduction in the time is by more than 70%. Still, unsaturated models, especially the UNS-FIN model, show a slight decrease in f s when increasing time, which means that in those cases, (Z, t) had not reached its maximum physical allowed value during the infiltration process. The previous statement is confirmed when analyzing the effect of d on Δf s due to the change in (Z, t) . In both unsaturated models, d positively affects f s . In these cases, an increase in d means that the water table is located deeper than the baseline value, and, as a result, the infiltration process is delayed and attenuated. Such a phenomenon is explained because the unsaturated zone acts as a filter that absorbs part of the water that infiltrates the ground surface; the remaining water passes through the unsaturated zone until it reaches the capillary fringe and accumulates at the base of the unsaturated zone above the initial d (Baum et al. 2010).
Finally, Fig. 3c shows the effect of varying the hydraulic and rainfall parameters on Δf s . There is a significant difference between saturated and unsaturated conditions. The only aspect in common is that variations in I ZLT do not modify f s in any of the models for the set of parameters used. Overall, it is interesting to note that the effect of these parameters on f s is different when it is a decrease or an increase of the value. In the SAT-FIN model, reduction in D 0 and I nZ by about one order of magnitude (OM) produces a substantial increase in f s by about 25%; a similar increase in f s occurs when K s increases by about one OM. The behavior of the SAT-INF model is quite similar to the SAT-FIN model, except that the increase in f s occurs when the D 0 and I nZ are reduced by less than one OM or when K s is increased by less than one OM. However, an increase in D 0 and I nZ or a decrease of K s does not produce any change in f s . In unsaturated models, these hydraulic and rainfall parameters variation affects differently the Δf s values. In these cases, variation in D 0 has no effect on f s . However, K s and I nZ may increase f s by about 7%, much less than observed for saturated conditions. Another interesting difference is that f s increases when K s is reduced, as well as when it is increased. Also, in the UNS-FIN model, there is a range in K s , at which f s is reduced by up to 7%.
Despite some tendencies were observed in OAT-SA regarding the hydraulic and the rainfall parameters, this approach may be considered limited since those parameters, along with d and Z max interact to compute (Z, t) and, in turn, f s . Therefore, Figs. 6 and 7 show an extended sensitivity analysis focused on the interaction among hydraulic parameters and its effect on Δf s ; the results are discussed in Sect. 4.3. Overall, results shown in Fig. 3 are surprising since the sensitivity of the model to each parameter is not symmetrical (i.e., it changes with the value of f s ) and may be significant only in a range of values. Thus, in the next section, the reference set of parameters will be changed.

The Effect of Changing the Baseline Set of Parameters
After detecting the most influential parameters in Δf s for the initial set of parameters P 0 evaluated in the previous section, they were varied to compose a modified set of parameters, P mod , that accomplishes two conditions: first, that f s evaluated at P mod ( f s, 0 | P mod ) was reduced by at least 25% compared to f s, 0 | P 0 , and second, that f s for the saturated models were close to, but no lower than unity. The modified parameters and their variations are recorded in Table 2. As the aim was to evaluate the model sensitivity with a distinctly different set of values, some parameters were increased while others were decreased. For instance, to evaluate the effect of the shear strength, the c ′ value was reduced by 50% while the ′ value was increased by 18.5%. Similarly, , is decreased by 16.7%, while Z max is increased by the same 16.7%. The f s, 0 for all models and the change between results for each set are shown in Table 3. Note in this table that the change in f s is greater than −25 % for all models, ranging between 26 and 29%, as required. The results for the unsaturated models are different for the finite depth model ( f s, 0 | P mod = 1.27) and for the infinite depth model ( f s, 0 | P mod =1.36). In addition, the differences are slightly lower between the saturated and unsaturated models with P mod than those for P 0 .
The results of the OAT-SA considering P mod are presented in Fig. 4. In this case, the hydraulic and rainfall parameters were not included, as they will be analyzed in Sect. 4.3. Continuous lines correspond to the same results presented in Sect. 4.1, while dashed lines evidence Δf s after variation in the parameters to which TRIGRS results are most sensitive ( Table 2).
The most contrasting behavior is observed for both saturated models in s , as it has a negative effect on Δf s if P = P 0 (continuous lines in Fig. 4b) but a positive effect on Δf s if P = P mod (dashed lines). As for the unsaturated models, s has little to no effect on Δf s with the modified parameters. It can be explained in part because, in P 0 , d was located at Z max = 3.0 m , while in P mod , d was located at 2.0m and Z max at 3.5 m. Thus, TRIGRS computes an averaged s to account for the lower unit weight of partially saturated soil above d (Baum et al. 2010), as a result, s becomes a function of the ratio between d and Z max . For the unsaturated models, the changes of d and Z max in P mod seem to considerably affect the influence of variations in d , t, , sat , and res on Δf s , in contrast to the little effect of variations in the same parameters on Δf s for P 0 . While TRIGRS remains most sensitive to changes in ′ , c ′ , Z max , and , despite the selected set of parameters, the difference in the magnitude of the effect of the parameters might be minor to significant. For instance, in the saturated models and for P 0 , Δf s is more sensitive to changes in c ′ than to changes in ′ , but for P mod , the sensitivity is inverted. In the unsaturated models, for P 0 , f s was almost equally affected by changes in both c ′ and ′ , but for P mod , Δf s is much more sensitive to changes in ′ than to changes in c ′ . The effect of Z max on Δf s was reduced when setting P mod instead of P 0 for all the range of changes in the parameter in the saturated models, but only for some increments in the parameter in the unsaturated models. Similarly, f s is slightly more sensitive to increments in for P mod than for P 0 .
Some parameters, such as d , t, sat , res , and , did not show significant modifications in the saturated models when changing P 0 to P mod for the same reasons explained in Sect. 4.1. However, some changes are seen in the unsaturated models, especially for the unsaturated finite model and when changes are smaller than or greater than 20%. Although the trend of most parameters is analogous to the observed for Fig. 4 Effect on Δf s when changing P 0 for P mod P 0 , it can be seen that the response of Δf s to changes in individual parameters relies on the selected initial set of parameters. It is expected that Δf s will be affected differently by a change in parameters if the baseline set of parameters is modified again. Thus, in the next section, the sensitivity analysis will be performed to evaluate the interaction of variations between parameters.

Sensitivity Analysis of Hydraulic and Rainfall Parameters
Wagener and Kollat (2007) and Zieher et al. (2017) stated that OAT-SA is a limited approach for multivariable and non-linear models with correlated parameters in hydrological modeling. This is the case of TRIGRS when calculating (Z, t) as part of the slope stability assessment. Hence, a specific sensitivity analysis was performed to evaluate the effects of hydraulic and rainfall parameters on Δf s . Figures 5, 6, 7, and 8 show the results of the sensitivity analysis focused on interaction among hydraulic parameters for the SAT-FIN, SAT-INF, UNS-FIN, and UNS-INF models, respectively. In all cases, rows and columns represent variation in K s and D 0 , respectively, with variation by two OM in each; colors are associated with different values of I ZLT , also varied by two OM; the line styles correspond to three combinations of d and Z max ; finally, abscissas represent a continuous increase in I nZ (constant during t = 24 h ) as the triggering mechanism of the shallow landslide.
The baseline case of f s, 0 evaluated at P 0 is represented with a red circle in the upper-middle facet.
The results confirmed that understanding the effect of hydraulic parameters on the change in f s is quite challenging. The f s calculation depends on (Z, t) , and this in turn depends on the soil hydraulic parameters, the rainfall features (intensity and duration), and the water table position regarding the slip surface depth. The variation in one of those parameters influences the effect of the others on f s . The following paragraphs describe the main tendencies identified in the four TRIGRS models based on the baseline set of parameters P 0 used in the analysis of Sect. 4.1. However, as concluded in Sect. 4.2, the results might be different if the baseline set of parameters is modified, including a shortening or lengthening in the rainfall duration, which was set constant to 24h (Table 1).
Overall, an increase in I nZ produces either a reduction in f s or has no effect on it (i.e., Δf s keeps constant). In the saturated models (Figs. 5, 6) and the UNS-INF model (Fig. 8), f s is not reduced below the baseline f s (Table 3), i.e., Δf s ≥ 0 within the I nZ range evaluated in the SA is equal to zero. Only in the UNS-FIN model, for K s = 4 × 10 −4 m s −1 and I nZ > 40 mm h −1 , f s is reduced by up to −26 % with respect to f s, 0 (Fig. 7), which is in accordance with a similar observation highlighted when analyzing Fig. 3c.
The saturated models (Figs. 5 and 6) showed outcomes very similar in terms of the shapes of the curves. The main difference between both saturated models is the value of I nZ at which Δf s starts to decrease. In these two models, the interaction between D 0 and K s strongly influences the I nZ value at which Δf s reaches its minimum value ( Δf s = 0 ). When D 0 is very low ( 4 × 10 −6 m 2 s −1 ), the drop in Δf s is not appreciated in the whole range of I nZ regardless of K s , but when D 0 is increased by 2 and 4 OM, Δf s reaches its minimum value for lower values of I nZ depending on K s . For instance, if K s is fixed to 4 × 10 −4 m s −1 , the drop of Δf s occurs at lower values of I nZ , which Unlike the saturated models, the outcomes obtained for the unsaturated models (Figs. 7, 8) are not alike in terms of the shape of the curves. On the other hand, in the unsaturated models D 0 is not as influential as in the saturated models, in this case, most of the changes are observed along rows, indicating that the influence of K s is stronger than the influence of D 0 on Δf s . Only for low to medium values of D 0 , Δf s shows a change when d = 1.5 m , Z max = 3.0 m , and K s = 4 × 10 −6 to 4 × 10 −4 m s −1 in the UNS-FIN model or K s = 4 × 10 −6 m s −1 in the UNS-INF model.
Although it is acknowledged that infiltration models are very sensitive to the steady seepage initial condition (Baum et al. 2010), in the present study Δf s only showed sensitivity to I ZLT for very low values of K s ( 4 × 10 −8 m s −1 ) in all the models, and little sensitivity when K s = 4 × 10 −6 m s −1 only in the unsaturated models. For medium to high values of K s , f s is not affected by changes in I ZLT . However, such a statement is not necessarily a generalization, as I ZLT might have more effect on Δf s for shorter rainfall events.
Finally, in all the models, both Z max and d as well as the ratio between them, showed to have a considerably high effect on Δf s in most of the combinations of the hydraulic parameters. There were no clear trends, Fig. 7 Extended SA of variation in hydraulic parameters for the UNS-FIN model though. As a result, and also based on the high sensitivity detected in Δf s to changes in Z max (Sect. 4.1, Fig. 3a), it is clear that the correct definition of those two parameters will strongly affect the slope assessment.
To sum up, Δf s sensitivity to individual variations in the model parameters depends on the set of parameters selected as the baseline and the combination of hydraulic and rainfall parameters. However, a general tendency is observed for any given initial set of parameters: Δf s is more affected by variations in , Z max , c ′ , and ′ than by variations in any of the other parameters. This is true except for the hydraulic parameters and the water table depth, whose interaction influences the Δf s response to variation of individual parameters. The next section will present the conclusions of this study and some perspectives to be addressed in future research.

Conclusions and Future Research
An extensive sensitivity analysis of the physicallybased model TRIGRS was conducted to evaluate the influence of its parameters on the stability assessment for rainfall-triggered shallow landslides. Four models which resulted from the combinations of saturated/ unsaturated conditions and finite/infinite depth of an impermeable basal boundary were considered. It was found that the effect of each parameter on Δf s when varying the parameter depends on three aspects: the initial set of parameters, the selected model itself (saturated/unsaturated, finite/infinite infiltration), and the interaction among physically correlated variables, especially those involved in the calculation of the pressure head. The last aspect is particularly applicable to the hydraulic parameters of the soil and the