Work flow and software
Figure 2 illustrates the general work flow of the study. We compute the slope failure susceptibility index (SFSI) (dimensionless number in the range 0–1) based on sets of factor of safety (FOS) values derived through the controlled variation of selected key parameters within a defined parameter sub-space. This procedure is repeated for various sub-spaces. The resulting SFSI values are evaluated against the inventory of observed landslides, and the findings are compared and interpreted.
In a first step, we vary the geotechnical parameters (tests A and B) and in a second step, we vary the geohydraulic parameters (test C). Test D uses a simple statistical model for the sake of comparison. Test A builds on the infinite slope stability model, test B on the sliding surface model of the tool r.slope.stability (Mergili et al. 2014a, b), designed as a raster module of the open source GRASS GIS software (Neteler and Mitasova 2008; GRASS Development Team 2016). Test C makes use of TRIGRS (Transient Rainfall Infiltration and Grid-Based Regional Slope-Stability Model; Baum et al. 2008), which is a grid-based tool simulating the permanent and transient rainfall influences on slope stability. Python scripting is used to derive SFSI, and the R Project for Statistical Computing (R Core Team, 2016) is employed for the evaluation of the results. Test D relies entirely on Python and R scripting.
Geotechnical model
Slope stability modelling commonly builds on the limit equilibrium theory (Duncan and Wright 2005): a factor of safety (FOS) is computed as the ratio between resisting forces R and driving forces T:
$$ \mathrm{FOS}=\frac{R}{T} $$
(1)
When FOS = 1, the slope is in static equilibrium. Values of FOS <1 indicate potential failure (in reality, such slopes do not exist), values of FOS >1 indicate stable slopes. The use of this method requires the prior definition of a slip surface, and the soil is considered as rigid material.
For GIS-supported catchment-scale analyses of slope stability, the infinite slope stability model is most commonly employed (Montgomery and Dietrich 1994; Pack et al. 1998; Xie et al. 2004a; Baum et al. 2008). It assumes (i) a uniform slope of infinite length, and (ii) a plane, slope-parallel failure surface. As inter-slice forces do not have to be considered, it is conveniently applied on a pixel-to-pixel basis. Based on Eq. (1), FOS can be expressed in various ways. For fully saturated soil, the equation may be formulated as follows (modified after Baum et al. 2008):
$$ \mathrm{FOS}=\frac{ \tan \phi}{ \tan \alpha}+\frac{c- u \tan \phi}{\gamma_s d \sin \alpha \cos \alpha} $$
(2)
where α is the slope angle, u (N m−2) is the pore water pressure, γ
s (N m−3) is the specific weight of the saturated soil and d (m) is the depth of the sliding surface.
In the present work, we use the infinite slope stability model implemented with r.slope.stability and with TRIGRS. Alternatively, we also apply the sliding surface model of r.slope.stability. Thereby, the slope stability is tested for a large number of randomly selected ellipsoid-shaped potential sliding surfaces, truncated at the depth of the soil. R and T are summarized over all pixels intersecting a given sliding surface, and FOS is computed for each surface in a way analogous to Eqs. 1 and 2, applying a modification of the Hovland (1977) model. Finally, the minimum value of FOS resulting from the overlay of all sliding surfaces is applied to each pixel. For a more detailed description of the sliding surface model of r.slope.stability, we refer to Mergili et al. (2014a, b).
Geohydraulic model
In TRIGRS, FOS is computed for one or more user-defined depths. The Richard’s equation is used to calculate the soil transient infiltration for saturated and unsaturated soil conditions (Iverson 2000):
$$ \frac{\partial \psi}{\partial t}\frac{d\theta}{d\psi}=\frac{\partial }{\partial x}\left[{K}_L\left(\psi \right)\left(\frac{\partial \psi}{\partial x}- \sin \alpha \right)\right]+\frac{\partial }{\partial y}\left[{K}_L\left(\psi \right)\left(\frac{\partial \psi}{\partial y}\right)\right]+\frac{\partial }{\partial z}\left[{K}_z\left(\psi \right)\left(\frac{\partial \psi}{\partial z}- \cos \alpha \right)\right] $$
(3)
where ψ (m) is pressure head, θ is soil volumetric water content, t (s) is time, K
L
(m s−1) is lateral soil conductivity and K
z
(m s−1) is soil conductivity in z direction.
To solve the Richards equation, TRIGRS uses an approach developed by Iverson (2000), considering homogeneous soil, isotropic flow, relatively shallow depth, one-dimensional vertical downslope flow and soil moisture close to saturated conditions (Baum et al. 2008; Park et al. 2013), following the heat conduction approach described by Carslaw and Jaeger (1959). We refer to Baum et al. (2008) for a detailed description of the procedure.
For computing the groundwater level, TRIGRS compares the infiltrated water volume V
I and the maximum drainage capacity of the soil V
D. If V
D ≥ V
I, the water table remains constant. Otherwise, the water table rises, depending on K
s and the transmissivity T. For unsaturated conditions, the maximum value of ψ is the new water level multiplied with β (value set according to the adopted flow condition). The amount of water exceeding the maximum infiltration rate is considered surficial runoff. However, surficial runoff is not taken over from one time step to the next (Baum et al. 2008).
Slope failure susceptibility index
The slope failure susceptibility index (SFSI) in the range 0–1 refers to the fraction of geotechnical and/or geohydraulic parameter combinations resulting in FOS <1, out of an arbitrary number of tested parameter combinations. This means that SFSI for a given pixel increases with each parameter combination where FOS <1 and, finally, low values of FOS correspond to high values of SFSI. The principal concept of the SFSI is identical to the concept of the slope failure probability yielded by r.slope.stability (Mergili et al. 2014a). However, we refer to it as a susceptibility index in the context of the present study as we simply use a uniform probability density function throughout all the computations. Such a distribution does not necessarily capture the real-world parameter distribution (which is unknown) and its use does therefore not justify applying the concept of probability in a strict sense.
Statistical model
In test D, a statistical model is applied for the purpose of comparison, employing the slope angle as the only predictor layer (Table 3). We keep the statistical model as basic as possible in order to evaluate the performance of a simplistic statistical approach in comparison to the physically based models (“Geotechnical model” to “Slope failure susceptibility index” sections). This allows us to conclude on the need of using more complex physically based models for catchment-scale landslide susceptibility analysis. Thereby, we overlay a classified slope map with the map of the observed landslide release areas (ORA; “Model evaluation” section) and, for each slope class, compute the fraction f
C of observed landslide release pixels related to all pixels. SFSI—referred to as release probability by Mergili and Chu (2015) who employed a comparable approach—is then computed by applying f
C to all pixels of the corresponding slope class. Thereby, it is important to use two different areas for the derivation of f
C and for the computation and evaluation of SFSI (“Test layout” section).
Table 3 Summary of all tests performed
Model evaluation
The landslide inventory for the Quitite and Papagaio watersheds displays the entire observed landslide impact areas (OIAs), i.e. the release, transit and deposition areas without any differentiation. We approximate the ORA as the upper third part of each OIA polygon. Depending on the test (“Test layout” section and Table 3), either the OIA map or the ORA map is overlaid with the corresponding SFSI map. When using the ORA map, the lower two-thirds portion of the OIA is not considered for evaluation. The true positive (TP), true negative (TN), false positive (FP) and false negative (FN) pixel counts are derived for selected levels of SFSI. An ROC curve is produced by plotting the true positive rates TP/(TP + FN) against the false positive rates FP/(FP + TN) derived with each combination of parameters. The area under the ROC curve AUROC indicates the predictive capacity of the model: AUROC = 1.0 (the maximum) means a perfect prediction, AUROC = 0.5 (corresponding to a straight diagonal line) indicates a random prediction, i.e. model failure. AUROC refers to the entire area used for model evaluation.
In addition, we introduce a conservativeness measure:
$$ \mathrm{FoC}=\frac{\mu_{\mathrm{SFSI}}}{r_{\mathrm{OP}}} $$
(4)
where μSFSI is the average of SFSI over the entire study area, and r
OP is the observed positive rate, i.e. the fraction of observed landslide pixels out of all pixels in the study area. If FoC >1, the model overestimates the landslide susceptibility, compared to the observation whilst values FoC <1 indicate an underestimation of the landslide susceptibility.
Test layout
Tables 3 and 4 summarize the main characteristics of each test and the parameter values and ranges considered.
Table 4 Geotechnical and geohydraulic parameter values and ranges applied for the tests A–C (Table 3), following Guimarães et al. (2003); Saxton and Rawls (2006); Hurtado Espinoza (2010); Conti (2012); Park et al. (2013)
In a first step (tests A1–A4 and B), the sensitivity of SFSI and the associated model performance to the geotechnical parameters c′ and φ′ and the shape of the sliding surface is explored, assuming fully water-saturated soils, and the depth of the sliding surface corresponding with the soil depth. The infinite slope stability model and the sliding surface model implemented in r.slope.stability are employed for this purpose. We introduce a two-dimensional parameter space constrained by lower boundaries of c′ = 0 kN m−3 and φ′ = 21°, and upper boundaries of c′ = 24 kN m−3 and φ′ = 45° (Fig. 3a; Table 4). This parameter space accounts for the full ranges of c′ and φ′ considered representative for the area (“Study area and data” section). We note that the resulting values of FOS vary according to φ′ and c′/d, so that the value of FOS obtained with d = 3 m and with a given value of c′ is identical (infinite slope stability model) or similar (sliding surface model) to the value of FOS with other values of c′ and d, but the same c′/d ratio. The dry specific weight of the soil γ
d = 13.5 kN m−2 and the volumetric saturated water content θ
s = 40 vol.% are set to constant values. We neglect the weight of the trees and the effects of their root systems on the cohesion: sliding surfaces are assumed to develop beneath the rooting depth.
The ranges of both c′ and φ′ are (i) considered in their entire extent; (ii) subdivided into two sub-ranges of equal extent and (iii) subdivided into three sub-ranges of equal extent (Fig. 4a, b). Considering all possible combinations of sub-ranges of the two parameters results in 36 partly overlapping parameter sub-spaces with 25 corner points. SFSI is computed for each parameter sub-space, with ten sampled parameters in each dimension (Fig. 4c). This procedure may be extended to three or more dimensions or repeated at a finer level by employing the sub-space with the best model performance as the entire space for the next level. For reasons to be explained in the “Results” section, only one level is applied in the present work. This work flow is repeated for two assumptions of soil depth and two versions of the landslide inventory used for evaluation, resulting in a total of four sub-tests (Table 3).
Test C explores the sensitivity of SFSI and the associated model performance to K
s
and the initial depth of the water table d
i (m). We introduce a two-dimensional parameter space constrained by lower boundaries of K
s
= 10−7 m s−1 and d
i = 0 m and upper boundaries of K
s
= 10−4 m s−1 and d
i = 3 m (Fig. 3b; Table 4). The ranges of values used are based on works of Saxton and Rawls (2006) and Guimarães et al. (2003). We set γ
s = 16 kN m−2, θ
s
= 40 vol.%, θ
r
= 5 vol.%, c′ = 4.5 kN m−2, φ′ = 45° and d = 3 m to constant values. The choice of these values is supported by data from Guimarães et al. (2003) and Hurtado Espinoza (2010). We further assume constant values of diffusivity (D = 200K
s
; Park et al., 2013) and initial infiltration rate (I
0 = 1.3 10−6 m s−1; Conti 2012).
In a way analogous to the geotechnical parameters, the ranges of both K
s
and d
i are (i) considered in their entire extent, (ii) subdivided into two sub-ranges of equal extent and (iii) subdivided into three sub-ranges of equal extent, resulting in 36 partly overlapping parameter sub-spaces with 25 corner points. SFSI is computed for each parameter sub-space, with five sampled parameters in each dimension. The landslide inventory used for evaluation is ORA.
This procedure is repeated for four combinations of rainfall duration and type of pluviograph (Table 3). We assume rainfall durations of 6 and 10 h and a total rainfall amount derived from the measurements at the Jacarepaguá and Boa Vista stations on 13 and 14 February 1996 (Conti 2012). The Thiessen method is applied for estimating the precipitation in the catchment, and 20% of interception are deduced (Coelho Netto 2005). The total rainfall considered for the analysis is 144 mm in all the scenarios C1–C4.
In test D, we apply the statistical model introduced in the “Statistical model” section for the purpose of comparison (Table 3). f
C is derived for one of the two catchments. SFSI is then computed for the other catchment and evaluated against the corresponding ORA. The entire procedure is repeated in the reverse way, so that a clear separation between the model development and model evaluation areas is ensured.