Modeling probabilistic-based reliability assessment of gridded rainfall thresholds for shallow landslide occurrence due to the uncertainty of rainfall in time and space

Wang, Xi-Jun; Wu, Shiang-Jen; Tsai, Tung-Lin; Yen, Keh-Chia

doi:10.2166/hydro.2023.124

Abstract

This study aims to model a probabilistic-based reliability assessment of the gridded rainfall thresholds for shallow landslide occurrence (RA_GRTE_LS) to quantify the effect of the uncertainty of rainfall in time and space on the rainfall thresholds under consideration of local soil properties. The proposed RA_GRTE_LS model is developed by coupling the uncertainty analysis with the logistic regression equation using a significant number of the landslide-derived rainfall thresholds of the specific warning times. The 30 historical gridded hourly rainstorms at 10 study grids in the study area (Jhuokou River watershed) are used in 1,000 simulations of rainfall-induced shallow landslides under an assumption of the soil layer of 310 cm. The results reveal that the shallow landslide in the study area probably occurs at the time step of less than the 36th hour around the bottom of the soil layer (about 275 cm) during a rainstorm; also, using the proposed RA_GRTE_LS model, the resulting rainfall thresholds and quantified reliabilities, especially for the warning time of less than 18 h, exhibit a sizeable varying trend in space due to the variations in rainfall and soil properties; accordingly, the short-term rainfall thresholds for shallow landslide occurrence could be locally determined under acceptable reliability.

HIGHLIGHTS

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This study aims to model a probabilistic-based reliability assessment of the gridded rainfall threshold estimates for shallow landslide occurrence.
The resulting rainfall thresholds and quantified reliabilities exhibit a sizeable varying trend in space due to the spatial variations in rainfall and soil properties.
The short-term thresholds for shallow landslides could be locally determined under acceptable reliability.

gridded rainfall threshold, logistic regression analysis, gridded rainstorms, shallow landslide, unsaturated soil, digitalwatercollection, digitalwatertechniques

INTRODUCTION

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Rainfall-induced shallow landslides are comprehensively treated herein as potentially damaging geomorphological disaster events in response to the triggering by various rainfall intensities during a given duration (Hong et al. 2006; Piegari et al. 2009; Wu et al. 2017); Unfortunately, with the increasing occurrence of extreme weather events caused by climate change, rainfall-induced landslides frequently and significantly raise the risks of economic, environmental and human losses (Salvati et al. 2010; Huang et al. 2015; Rosi et al. 2016; Zhao et al. 2022). Since the rainfall-induced landslide commonly takes place at the shallow soil layer of less than 3–5 m, the rainfall thresholds in reaction to shallow landslide occurrence should be determined as a result of the construction of an early warning system, thereby saving lives and property (Vennari et al. 2014; Wu et al. 2017).

To effectively achieve the goal of early warning for shallow landslide occurrence, the well-known rainfall thresholds could be classified into two types: rainfall intensity-depth (I–D) (e.g., Aleotti 2004; Rosso et al. 2006; Guzzetti et al. 2008; Segoni et al. 2014) and event-based accumulated depth (E–D) (Aleotti 2004; Tsai 2008; Brunetti et al. 2010; Vennari et al. 2014; Gariano et al. 2015; Schiliro et al. 2015; Wu et al. 2017; Segoni et al. 2018b). The I–D-based rainfall thresholds are widely applied for early warning shallow landslide operations via the nonlinear relationship (e.g., the power law function) between the rainfall intensity and duration, mostly being an exponential function (Huang et al. 2015; Roccati et al. 2020). In addition, the numerous relevant investigations pay attention to the E–D-based rainfall thresholds in terms of various accumulated rainfall amounts under consideration of the different warning periods (named warning times) (Gariano et al. 2015; Wu et al. 2017). In detail, the I–D-based rainfall threshold could be employed via the equation of the rainfall intensity with the corresponding accumulated rainfall depth without considering the warning time; on the contrary, the E–D-based rainfall threshold should come with a given warning time. Conversely, regardless of the I–D and E–D rainfall thresholds, they are commonly estimated by means of data-driven approaches, (i.e., the empirical rainfall thresholds). Note that the aforementioned empirical rainfall thresholds are frequently formulated via statistical methods (e.g., regression analysis) with the historical gauged rainfall measurements (Guzzetti et al. 2008; Gariano et al. 2015; Hong et al. 2015). Although the empirical rainfall thresholds could result from the practical rainfall-induced shallow landslide events, their early-warning effectiveness might be impacted due to the uncertainties in the quality of the insufficiently available data regarding the recorded rainfall events that practically trigger the shallow landslides (Hong et al. 2015; Rosi et al. 2016).

To reduce uncertainty regarding the record length of the available data, the simulations of the rainfall-induced shallow landslides are achieved using the slope-stability numerical models with rainfall events. The well-known slope-stability numerical model TRIGRS (grid-based regional slope-stability) is comprehensively employed herein to simulate the gridded safety factor (FS) sequences triggered by the rainfall events in order to identify the corresponding rainfall thresholds (named physical thresholds); it could be obtained based on the safety factors lower than critical values (Aleotti 2004; Schiliro et al. 2015; Wu et al. 2017; Zhang et al. 2022). For example, Zhang et al. (2022) utilized the screening methods to extract the I–D-based rainfall thresholds from the slope-stability simulations carried out by the TRIGRS model with a specific increasing sequence of I–D data. Wu et al. (2017) detected the time steps of the resulting safety factors lower than the critical values from the TRIGRS model with the simulated hourly rainstorms; the corresponding E–D-based rainfall thresholds were then obtained under consideration of the warning periods. In general, the rainfall-induced shallow landslide simulations via the TRIGRS model with the soil conditions given are implemented under the assumption of the uniform groundwater table depth, namely, the initial condition of saturated soil (Tsai & Chen 2010; Liao et al. 2011). However, in reality, the ground is supposed to transmute from unsaturation to saturation during a rainfall event; thus, the pore water pressures then exhibit a significant increase in the slope to reduce effective stress and slope-stability (Schnellmann et al. 2010), implying that the soil moisture significantly influences the early warning effectiveness of shallow landslides (Marino et al. 2020). Hence, soil moisture plays an important role in estimating the rainfall thresholds used in the early warning system for shallow landslide occurrence (Segoni et al. 2018a; Marino et al. 2020). As a result, regarding the estimations of the landslide-triggering rainfall thresholds, a slope-stability numerical simulation model accounting for the unsaturated soil might be advantageous in evaluating the effect of uncertainty on the soil moisture to the simulation of slope-stability.

Additionally, the extreme rainfall trend in time and space significantly damages the geo-hydrological process from inducing the landslide with a high likelihood (Wu et al. 2017; Roccati et al. 2020). By so doing, the uncertainty of rainfall in time and space might influence the effectiveness and reliability of early issuing the shallow landslide alert in accordance with the I–D-based and E–D-based rainfall thresholds (Wu et al. 2017; Peres et al. 2018). Therefore, it is necessary to quantify the effect of spatiotemporal variations in rainfall on estimating the I–D-based and E–D-based rainfall thresholds. Generally speaking, the landslide-related rainfall thresholds are achieved using the gauged rainfall data (Wu et al. 2017); however, they are frequently estimated with a lack of effective rainfall data, especially in the mountainous areas with a high occurrence risk of shallow landslides (Nikolopoulos et al. 2014). Therefore, the grid-based quantitative precipitations of fine spatiotemporal resolution (e.g., radar and satellite rainfall data) should be used to determine the landslide-triggering thresholds (Montrasio & Valentino 2008; Nikolopoulos et al. 2017; Zhao et al. 2022). Furthermore, due to the extreme rainfall events and climate change, the resulting uncertainty in the grid-based precipitation probably influences the resulting hydrological process (Wu et al. 2017). Therefore, to facilitate the early-warning performance of the rainfall threshold, the reliability of the landslide-triggering rainfall thresholds should be assessed via the well-known uncertainty and risk analysis by considering the uncertainty of rainfall in time and space under different precipitation conditions (Melchiorre & Frattini 2012; Wu et al. 2017).

In spite of the reliability assessment for the landslide-triggering rainfall threshold achieved in numerous investigations (Talebi et al. 2008; Berti et al. 2012; Huang et al. 2015; Lee & Park 2015; Wu et al. 2017), they mostly focused either on the zone-based rainfall thresholds due to the temporal uncertainty in the gauged rainfall data or on the simulations of the rainfall-induced shallow landslide via the slope-stability numerical model under the initial condition of the saturated soil. Accordingly, this study models a probabilistic-based reliability analysis for the gridded rainfall thresholds for shallow landslide occurrence attributed to the uncertainties of rainfall in time and space, named the RA_GRTE_LS model. To develop the proposed RA_GRTE_LS model, a considerable number of the rainfall events of high spatiotemporal resolution (called gridded rainstorms) could be reproduced in advance under consideration of uncertainties in rainfall in time and space; after that, a significant number of the simulations of the rainfall-induced shallow landslide could be implemented using a slope-stability numerical model under the initial condition of unsaturated soil with the gridded rainstorms in a potentially rainfall-induced shallow landside zone. Note that the soil-related parameters adopted in the aforementioned slope-stability model are assigned based on the local soil properties in response to the spatial variation in the topography. After that, the landslide-triggering rainfall thresholds corresponding to the desired warning times can result from the rainfall-induced shallow landslide simulation cases; their reliabilities could be quantified via the well-known uncertainty-risk analysis. It is anticipated that the proposed RA_GRTE_LS model can quantify the corresponding reliabilities to the landslide-triggering rainfall thresholds under the condition of rainfall in time and space, along with providing the estimated rainfall thresholds with an acceptable likelihood (i.e., reliability).

METHODOLOGY

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Model concept

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This study aims to develop a probabilistic-based reliability assessment of the gridded rainfall thresholds regarding shallow landslide occurrence by coupling the uncertainty analysis with the slope-ability numerical simulation model with unstructured soil (named the RA_GRTE_LS model). The proposed RA_GRTE_LS is derived by modifying the PRTE_LS model (Wu et al. 2017) under consideration of uncertainties in the rainfall in time and space and the initial condition of unsaturated soil. Moreover, the advance first-order and second-moment (AFOSM) and Monte Carlo simulation (MCS) approaches are comprehensively applied in the relevant hydrological analysis, especially for floods and rainfall-induced disasters (Lee & Park 2015; Wu et al. 2017). Therefore, in this study, a significant number of the gridded rainstorm events in the watershed are generated via the multivariate MCS based on their spatial and temporal correlations; they are then employed in the shallow landslide simulation by means of the slope-stability unsaturated soil numerical model for unsaturated soil to obtain the corresponding safety factors at various soil depths within the study area. Accordingly, the time steps to the safety factors lower than critical values (called failure time steps) at the corresponding soil depths (named failure soil depths) for all simulations are identified to calculate the corresponding rainfall thresholds of the specific warning times. After that, the probabilities of the estimated rainfall thresholds exceeding the specific thresholds could be quantified through the uncertainty analysis with a considerable number of simulated rainfall characteristics at different locations and failure soil depths. Eventually, a nonlinear functional relationship between the reliability of the estimated rainfall thresholds of specific warning times and the uncertainty factors identified can be established via the regression analysis (defined as the exceedance-probability calculation equation).

In detail, the proposed RA_GRTE_LS model could evaluate the reliability of the specified landslide-triggering rainfall thresholds of the warning times due to the variation of rainfall in time and space using the exceedance-probability calculation equations adapted in the proposed RA_GRTE_LS model; additionally, by using the exceedance-probability calculation equations, the gridded landslide-triggering rainfall thresholds corresponding to signed reliability could be accordingly estimated within a potential rainfall-induced shallow landslide region. In summary, the proposed RA_GRTE_LS model is comprised of six parts:

Part [1]: Determination of rainfall thresholds for shallow landslide occurrence

Part [2]: Generation of gridded rainstorms

Part [3]: Configuration of the slope-stability numerical model for the saturated soil initial condition

Part [4]: Simulation of the rainfall-induced shallow landslide using generated gridded rainstorms

Part [5]: Reliability assessment of the landslide-triggering rainfall thresholds of various warning times

Part [6]: Establishing the landslide-triggering rainfall threshold estimation and corresponding exceedance-probability calculation equation.

The detailed methods and concepts utilized in the development of the proposed RA_GRTE_LS model are introduced below.

Definition of landslide-triggering rainfall thresholds and associated reliability

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In general, the E–D-based and I–D-based rainfall thresholds are commonly used in the early warning of shallow landslides. In this study, the landside-triggering rainfall thresholds of the various warning times could be estimated via the following equation (Wu et al. 2017):

(1)

where

is the time step (h) to a shallow landslide with the corresponding safety factor (FS) smaller than the critical value (⁠

⁠) (called the failure time step); t_w is the warning time (h); and r_t is the rainfall (mm) at time step t (h). Figure 1 shows a schematic illustration of estimating the rainfall thresholds for various warning times. In detail, when estimating the rainfall thresholds via Equation (), the failure time step associated with the corresponding safety factor being lower than the critical

value, commonly assigned as 1.0 (e.g., Bromhead 1992; Piegari et al. 2009; Wu et al. 2017), should be given in advance; the landslide-triggering rainfall threshold for a specific warning time t_w could be then obtained by accumulating the rainfall amount forward from t* to t* − t_w + 1. In summary, the landslide-triggering rainfall threshold could be determined based on the rainfall amount cumulated forward from the failure time steps for a specific period (i.e., warning time). Accordingly, the uncertainty factors that impact the estimation of the landslide-triggering rainfall thresholds probably are mainly related to the gridded rainfall characteristics and safety factors.

Since the rainfall threshold is widely employed in the early warning of shallow landslide occurrence, its reliability should influence the corresponding effectiveness and performance of alerting the shallow landslide as the cumulative rainfall exceeds the threshold. Therefore, Wu et al. (2017) quantified the reliability of the zone-based landslide-triggering rainfall thresholds of various warning times in terms of the probability of the rainfall thresholds exceeding the announced ones (i.e., underestimated risk) by the following equation:

(2)

where

(mm) is the landslide-triggering rainfall threshold for the warning time t_w (h); and

(mm) serves as the specific threshold. Referring to Equation (), if the exceedance probability (i.e., underestimated risk) approaches 0.0, the corresponding reliability reaches 1.0; that is to say, the exceedance probability is negatively related to the reliability. In detail, the aforementioned reliability mainly accounts for the performance of the landslide-triggering rainfall threshold applied in the early warning for the shallow landslide. when the cumulative rainfall depth of a warning time exceeds the specific threshold, the shallow landslide takes place with a high likelihood. As a result, within the proposed RA_GRTE_LS model, the reliability analysis for the rainfall threshold of the shallow landslide at the different locations and failure soil depths could be achieved by calculating the exceedance probabilities of the gridded rainfall thresholds attributed to the uncertainties in the rainfall fields under the local soil properties adopted.

Simulation of gridded rainstorm events

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To quantify the uncertainty of rainfall in time and space, a significant number of the rain fields consisting of the rainstorms at all grids in the study reproduced by the SM_GSTR model (Wu et al. 2021) are used in the simulation of the rainfall-induced shallow landslide in a region by means of the slope-stability numerical under the initial condition of unsaturated soil.

Within the SM_GSTR model, the event-based rainstorm is characterized in terms of three rainfall characteristics, the event-based rainfall duration, gridded rainfall depths (regarded as the spatial variates), and gridded storm depths comprised of the dimensionless rainfalls at the various dimensionless times (treated as the spatiotemporal correlated variates); as for the gridded storm pattern, it can be grouped into two components, the areal average of the dimensionless rainfalls (i.e., the storm pattern) and the associated deviations at the various dimensionless times. Figure 2 graphically illustrates the process of characterizing the gridded rainstorms in terms of the five gridded rainfall characteristics.

Figure 1

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Definition of the landslide-triggering rainfall threshold corresponding to the warning time t_w = d h (Wu et al. 2017).

Figure 2

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Schematic process of extracting the gridded rainfall characteristics from observed photographs of rainstorm events (Wu et al. 2021).

After that, the statistical analysis for the gridded rainfall characteristics is implemented to quantify their uncertainties in time and space, including the first four statistical moments, correlation coefficients and the appropriate probability functions. However, a significant number of the gridded rainfall characteristics are reproduced using the correlated multivariate MCS method (Chang et al. 1997) with the normalized-based algorithms, including the standardized, orthogonal and inverse transformations, whose correlations are calculated via the Nataf distribution (Nataf 1962):

(3)

where

and

are the correlated variables at the points i and j, respectively, with the means

and

⁠, the standard deviations

and

and the correlation coefficient

⁠;

and

are the corresponding bivariate standard normal variables to the variable

and

with the correlation coefficient and the joint standard normal density function

⁠. Eventually, the simulations of the gridded storm patterns are reproduced by combining the simulations of the areal averages of dimensionless cumulative rainfalls and the associated gridded bias; the gridded rainstorms are then emulated by coupling the simulated storm patterns at all the grids with the simulations of the gridded rainfall depths for the simulated event-based duration. The thorough introduction to the SM_GSTR model and the results from the model demonstration can be referred to in the investigation by Wu et al. (2021).

Slope-stability numerical model for unsaturated soil

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Recently, a group of numerical models for simulating the shallow landslide has been proposed to derive the shallow landslide susceptibility maps, such as the SHALSTAB (Shallow Landslide Slope-Stability) model (Montgomery & Dietrich 1994), SINMAP (Stability Index Mapping) model (Tarboton 1997), SLIP (Shallow Landslides Instability Prediction) model (Montrasio & Valentino 2008) and TRIGRS (Transient Rainfall Infiltration and Grid-based Regional Slope-stability) model (Baum et al. 2008). From among the relevant numerical models, the TRIGRS model is comprehensively applied in the slope-stability analysis to simulate the process of the rainfall-induced landslide; accordingly, the temporal sequences of the safety factors could be calculated in response to the effect of hydrological and mechanical characteristics (Chien et al. 2015; Wu et al. 2017; Schiliro et al. 2021). However, the TRIGRS model can only account for the effect of rainfall in time on the regional safety factors within a study area with saturated soil. This study focuses on the reliability assessment of the rainfall thresholds regarding the shallow landslides at the different locations under the initial condition of unsaturated soil. Tsai & Chen (2010) improved the Iverson equation to carry out rainfall-triggered shallow landslides by considering the effects of the unit weight and the unsaturated shear strength as a function of the degree of saturation. In Tsai's model, under an assumption of the pore water pressure being equal to the air pressure, the FS sequences at the different soil depths can be calculated by the following equations:

(4)

(5)

where

is the safety factor at the particular soil depth (⁠

⁠) (cm) and time step;

accounts for the unit weight of water (⁠

⁠);

serves as the depth-averaged unit weight of soil (⁠

⁠);

represents the effective cohesion (⁠

⁠);

is the slope angle;

denotes specific gravity of soil solid;

is the effective friction angle (°);

represents the slope angle (°);

denotes the parameter for shear strength of unsaturated soil; and

and

are the groundwater pressure heads (cm) under the different conditions.

Consequently, by means of a significant number of simulations of the gridded rainstorms, the resulting information on the rainfall-induced shallow landslide, including the safety factors and associated failure time steps and soil depths within the study area, can be accordingly achieved via Tsai's model; and the corresponding underestimated risk can be quantified in terms of the exceedance probability regarding the specific threshold for the development of the proposed RA_GRTE_LS model.

Reliability quantification of the landslide-triggering threshold rainfall

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In developing the proposed RA_GRTE_LS model, the probabilities of the estimated landslide-triggering rainfall thresholds exceeding the specific values (i.e., the exceedance probability) could be obtained via the existing uncertainty analysis method, i.e., the advanced first-order and second-moment approach (AFOSM) (Wu et al. 2017; Tung 2018), frequently employed in the hydrological-related risk analysis (e.g., Hassan & Wolff 2000; Ganji & Jowkarshorijeh 2012; Wu et al. 2017).

AFOSM is widely and successfully employed in water resources reliability analysis to assess the safety of structural components and structural systems by calculating the exceedance probability based on the standard normal distribution with the first two statistical moments (mean and variance). In general, the AFOSM quantifies the reliability by calculating the probability of the resistance (⁠

⁠) exceeding the load variable (⁠

⁠) in the following equation:

(6)

where

is defined as the exceedance probability of the landslide-triggering rainfall threshold, which could be denoted as the underestimated risk. Hence, as compared to Equation (), the reliability of the landslide-triggering rainfall threshold is equal to

via Equation (). Thereby, in this study, in employing the AFOSM method to quantify the reliability of the landslide-triggering rainfall threshold (⁠

⁠), its corresponding exceedance probability of the warning time (t_w) could be obtained in advance through the following equation:

(7)

in which Z is the performance function;

serve as the mean and standard deviation of Z, respectively;

denotes the standard normal distribution and

is the reliability index. Note that in the AFSOM method,

and

can be then computed via the following equations:

(8)

(9)

where

is the failure point of the ith uncertainty factor when the performance function z = 0;

and

are denoted as the mean and standard deviation of the ith uncertainty factor, respectively;

serves as the landslide-triggering rainfall threshold (mm) using the uncertainty factors’ failure points

⁠; and

represents the sensitivity coefficient of the ith uncertainty factor. In the case of the mean and variance of the given uncertainty factors, the exceedance probability of the landslide-triggering rainfall threshold

could be computed accordingly.

Note that by using the AFOSM to carry out the risk quantification, the functional relationship between the model outputs and inputs is supposed to be required in advance. Thus, referring to Equations () and (), the hyetographs of the gridded rainstorms and the soil-related factors, including the failure time steps and soil depths make a big contribution to the estimation of the landslide-triggering rainfall thresholds of the warning times for shallow landslide occurrence; namely, the uncertainty factors regarding the estimation of the landslide-triggering rainfall thresholds are comprised of two features: rainfall and safety factors. Consequently, the equation of the landslide-triggering rainfall thresholds with the corresponding uncertainty factors could be established through the multivariate regression analysis as:

(10)

where

represents the rainfall threshold (mm) of the warning time

(h);

stands for the rainfall features and

stands for the soil-related factors. The above regression coefficients adopted in Equation () could be calibrated through the multivariate regression analysis with a significant number of simulations of the rainfall-induced shallow landslide with the correspondingly generated uncertainty factors (i.e.,

⁠).

Thereby, within the proposed RA_GRTE_LS model, the reliability of the gridded landslide-triggering rainfall threshold could be quantified via the AFOSM (i.e., Equations (7)–(9) with its functional relationship (i.e., Equation (10)) in the case of the rain-related and soil-related uncertainty factors of interest.

Derivation of the exceedance-probability calculation equation

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Although the Monte Carlos simulation coupled with the uncertainty-risk approach of AFOSM can provide the requisite results from the reliability analysis for the rainfall thresholds for shallow landslide occurrence, a complicated process of calculating the exceedance probability through Equations ()–() should be proceeded. To effectively carry out reliability analysis for the landslide-triggering rainfall thresholds to quantify the exceedance probability, this study utilizes the logistic regression analysis to establish an equation for calculating the corresponding exceedance probability

with the uncertainty factors identified. Note that the logistic regression analysis mainly derives the relationship between the numerous independent variables and the occurrence probability corresponding to a dependent variable as follows:

(11)

where X_i and P represent the ith independent variables and the corresponding occurrence probability, respectively.

(i = 1,2,…n) stands for the regression coefficients and

serves as the intercept. Accordingly, the probability P in Equation () is treated as the exceedance probability of the landslide-triggering rainfall threshold regarding the warning time (⁠

⁠) (h)

⁠; and the corresponding reliability is treated as

⁠.

Therefore, within the proposed RA_GRTE_LS model, the exceedance-probability calculation equation derived based on the logistic regression equation is expected to robustly provide regional stochastic information on the landslide-triggering rainfall thresholds of the various warning times within a potential landslide-induced disaster region.

Model framework

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To sum up, the concepts and methods introduced, the landslide-triggering rainfall threshold estimation equations, and the resulting exceedance-probability calculation equations should be established in advance within the proposed RA_GRTE_LS model. In detail, within the proposed RA_GRTE_LS, the landslide-triggering rainfall threshold estimation equations are coupled with a multivariate MCS approach to quantify the exceedance probabilities of a number of the landslide-triggering rainfall thresholds given under consideration of the uncertainties in the known rain-related and soil-related factors; the resulting reliabilities would be employed in establishing the exceedance-probability calculation equation. After that, the exceedance-probability calculation equations would be then applied in the reliable quantification of the gridded landslide-triggering rainfall thresholds due to the uncertainty of rainfall in time and space and the spatial variations of the soil properties. Thereby, the thorough flowchart of model development and application could be addressed as follows:

Model development

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Step [1]: Collect the soil properties information at various locations of concern to configure the shallow landslide numerical model for the unsaturated soil (i.e., the Tsai's model) in the study.

Step [2]: Collect the historical hourly rainfall data at the various locations in the study area and extract the corresponding gridded rainfall characteristics, and calculate their statistical properties in time and space.

Step [3]: Generate a noticeable number of the gridded rainfall characteristics to simulate the corresponding hyetographs at various locations within the study area.

Step [4]: Import the simulated hyetographs of the gridded rainstorms into the Tsai's model to carry out the shallow landslide simulation to estimate the FS sequences at different soil depths regarding the various locations within the study area.

Step [5]: Extract the time-to-shallow landslide (i.e., ⁠) (i.e., failure time step) and corresponding soil depth (i.e., failure soil depth) at various locations from the numerous simulation cases of rainfall-induced shallow landslides obtained at Step [4] based on the specific critical safety factor (FS_cri = 1.0) and then calculate the landslide-triggering rainfall thresholds for the particular warning times (t_w) via Equation (1).

Step [6]: Conduct the AFOSM method to calculate the exceedance probability and corresponding reliability of the gridded landslide-triggering rainfall thresholds under consideration of the various warning times (t_w).

Step [7]: Perform the logistic analysis for establishing the exceedance-probability calculation equations corresponding to the landslide-triggering rainfall threshold estimates at various locations in the study area.

Model application

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Step [1]: Identify the rainfall uncertainty factors related to the gridded rainfall thresholds for shallow landslide occurrence required in the proposed RA_GRTE_LS model.

Step [2]: Calculate the exceedance probabilities of the specific rainfall thresholds of concerned warning times at various locations for shallow landslides occurrence.

Step [3]: Estimate the landslide-triggering rainfall thresholds of specific warning times at various locations corresponding to the desired exceedance probability (i.e., reliability).

STUDY AREA AND DATA

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To express the development and applicability of the proposed RA_GRTE_LS model, a mountainous catchment Jhuokou River watershed in southern Taiwan, as shown in Figure 3, is selected as the study area, in which the main river length and corresponding catchment extent are nearly 55 km and 529 km², respectively. Since the soil in the Jhoukou River watershed is mostly slate, a shallow landslide is frequently attributed to heavy rainfall events. Thus, as a result of the effect of the rainfall in time and space on the gridded landslide-triggering rainfall thresholds, 10 locations within the Jhuokou River watershed are selected as the study grids subject to the slopes (⁠

⁠) and soil properties (see Figure 3); this implies that the proposed RA_GRTE_LS model could take into account the effect of the uncertainty of the rainfall in time and space as well as the spatial variations in the soil properties on the estimation of the landslide-triggering rainfall thresholds.

Figure 3

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Map of the study area Jhuokou River watershed and locations of the 10 study grids selected.

Figure 4

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Event-based rainfall depths of 30 historical rainstorms at the 10 study grids within the study area.

In developing the proposed RA_GRTE_LS model, the grid-based hourly radar rainfall data of 30 historical rainstorms at the 10 study grids provided by the Center Weather Bureau (CWB) in Taiwan were adopted as the study data (see Table 1); Table 1 lists the occurrence periods, rainfall durations and areal average depths of 30 historical events. Figures 4 and 5 represent the gridded rainfall depths of 30 rainstorms and corresponding storm patterns (i.e., the temporal distribution of rainfall). By summarizing Table 1 and Figures 4 and 5, the event-based rainfall depths have an obvious change with the locations during a rainstorm in the study area. For example, as for the EV2 of 76 h, the maximum rainfall depth can be found at the study grid PT2 (1,822 mm); whereas the study grid PT10 has the minimum depth of 450 mm; also, the storm pattern shows a noticeable change related to the event and location; also, the type of the storm pattern could be identified based on the maximum dimensionless rainfall and its occurrence time step. For illustration, the storm pattern of EV3 can be denoted as the central type due to the maximum dimensionless rainfall occuring at the middle of the dimensionless times. In contrast with EV3, EV5 can be treated as the advanced type based on the maximum dimensionless rainfall at the early dimensionless time.

Table 1

Information on 30 historical rainstorm events within the study area used in the model development and demonstration

Rainstorm event		Beginning time	Ending time	Rainfall duration (h)	Areal average rainfall depth (mm)
EV1	HAITANG	2005/7/18 01:00	2005/7/22 00:00	96	1,743.5
EV2	MATSA	2005/8/4 05:00	2005/8/7 08:00	76	697.9
EV3	TALIM	2005/8/31 13:00	2005/9/1 23:00	35	596.8
EV4	LONGWANG	2005/10/2 03:00	2005/10/2 23:00	21	222.6
EV5	BILIS	2006/7/13 13:00	2006/7/17 02:00	84	765.4
EV6	KAEMI	2006/7/24 14:00	2006/7/27 15:00	74	461.6
EV7	SEPAT	2007/8/17 13:00	2007/8/21 20:00	104	952.1
EV8	KROSA	2007/10/6 06:00	2007/10/8 05:00	48	680.1
EV9	KALMAEGI	2008/7/17 09:00	2008/7/19 21:00	61	820.4
EV10	FUNG-WONG	2008/7/28 01:00	2008/7/29 17:00	41	623.8
EV11	SINLAKU	2008/9/12 18:00	2008/9/15 21:00	76	830.0
EV12	JANGMI	2008/9/28 02:00	2008/9/30 10:00	57	542.3
EV13	LINFA	2009/6/20 16:00	2009/6/23 01:00	58	269.7
EV14	MORAKOT	2009/8/6 14:00	2009/8/11 00:00	107	2,498.6
EV15	FANAPI	2010/9/18 23:00	2010/9/20 21:00	47	578.0
EV16	NANMADOL	2011/8/27 21:00	2011/8/31 21:00	97	561.6
EV17	Rainfall event	2012/6/9 12:00	2012/6/13 07:00	92	1,354.5
EV18	TALIM	2012/6/18 23:00	2012/6/22 06:00	80	582.9
EV19	SAOLA	2012/8/1 14:00	2012/8/3 15:00	50	285.1
EV20	TEMBIN	2012/8/27 12:00	2012/8/28 19:00	32	128.5
EV21	SOULIK	2013/7/12 18:00	2013/7/13 20:00	27	421.5
EV22	TRAMI	2013/8/21 14:00	2013/8/24 01:00	60	438.7
EV23	KONG-REY	2013/8/29 00:00	2013/9/1 14:00	87	828.4
EV24	USAGI	2013/9/20 20:00	2013/9/22 08:00	37	303.7
EV25	MATMO	2014/7/22 14:00	2014/7/24 03:00	38	507.5
EV26	FUNG-WONG	2014/9/20 17:00	2014/9/22 06:00	38	112.1
EV27	SOUDELOR	2015/8/8 03:00	2015/8/9 15:00	37	607.9
EV28	DUJUAN	2015/9/28 14:00	2015/9/29 11:00	22	273.3
EV29	MERANTI	2016/9/13 23:00	2016/9/15 18:00	44	304.0
EV30	MEGI	2016/9/27 03:00	2016/9/29 17:00	63	683.2

Rainstorm event		Beginning time	Ending time	Rainfall duration (h)	Areal average rainfall depth (mm)
EV1	HAITANG	2005/7/18 01:00	2005/7/22 00:00	96	1,743.5
EV2	MATSA	2005/8/4 05:00	2005/8/7 08:00	76	697.9
EV3	TALIM	2005/8/31 13:00	2005/9/1 23:00	35	596.8
EV4	LONGWANG	2005/10/2 03:00	2005/10/2 23:00	21	222.6
EV5	BILIS	2006/7/13 13:00	2006/7/17 02:00	84	765.4
EV6	KAEMI	2006/7/24 14:00	2006/7/27 15:00	74	461.6
EV7	SEPAT	2007/8/17 13:00	2007/8/21 20:00	104	952.1
EV8	KROSA	2007/10/6 06:00	2007/10/8 05:00	48	680.1
EV9	KALMAEGI	2008/7/17 09:00	2008/7/19 21:00	61	820.4
EV10	FUNG-WONG	2008/7/28 01:00	2008/7/29 17:00	41	623.8
EV11	SINLAKU	2008/9/12 18:00	2008/9/15 21:00	76	830.0
EV12	JANGMI	2008/9/28 02:00	2008/9/30 10:00	57	542.3
EV13	LINFA	2009/6/20 16:00	2009/6/23 01:00	58	269.7
EV14	MORAKOT	2009/8/6 14:00	2009/8/11 00:00	107	2,498.6
EV15	FANAPI	2010/9/18 23:00	2010/9/20 21:00	47	578.0
EV16	NANMADOL	2011/8/27 21:00	2011/8/31 21:00	97	561.6
EV17	Rainfall event	2012/6/9 12:00	2012/6/13 07:00	92	1,354.5
EV18	TALIM	2012/6/18 23:00	2012/6/22 06:00	80	582.9
EV19	SAOLA	2012/8/1 14:00	2012/8/3 15:00	50	285.1
EV20	TEMBIN	2012/8/27 12:00	2012/8/28 19:00	32	128.5
EV21	SOULIK	2013/7/12 18:00	2013/7/13 20:00	27	421.5
EV22	TRAMI	2013/8/21 14:00	2013/8/24 01:00	60	438.7
EV23	KONG-REY	2013/8/29 00:00	2013/9/1 14:00	87	828.4
EV24	USAGI	2013/9/20 20:00	2013/9/22 08:00	37	303.7
EV25	MATMO	2014/7/22 14:00	2014/7/24 03:00	38	507.5
EV26	FUNG-WONG	2014/9/20 17:00	2014/9/22 06:00	38	112.1
EV27	SOUDELOR	2015/8/8 03:00	2015/8/9 15:00	37	607.9
EV28	DUJUAN	2015/9/28 14:00	2015/9/29 11:00	22	273.3
EV29	MERANTI	2016/9/13 23:00	2016/9/15 18:00	44	304.0
EV30	MEGI	2016/9/27 03:00	2016/9/29 17:00	63	683.2

Figure 5

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Dimensionless rainfalls of the storm patterns regarding 30 historical rainstorms at 10 study grids within the study area.

In summary, 30 historical rainstorms at the 10 study grids within the study area used in the model development exhibit high variation in time and space. The proposed RA_GRTE_LS model could be developed in order to reflect the effect of rainfall variation in time and space on the gridded rainfall threshold for shallow landslide occurrence.

RESULTS AND DISCUSSIONS

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The rainfall exhibits variations in time and space due to extreme events and climate change, possibly influencing the early warning effectiveness of shallow landslides based on the issued rainfall thresholds. Therefore, this study develops the RA_GRTE_LS model to quantify and evaluate the reliability of the landslide-triggering rainfall thresholds of interest at various locations (i.e., gridded rainfall threshold). According to the framework of the model development as mentioned in section 2.7 with the gridded rainfall characteristics of 30 historical hourly rainstorms, the 1,000 simulations of the rainstorm events at the 10 study grids would be reproduced and the corresponding shallow landslide simulation could then be carried out via the slope-ability numerical model under the initial condition of unsaturated soil; accordingly, the rainfall thresholds of the specific warning durations can be estimated using Equation (2) subject to the corresponding FS estimates lower than the critical value; also, the exceedance probabilities corresponding to the landslide-triggering rainfall thresholds of different warning times at the 10 study grids can be calculated using the AFOSM approach for deriving the exceedance-probability calculation equations. Eventually, the proposed RA_GRTE_LS model applied in the reliability assessment of the landslide-triggering rainfall thresholds would be implemented using the numerical experiments, including the reliability quantifications of the rainfall thresholds given with a variety of rainfall factors concerned and the estimation of the gridded rainfall thresholds with designed reliability. Note the warning times and critical safety factors used in this study are assigned as 1–36 h and 1.0, respectively. The relevant results and discussion are addressed below.

Simulation of gridded rainstorms

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To quantify the uncertainty of rainfall in time and space, this study employs the SM_GSTR model (Wu et al. 2021) to generate the 1,000 simulations of gridded rainstorms based on the statistical properties of the gridded rainfall characteristics calculated from 30 historical rainstorms within the study area referred to in Table 1 and Figures 4 and 5. By combining the gridded rainfall characteristics, the simulated gridded hyetographs could be obtained; Figure 6 illustrates the resulting hyetographs at the 10 study grids from the simulated first event of 92 h, indicating that the corresponding hyetographs to the 10 study grids show a similar varying trend with the different rainfall amounts; also, the maximum rainfall intensity nearly reaches 60 mm/h at the study grid PT7; whereas, the minimum one (about 38 mm/h) could be seen at the study grid PT9.

Figure 6

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Graphical illustration of the hyetographs of the first simulated event at 10 study grids within the study area.

The above results reveal that the simulated rainstorms at the 10 study grids generated by means of the SM_GSTR model have an ability to preserve the spatial and temporal statistical properties of rainfall in the study area. It is advantageous in the regional simulation of the rainfall-induced shallow landslide in response to the uncertainty of rainfall in time and space, under consideration of the appropriate locally-based soil properties. By so doing, the 1,000 simulated rainstorms at the 10 study grids would be treated as the input data for the Tsai's model to achieve the corresponding temporal sequences of the safety factors regarding the various soil depths at the locations of interest.

Simulation of a gridded rainfall-induced shallow landslide

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According to the framework of developing the proposed RA_GRTE_LS model, the rainfall-induced shallow landslides at the different locations could be carried out via the Tsai's model with 1,000 simulations of the gridded rainfall events. Since the soil-related parameters required by the Tsai's model should be known in advance, this study refers to the types of soil and slopes at the 10 study grids to assign the appropriate values of the relevant parameters as listed in Table 2.

Table 2

Soil-related parameters of the Tsai's model adopted at 10 study grids within the study area

Study grid	Slope	Saturated hydraulic conductivity (cm/s)	Saturated volumetric water content	Residual volumetric water content	Effective friction angle (°)	Effective cohesion (pa)
PT1	34.6	0.0015	0.45	0.2	30	1,800
PT2	41.7	0.000868	0.47	0.16	36	2,300
PT3	32.2	0.00168	0.47	0.16	27	1,850
PT4	30.9	0.00168	0.47	0.16	26	1,700
PT5	24.8	0.00868	0.45	0.2	21	1,250
PT6	36.3	0.000868	0.45	0.18	31	1,800
PT7	28.1	0.00868	0.47	0.16	25	1,680
PT8	31.3	0.00168	0.47	0.16	27	1,740
PT9	45.9	0.00308	0.47	0.16	36	5,100
PT10	28.6	0.00868	0.47	0.16	25	1,680

Study grid	Slope	Saturated hydraulic conductivity (cm/s)	Saturated volumetric water content	Residual volumetric water content	Effective friction angle (°)	Effective cohesion (pa)
PT1	34.6	0.0015	0.45	0.2	30	1,800
PT2	41.7	0.000868	0.47	0.16	36	2,300
PT3	32.2	0.00168	0.47	0.16	27	1,850
PT4	30.9	0.00168	0.47	0.16	26	1,700
PT5	24.8	0.00868	0.45	0.2	21	1,250
PT6	36.3	0.000868	0.45	0.18	31	1,800
PT7	28.1	0.00868	0.47	0.16	25	1,680
PT8	31.3	0.00168	0.47	0.16	27	1,740
PT9	45.9	0.00308	0.47	0.16	36	5,100
PT10	28.6	0.00868	0.47	0.16	25	1,680

Table 3

Summary of uncertainty factors adopted in the proposed RA_GRTE_LS model

Symbol of uncertainty factor	Type of uncertainty factor	Definition
	Soil-related factor	Failure time step
	Soil-related factor	Failure soil depth
	Rain-related factor	Rainfall duration
		Rainfall depth
		Maximum rainfall intensity
		Time to the maximum rainfall intensity
		Cumulative rainfall at time to the maximum rainfall intensity

Symbol of uncertainty factor	Type of uncertainty factor	Definition
	Soil-related factor	Failure time step
	Soil-related factor	Failure soil depth
	Rain-related factor	Rainfall duration
		Rainfall depth
		Maximum rainfall intensity
		Time to the maximum rainfall intensity
		Cumulative rainfall at time to the maximum rainfall intensity

In addition to the locally based soil parameters adopted, the boundary conditions, including the simulation period and thickness of the soil layer (i.e., soil thickness), should be given in advance for numerically simulating the shallow landslide via the Tsai's model. Since the durations of simulated rainstorms mostly range from 30 to 120 h, the simulation period corresponding to shallow landslide is assigned 120 h; also, as a result of the slopes at the 10 study grids being between and ⁠, the corresponding soil depths are approximately between 237 and 289 cm, meaning that the soil thicknesses at the 10 study grids are assigned 310 cm under an assumption of the bottoms of the soil layers being permeable. Moreover, as for the remaining simulation conditions, the resolutions in time and space are assigned 1 h and 1 cm, respectively.

Using the Tsai's model with the appropriate soil parameters at the 10 study grids (see Table 2), the resulting safety factors during 120 h at the various soil depths could be estimated from the 1,000 simulations of rainfall-induced shallow landslides. Figure 7 shows the safety factor sequences at the soil depths of 50, 150 and 310 cm regarding 10 study grids induced by the first simulated rainstorm of 80 h; this implies that safety factors exhibit a similar varying trend with time at the specific soil depths; namely, the safety factors are proportional to the accumulating rainfall. That is to say, when the cumulative rainfall depth, on average, rises from 90 (the 18th hour) to 1,450 mm (the 80th hour), the safety factor at the soil depth of 150 cm decreases from 2.33 to 1.28 at the 30th hour and then rises to 1.88. According to Equation (), the groundwater pressure heads (i.e.,

and

⁠) adversely change with the safety factor under consideration of the absorbent ground adopted in the Tsai's model; thus, increasing the rainfall depth possibly results in the higher safety factor. Instead, the safety factor could decline depending on the rainfall amount. A similar conclusion could be made in the remaining simulation cases.

Figure 7

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Change in the safety factors with the cumulative rainfalls at the various soil depths regarding the first simulation.

Simulation of the rainfall threshold for shallow landslide

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According to Equation (), the corresponding time step and soil depth of the estimated safety factor being lower than the critical value (i.e., 1.0) are denoted as the failure time step and soil depth, respectively; this should be determined in advance to estimate the gridded landslide-triggering rainfall thresholds corresponding to various warning times. Thereby, using the resulting 1,000 simulations of the safety factor sequences from the generated rainstorms at the 10 study grids, the failure time steps and soil depths can be accordingly detected as shown in Figure 8; thus, the failure soil depths are mainly located between 150 and 300 cm, except for the study grid PT5, in which the failure soil depths are nearly around the bottom of the soil layer (i.e., 310 cm); also, the failure time steps range from the 20th to the 80th hour.

Figure 8

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1,000 simulation cases of the corresponding failure time steps and soil depths to the safety factors lower than 1.0 at the 10 study grids within the study area.

By employing uncertainty analysis for the simulations of the failure time steps and soil depths at the 10 study grids, their statistical properties can be quantified, as shown in Figure 9. From Figure 9, the averages of the failure time steps and soil depths are nearly 34 h and 215 cm, revealing that the assumption of the simulation period and total soil thickness being 120 h and 310 cm is reasonably applied in the shallow landslide simulation within the study area. In addition, the failure time steps, on average, were approximately 35 h, indicating that the warning times for the early warning operation of a sallow landslide could be appropriately assigned as 1, 3, 12, 18, 24, 30 and 36 h. Consequently, the corresponding landslide-triggering rainfall thresholds could be accordingly estimated using Equation () as shown in Figure 10, which reveals that the gridded landslide-triggering rainfall thresholds are primarily located between 40 and 1,500 mm; however, at the study grid PT5, the resulting rainfall thresholds change from 40 to 1,000 mm due to their subdued slopes, implying that more cumulative amount of rainfall required could induce shallow landslide occurrence.

Figure 9

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Summary for statistical properties of the failure time steps and soil depths at the 10 study grids within the study area: (a) Failure time step and (b) failure soil depth.

Figure 10

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1,000 simulations of the landslide-triggering rainfall thresholds of the various warning times concerned at the 10 study grids within the study.

Hence, to quantify the effect of the rainfall in time and space on the estimation of the gridded landslide-triggering rainfall thresholds, the uncertainty analysis for the estimated rainfall threshold is implemented to calculate their statistical properties as shown in Figure 11, indicating that all the study grids have a similar average of landslide-triggering rainfall thresholds (about 520 mm) with a markedly considerable variation (63–162 mm); in particular, their standard deviations at the study grids PT3 and PT7 are significantly greater than those at the remaining study grids so as to obtain the wider 95% confidence intervals.

Figure 11

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Summary for statistical properties of the landslide-triggering rainfall thresholds of the various warning times concerned at 10 study grids within study.

In summary, the estimations of the landslide-triggering rainfall thresholds of various warning times exhibit a significantly varying trend in space attributed to spatial and temporal variation in rainfalls; also, they might be influenced due to the failure time steps and soil depths. Consequently, in addition to the variation in the rainfall factors extracted from the gridded characteristics, the uncertainties in the gridded rainstorms, the failure time steps and the soil depths should be treated as the soil-related factors regarding the development of the proposed RA_GRTE_LS model.

Development of the proposed RA_GRTE_LS model

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According to the model framework introduced in Section 2.7, the results of 1,000 simulations of the rainfall-induced shallow landslide were utilized in developing the proposed RA_GRTE_LS model. In detail, the resulting rainfall thresholds of various warning times from the 1,000 simulation cases of the rainfall-induced shallow landslides at the study grids could be used in establishing the rainfall threshold estimation equations, such as Equation (10) for calculating the corresponding exceedance probabilities via the AFOSM approach; the resulting exceedance probabilities of the gridded landslide-triggering rainfall thresholds are used to derive the exceedance-probability calculation equations using the logistic regression equations.

Finally, in assessing the reliability of the specific rainfall thresholds for the shallow landslide via the proposed RA_GRTE_LS model, the derived exceedance-probability calculation equations could be employed in the case of the uncertainty factors of interest. The process of model development and demonstration is described as follows:

Identification of uncertainty factors

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Before developing the proposed RA_GRTE_LS model, the uncertainty factors corresponding to the estimation of the gridded rainfall thresholds for shallow landslide occurrence should be identified in advance. According to the above results from the 1,000 simulations of the rainfall-induced shallow landslides at the 10 study grids, the gridded rainfall characteristics, including the event-based rainfall duration, rainfall depths and storm patterns, significantly impact the estimation of the landslide-triggering rainfall thresholds. Apart from the soil-related factors (i.e., failure time steps and soil depths), the gridded characteristics, failure time steps and soil depth could be treated as the uncertainty factors.

The gridded rainfall characteristics are briefly comprised of the event-based durations, rainfall depths and storm patterns combined as the hyetographs. Of the rainfall characteristics, the storm pattern mainly contributes to the distribution of rainfall in time; their types could be recognized based on the dimensionless time step and corresponding forward cumulative dimensionless rainfall as well as the maximum dimensionless rainfall (Wu et al. 2006). Therefore, the maximum rainfall intensity and the associated time to the maximum rainfall and the forward cumulative rainfall at the time to the maximum rainfall intensity are supposed to be treated as the uncertainty factors for the areal landslide-triggering rainfall threshold (Wu et al. 2017).

To summarize the above results, the uncertainty factors related to the reliability of the rainfall thresholds for shallow landslide occurrence could be grouped into rain-related and soil-related factors. The rain-related factors include the rainfall duration, rainfall depth, maximum rainfall intensity and its occurrence time step (i.e., time to the maximum rainfall intensity) as well as the forward cumulative rainfall at the time to the maximum rainfall intensity. Furthermore, the failure time steps and soil depths are regarded as the soil-related factors.

Establishment of the rainfall threshold estimation equation

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In developing the proposed RA_GRTE_LS model via the AFOSM approach under consideration of the rain-related and soil-related uncertainty factors, the functional relationship between the rainfall thresholds and uncertainty factors is required. Accordingly, the nonlinear relationship of the landslide-triggering rainfall thresholds with the rainfall and soil-related uncertainty factors could be defined as in the following equation:

(12)

where

is the warning time (h);

serves as the number of uncertainty factors of interest;

and

are the regression coefficients; and

stands for the uncertainty factors as listed in Table 3. Table 4 illustrates the parameters of the uncertainty factors adopted in the rainfall threshold estimation equation at the 10 study grids; the corresponding determination coefficient R², on average, approaches 0.6, indicating that Equation () can reasonably describe the varying trend of the landslide-triggering rainfall thresholds estimated with the change in the rain-related and soil-related uncertainty factors; that is to say, using Equation (), the landslide-triggering rainfall threshold estimates of the different warning times could be effectively provided under consideration of the uncertainty factors given with a high likelihood.

Table 4

Summary of the regression coefficient of uncertainty factors adopted in the rainfall threshold estimation equations at the 10 study grids

Study grid	Warning time (h)	Uncertainty factors								Determination coefficient (R²)
Study grid	Warning time (h)									Determination coefficient (R²)
PT1	1	38.702	−0.468	−1.094	0.334	0.233	0.610	0.423	−0.117	0.385
	3	113.865	−0.079	−0.711	0.156	0.051	0.182	0.571	−0.032	0.703
	6	233.205	0.113	−0.518	0.115	0.097	0.074	0.407	−0.095	0.866
	12	163.846	0.016	−0.228	0.187	−0.022	0.064	0.081	0.059	0.423
	18	93.464	−0.161	0.057	0.074	−0.270	0.033	0.060	0.321	0.294
	24	30.168	0.140	0.176	0.049	−0.572	0.007	0.079	0.465	0.431
	30	54.311	0.405	0.050	0.029	−0.739	−0.047	−0.033	0.576	0.487
	36	32.780	0.624	0.019	−0.006	−0.734	−0.046	−0.052	0.592	0.587
PT2	1	4.309	−0.875	−1.039	1.270	−0.315	0.559	0.554	0.072	0.592
	3	135.380	−0.359	−0.891	0.672	−0.338	0.112	0.742	0.098	0.655
	6	857.024	−0.061	−0.810	0.397	−0.194	−0.097	0.710	0.026	0.650
	12	331.528	−0.291	−0.316	0.203	0.248	0.128	0.374	−0.219	0.514
	18	414.005	−0.255	−0.202	−0.067	−0.135	0.265	0.246	0.006	0.496
	24	376.054	−0.032	−0.215	−0.054	−0.464	0.200	0.204	0.213	0.592
	30	104.798	0.323	−0.031	−0.019	−0.739	0.044	0.157	0.404	0.601
	36	35.079	0.709	−0.004	−0.142	−0.844	0.077	0.095	0.483	0.688
PT3	1	2.185	−0.313	−1.162	0.907	−0.145	0.844	0.563	−0.219	0.430
	3	21.753	0.050	−0.627	0.194	−0.177	0.347	0.758	−0.104	0.676
	6	71.283	0.218	−0.389	0.115	−0.042	0.099	0.587	−0.117	0.870
	12	25.450	0.085	−0.084	0.006	0.121	0.307	0.347	−0.180	0.544
	18	111.497	−0.094	−0.168	−0.141	−0.278	0.398	0.180	0.130	0.469
	24	71.053	0.098	−0.071	−0.058	−0.656	0.280	0.078	0.394	0.460
	30	38.313	0.349	−0.007	0.045	−0.838	0.119	0.034	0.547	0.494
	36	33.886	0.665	−0.075	−0.042	−0.974	0.091	−0.065	0.676	0.686
PT4	1	0.002	0.040	0.000	−0.982	0.952	2.162	−0.497	−0.549	0.355
	3	0.039	−0.660	0.000	−0.893	1.119	2.230	−0.440	−0.712	0.356
	6	1.562	−1.567	0.000	−0.911	1.272	2.299	−0.475	−0.769	0.445
	12	211.180	−2.002	0.000	−1.553	1.529	2.461	−0.401	−1.077	0.673
	18	195.879	−1.910	0.000	−1.043	1.309	2.008	−0.028	−0.983	0.771
	24	170.457	−1.501	0.000	−0.729	0.720	1.527	0.104	−0.578	0.793
	30	30.311	−0.594	0.000	−0.409	0.071	1.011	0.129	−0.113	0.844
	36	6.263	0.162	0.000	−0.380	−0.286	0.775	0.118	0.146	0.915
PT5	1	0.002	0.040	0.000	−0.982	0.952	2.162	−0.497	−0.549	0.355
	3	0.039	−0.660	0.000	−0.893	1.119	2.230	−0.440	−0.712	0.356
	6	1.562	−1.567	0.000	−0.911	1.272	2.299	−0.475	−0.769	0.445
	12	211.180	−2.002	0.000	−1.553	1.529	2.461	−0.401	−1.077	0.673
	18	195.879	−1.910	0.000	−1.043	1.309	2.008	−0.028	−0.983	0.771
	24	170.457	−1.501	0.000	−0.729	0.720	1.527	0.104	−0.578	0.793
	30	30.311	−0.594	0.000	−0.409	0.071	1.011	0.129	−0.113	0.844
	36	6.263	0.162	0.000	−0.380	−0.286	0.775	0.118	0.146	0.915
PT6	1	2.899	0.098	−1.229	1.248	−1.259	0.290	0.351	0.712	0.467
	3	39.488	0.282	−0.907	0.592	−0.815	0.018	0.686	0.396	0.724
	6	72.354	0.266	−0.551	0.354	−0.253	−0.069	0.784	−0.017	0.822
	12	17.822	0.018	−0.057	0.115	0.079	0.265	0.494	−0.221	0.676
	18	53.960	0.000	−0.045	−0.078	−0.329	0.331	0.279	0.077	0.673
	24	92.320	0.282	−0.071	−0.189	−0.536	0.230	0.182	0.241	0.586
	30	66.783	0.623	−0.089	−0.171	−0.780	0.103	0.110	0.434	0.610
	36	54.629	0.853	−0.145	−0.178	−0.881	0.081	0.067	0.497	0.689
PT7	1	0.402	−0.413	−0.639	0.784	−0.198	0.566	0.417	0.158	0.329
	3	35.957	0.055	−0.754	0.225	−0.297	0.152	0.656	0.223	0.491
	6	2,916.726	0.145	−1.138	0.089	−0.035	−0.043	0.762	0.011	0.585
	12	4,871.268	−0.171	−1.008	0.074	0.462	0.226	0.566	−0.375	0.491
	18	3,612.348	−0.293	−0.745	−0.183	0.241	0.416	0.468	−0.282	0.581
	24	2,402.152	−0.018	−0.697	−0.118	−0.236	0.296	0.359	0.042	0.595
	30	5,786.966	0.213	−0.796	−0.170	−0.384	0.196	0.252	0.166	0.528
	36	1,493.026	0.536	−0.684	−0.221	−0.537	0.179	0.190	0.283	0.605
PT8	1	7.551	−0.406	−0.944	0.781	0.176	0.257	0.450	0.097	0.793
	3	49.752	−0.197	−0.571	0.327	0.114	0.032	0.636	0.032	0.908
	6	206.390	−0.084	−0.311	0.157	0.136	−0.069	0.528	−0.092	0.916
	12	104.843	−0.166	0.007	0.092	0.183	0.226	0.141	−0.223	0.354
	18	428.210	−0.175	−0.025	−0.301	−0.054	0.270	0.082	0.023	0.330
	24	71.574	0.044	0.053	0.001	−0.380	0.140	0.093	0.253	0.350
	30	86.216	0.196	−0.010	0.012	−0.516	0.049	0.011	0.425	0.423
	36	194.850	0.392	−0.089	−0.263	−0.594	0.081	−0.098	0.525	0.575
PT9	1	0.117	−0.628	−0.529	1.648	−0.507	0.094	0.613	0.395	0.299
	3	4.565	−0.112	−0.400	0.701	−0.343	−0.056	0.774	0.227	0.466
	6	333.697	0.080	−0.784	0.318	−0.066	−0.065	0.827	−0.042	0.778
	12	405.379	−0.059	−0.715	0.237	0.226	0.218	0.574	−0.264	0.671
	18	126.101	−0.138	−0.095	−0.011	−0.026	0.020	0.416	0.100	0.424
	24	146.709	0.090	−0.183	0.175	−0.397	−0.056	0.253	0.321	0.327
	30	301.313	0.384	−0.347	0.088	−0.562	−0.065	0.162	0.412	0.416
	36	291.946	0.634	−0.477	0.018	−0.685	−0.035	0.127	0.499	0.622
PT10	1	4,120.738	−0.556	−2.572	0.582	−0.101	1.425	−0.778	0.226	0.349
	3	1,730.086	−0.178	−1.561	0.250	−0.159	0.539	0.196	0.160	0.550
	6	767.111	−0.078	−0.920	0.258	−0.034	0.084	0.549	0.022	0.742
	12	108.581	−0.432	−0.313	0.432	0.414	0.165	0.505	−0.337	0.536
	18	530.816	−0.611	−0.142	−0.093	0.252	0.334	0.403	−0.247	0.558
	24	453.564	−0.424	−0.027	−0.151	−0.021	0.242	0.312	−0.040	0.589
	30	229.512	−0.082	0.006	−0.179	−0.174	0.208	0.196	0.070	0.456
	36	58.915	0.298	0.045	−0.227	−0.334	0.232	0.138	0.173	0.488

Study grid	Warning time (h)	Uncertainty factors								Determination coefficient (R²)
Study grid	Warning time (h)									Determination coefficient (R²)
PT1	1	38.702	−0.468	−1.094	0.334	0.233	0.610	0.423	−0.117	0.385
	3	113.865	−0.079	−0.711	0.156	0.051	0.182	0.571	−0.032	0.703
	6	233.205	0.113	−0.518	0.115	0.097	0.074	0.407	−0.095	0.866
	12	163.846	0.016	−0.228	0.187	−0.022	0.064	0.081	0.059	0.423
	18	93.464	−0.161	0.057	0.074	−0.270	0.033	0.060	0.321	0.294
	24	30.168	0.140	0.176	0.049	−0.572	0.007	0.079	0.465	0.431
	30	54.311	0.405	0.050	0.029	−0.739	−0.047	−0.033	0.576	0.487
	36	32.780	0.624	0.019	−0.006	−0.734	−0.046	−0.052	0.592	0.587
PT2	1	4.309	−0.875	−1.039	1.270	−0.315	0.559	0.554	0.072	0.592
	3	135.380	−0.359	−0.891	0.672	−0.338	0.112	0.742	0.098	0.655
	6	857.024	−0.061	−0.810	0.397	−0.194	−0.097	0.710	0.026	0.650
	12	331.528	−0.291	−0.316	0.203	0.248	0.128	0.374	−0.219	0.514
	18	414.005	−0.255	−0.202	−0.067	−0.135	0.265	0.246	0.006	0.496
	24	376.054	−0.032	−0.215	−0.054	−0.464	0.200	0.204	0.213	0.592
	30	104.798	0.323	−0.031	−0.019	−0.739	0.044	0.157	0.404	0.601
	36	35.079	0.709	−0.004	−0.142	−0.844	0.077	0.095	0.483	0.688
PT3	1	2.185	−0.313	−1.162	0.907	−0.145	0.844	0.563	−0.219	0.430
	3	21.753	0.050	−0.627	0.194	−0.177	0.347	0.758	−0.104	0.676
	6	71.283	0.218	−0.389	0.115	−0.042	0.099	0.587	−0.117	0.870
	12	25.450	0.085	−0.084	0.006	0.121	0.307	0.347	−0.180	0.544
	18	111.497	−0.094	−0.168	−0.141	−0.278	0.398	0.180	0.130	0.469
	24	71.053	0.098	−0.071	−0.058	−0.656	0.280	0.078	0.394	0.460
	30	38.313	0.349	−0.007	0.045	−0.838	0.119	0.034	0.547	0.494
	36	33.886	0.665	−0.075	−0.042	−0.974	0.091	−0.065	0.676	0.686
PT4	1	0.002	0.040	0.000	−0.982	0.952	2.162	−0.497	−0.549	0.355
	3	0.039	−0.660	0.000	−0.893	1.119	2.230	−0.440	−0.712	0.356
	6	1.562	−1.567	0.000	−0.911	1.272	2.299	−0.475	−0.769	0.445
	12	211.180	−2.002	0.000	−1.553	1.529	2.461	−0.401	−1.077	0.673
	18	195.879	−1.910	0.000	−1.043	1.309	2.008	−0.028	−0.983	0.771
	24	170.457	−1.501	0.000	−0.729	0.720	1.527	0.104	−0.578	0.793
	30	30.311	−0.594	0.000	−0.409	0.071	1.011	0.129	−0.113	0.844
	36	6.263	0.162	0.000	−0.380	−0.286	0.775	0.118	0.146	0.915
PT5	1	0.002	0.040	0.000	−0.982	0.952	2.162	−0.497	−0.549	0.355
	3	0.039	−0.660	0.000	−0.893	1.119	2.230	−0.440	−0.712	0.356
	6	1.562	−1.567	0.000	−0.911	1.272	2.299	−0.475	−0.769	0.445
	12	211.180	−2.002	0.000	−1.553	1.529	2.461	−0.401	−1.077	0.673
	18	195.879	−1.910	0.000	−1.043	1.309	2.008	−0.028	−0.983	0.771
	24	170.457	−1.501	0.000	−0.729	0.720	1.527	0.104	−0.578	0.793
	30	30.311	−0.594	0.000	−0.409	0.071	1.011	0.129	−0.113	0.844
	36	6.263	0.162	0.000	−0.380	−0.286	0.775	0.118	0.146	0.915
PT6	1	2.899	0.098	−1.229	1.248	−1.259	0.290	0.351	0.712	0.467
	3	39.488	0.282	−0.907	0.592	−0.815	0.018	0.686	0.396	0.724
	6	72.354	0.266	−0.551	0.354	−0.253	−0.069	0.784	−0.017	0.822
	12	17.822	0.018	−0.057	0.115	0.079	0.265	0.494	−0.221	0.676
	18	53.960	0.000	−0.045	−0.078	−0.329	0.331	0.279	0.077	0.673
	24	92.320	0.282	−0.071	−0.189	−0.536	0.230	0.182	0.241	0.586
	30	66.783	0.623	−0.089	−0.171	−0.780	0.103	0.110	0.434	0.610
	36	54.629	0.853	−0.145	−0.178	−0.881	0.081	0.067	0.497	0.689
PT7	1	0.402	−0.413	−0.639	0.784	−0.198	0.566	0.417	0.158	0.329
	3	35.957	0.055	−0.754	0.225	−0.297	0.152	0.656	0.223	0.491
	6	2,916.726	0.145	−1.138	0.089	−0.035	−0.043	0.762	0.011	0.585
	12	4,871.268	−0.171	−1.008	0.074	0.462	0.226	0.566	−0.375	0.491
	18	3,612.348	−0.293	−0.745	−0.183	0.241	0.416	0.468	−0.282	0.581
	24	2,402.152	−0.018	−0.697	−0.118	−0.236	0.296	0.359	0.042	0.595
	30	5,786.966	0.213	−0.796	−0.170	−0.384	0.196	0.252	0.166	0.528
	36	1,493.026	0.536	−0.684	−0.221	−0.537	0.179	0.190	0.283	0.605
PT8	1	7.551	−0.406	−0.944	0.781	0.176	0.257	0.450	0.097	0.793
	3	49.752	−0.197	−0.571	0.327	0.114	0.032	0.636	0.032	0.908
	6	206.390	−0.084	−0.311	0.157	0.136	−0.069	0.528	−0.092	0.916
	12	104.843	−0.166	0.007	0.092	0.183	0.226	0.141	−0.223	0.354
	18	428.210	−0.175	−0.025	−0.301	−0.054	0.270	0.082	0.023	0.330
	24	71.574	0.044	0.053	0.001	−0.380	0.140	0.093	0.253	0.350
	30	86.216	0.196	−0.010	0.012	−0.516	0.049	0.011	0.425	0.423
	36	194.850	0.392	−0.089	−0.263	−0.594	0.081	−0.098	0.525	0.575
PT9	1	0.117	−0.628	−0.529	1.648	−0.507	0.094	0.613	0.395	0.299
	3	4.565	−0.112	−0.400	0.701	−0.343	−0.056	0.774	0.227	0.466
	6	333.697	0.080	−0.784	0.318	−0.066	−0.065	0.827	−0.042	0.778
	12	405.379	−0.059	−0.715	0.237	0.226	0.218	0.574	−0.264	0.671
	18	126.101	−0.138	−0.095	−0.011	−0.026	0.020	0.416	0.100	0.424
	24	146.709	0.090	−0.183	0.175	−0.397	−0.056	0.253	0.321	0.327
	30	301.313	0.384	−0.347	0.088	−0.562	−0.065	0.162	0.412	0.416
	36	291.946	0.634	−0.477	0.018	−0.685	−0.035	0.127	0.499	0.622
PT10	1	4,120.738	−0.556	−2.572	0.582	−0.101	1.425	−0.778	0.226	0.349
	3	1,730.086	−0.178	−1.561	0.250	−0.159	0.539	0.196	0.160	0.550
	6	767.111	−0.078	−0.920	0.258	−0.034	0.084	0.549	0.022	0.742
	12	108.581	−0.432	−0.313	0.432	0.414	0.165	0.505	−0.337	0.536
	18	530.816	−0.611	−0.142	−0.093	0.252	0.334	0.403	−0.247	0.558
	24	453.564	−0.424	−0.027	−0.151	−0.021	0.242	0.312	−0.040	0.589
	30	229.512	−0.082	0.006	−0.179	−0.174	0.208	0.196	0.070	0.456
	36	58.915	0.298	0.045	−0.227	−0.334	0.232	0.138	0.173	0.488

To evaluate the sensitivity of the estimated landslide-triggering rainfall thresholds to the uncertainty factors, the average of the regression coefficients adopted in Equation () at the 10 study grids is calculated as shown in Figure 12. With the results from Figure 12, the negative average of the regression coefficients regarding the soil-related factors is obtained; this implies that the landslide-triggering rainfall threshold exhibits an adverse change with the soil-related factors. In particular, the absolute value of the regression coefficient for the failure soil depth is greater than the failure time step, implying that a large rainfall threshold might be needed in the case of the shallow landslide taking place at a deep soil depth in response to the corresponding safety factor being lower than 1.0.

Figure 12

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The averages of regression coefficients regarding the rain-related and soil-related uncertainty factors regarding the estimation of the landslide-triggering rainfall thresholds.

With the regression coefficients, the rain-related factors have a positive correlation with the landslide-triggering rainfall thresholds, except for the time to the maximum rainfall intensity; namely, the large rainfall depths trigger the high rainfall thresholds with a high likelihood. Specifically, the negative regression coefficient of the time to the maximum rainfall intensity indicates that as the shallow landslide is triggered at the beginning of a rainfall event, the heavy rainfall amount might be measured in the early time steps. Thereby, the maximum rainfall intensity and associated forward cumulative rainfall make a positive contribution to the estimation of the landslide-triggering rainfall thresholds.

In conclusion, the rain-related and soil-related uncertainty factors considered in this study are proven to influence the estimation of rainfall thresholds for shallow landslide occurrence. As a result, reliability analysis for the rainfall threshold of various warnings should be performed under consideration of the variations in the rainfall and soil-related factors.

Calculation of exceedance probabilities of rainfall thresholds

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In this section, to quantify the reliability of the landslide-triggering rainfall threshold, its exceedance probability should be calculated in advance via the AFOSM approach with the derived rainfall threshold estimation equations at the 10 study grids with the regression coefficients of rainfall and soil-related factors used in Equation (12) (see Table 4).

Figure 13 presents the resulting exceedance probabilities of the specific landslide-triggering rainfall thresholds (10–2,000 mm) of eight warning times (1, 3, 6, 12, 18, 24, 30 and 36 h) at the 10 study grids, indicating that the exceedance probability has an apparent increase with the warning duration for a specific threshold; and the landslide-triggering rainfall thresholds are positively related to the warning times under an exceedance probability of interest; namely, the reliability of the landslide-triggering rainfall threshold has a negative relationship with the warning time. For illustration, at the study grid PT1, the exceedance probability corresponding to the rainfall threshold of 200 mm markedly increases from 0.0 (1 h) to 0.99 (36 h); as for the study grid PT10, in the case of the exceedance probability being 0.9, the resulting rainfall thresholds raise from 10 (1 h) to 400 mm (35 h).

Figure 13

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Exceedance probabilities of the landslide-triggering rainfall thresholds of the various warning durations at the 10 study grids via the AFOSM approach.

Additionally, focusing on the 3 h rainfall threshold of 100 mm, the corresponding exceedance probabilities significantly changed with the study grids, i.e., 0.85 (PT1), 0.84 (PT2), 0.81 (PT3), 0.96 (PT4), 0.12 (PT5), 0.74 (PT6), 0.86 (PT7), 0.98 (PT8), 0.57 (PT9) and 0.33 (PT10); this concludes that the corresponding exceedance probabilities to a specific rainfall threshold exhibit a noticeable variation in space. That is to say, during a rainstorm, the reliability of a specific landslide-triggering rainfall threshold markedly changes with the location.

In conclusion, the exceedance probability of the landslide-triggering rainfall threshold shows spatial and temporal variations, revealing that the results from the reliability analysis for the landslide-triggering rainfall thresholds change with time and space. Therefore, it is demonstrated that the rainfall thresholds and associated effectiveness for the early warning of the shallow landslide should be impacted due to the uncertainty of rainfall in time and space and the spatial variation in the soil properties adopted; namely, the landslide-triggering rainfall threshold should be regarded as the spatial variate; it is supposed to be determined according to the gridded rainfall characteristics and soil properties. Therefore, the inherent spatial variations of the rainfall thresholds should be considered in the early warning performance of the rainfall thresholds for shallow landslide occurrence.

Derivation of the exceedance-probability calculation equation

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Although the underestimated risk of the rainfall thresholds for shallow landslide can be achieved via the AFOSM approach, expensive computation time is possibly required (Wu et al. 2017). Therefore, to efficiently carry out the reliability assessment of the landslide-triggering rainfall thresholds using the AFOSM approach, this study derives the exceedance-probability calculation equation by means of logistic regression analysis based on Equation (). In referring to Equation (), the uncertainty factors corresponding to the exceedance probability of the landslide-triggering rainfall thresholds should be selected among the rain-related and soil-related factors. Since the early warning of the shallow landslide is commonly announced according to the precipitation observations and forecasts (Wu et al. 2017), the rainfall factors are adopted in the desired exceedance-probability calculation equations, i.e., the rainfall depths, maximum rainfall intensity and the corresponding forward cumulative rainfall as in the following equation:

(13)

where

accounts for the landslide-triggering rainfall threshold (mm) of the warning time

(h);

is the specific rainfall threshold (⁠

⁠) (mm); and

stands for the rainfall depth (mm);

serves as the cumulative rainfall depth (mm) at the time to the maximum rainfall intensity (mm/h) and

are the regression coefficients used in Equation (). Note that after calculating the exceedance probability of the landslide-triggering rainfall threshold (mm), the corresponding reliability can be obtained through Equation (), that is,

⁠.

In this study, the above regression coefficients are calibrated using the exceedance probabilities of the specific rainfall thresholds of the various warning times under consideration of the rainfall factors given (the rainfall depths, maximum rainfall intensity and corresponding forward cumulative rainfall) and the statistics of the remaining rainfall factors (see Tables 5 and 6). Figure 14 illustrates the exceedance probabilities of the specific rainfall thresholds of various warning times at the 10 study grids, meaning that the resulting exceedance probability of the specific rainfall thresholds is induced due to the variations in the uncertainty factors as listed in Table 3. There, the regression coefficients of the rainfall factors adopted in Equation () could be determined, as referred to in Table 7.

Table 5

Summary of the given rain-related factors used in the derivation of the exceedance-probability calculation equations

Study site	Simulation cases	Rainfall depth (mm	Max rainfall intensity (mm)	Cumulative rainfalls to maximum rainfall intensity (mm)
PT1	Case 1–11	689.63–2,280.6	49.99	645.32
	Case 12–22	1,135.62	33.54–105.6	645.32
	Case 23–33	1,135.62	49.99	178.09–1,183.3
PT2	Case 1–11	557.21–2,644.9	49.66	657.10
	Case 12–22	1,145.87	29.59–101.8	657.10
	Case 23–33	1,145.87	49.66	197.25–1,409.1
PT3	Case 1–11	746.3–1,970.6	62.23	832.94
	Case 12–22	1,276.53	34.8–127.396	832.94
	Case 23–33	1,276.53	62.23	230.9–1,970.6
PT4	Case 1–11	715.2–2,497.01	62.23	832.94
	Case 12–22	1,276.53	46.61–104.9	832.94
	Case 23–33	1,276.53	62.23	394.38–1,902.5
PT5	Case 1–11	211.16–1,578.3	39.42	413.97
	Case 12–22	863.57	11.41–83.8	413.97
	Case 23–33	863.57	39.42	80.37–1,173.1
PT6	Case 1–11	510.65–1,713.8	47.60	613.77
	Case 12–22	1,043.33	29.28–82.3	613.77
	Case 23–33	1,043.33	47.60	195.05–1,383.8
PT7	Case 1–11	612.33–1,927.7	58.18	676.15
	Case 12–22	1,261.21	32.26–121.62	676.15
	Case 23–33	1,261.21	58.18	178.15–1,649.9
PT8	Case 1–11	662.87–29,147	55.02	601.33
	Case 12–22	1,069.27	38.10–142.8	601.33
	Case 23–33	1,069.27	55.02	261.05–1,154.7
PT9	Case 1–11	543.23–1,604.4	45.57	555.86
	Case 12–22	952.49	33.23–79.3	555.86
	Case 23–33	952.49	45.57	224.79–1,007.9
PT10	Case 1–11	602.91–1,620.9	43.59	527.24
	Case 12–22	947.20	26.40–108.2	527.24
	Case 23–33	947.20	43.59	163.84–1,249.4

Study site	Simulation cases	Rainfall depth (mm	Max rainfall intensity (mm)	Cumulative rainfalls to maximum rainfall intensity (mm)
PT1	Case 1–11	689.63–2,280.6	49.99	645.32
	Case 12–22	1,135.62	33.54–105.6	645.32
	Case 23–33	1,135.62	49.99	178.09–1,183.3
PT2	Case 1–11	557.21–2,644.9	49.66	657.10
	Case 12–22	1,145.87	29.59–101.8	657.10
	Case 23–33	1,145.87	49.66	197.25–1,409.1
PT3	Case 1–11	746.3–1,970.6	62.23	832.94
	Case 12–22	1,276.53	34.8–127.396	832.94
	Case 23–33	1,276.53	62.23	230.9–1,970.6
PT4	Case 1–11	715.2–2,497.01	62.23	832.94
	Case 12–22	1,276.53	46.61–104.9	832.94
	Case 23–33	1,276.53	62.23	394.38–1,902.5
PT5	Case 1–11	211.16–1,578.3	39.42	413.97
	Case 12–22	863.57	11.41–83.8	413.97
	Case 23–33	863.57	39.42	80.37–1,173.1
PT6	Case 1–11	510.65–1,713.8	47.60	613.77
	Case 12–22	1,043.33	29.28–82.3	613.77
	Case 23–33	1,043.33	47.60	195.05–1,383.8
PT7	Case 1–11	612.33–1,927.7	58.18	676.15
	Case 12–22	1,261.21	32.26–121.62	676.15
	Case 23–33	1,261.21	58.18	178.15–1,649.9
PT8	Case 1–11	662.87–29,147	55.02	601.33
	Case 12–22	1,069.27	38.10–142.8	601.33
	Case 23–33	1,069.27	55.02	261.05–1,154.7
PT9	Case 1–11	543.23–1,604.4	45.57	555.86
	Case 12–22	952.49	33.23–79.3	555.86
	Case 23–33	952.49	45.57	224.79–1,007.9
PT10	Case 1–11	602.91–1,620.9	43.59	527.24
	Case 12–22	947.20	26.40–108.2	527.24
	Case 23–33	947.20	43.59	163.84–1,249.4

Table 6

Summary of the statistical properties of the uncertainty factors used in the derivation of the exceedance-probability calculation equations

Study grid	Statistics	Failure time step (h)	Failure soil depth (cm)	Rainfall duration (h)	Time to max rainfall intensity (h)
PT1	Mean	44.29	206.83	80.57	43.85
PT1	Coefficient of Variance (CV)	0.18	0.19	0.14	0.26
PT2	Mean	43.45	215.34	80.81	44.21
PT2	CV	0.16	0.05	0.14	0.26
PT3	Mean	46.35	196.99	82.66	47.04
PT3	CV	0.24	0.16	0.14	0.32
PT4	Mean	48.91	193.77	80.39	48.53
PT4	CV	0.22	0.14	0.13	0.23
PT5	Mean	44.74	310.00	71.54	30.57
PT5	CV	0.11	0.00	0.22	0.50
PT6	Mean	44.80	191.92	78.84	45.80
PT6	CV	0.22	0.10	0.16	0.26
PT7	Mean	44.18	209.96	78.09	41.21
PT7	CV	0.16	0.03	0.16	0.31
PT8	Mean	46.21	202.79	80.91	44.34
PT8	CV	0.21	0.18	0.10	0.22
PT9	Mean	44.78	236.92	81.67	43.57
PT9	CV	0.20	0.03	0.12	0.26
PT10	Mean	45.54	191.92	80.39	41.66
PT10	CV	0.18	0.11	0.12	0.29

Study grid	Statistics	Failure time step (h)	Failure soil depth (cm)	Rainfall duration (h)	Time to max rainfall intensity (h)
PT1	Mean	44.29	206.83	80.57	43.85
PT1	Coefficient of Variance (CV)	0.18	0.19	0.14	0.26
PT2	Mean	43.45	215.34	80.81	44.21
PT2	CV	0.16	0.05	0.14	0.26
PT3	Mean	46.35	196.99	82.66	47.04
PT3	CV	0.24	0.16	0.14	0.32
PT4	Mean	48.91	193.77	80.39	48.53
PT4	CV	0.22	0.14	0.13	0.23
PT5	Mean	44.74	310.00	71.54	30.57
PT5	CV	0.11	0.00	0.22	0.50
PT6	Mean	44.80	191.92	78.84	45.80
PT6	CV	0.22	0.10	0.16	0.26
PT7	Mean	44.18	209.96	78.09	41.21
PT7	CV	0.16	0.03	0.16	0.31
PT8	Mean	46.21	202.79	80.91	44.34
PT8	CV	0.21	0.18	0.10	0.22
PT9	Mean	44.78	236.92	81.67	43.57
PT9	CV	0.20	0.03	0.12	0.26
PT10	Mean	45.54	191.92	80.39	41.66
PT10	CV	0.18	0.11	0.12	0.29

Table 7

Summary of calibrated regression coefficients of the rain-related factors in the exceedance-probability calculation equations

Study grid	Constant	Warning time	Rainfall depth (mm)	Maximum rainfall intensity (mm)	Forward cumulative rainfall at the time to (mm)	Rainfall threshold (mm)
Study grid
PT1	−3.3039	0.2478	−0.0004	0.0014	0.0056	−0.0065
PT2	−4.6827	0.3121	−0.0001	−0.0005	0.0046	−0.0064
PT3	0.1391	0.1766	−0.0007	0.0023	0.0030	−0.0053
PT4	−0.9448	0.2343	−0.0003	−0.0058	0.0033	−0.0068
PT5	−6.5921	0.4332	−0.0015	−0.0046	0.0043	−0.0039
PT6	−3.0425	0.2840	−0.0012	−0.0034	0.0044	−0.0062
PT7	−2.1765	0.2855	−0.0011	0.0012	0.0039	−0.0056
PT8	−2.8428	0.2361	−0.0001	0.0001	0.0049	−0.0072
PT9	−4.4539	0.3558	−0.0016	−0.0111	0.0049	−0.0063
PT10	−3.0102	0.2894	−0.0013	−0.0015	0.0045	−0.0058

Study grid	Constant	Warning time	Rainfall depth (mm)	Maximum rainfall intensity (mm)	Forward cumulative rainfall at the time to (mm)	Rainfall threshold (mm)
Study grid
PT1	−3.3039	0.2478	−0.0004	0.0014	0.0056	−0.0065
PT2	−4.6827	0.3121	−0.0001	−0.0005	0.0046	−0.0064
PT3	0.1391	0.1766	−0.0007	0.0023	0.0030	−0.0053
PT4	−0.9448	0.2343	−0.0003	−0.0058	0.0033	−0.0068
PT5	−6.5921	0.4332	−0.0015	−0.0046	0.0043	−0.0039
PT6	−3.0425	0.2840	−0.0012	−0.0034	0.0044	−0.0062
PT7	−2.1765	0.2855	−0.0011	0.0012	0.0039	−0.0056
PT8	−2.8428	0.2361	−0.0001	0.0001	0.0049	−0.0072
PT9	−4.4539	0.3558	−0.0016	−0.0111	0.0049	−0.0063
PT10	−3.0102	0.2894	−0.0013	−0.0015	0.0045	−0.0058

Figure 14

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Illustrations of the exceedance probabilities of the specific landslide-triggering rainfall thresholds at the 10 study grids regarding the first simulation case of rainfall factors used in establishing the exceedance-probability.

In addition to the reliability quantification, Equation () can be rewritten to estimate the rainfall threshold of the specific warning time

corresponding to designed reliability

as:

(14)

Thus, within the proposed RA_GRTE_LS model, Equation (14), the rainfall thresholds (mm) of the various warning times (h) for the early warning of shallow landslide occurrence at the different locations in a watershed could be estimated through Equation (14) with the rainfall factors given under a designed reliability.

Model demonstration

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Within the proposed RA_GRTE_LS model, the results from the reliability assessment of the desired landslide-triggering rainfall thresholds of the specific warning times could be conducted by calculating their corresponding exceedance probabilities via the exceedance-probability calculation Equation (13); also, the landslide-triggering rainfall threshold of the specific warning time could be estimated under an exceedance probability (i.e., reliability) given through Equation (14). To demonstrate the applicability of the proposed RA_GRTE_LS model on the reliability assessment of the gridded rainfall thresholds, five study cases are analyzed based on an assumption of the rainfall factors given as listed in Table 8. The relevant results and discussion are addressed below.

Table 8

Conditions of rain-related factors used in the application of the proposed RA_GRTE_LS model for the reliability assessment of the landslide-triggering rainfall thresholds

Study case	Warning time (h)	Rainfall depth (mm)	Max rainfall intensity (mm)	Forward cumulative rainfall at the time to the maximum rainfall intensity (mm)	Rainfall threshold (mm)
I	10	1,000	100–600	700	150
II	20	1,000	100	300–800	150
III	10	500–1,500	75	500	100
VI	1, 12,24 and 36	1,000	100	550	Corresponding to the exceedance probability of 0.2

Study case	Warning time (h)	Rainfall depth (mm)	Max rainfall intensity (mm)	Forward cumulative rainfall at the time to the maximum rainfall intensity (mm)	Rainfall threshold (mm)
I	10	1,000	100–600	700	150
II	20	1,000	100	300–800	150
III	10	500–1,500	75	500	100
VI	1, 12,24 and 36	1,000	100	550	Corresponding to the exceedance probability of 0.2

Study case I

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In this section, the effect of variations in the maximum rainfall intensity on the reliability of the 10 h rainfall threshold of 150 mm is quantified based on the rainfall depth of 1,000 mm as shown in Figure 15(a); in increasing the maximum rainfall intensity from 100 to 600 mm, the exceedance probabilities of the rainfall thresholds over 150 mm slightly decline with the maximum rainfall intensity by 0.1, especially for the study grid PT4, in which the corresponding exceedance probabilities significantly decrease from 0.9 to 0.5. This decline reveals that the shallow landslide was merely triggered due to the maximum rainfall intensity. Nevertheless, the spatial uncertainty in the landslide-triggering rainfall threshold during a rainstorm event still exists; thus, a magnitude of the landslide-triggering rainfall threshold adapted to the region possibly causes the different reliabilities at the various locations; in other words, a signal landslide-triggering rainfall threshold of a specific warning time is hardly proper for the early warning operation of shallow landslide occurrence in a region with varied topography.

Figure 15

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Model demonstration results from the reliability assessment of the gridded landslide-triggering rainfall thresholds via the proposed RA_GRTE_LS model under various conditions of rain-related factors. (a) Study case I; (b) Study case II; (c) Study case III; and (d) Study case IV.

Figure 16

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Comparison between the 12 h rainfall thresholds at the study grids and corresponding slopes.

Study case II

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In study case II, the exceedance probabilities of the threshold of 150 mm are calculated for the different forward cumulative rainfalls at the time to the maximum rainfall intensity subject to the identical rainfall depth and maximum rainfall intensity, 1,000 and 100 mm, respectively; Figure 15(b) presents that the resulting exceedance probabilities noticeably increase with the forward cumulative rainfalls at the time to the maximum rainfall intensity. In detail, the study grids PT5 and PT9 exhibit the most and slightest increasing trends, 0.85–0.96 and 0.75–0.96, respectively. Since the cumulative rainfall features the storm pattern at the time of the maximum rainfall, it proves that the reliability of the landslide-triggering rainfall thresholds might be attributed to the temporal distribution of the rainfall; this is because the landslide-triggering threshold could be treated as the spatial variate.

Study case III

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In addition to the maximum rainfall intensity and corresponding cumulative rainfall depths, the rainfall depths are supposed to influence the reliability of the gridded rainfall thresholds for shallow landslide occurrence. Hence, the probabilities of the 10 h rainfall thresholds exceeding 100 mm are calculated with consideration of the various rainfall depths (500–1,000 mm), as shown in Figure 15(c); this implies that the resulting exceedance probabilities at the 10 study grids merely drop, on average, from 0.76 to 0.64, in which the changes in the exceedance probabilities at the study grids PT6 and PT9 reach 0.1. However, as compared to the results from study cases I and II, the effect of the variation in the rainfall depths on the reliability of the rainfall threshold is significantly less than the remaining featured rainfall factors in the storm pattern.

Study case IV

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Apart from the temporal change in rainfall, the variation of the landslide-triggering rainfall threshold in space should be evaluated under the condition of the same rainfall factors. Figure 15(d) shows the estimated 1, 12, 24 and 36 h rainfall thresholds at the 10 study grids for the exceedance probability of 0.2 (i.e., reliability = 0.8) using Equation (14), implies that the landslide-triggering rainfall thresholds exhibit a significant variation for durations shorter than 24 h; whereas, the 24 and 36 h rainfall thresholds reach the constant, around 850 and 950 mm, respectively. That is to say, the landslide-triggering rainfall threshold has a significant change in relation to the locations.

Also, there exists a large variation in the slope of the study grid in the study area; accordingly, the landslide-triggering rainfall threshold is supposed to vary with the slope. In detail, as compared to the results from Figure 16 for the 12 h warning time subject to the gridded slope, the corresponding rainfall thresholds at the 10 study grids adversely change with their slopes. This change reveals that the low gridded landslide-triggering rainfall threshold results from a steep slope, rapidly causing the safety factor smaller than 1.0; for example, the slope at the study grid PT9 (about 46°) with the minimum threshold (330 mm) is steeper than those at the remaining study grids; thus, a shallow landslide occurs at the study grid PT9 with a high likelihood of leading to the low rainfall threshold. Instead, the maximum rainfall threshold (805 mm) could be found at the study grid PT3 with a relatively smooth slope of 30°.

Summary

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The above results conclude that within a potential rainfall-induced shallow landslide region, using Equation (13), the effect of the variations in the rainfall factors concerned in Equation (13), especially for the forward cumulative rainfall at the time to the maximum rainfall intensity, on the reliability of the landslide-triggering rainfall thresholds could be quantified via Equation (13). Additionally, it is proven that the landslide-triggering rainfall thresholds should be denoted as the spatial variate (i.e., gridded rainfall thresholds) for shallow landslide occurrence, especially for short-term rainfall (⁠⁠). Accordingly, it is necessary that the landslide-triggering rainfall thresholds are supposed to be locally determined in a large zone under consideration of the apparent variations in the rainfall characteristics and soil properties (Rosi et al. 2016; Segoni et al. 2018a). As a result, the proposed RA_GRTE_LS model provides the stochastically-based landslide-triggering rainfall thresholds subject to the spatiotemporal variation in rainfall and soil properties and responds to the effect of the varied topography on the estimation of the gridded rainfall thresholds.

5. CONCLUSIONS

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This study aims to develop a probabilistic-based reliability assessment of the gridded rainfall thresholds regarding the early warning of shallow landslide occurrence, named the RA_GRTE_LS model, due to the uncertainties of rainfall in time and space and spatial variation in the soil properties. In detail, the identified uncertainty factors consist of the rainfall depths and maximum rainfall intensity as well as the forward cumulative rain-related and soil-related features, including the failure time steps and soil depths. In developing the proposed RA_GRTE_LS model, a considerable number of the rainfall-induced shallow landslide are simulated by coupling the stochastic modeling of gridded rainstorms (Wu et al. 2021) with the slope-stability numerical model under the initial condition of unsaturated soil (Tsai & Chen 2010); the resulting gridded rainfall thresholds of various warning durations and corresponding exceedance probabilities are achieved via the advanced first-order second moment (AFOSM); after that, the exceedance-probability calculation equations regarding the landslide-triggering rainfall thresholds with the rainfall factors at the different grids of interest are established using the logistic analysis. Eventually, within the proposed RA_GRTE_LS model, the regional reliability analysis for the rainfall thresholds for shallow landslide occurrence could be achieved by computing the exceedance probabilities of the specific thresholds via the exceedance-probability calculation equations under the given conditions of rain-related factors.

A total of 1,000 simulation cases of rainfall-induced shallow landslides are reproduced in advance from the gridded rainfall characteristics of the 30 historical rainstorms and soil parameters adopted in the Tsai's model at 10 locations (i.e., study grids) within the Jhuokou River watershed (study area) and then used in the model development and demonstration; the results indicate that safety factor sequences at different soil depths significantly vary with the change in the accumulated rainfall as a result of directly impacting the groundwater pressure heads, during a rainfall event. Additionally, the failure time steps corresponding to the safety factors lower than 1.0 mostly range from the 20th hour and 80th hour (on average, the 34th hour) at the soil depths of around 150 and 300 cm; accordingly, the warning times in the study area are suggested to be 1, 3, 12, 18, 24 and 36 h. Also, the exceedance probabilities of the gridded rainfall thresholds of various warning times significantly result from the effect of the spatiotemporal variations of rain-related and soil-related uncertainty factors, especially for short-term rainfall (⁠⁠). In addition to the uncertainties in the rain-related and soil-related factors, through the proposed RA_GRTE_LS model, the stochastically based landslide-triggering rainfall thresholds at the various locations could be achieved in response to the effect of the varied topography.

Although the proposed RA_GRTE_LS model can reasonably quantify the effects of the uncertainty in rainfall in time and space based on the soil parameters adopted based on the grid-based soil properties; the resulting safety factor sequences from the unsaturated soil slope-stability numerical modeling are proven to be impacted by the storm pattern and soil properties in space (Tsai & Chen 2010). Thereby, future work would be done by improving the proposed RA_GRTE_LS model by considering the spatial uncertainties in the soil properties. Additionally, a significant number of rainfall-induced shallow landslide simulations can be applied in stochastically modeling the physical-based slope-stability numerical model based on the well-known artificial intelligence (AI) model, which is anticipated to efficiently provide the grid-based safety factor sequences and induced failure time steps as well as soil depths under consideration of the rain-related and soil-related factors in time and space.

DATA AVAILABILITY STATEMENT

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Data cannot be made publicly available; readers should contact the corresponding author for details.

CONFLICT OF INTEREST

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The authors declare there is no conflict.

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This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits copying, adaptation and redistribution, provided the original work is properly cited (http://creativecommons.org/licenses/by/4.0/).

Modeling probabilistic-based reliability assessment of gridded rainfall thresholds for shallow landslide occurrence due to the uncertainty of rainfall in time and space

Abstract

HIGHLIGHTS

INTRODUCTION

METHODOLOGY

Model concept

Definition of landslide-triggering rainfall thresholds and associated reliability

Simulation of gridded rainstorm events

Slope-stability numerical model for unsaturated soil

Reliability quantification of the landslide-triggering threshold rainfall

Derivation of the exceedance-probability calculation equation

Model framework

Model development

Model application

STUDY AREA AND DATA

RESULTS AND DISCUSSIONS

Simulation of gridded rainstorms

Simulation of a gridded rainfall-induced shallow landslide

Simulation of the rainfall threshold for shallow landslide

Development of the proposed RA_GRTE_LS model

Identification of uncertainty factors

Establishment of the rainfall threshold estimation equation

Calculation of exceedance probabilities of rainfall thresholds

Derivation of the exceedance-probability calculation equation

Model demonstration

Study case I

Study case II

Study case III

Study case IV

Summary

5. CONCLUSIONS

DATA AVAILABILITY STATEMENT

CONFLICT OF INTEREST

REFERENCES

Cited by

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