Does parameterization influence the performance of slope stability model results? A case study in Rio de Janeiro, Brazil

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.

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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.

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.

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).

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).

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.

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.

where μ_SFSI 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.

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.

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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).

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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 (I₀ = 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.

Figure 5 illustrates the results of test A in terms of model performance (AUROC) and conservativeness (FoC). Assuming a constant soil depth, the model performs significantly better when considering only the ORA (test A2; AUROC ≤ 0.741; Fig. 5b) instead of the entire OIA (test A1; AUROC ≤ 0.691; Fig. 5a). This result clearly indicates that the OIA is unsuitable as reference for evaluation, and an appropriate inventory sub-setting is essential. Focusing on Fig. 5b, we note that the model performance in terms of AUROC is insensitive to the variation of the geotechnical parameterization within much of the tested ranges. In particular, the sub-spaces along a diagonal line from medium-high values of c′ and low values of φ′ to low values of c′ and high values of φ′ display almost identical AUROC values to the entire parameter space and to those sub-spaces including broad ranges of c′ or broad ranges of φ′ with medium-low values of c′. Only those sub-ranges limited to high values of c′ or low values of c′ and φ′ yield significantly lower AUROC values. These sub-ranges result in poorly patterned relatively non-conservative and extremely conservative predictions, i.e. they display very low and very high FoC values, respectively. In general, the model results are very conservative, indicated by FoC > > 1. At a lower level of AUROC—and a lower level of FoC caused by a higher number of OP pixels—similar patterns are observed in Fig. 5a.

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Varying d as a function of the topographic wetness index exerts contrasting effects on the patterns of AUROC, depending on whether the OIA or the ORA is used as reference. With the ORA as reference (Test A4; Fig. 5d), the sub-spaces with low values of c′ perform comparable to test A2 (Fig. 5b). This is not surprising as the influence of d on FOS increases with c′ (with c′ = 0, d has no influence). However, AUROC and also FoC decrease significantly with increasing c′, resulting in a very poor performance associated to those sub-spaces with high c′, and a reduced performance associated to those sub-spaces with broad ranges of c′, compared to Fig. 5b. This trend clearly indicates that most ORA pixels spatially coincide with areas of relatively low topographic wetness index and therefore low values of d (Table 3) resulting in high values of FOS and low values of SFSI in cohesive soils.

The reverse effect occurs when using the entire OIA as reference (test A3; Fig. 5c): many pixels in the lower portions of the landslide polygons coincide with high values of the topographic wetness index. Consequently, d and the resulting values of SFSI are comparatively high for many of the OP pixels, resulting in an improved model performance, compared to the tests A1 – A3 (AUROC ≤ 0.742; Fig. 5b). However, since most of the lower parts of the landslide polygons do most likely not represent release areas, the increased performance represents an artefact of inappropriate assumptions rather than an indicator for model success.

Considering the findings outlined, we identify test A2 as most representative. Even though the full parameter space yields an insignificantly lower value of AUROC than do some of the sub-spaces, there is no basis to support the choice of a particular sub-space in this specific case. The parameter values used and optimized by Guimarães et al. (2003) are mostly located within the parameter sub-spaces with the higher values of AUROC, indicating a certain plausibility of the results (Fig. 5b). Figure 6a shows the spatial patterns of SFSI derived in the tests A1 and A2 with the full parameter space of c′ and φ′. We note that the results of those tests are similar in terms of SFSI, as only the reference information for validation is varied. The same is true for the SFSI maps derived through the tests A3 and A4 (Fig. 6b).

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The spatial patterns of SFIS derived with the sliding surface model of r.slope.stability (test B) are illustrated in Fig. 6c. Applying the full parameter space of c′ and φ′ along with constant soil depth and the ORA as reference, the associated value of AUROC is almost identical to the value yielded with the infinite slope stability model (0.735 vs. 0.734 in test A2). Thereby, the results yielded with the sliding surface model are more conservative: FoC = 59.5, compared to a value of 48.3 yielded with the infinite slope stability model (Fig. 5b).

Figure 7 illustrates the performance (AUROC) and conservativeness (FoC) of the model results for the various parameter sub-spaces of K_s and d_i. Firstly, we note that the results are largely insensitive to the four assumptions of rainfall duration and hydrograph shape (C1–C4): the patterns yielded are identical for all four scenarios, even though the numbers vary slightly. Within each scenario, the model performance responds highly sensitive to variations of K_s and d_i: it peaks at AUROC = 0.719–0.724 for the upper sub-range of the hydraulic conductivity (K_s = 10⁻⁵–10⁻⁴ m s⁻¹) and the lower sub-range of the initial depth of the water table (d_i = 0–1 m). However, the model performance drops only slightly when the full range of both parameters K_s and d_i is applied (AUROC = 0.711–0.712). Figure 8 presents the SFSI maps produced in test C1 with the full space of K_s and d_i. The SFSI maps resulting from tests C2, C3 and C4 are almost similar to the map resulting from test C1 and are therefore not shown.

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Constraining the model input to the lower ranges of hydraulic conductivity or to deeper initial water tables leads to a significant drop in the model performance. Considering K_s ≤ 10^–5.5 leads to model failure (AUROC = 0.494), independently of the range applied for d_i and the rainfall scenario. In this case, FoC = 3.9 (blue font colour in Fig. 7). As expected, FoC is highest for the configurations with high K_s and shallow d_i and lowest for the configurations with low K_s and deep d_i. Its maximum coincides with the best model performance (FoC = 48.0–48.9).

These outcomes reflect the fact that, with K_s ≤ 10^–5.5, too little water propagates through the soil to substantially influence slope stability. The effect is similar with higher values of K_s if the initial water table is too deep. A shallower initial water table and higher values of K_s facilitate increased values of u over broad parts of the study area and, consequently, lead to less stable slopes (Eq. 2) and higher values of FoC. Only combinations of high K_s and deep di lead to a sufficient signal to reproduce the observed landslide release patterns with a fair performance. As for tests A and B, all results are very conservative also for test C (FoC > > 1).