Rainfall-triggered shallow landslides at catchment scale: Threshold mechanics-based modeling for abruptness and localization

Authors


Abstract

[1] Rainfall-induced shallow landslides may occur abruptly without distinct precursors and could span a wide range of soil mass released during a triggering event. We present a rainfall-induced landslide-triggering model for steep catchments with surfaces represented as an assembly of hydrologically and mechanically interconnected soil columns. The abruptness of failure was captured by defining local strength thresholds for mechanical bonds linking soil and bedrock and adjacent columns, whereby a failure of a single bond may initiate a chain reaction of subsequent failures, culminating in local mass release (a landslide). The catchment-scale hydromechanical landslide-triggering model (CHLT) was applied to results from two event-based landslide inventories triggered by two rainfall events in 2002 and 2005 in two nearby catchments located in the Prealps in Switzerland. Rainfall radar data, surface elevation and vegetation maps, and a soil production model for soil depth distribution were used for hydromechanical modeling of failure patterns for the two rainfall events at spatial and temporal resolutions of 2.5 m and 0.02 h, respectively. The CHLT model enabled systematic evaluation of the effects of soil type, mechanical reinforcement (soil cohesion and lateral root strength), and initial soil water content on landslide characteristics. We compared various landslide metrics and spatial distribution of simulated landslides in subcatchments with observed inventory data. Model parameters were optimized for the short but intense rainfall event in 2002, and the calibrated model was then applied for the 2005 rainfall, yielding reasonable predictions of landslide events and volumes and statistically reproducing localized landslide patterns similar to inventory data. The model provides a means for identifying local hot spots and offers insights into the dynamics of locally resolved landslide hazards in mountainous regions.

1. Introduction

[2] Rainfall-induced shallow landslides are important natural geomorphic processes that shape landscape evolution [Hovius et al., 1997; Korup et al., 2010], but they also represent a significant natural hazard in mountainous regions. Prediction of susceptible locations and hydrological scenarios conducive for landslides is essential for developing landslide mitigation strategies. Such predictions are often based on statistical or physically based (often deterministic) models. Statistical models attempt to link geomorphological and topographic variables such as slope, curvature, and vegetation cover with the likelihood of landslide occurrence assuming that (i) future landslides are likely to occur under similar conditions as observed ones [Guzzetti et al., 1999; Van Westen et al., 2008], and (ii) properties deduced from digital elevation models (DEMs) can provide surrogate information for hydrological and mechanical hillslope processes.

[3] Many physically based models explicitly account for hydrological and mechanical processes, such as by considering hydrologic pathways associated with infiltration and subsurface water flow within the context of an infinite slope stability model. These models may compute a factor of safety (FOS; i.e., a ratio between resisting and driving forces) as an indicator of the mechanical state and landslide susceptibility [O'Loughlin and Pearce, 1976; Wu et al., 1979; Casadei et al., 2003]. Hydrological effects on FOS are typically expressed as functions of seepage flow and rainfall rate [Montgomery and Dietrich, 1994; Fernandes et al., 2004] or by explicit solution of the Richards equation (for unsaturated flow) to estimate effects of pore pressure evolution [Iverson, 2000; Simoni et al., 2008; Arnone et al., 2011]. Several models have been used to study landslide occurrences at a catchment scale such as the SHALSTAB model, which uses the infinite slope approach with steady state hydrological processes [Dietrich and Montgomery, 1998], and the SINMAP model by Pack et al. [1998], which incorporates parameter uncertainty and was implemented in a raster-based Geographic Information System (GIS) framework. Other physically based models, such as SHETRAN [Burton and Bathurst, 1998] and TRIGRS [Baum et al., 2002], include spatiotemporal modeling of hydrological processes of the saturated as well as the unsaturated zone across the soil mantle, whereas GEOtop-FS [Simoni et al., 2008] describes the soil mantle with individual layers, and STARWARS+PROBSTAB [Malet et al., 2005; Kuriakose et al., 2009] implements the important process of preferential flow paths (e.g., cracks and fissures).

[4] In contrast with the gradual evolution of hillslope-scale FOS during a rainfall event and the generally nonlocal description of hillslope mechanical status (large regions with similar FOS), evidence suggests that shallow landslides tend to be abrupt and highly localized, whereby a large mass of soil mantle suddenly mobilizes without visible warning signs. Iverson et al. [2000] found that thin, sandy soils with large effective hydraulic diffusivities fail abruptly, whereas Petley et al. [2005] attributed the observed failure abruptness to sudden coalescence of local failure crack-line planes into a continuous failure plane. Generally, FOS-based models do not reproduce observed size/volume-frequency distributions of landslides, which often follow a power law [Stark and Hovius, 2001; Brunetti et al., 2009; Guzzetti et al., 2002].

[5] The observed power law in landslide size-frequency implies strong localization (with many small landslides and less likely large ones) and is also indicative for systems in a critical state (this term implies that a rainfall event may release shallow landslides with a wide range of sizes). As demonstrated by the self-organized criticality (SOC) framework [Bak et al., 1987], a critical state can be attained through interactions (load or mass redistribution) among many elements, often involving cascading chain reactions [Turcotte, 1999]. These concepts have been previously applied to modeling a wide range of abrupt mass release phenomena, such as glacier breakdown [Faillettaz et al., 2010] and landslides [Piegari et al., 2006; Hergarten and Neugebauer, 2000]. Recently, Lehmann and Or [2012] translated concepts of SOC into physically based hydromechanical modeling of rainfall-triggered landslides on hillslopes consisting of many interacting soil columns. Briefly, when the (hydromechanical) load exerted on a column exceeds the strength of mechanical bonds linking soil and underlying bedrock (as prescribed failure plane), these bonds fail, and the column's load is redistributed to different lateral mechanical bonds with neighboring columns (the strength thresholds of these bonds depend on lateral root reinforcement and capillary forces). In their study, Lehmann and Or [2012] have shown that subsequent to soil base failure, failure of mechanical bonds interconnecting soil columns may cascade and trigger an abrupt release of landslides. Mechanical bonds could be represented as bundles of fibers [Cohen et al., 2009], with gradual failure revealing aspects of precursor events preceding mass release. The model of Lehmann and Or [2012] was limited to small systems (a hillslope ∼50 m) due to the computational burden of modeling of precursor events and details of load redistribution cascades for large systems (such as catchments).

[6] The objective of this study was to extend a simplified version of the landslide hydromechanical triggering model (LHT) of Lehmann and Or [2012] to larger scales encompassing an entire catchment. The extended model links hydrological pathways with threshold mechanics required for reproducing localized landslide patterns, abrupt mass release, and wide landslide size-frequency distributions. Two rainfall event-based landslide inventories within the same region are used to test model predictions and to systematically evaluate influences of key attributes such as soil type, initial water content, soil cohesion, and lateral root reinforcement on landslide triggering, temporal evolution, and sizes of mass release events. In section 2, the catchment-scale hydromechanical landslide-triggering model (CHLT) is presented. In section 3, the landslide inventories and required data sets (DEM, rainfall map, soil depth map) are introduced. In section 4, the model performance for a wide range of key factor values is evaluated by comparing modeled properties with data of the first inventory. In section 5, parameter values that represent best the first landslide inventory (in a statistical sense) are used to predict landslide patterns for the second inventory. Model performance (and limitations) are discussed in section 6.

2. Catchment-Scale Hydromechanical Landslide Triggering Model (CHLT)

[7] The hydrological triggering model represents the soil mantle overlying a mountainous catchment as an assembly of hexagonal-shaped soil columns interconnected by mechanical bonds. We have used a hexagonal lattice for the isotropic geometry with equal distance to all neighboring columns. A mechanical bond links the base of a soil column with bedrock surface (considered to represent a potential failure plane). During a rainfall event, infiltrating water gradually modifies the weight and strength of soil columns. Detailed spatial and temporal distributions of soil water consider the following hydrological processes: (i) water infiltration, (ii) subsurface flow in soil matrix, (iii) fast flow along a saturated soil-bedrock interface, and (iv) multidirectional overland flow (all simulated hydrological processes are described in detail in Appendix Hydrological Model, and the validation of water-dependent soil strength is described in Appendix Parameterization of Soil Mechanical Properties). In the present model, we neglect several additional flow processes, such as rapid macropore flow and infiltration into bedrock.

2.1. Evolution of Forces at Soil Column Base

[8] In a first step, we assess the slope mechanical status using local FOS-like estimate applied individually to each column. We compare basal soil strength with driving forces along the maximal (local) elevation gradient zc = 1/2(zsu + zbr) between neighboring soil columns, where zsu is the elevation at the surface and zbr at bedrock. The total mass M of a soil column and its weight FZ are defined as follows:

display math(1)

with gravity acceleration g, soil depth Hsd = zsu − zbr, volumetric water content θ, soil porosity ϕ, density of water ρw, soil mineral density ρr, and cross section of hexagonal column Ah. The weight of a soil column FZ is divided into two components: (i) a driving force FG = FZ sin β and (ii) a normal force FN = FZ cos β, with β as the slope angle along the maximal elevation drop. Relating the forces to the cross section at the base along the slope AG = Ah/cos β, the normal stress σN and the downslope force component W are expressed as

display math(2)
display math(3)

[9] The shear stress τS opposing the driving force FG is a function of the normal stress and the internal friction angle γ = 30°. Increasing soil water may result in the formation of a saturated water table at the soil-bedrock interface. For a saturated water table of height Hsat, a positive pore pressure acts to reduce the effective stress according to Iverson et al. [2000]:

display math(4)

with (internal) soil cohesion c′. For unsaturated conditions, capillary forces provide an additional strength term τh as introduced by Bishop [1960] through the effective stress formulation:

display math(5)

[10] In the following, we use the term “matric suction” to express reinforcing capillary forces in the effective strength formulation. We express matric suction in units of “head” h (<0) (pressure per density and gravity acceleration) and apply the model of Lu et al. [2010] to quantify effect of matric suction on shear strength:

display math(6)

with the water saturation Θ (a water content normalized by minimum and maximum water saturation; see definition in Appendix Hydrological Model, equation (B1)).

2.2. Tensile and Compressive Mechanical Bonds

[11] The load of a failed soil column is redistributed to neighboring columns via mechanical “bonds” with a prescribed strength value. We considered two types of load redistributions: (1) along tensile bonds that connect a soil column to its side or upslope neighbors and (2) as compression in a downslope direction. Water content dependent tensile bond strength τt is quantified with the model of Lu et al. [2010]:

display math(7)

where α is the inverse of characteristic matric suction and n the shape parameter of the soil water characteristic curve according to Van Genuchten [1980] model. In addition to tensile strength imparted by capillary forces, lateral root reinforcement τr [Schmidt et al., 2001; Schwarz et al., 2011; Cohen et al., 2011] could significantly contribute to the tensile bond strength:

display math(8)

[12] This lateral root mechanical reinforcement is added to tensile bonds for soil columns located in forested areas, whereas in meadow, lateral root reinforcement is zero (τr = 0, and equation (8) equals equation (7)). No explicit root geometry, depth distribution, or root size information such as explored by Schwarz et al. [2011] and Cohen et al. [2011] was introduced to quantify root reinforcement. When all tensile bonds of a column fail, a soil column may “lean” downward and apply compressive stress on its neighbors. Water content compressive strength τC is quantified based on the approach of Mullins and Panayiotopoulos [1984]:

display math(9)

2.3. Load Redistribution and Failure

[13] When the driving forces at the base of a column exceeds its basal strength, the mechanical bond at the base fails. We then reduce the contribution by matric suction to soils' shear strength to a third of the prefailure value. The choice of the residual value was based on analysis by Cohen et al. [2009] and measurements of Fannin et al. [2005] concerning residual shear strength of a colluvium with roots. Subsequent to base failure, a column's excess load is redistributed to neighboring soil columns along tensile bonds first. Redistributed load that exceeds the strength of tensile bonds results in failure (interpreted as the formation of an upslope tensile crack). After all tensile bonds of a soil column failed, the load may then be redistributed as compression in a downslope direction. When the compressive stress applied by upslope columns exceeds the compressive strength of a downslope column, we assume that the latter loses its structural integrity and fails (which Lehmann and Or [2012] termed as fluidization). A fluidized column may destabilize neighboring columns and result in a landslide. For simplicity, we assumed that the hydrology is not affected by soil columns classified as unstable.

3. Inventory Data and Model Input Parameters

[14] The catchment-scale hydromechanical landslide triggering (CHLT) model was applied to two landslide inventories mapped by the Swiss Federal Institute for Forest, Snow and Landscape Research (WSL) by field surveys after the respective rainfall events [Rickli et al., 2004; Raetzo and Rickli, 2007; Rickli et al., 2008; Rickli and Graf, 2009]. The first inventory (Napf 2002) will be used to systematically explore the influence of various model parameters, whereas the second inventory (Napf 2005) will be used for model testing in a predictive mode.

3.1. Landslide Inventories Napf 2002 and 2005

[15] The landslide inventories were obtained from catchments located at the northern foothills of the Swiss Alps (inset in Figure 1b). The event-based inventories “Napf 2002” and “Napf 2005” are named after the geographical region “Napf” and the corresponding year of the triggering event. The rainfall event of 15–16 July 2002 was a local summer storm with high rainfall intensities (15–25 mm/h) and a total rainfall amount of 53 mm that triggered 51 documented shallow landslides (see Table 1). The 2005 event was a relatively long rainfall event (lasting from 18 to 23 August) with a rainfall amount of 229 mm triggering 36 shallow landslides in the study area and over 5000 landslides across the northern part of the Swiss Alps [Raetzo and Rickli, 2007]. In both inventories, it was observed that the majority of shallow landslides were translational with failure planes at the soil-bedrock interface. The catchments of the two inventories are only a few kilometers apart (inset in Figure 1b), with similar topographic features, land use, and geological formations based on findings of a landslide susceptibility study from von Ruette et al. [2011]. The landslide density (number of landslides per area) and median of landslide volume were higher for Napf 2005. Additionally, the percentage of landslides triggered in forested areas is higher in the Napf 2002 inventory (more details about the differences are presented in Rickli and Graf [2009]).

Figure 1.

Maps of the landslide inventories (a) Napf 2002 and (b) 2005 with mapped landslides presented with yellow points. The inset in Figure 1b shows the location of both landslide inventories within Switzerland and reveals the spatial proximity of the two inventories.

Table 1. Characteristics of the Landslide Inventoriesa
InventoryCatchment Area (km2)Number of SlidesNumber in ForestDensity (km−2)Median Volume (m3)Rainfall Amount (mm)Rainfall Duration (d)
  1. a

    For more details, see Rickli and Graf [2009].

Napf 20022.4513021.362531/8
Napf 20051.4361425.71162294

3.2. CHLT Input Model Parameters

[16] The catchment-scale hydromechanical landslide triggering model requires information on (i) surface elevation for computing surface runoff, (ii) soil depth to determine average water content and soil column mass, (iii) hydrological properties of the soil columns to compute water flow, (iv) vegetation patterns for assessment of lateral root reinforcement, and (v) initial water content.

3.2.1. Topography and Rainfall Data

[17] Original topographic data were taken from a 2.5 × 2.5 m DEM based on Lidar data from Swisstopo [2005]. Vegetation type consists of two classes: grassland and forest. These were determined by subtracting a DEM of terrain without vegetation from a DEM including vegetation. The definitive vegetation map was verified with aerial images (1 × 1 m resolution). These initial data sets were subsequently interpolated to a hexagonal grid with a distance LG of 2.5 m between centers of hexagonal soil columns, resulting in 455,534 soil columns to represent Napf 2002 and 266,563 for Napf 2005. Preprocessing of the hexagonal DEM included the filling of topographic depressions based on the algorithm of Planchon and Darboux [2002]. Rainfall data sets CH02H (SwissMeteo©) were used for the simulations, based on hourly radar rainfall intensities with a spatial resolution of 2 km and daily rain gauge records [Wüest et al., 2010]. The rainfall values on the original grid were interpolated for the hexagonal lattice using an inverse distance-weighting method.

3.2.2. Soil Depth Model

[18] Spatially distributed soil depth information at catchment scale is needed for hydrological modeling and for application of physically based landslide models. Presently, there is no acceptable method capable of estimating soil depth distribution at the required resolution and at a catchment scale [Pelletier and Rasmussen, 2009]. We thus resort to modeling guided by anecdotal observations (such as gleaned from landslide inventory data). Dietrich et al. [1995] and Roering [2008] developed geomorphologically based models for calculating soil depth distribution at a catchment scale. Dietrich et al. [1995] used a colluvium-diffusion equation where the sediment flux is described as a slope-dependent diffusion flux. Soil production can be modeled as an exponentially decreasing function with increasing soil depth or as a “humped” (bell-shaped) function with maximum soil production below the surface [Pelletier and Rasmussen, 2009]. In this study, we use a colluvium-diffusion equation, but we simplify the problem by assuming that the landscape topography is at steady state with soil depth defined by a balance of soil production and transport processes (see Appendix Soil Depth Model for details). A series of simulations were carried out by varying the parameters of the soil depth model (soil production rate, diffusion coefficient of soil flux, and a weight for erosion through surface runoff). We chose the result with shape of soil depth distribution function similar to values measured in the landslide inventory (see Figure 2). Because the simulated depths were consistently deeper than measured values, a constant value was subtracted from modeled soil depth to obtain a reasonable representation of average and variance of inventory data (see Appendix Soil Depth Model).

Figure 2.

Maps of simulated soil depth for the landslide inventories (a) Napf 2002 and (b) 2005 with the landslide locations shown with black points.

3.2.3. Soil Hydrological Properties

[19] Five soil samples (black crosses in Figure 3a) were taken in the field during the inventory and were classified based on the particle density distribution as loam and sandy loam (according to the U.S. Department of Agriculture's classification system). Since no more detailed information on soil type was available, we explored five different soil types covering the textural range of observed soil samples. These soil types are summarized in Table 2. The hydraulic properties presented in Figure 3 were parameterized according to the models of Mualem [1976] and Van Genuchten [1980], with values (Table 2) deduced from the ROSETTA pedotransfer function model [Schaap et al., 2001].

Figure 3.

Hydromechanical properties of soil materials of the two catchments. (a) Soil texture diagram with field samples shown with black crosses and four soil types in different colors marking the envelope of expected soil types. (b) The soil water characteristic curves are given with a range of initial matric suctions and water contents used in the simulations (see section 4) for meadow (squares) and forest (triangles). (c) Unsaturated hydraulic conductivity with parameterization according to Mualem [1976] and Van Genuchten [1980]. (d) Suction stress (strength) of soil material as deduced from Lu et al. [2010].

Table 2. Overview of Soil Types and Properties Used in This Studya
Soil Typeθrϕnα (m−1)KS (m/h)
  1. a

    Based on pedotransfer functions implemented in ROSETTA [Schaap et al., 2001] with θr as residual water content, ϕ porosity, KS saturated hydraulic conductivity, n the shape parameter of the soil water characteristic curve and α the inverse of a characteristic matric suction according to Van Genuchten [1980].

Sandy clay loam0.0750.411.341.9330.00289
Clay loam0.0810.441.490.8620.00511
Silt loam0.0410.421.660.5160.02459
Sandy loam0.0350.391.422.200.01866
Loam0.0780.431.563.600.01040

3.3. CHLT Model Assumptions

[20] Before presenting the results of CHLT model, we summarize the simplifying assumptions implemented in the CHLT model and in this study (note that some of these assumptions were not yet introduced but will be stated in the next section):

[21] 1. The soil type and vegetation maps are spatially uniform and do not include spatial correlation or spatial heterogeneity.

[22] 2. The friction angle was set constant to 30° as the average value listed in Lu et al. [2010]; the friction angle would be an additional property of high sensitivity for landslide triggering.

[23] 3. Root reinforcement was considered only for areas overlain by a forest without detailed consideration of root type, geometry, or depth distribution [Schwarz et al., 2011; Cohen et al., 2011].

[24] 4. When shear stress at the soil-bedrock interface exceeds it strength, the contribution of matric suction to effective stress is reduced by a third of the strength at failure.

[25] 5. A soil column fails (fluidized) when compressive stress exceeds compressive strength; afterward, the column becomes mechanically inactive (may not carry a load) but remains hydrologically intact.

[26] 6. The soil columns do not include any layering, and the failure plane is predefined at the soil-bedrock interface.

[27] 7. Preferential flow path and its effect on decreasing soil strength at the failure plane were not included.

[28] 8. At the beginning of the triggering rainfall event, we assume that the system is mechanically pristine, no prior damage occurred by previous rainfalls.

[29] 9. Based on various field observations with smaller water contents in forests, the initial soil moisture content for forested areas was defined as two-thirds of the water content in meadows.

[30] 10. The computations were carried out with grid resolution of 2.5 m, and the effect of different grid resolutions was not systematically tested (but first results reproduced the same trend).

[31] 11. Routing of surface runoff and subsurface flows was described until water reached the river network, which was taken as the boundary condition.

[32] 12. Discussion of simulation results and landslide statistics were done only for landslide volumes larger than 30 m3, which was used as a mapping criterion for the landslide inventories Napf 2002 and 2005.

[33] Accordingly, the results and conclusions presented below are restricted by these assumptions, and effects excluded in this study will be analyzed in forthcoming studies (e.g., spatial heterogeneity and variable failure plane).

4. Modeling Results

[34] To compute the hydromechanical loading and failure patterns using the catchment-scale hydromechanical landslide triggering model (CHLT), not only the data discussed in section 3 were required, but additional information concerning initial water content, lateral root strength, and cohesive forces was needed. For lateral root strength, we used typical values of 0, 2.5, and 5 kPa, as reported in back analyses of landslides [Sidle and Ochiai, 2006; Schwarz et al., 2011]. Most of the landslides in both inventories were translational shallow landslides (characteristic of cohesion-less soils) requiring a narrow range of cohesion values between 0 and 5 kPa. The soil-mechanical parameter internal friction angle γ was held constant (based on a representative value reported by Lu et al. [2010]). The initial volumetric water content θ0 was defined by choosing a narrow range of initial absolute matric suction values between |h| = 0.33 m and |h| = 0.8 m (see equation (B1) which relates matric suction with volumetric water content) according to the following constraints: For drier suction values (|h| > 0.8 m), we found that no landslides were triggered by the rainfall event, whereas the lower (wetter) bound was defined by running additional simulations of the period preceding the triggering rainfall event without simulating landslide occurrence. To account for drier conditions in forested areas observed in (case-independent) field studies, we lowered the water content in forest relative to meadows (forested soil water content was two thirds of meadow).

[35] We varied systematically the values of these primary parameters to explore their sensitivity on landslide triggering; additionally, we attempted to identify a parameter set most representative of landslide inventory data of Napf 2002. We tested 150 parameter combinations, out of which 132 resulted in simulated landslides (18 combinations with no landslides). The computation time for each parameter set was between 3 and 10 h, depending on the number of load redistributions and chain reactions occurring during the rainfall event (note that computation time for Napf 2005 was about 1.5 days due to the long duration of the rainfall event). In Figure 4, we show an illustrative example of one simulation for the Napf 2002 catchment for a selected simulation with soil type loam, cohesion-less soil, no lateral root strength, and initial matric suction of |h| = 0.33 m. The model not only generates localized patterns of mass release but also provides information of time of failure. Spatial and temporal properties of the simulated landslides are compared with landslide inventory in the next subsections.

Figure 4.

Landslide map of the study area Napf 2002 for the simulation with soil type loam, cohesion-less soil, no lateral root reinforcement, and an initial matric suction of |h| = 0.33 m. Colored polygons are the simulated landslides, where the color indicates time of failure (from dark green for the first two hours and up to red for the last two hours of the rainfall event). Black points indicate landslides of the inventory, areas in gray represent forested areas, and the black rectangle located in the northeast marks the location of the inset revealing more details of the spatiotemporal landslide pattern. The ellipse in the inset shows the measured width and length of the inventory landslide reported at this location.

4.1. Landslide Metrics

[36] The 132 scenarios that resulted in simulated landslides were quantified with respect to (i) total landslide volume, (ii) total number of landslides, (iii) median of slope of landslides, (iv) median of landslide elevation, and (v) percentage of landslides located in the forest. To simplify the presentation of the results (Figures 5-7) and the effects of changing attributes on landslide properties, we keep in the following one parameter of the mechanical bonds constant (cohesion is zero) and change systematically initial water content (as determined by matric suction), soil type, and the other mechanical bond strength parameter (lateral root reinforcement).

Figure 5.

Landslide metrics of simulated landslides compared to inventory data, which are represented with black horizontal lines (the true initial matric suction of the 2002 event is unknown). (a) Total landslide volume, (b) total number of landslides, (c) median of landslide angle, (d) median of landslide elevation, and (e) percentage of landslides located in forest areas. The different colors of the symbols indicate the soil types and their shape the value of lateral root reinforcement. For details, see section 4.1.

4.1.1. Effect of Initial Matric Suction and Corresponding Water Content

[37] In general, the size and number of simulated landslides decrease with decreasing initial soil moisture and with increasing absolute values of matric suction (Figures 5a and 5b). This could be expected due to higher shear strengths under dry (unsaturated) conditions, reducing the likelihood of landslide triggering (the different trend found for silt loam is discussed below). The positive correlation between initial matric suction and landslide slope angle (Figure 5c) indicates that for drier soils (higher absolute values of matric suction) only at steep slopes, the resulting driving forces may exceed stabilizing shear strength and landslides occur preferentially in steep slopes. A similar argument holds for the trend of increasing landslide elevation with increasing matric suction for soil type loam (Figure 5d) because greater slope angles are associated with higher elevations. For most soil types (except loam), the fraction of landslides occurring in forested areas (Figure 5e) decreases with increasing absolute matric suction because the drier forested area compared to meadow becomes too strong to be released.

4.1.2. Root Reinforcement in Forested Areas

[38] For zero lateral root strength, the volume and numbers of landslides were larger than for reinforced soil (Figures 5a and 5b). As shown in Figure 5e, the fraction of landslides in the forest increased with decreasing lateral root reinforcement. Because forests were preferentially located in steep regions of the catchment, less landslides were triggered in steep hillslopes with increasing reinforcement, therefore slope angles of landslides shown in Figure 5c slightly decrease with increasing reinforcement.

4.1.3. Soil Type

[39] The largest volumes and highest number of landslides were triggered for silt loam soil and were rather insensitive to the initial conditions. This is attributed primarily to the soil hydrological properties as shown in Figure 3. For silt loam, the initial soil water contents (in meadow) for all analyzed matric suctions were already close to saturation, resulting in a marginal difference for the initial conditions. Additionally, silt loam has the highest hydraulic conductivity (Table 2) and hence high water infiltration capacity that results in a rapid increase in water content (and even formation of perched water tables at the soil-bedrock interface). In comparison with clay loam, which exhibited similar soil water characteristics, its lower hydraulic conductivity prevented similar infiltration rates. Accordingly, more landslides were triggered for silt loam than for clay loam. The most stable soil type was sandy clay due to its small infiltration capacity and relatively low initial water content (compared to clay loam and silt loam), preventing the building of a free water table. Another remarkable feature of the effect of soil properties on landslide metrics was the high fraction of landslides in forest for loamy soils. Loam has the lowest matric suction and hence the lowest contribution to the effective stress, which leads to trigger landslide preferentially in the steep forested slopes areas. In simple terms, for loam also, the mechanical bonds in the forest were rather weak, and landslide occurrence was controlled by slope angle (which increases with elevations).

4.2. Landslide Temporal Dynamics

[40] In this section, we address the temporal evolution of landslide failures and discharge to the catchment's stream network as affected by rainfall intensities and soil hydraulic properties (infiltration capacity). Figure 6 shows the effect of soil type on the response of water discharge and evolution of landslide triggering. In all cases, the discharge was primarily in the form of surface runoff, with only marginal contribution from matrix flow. Accordingly, the resulting discharge is a strong function of soil infiltration capacity (affected by soil hydraulic conductivity and initial water content). The discharge was high for sandy clay loam and clay loam (low hydraulic conductivity), whereas for silt loam and sandy loam (high hydraulic conductivity) the simulated discharge was low. Landslide volumes were smallest in sandy clay loam due to low infiltration rates and relatively high suction strength of this soil material. Although the clay loam soil had similarly low infiltration capacity, the initially high water content (for the same initial matric suction) resulted in more frequent occurrences of perched water tables, perched water tables form more frequently resulting in dramatic reduction in soil strength (decrease of effective stress) and consequently higher landslide volumes. The infiltration capacity of sandy loam and silt loam was relatively high, yielding low and delayed discharge into the catchment's stream network. For a catchment with silt loam, we simulated a larger number of triggered landslides primarily due to more frequent formation of high water tables, thus reducing effective stress at the potential failure planes. The effect of soil type on landslide volume is also evident in landslide maps presented in Figure 7. Due to the relatively high initial water content in the silt loam and clay loam (for a prescribed initial matric suction), large landslides were triggered relatively early during the rainfall event, with most landslides triggered between 4 and 6 h. For the sandy loam, more landslides were triggered abruptly toward the end of the rainfall event (6–8 h) because high water contents and possibility of perched water table were obtained toward the end of the rainfall event.

Figure 6.

Relationship between rainfall intensities (black lines), inflow to stream network (blue lines), and failure dynamics (red points) for (a) sandy clay loam, (b) clay loam, (c) silt loam, and (d) sandy loam soils. In all four cases, the simulation parameters were as follows: (i) initial matric suction of |h| = 0.33 m, (ii) no lateral root reinforcement, and (iii) no cohesion. Main differences between the four cases are the hydraulic soil properties (see Figure 3) where the infiltration capacity is low (large runoff and moderate weakening of soil strength) for Figures 6a and 6b and high (large infiltration with effective weakening) for Figures 6c and 6d. The red points indicate the number of soil columns failing in a certain time step. Due to wet initial state of silt loam and clay loam, perched water tables are formed and weaken the soil strength (more failing soil columns).

Figure 7.

Landslide maps for simulations with no lateral root reinforcement, cohesion-less soil with initial matric suction of |h| = 0.33 m, and for four different soil types, Figures 7a–d. Forested regions in the study area Napf 2002 are shown in gray, black points are the location of landslides from the inventory, and the colored polygons are the simulated landslides. The four different colors indicate at which stage of the rainfall event a soil column fails, beginning with dark green for the first two hours and ending in red for the last two hours of the rainfall event. The ellipse in the inset shows the measured width and length of the inventory landslide reported at this location.

4.3. Simulated Landslide Spatial Distribution Over a Catchment

[41] It is impractical to expect detailed knowledge of the spatial distribution of key parameters (water content, friction angle, soil type and depth, lateral root strength and cohesion), nor knowledge of the mechanical history required for reproducing landslides at the exact positions of observed ones. Instead, the research was aimed at enhancing our understanding of the conditions contributing to the dynamics, volumes, and number of landslides with characteristics and metrics that are as similar as possible (in a statistical sense) to inventory results for a given catchment. To compare the spatial patterns of simulated landslides with the inventory, the Napf 2002 study area was divided into subcatchments using local drainage divides as boundaries (Figure 8). This allows comparison of simulated and observed landslides occurring within a subcatchment as functions of soil type, initial water content, and strength of mechanical bonds.

Figure 8.

Map of 18 subcatchments of the landslide inventory Napf 2002 defined along local watersheds to quantify the spatial distribution of modeled landslides (black points for inventory landslides).

[42] Figure 9 shows the spatial distribution of simulated landslides for the five soil types, initial matric suction of |h| = 0.33 m, and considering 0 to 5 kPa lateral root reinforcement values. Sandy clay loam (Figure 9a) with 0 and 5 kPa lateral root reinforcement and loam (Figure 9e) with 5 kPa lateral root reinforcement exhibited the closest similarity in landslide distribution relative to inventory data. For example, subcatchment 3 has the highest number of landslides in the inventory, and a similar landslide number was reproduced by lateral root reinforced loam soil. On the other hand, in subcatchment 14 (one of the largest subcatchments) only five landslides were triggered in 2002, as reproduced by the simulations with sandy clay loam.

Figure 9.

Landslide densities in the different 18 subcatchments (values on the x axis define subcatchment shown in Figure 8) for the five different soil types with lateral root reinforcement of 5 kPa (green) and without lateral root reinforcement (blue). The black line represents landslide distribution of the inventory Napf 2002. Note that the lines are used to guide the eye without any (physical) meaning.

4.4. Evaluating Model Performance

[43] To determine the model parameter values that best represent (in a statistical sense) the landslide inventory of Napf 2002, we analyzed similarities between simulations and landslide inventory using several landslide metrics as presented in Figure 5. These metrics included total landslide volume, the number of landslides, median of landslide slope angle, median of landslide elevation, and the fraction of landslides in forested areas.

[44] Another potential landslide metric would have been the exponent of the power law for the volume/frequency distribution of landslides [e.g., Stark and Hovius, 2001; Lehmann and Or, 2012]. However, fitting a power law to small data sets can heavily influence the resulting exponent. Hence, we decided to omit the power law exponent as a landslide metric. However, we determined the exponent for the landslide inventory of Napf 2002 (|ζ| = 1.9) and of simulated landslides, which resulted in an average power law exponent of |ζ| = 1.9 ± 0.4, which is in agreement with literature data [Brunetti et al., 2009; Guzzetti et al., 2009; Lehmann and Or, 2012] (further details and discussion are presented in Appendix Power Law Distributions of Landslide Volumes).

[45] In an attempt to reduce the comparison into a single measure, we defined normalized quadratic sum ɛ1 (equation (10)) for the five landslide metrics and quadratic sum ɛ2 (equation (11)) for the spatial distribution across the 18 subcatchments. The difference between simulation pS,j and inventory pI,j for each landslide metric j is normalized by the maximal present value (either simulations or inventory) pmax,j since it involves different quantities and dimensions:

display math(10)
display math(11)

[46] Figure 10 presents the total errors (ɛ1 and ɛ2) resulting from the various simulations: as anticipated, soil type plays a dominant role in model results, with soil loam and sandy loam type providing the best match considering the chosen landslide metrics and spatial distributions of landslide occurrence in the various subcatchments. The silt loam, which resulted in the highest amount of simulated landslides and volumes (Figures 5a and 5b), also shows the highest error values.

Figure 10.

Error in model predictions quantified by landslide metrics (y axis) and spatial distribution in subcatchments (x axis). The color of the symbol represents the soil type. The larger symbols are the simulations with the lowest errors also summarized in Table 3.

[47] The resulting parameter values for the 10 “best” simulations (represented with larger points in Figure 10) with respect to landslide metrics and spatial distribution into subcatchments are listed in Table 3. We have selected these 10 parameter sets to test how the criteria of landslide metrics and spatial distributions of landslides worked when used to simulate landslide occurrence in study area Napf 2005, as presented next.

Table 3. Parameters of Simulations With Best Performance Compared to the Landslide Inventory Napf 2002a
No.Soil TypeInitial Matric Suction (m)Root Strength (kPa)Cohesion (kPa)Error 1 Equation (10)Error 2 Equation (11)
  1. a

    The 10 simulations in bold have the lowest error related to landslide metrics (top half) and landslide distribution in the subcatchments (bottom half), respectively.

1Loam0.46500.040202
2Loam0.39500.063185
3Loam0.52500.078185
4Loam0.33500.123242
5Loam0.462.500.167294
6Sandy loam0.392.500.07788
7Sandy loam0.5202.50.100101
8Sandy loam0.46500.080102
9Sandy loam0.46050.062112
10Clay loam0.46500.078118

5. Testing Model's Predictive Capabilities Using Napf 2005

[48] For the simulation of Napf 2005, we used parameter values for soil type, lateral root strength, and cohesion according to the model evaluation presented in section 4.4 and summarized in Table 3. However, the initial water content (expressed with matric suction) used for simulating the Napf 2002 inventory would be too high for the 2005 event due to the higher rainfall amount recorded in the period preceding the 2002 event (105 mm in the 2 weeks before triggering in 2002, including an event with high intensity; 71 mm rainfall in 2 weeks before triggering in 2005 with no rainfall the day before the onset of triggering rainfall). Hence, we selected a higher matric suction value of |h| = 1.75 m (drier initial conditions) to define initial soil moisture conditions for the Napf 2005 simulations (based on equation (B2)).

[49] Figure 11 depicts the relationship between rainfall intensities, discharge in stream network, and landslide failure dynamics for the parameters with the lowest errors based on landslide metrics and subcatchment distribution (parameter sets 1 and 6 from Table 3). The resulting discharge was limited to only a few hours of the total 100 hours of rainfall, and most of the precipitation infiltrates into the soil, increasing total stress at the failure plane. Despite the larger rainfall amount in 2005 of 229 mm, only 36 mm ended up as discharge into the stream network for the soil type loam (in 2002, with 53 mm rainfall, the discharge was 21 mm for the same soil type). Figure 11 reveals that landslides were triggered during the second half of the event, a finding that was confirmed by observations during the event in 2005 [Rickli et al., 2008]. The second simulation using the parameter values deduced from the analysis of landslide occurrence in the 18 subcatchments (using sandy loam instead of loam) predicted a discharge of 9 mm (10 mm for sandy loam in simulation for 2002). In comparison to loam, the infiltration capacity is higher for sandy loam (see hydraulic properties in Figure 3c), and thus less surface runoff was generated. The simulated landslides are depicted with inventory data for loam and sandy loam in Figure 12. The results in the figure support the observation that landslides were triggered during the second period of rainfall event, and that more landslides and larger volumes were released for simulations with loam. Despite the smaller amount of infiltrating water in loam (larger runoff in Figure 11a), the suction strength of the sandy loam is higher for the given initial matric suction, and less landslides are triggered for sandy loam. The main characteristics of simulated and measured landslides are listed in Table 4 for the 10 best simulations listed in Table 3. Note that for simulations with clay loam, no landslides were triggered for this initial state because no perched water table occurred, and unsaturated strength in clay loam is higher.

Figure 11.

Relationship between rainfall intensities (black line), discharge (blue line), and failure dynamics (red points) for the landslide inventory Napf 2005. The two parameter sets providing the smallest error for Napf 2002 event were chosen for the simulations (best model with respect to landslide metrics for loam in Figure 11a, for subcatchment distribution with sandy loam in Figure 11 b).

Figure 12.

Landslide maps for Napf 2005 computed with the two parameter sets providing best performance for Napf 2002: (a) based on landslide metrics with soil type loam and (b) based on subcatchment distribution with soil type sandy loam. Locations of inventory data are shown as black points with ellipses (in the insets) representing length and width of the landslides. Simulated landslide in colored polygons indicates the time interval of mass release.

Table 4. Overview of the Simulations of Napf 2005 Based on the Parameters From Table 3 Summarized With the Same Landslide Metrics Used in Figure 5 as Well as for the Landslides From the Inventorya
No.SimulationTotal Volume (m3)No. of LandslidesMedian Slope (°)Median Elevation (m)Landslides in Forest (%)Maximum Volume (m3)Error 1 Equation (10)
  1. a

    Note that simulations 1 to 4 from Table 3 have the same simulation parameters except for different initial soil moisture conditions, which for Napf 2005 are the same, resulting in a single simulation.

1Loam39,0008646.9982.858 (67)28952.3
5Loam32,1167150.1990.045 (63)30861.9
6Sandy loam21,2185245.0988.143 (82)13661.3
7Sandy loam14,2784056.01000.034 (85)11162.2
8Sandy loam42,1897348.1987.960 (82)23962.4
9Sandy loam6,3692653.8998.024 (92)9232.1
10Clay loam000.00.000
 Inventory11,9923629.0945.014 (38)5443

[50] For the best parameter sets related to landslide metrics and subcatchment distribution (numbers 1 and 6 in Table 4), the simulated landslide numbers were 86 for loam and 52 for sandy loam values that are comparable with the observed 36 inventory landslides. Note that the landslide volume in the inventory data was dominated by a single large landslide of more than 3000 m3. The cumulated volume of the other 35 inventory landslides was 2443 m3, and a similar value resulted for simulations with loam (2895 m3). The total landslide volume predicted with sandy loam was lower (1366 m3), but both simulations provided a reasonable estimate of expected volume and number of landslides.

[51] For both simulations, we quantified the landslides in more detail and present the findings as cumulative distributions in Figure 13. While the distribution of landslide volume was reasonable (more for sandy loam with smaller landslide volumes), the simulated landslides tend to occur in steeper slopes than reported in the inventory. Accordingly, also the simulated landslide elevation distribution deviates from the inventory because the steep slopes (where landslides were preferentially triggered in the model) are at higher elevations. Consulting Table 4 with information on landslides in the forest, we note that the model overestimated the fraction of landslides triggered in the forest (82% for sandy loam and 67% for loam) that was only 38% in the inventory. Possible factors contributing to this difference are underestimation of lateral root reinforcement by the model (for Napf 2005), overestimation of initial water content in the forested regions, or an overestimation of the modeled soil depth distribution for the study area, as discussed in section 3.2.2.

Figure 13.

Cumulative distribution of landslide (a) volume, (b) slope, and (c) elevation compared to landslide inventory Napf 2005 (black line) for the two parameter sets with best model performance for Napf 2002 (soil type loam in blue and sandy loam in green).

[52] Another possibility (that could not be simulated with present model) would be a dominant influence of the bedrock with water flowing out of the bedrock into the soil mantle in downslope regions (that are less steep) that would reduce drastically the strength of the soil material and would explain the observed pattern of frequent landslides in less steep regions.

6. Summary and Conclusions

[53] We developed a catchment-scale hydromechanical landslide triggering model (CHLT) to simulate rainfall-induced spatial and temporal hillslope loading patterns. The model considers abrupt failure of mechanical bonds between adjacent soil columns and between soil and bedrock. The inherent threshold-mechanics combined with interactions among many mechanical elements introduce localization, whereby a failure of a single column under certain conditions may initiate a chain reaction culminating in a landslide release of a wide range of sizes. The model requires spatial information on surface elevation (here deduced from lidar-based DEM), surface cover (meadow or forest to account for lateral root reinforcement), soil type (defining hydromechanical properties), soil depth (here modeled as a function of surface elevation), and rainfall data. The model was applied for two event-based landslide inventories (Napf 2002 and Napf 2005) mapped in the very same region of the Prealps in Switzerland. We analyzed the effects of initial water content, soil type, and soil strength (cohesion and lateral root reinforcement) on landslide properties in comparison to field data. The following are the main findings of these case studies:

[54] 1. With decreasing initial soil moisture, the soil strength increases and consequently the amount and volume of triggered landslides decreases; additionally, landslides occur preferentially in steeper slopes with highest downslope driving forces.

[55] 2. The hydromechanical properties of the different soil types play a major role in the spatiotemporal patterns of failure dynamics, with the volume of landslides increasing with infiltration capacity (more effective soil weakening) and with increasing soil water content for decreasing matric suction (absolute values), resulting in the formation of water tables above the bedrock.

[56] 3. For silt loam and loamy soils, larger landslides were observed due to high infiltration rate, weakening of capillary forces (loam), and higher occurrence of perched water tables (silt loam), whereas sandy clay and clay loam soils were more stable due to high surface runoff limiting the loading of the slopes.

[57] 4. Increasing mechanical reinforcement due to basal soil cohesion or lateral root strength resulted in a decrease in number and volumes of landslides.

[58] Because an independent and experimental evaluation of such model is presently not possible at the required scale and detail, we opted for partial evaluation based on certain landslide metrics and spatial distributions of landslides in 18 subcatchments. The model was calibrated using information and optimizing performance for the first inventory (Napf 2002) and was subsequently evaluated for the second inventory (Napf 2005). The number and volumes of predicted landslides were in fair agreement with the inventory data. However, the model overpredicted the number of landslides in the forest compared to meadow, resulting in more simulated landslides in steeper slopes at higher elevations compared to inventory. These findings can be explained by underestimating the lateral root strength, by overestimating soil depth and/or the initial water content in the forest, or by the effect of subsurface with water intrusion into the soil mantle.

[59] The case studies reveal high sensitivity of model outcomes to values and distribution of soil depth, soil type, initial water content, and friction angle (which was not analyzed in this study). Such information is often not available as measurements at catchment scale with the necessary spatial resolution; hence, we find it unrealistic to expect predictions of exact landslide locations, and instead we focused on predicting time windows and identify slopes prone to landslide initiation. This new catchment-scale hydromechanical landslide triggering model was capable of describing reasonably well the timing of the most severe landslides in Napf 2005. More important, the model reproduced several general characteristics concerning landslide initiation:

[60] 1. Landslides are localized phenomena (similar to a point pattern) in the sense that only small volumes within a large catchment are released, despite the fact that similar conditions can be found in larger regions of the catchment.

[61] 2. A landslide can be triggered abruptly, with large volumes released at an instance.

[62] 3. There is no characteristic size of a landslide for a catchment and rainfall event but a wide size-frequency range that can be represented by a power law.

[63] Despite the model's capacity to reproduce key features of rainfall-induced shallow landslides, numerous shortcomings remain in the present model with a partial list given in section 3.3 and will require future modifications of the CHLT model. However, the presented approach is well suited to be adapted to more realistic values and patterns of soil properties and rainfall. Additionally, the local rules implemented in this model to describe load redistribution and failure can be extended to more appropriate mechanical rules and could even be represented by fiber bundle models to compute precursor statistics that may indicate the risk of forthcoming mass release.

Appendix A:: Soil Depth Model

[64] The soil depth model and the code are based on the report from Stothoff [2008] and personal communications. It assumes steady state conditions between soil production rate and erosive processes. Soil depth is solved through a routing algorithm starting with the soil column with highest elevation zi (for simplicity, soil surface elevation zsu is denoted just as z in Appendix A) and progressively going downslope. For each column i, the volume balance (expressed volume per cross-sectional area) is computed by varying soil depth bi (we denote iteratively changing soil depth as bi to separate from final soil depth Hsd), with the final soil depth chosen for the smallest volume balance (“error”) ɛi:

display math(A1)

where the first term is the local soil production for column i across its cross-sectional area Ah (Q0 is the bedrock weathering rate, and b0 is a weathering-protection thickness); the second term is the outgoing erosion for column i in flow direction j, with a diffusion coefficient k across the column border length LH (in accordance with film-flow theory [Stothoff, 2008]); the third term describes the inflow of erosion from upslope columns with soil depth bj and surface elevation zj; and the fourth term is a proxy for erosion through surface runoff, where CAi is the value of contributing area of the target column and CAi/CAmax is the normalized contributing area being a proxy for surface runoff, with CAmax the largest contributing area value in the study area. The contributing area CAi is determined from the highest column down to the lowest one, using a multiflow direction algorithm. The contribution of one column CAi to its downslope neighbor CAj depends on the elevation difference, which is normalized to maintain mass balance:

display math(A2)

[65] Since no detailed knowledge of soil depth distribution for the study areas Napf 2002 and 2005 was available, we used the reported landslide thicknesses from the inventory as a measure to find the best fit of soil depth distributions. Figure A1 shows the cumulative distribution functions for the landslide thicknesses in the two study areas with the chosen soil depth distributions.

Figure A1.

Cumulative distribution of soil depth maps based on the soil depth model of the two study areas Napf 2002 (black line) and 2005 (gray line) together with the corresponding thicknesses of the landslides (points) from the inventories.

Appendix B:: Hydrological Model

B1. Soil Parameterization

[66] Estimation of hydraulic soil parameters was based on the pedotransfer functions provided by the software ROSETTA [Schaap et al., 2001] model, and hydraulic properties were described with Van Genuchten's [1980] parameterization:

display math(B1)

with volumetric water content θ, effective saturation Θ, saturated water content θS here chosen as soil porosity, residual volumetric water content θr, matric suction h, and the Van Genuchten parameters α[m−1] as the inverse of characteristic matric suction and shape parameter n defining pore size distribution. Initial soil water content θ0 for all simulations was determined using the initial matric suction h0, thereby ensuring identical water potential for all selected soil types:

display math(B2)

[67] The unsaturated hydraulic conductivity K(Θ) (units m/h) is determined as

display math(B3)

with m = 1 − 1/n. The matric suction h is

display math(B4)

B2. Infiltration Capacity

[68] In each time step Δt, potential infiltration water is Ah·r(t)·Δt depending on the rainfall intensity r(t) with cross-sectional area of hexagon Ah. Infiltration rate f(t) as a function of time t is determined by the infiltration theory of Philip [1957] using time compression approximation and assuming that infiltration rate converges to half of saturated hydraulic conductivity:

display math(B5)

with sorptivity S (capacity to adsorb water based on capillarity) determined according to Parlange and Smith [1976] as:

display math(B6)

where θ0 is the initial water content and D(θ) = K(θ)/CW(θ) is soil water diffusivity with soil water capacity CW as a derivative of a water retention curve. The time tP relates to the infiltration rate equal to rainfall intensity r, and tC is the correction due to time compression approximation. To account for time variant rainfall rate, we followed concept of Assouline et al. [2007].

B3. Surface Runoff

[69] Surface runoff occurs when infiltration capacity reaches is limited and not all available ponding water is infiltrated Vpond (m3). Surface water routing is then described using Manning equation to determine runoff velocity v(i, j) for each column i and in any downslope direction j:

display math(B7)

where nM = 0.03 is the Gauckler-Manning coefficient, SM is the hydraulic gradient, and hydraulic radius Hpond =  Vpond/Ah is the thickness of ponding water of the target column i. The model allows surface runoff in all downslope directions, where the ponding water at the target column is distributed to all possible downslope according to their normalized flow velocities.

B4. Subsurface Flow

[70] Subsurface flow in the soil matrix is described by Buckingham-Darcy law using differences in gravitational height (elevation of centers zc) and matric suction h as driving forces. The flow Jh (m3/h) equals

display math(B8)

with area of intersection Ai of two adjacent soil columns. Saturated water flow Jsat at the soil-bedrock interface occurs when a perched water table is present at the soil-bedrock interface. A perched water table occurs when the soil water volume in a column defined by V(t) = Hsd·Ah·θ is higher than a critical water volume (Vcrit) defined next. The critical water volume is the soil water volume distributed according to soil water characteristics with zero head at the soil-bedrock interface (just before perched water starts to build up):

display math(B9)

[71] The height of the saturated water table Hsat is determined by solving iteratively the following equation by varying Hsat from 0 to Hsd:

display math(B10)

until the volume equals the required water volume of the soil column. Water flux Jsat to the adjacent column is described by Darcy's law as flow across the saturated part of the cross section of interfacial area:

display math(B11)

where KIS = 10Ksat, which is a rough estimate motivated by observations of Tromp-van Meerveld and McDonnell [2006].

Appendix C:: Parameterization of Soil Mechanical Properties

[72] The relationship between matric suction h and soil's suction stress can be described by equation (C1) based on equation (21b) in Lu et al. [2010]:

display math(C1)

which is validated using experimental data from Blight [1967], Gan et al. [1988], and Vanapalli et al. [1996], as presented in Figure C1a.

Figure C1.

Comparison of the soil mechanics model used in this study with results from Gan et al. [1988], Vanapalli et al. [1996], and Lu et al. [2010]. Relationship between matric suction and (a) suction stress and (b) shear stress for glacial till. Dashed lines in Figure C1b are based on parameters from Lu et al. [2010] otherwise extracted from the soil water retention curves published in the respective studies.

[73] In addition, the validation of determining shear strength based on equation (6) is presented in Figure C1b, revealing a good prediction of shear strength values for glacial till from Gan et al. [1988] and Vanapalli et al. [1996].

Appendix D:: Power Law Distributions of Landslide Volumes

[74] Distribution of measured and modeled landslide volumes VLS were determined as described in Lehmann and Or [2012] with a probability distribution [Malamud et al., 2004]:

display math(D1)

where NLS is the total number of landslides, and δNLS is the number of landslides within a range of landslide volume δVLS that increases with larger volumes (logarithmic binning). For landslides larger than a cutoff value Vmin, the landslide magnitude distribution can often be described by the following power law [Stark and Guzzetti, 2009]:

display math(D2)

with a power law exponent of |ζ| 1's. In Figure D1a, the probability density p for the inventory is shown than can be described by a power law of exponent 1.9 (the figure reveals as well that the accurate determination of an exponent is difficult for a rather small number of landslides). The frequency-magnitude relationship obtained for simulation with sandy loam (simulation no. 6 in Table 3 with the best performance related to landslide distribution in subcatchments) covers a similar range of values as the inventory and can be characterized by a power law exponent of 1.4. In Figure D1b, the exponent values for all simulations of Napf 2002 with a minimum of 10 landslides are shown using a cutoff Vmin between 30 and 95 m3. Most exponents of landslide simulations are between 1.0 and 2.2. Note that simulations with exponents close to inventory (1.9) are not those with best performance related to the landslide metrics defined in section 4.4. Accordingly, at least for frequency/magnitude statistics that are rather poorly defined due to the small number of landslides, a detailed analysis of power law exponent cannot be used to determine model performance but reveals the validity of the power law with similar exponents as inventory data.

Figure D1.

Landslide frequency and magnitude statistics for landslide inventory Napf 2002 (gray points) and a simulation with sandy loam type (blue triangles) fitted by power laws (lines) (a). Relationship between total landslide volume and exponent of the fitted power law for Napf 2002 inventory and all simulations with more than 10 landslides (b). Colors represent the different soil types and the larger points are the simulations with the highest performance as discussed in section 4.4 and summarized in Table 3.

Notation
AG

cross section along the slope, m2.

Ah

cross section of hexagonal cell, m2.

Ai

interfacial area between two cells, m2.

bi

soil depth of a cell, m.

b0

bedrock weathering-protection thickness, m.

CAi

contributing area.

CAmax

maximal contributing area of study area.

c

basal soil cohesion, kg/(m s2).

CW

soil water capacity, m−1.

D

soil water diffusivity, m2 s−1.

f(t)

time-dependent infiltration rate, m h−1.

FG

weight component along slope, kg m s−2.

FN

weight component normal to slope direction, kg m s−2.

FZ

weight (force) along gravity, kg m s−2.

g

gravitational acceleration, m s−2.

h

capillary head, m.

h0

initial capillary head, m.

Hsd

soil depth, m.

Hsat

height of free water table in soil column, m.

Hpond

hydraulic radius, m.

i

index of target cell.

j

index of neighbor of target cell.

Jh

water flow in the soil matrix, m3 h−1.

Jsat

water flow along soil-bedrock interface, m3 h−1.

k

diffusion coefficient, m−1.

K

hydraulic conductivity for water unsaturated conditions, m h−1.

KIS

hydraulic conductivity at soil-bedrock interface, m−1.

Ksat

hydraulic conductivity under water saturated conditions, m h−1.

LG

horizontal distance between centers of hexagonal cells, m.

LH

side length of hexagonal cell, m.

m

exponent in Van Genuchten model, m = 1 − 1/n.

M

mass of soil column, kg.

n

shape parameter of water retention curve for the Van Genuchten model.

NLS

total number of landslides.

nM

surface roughness coefficient.

pI

value of landslide metric of the landslide inventory.

pmax

maximal value of landslide metric.

pmin

minimal value of landslide metric.

Pn

height of ponding water of a soil column, m.

pS

value of landslide metric of a simulation.

p(VLS)

landslide volume probability density, m−3.

Q0

bedrock weathering rate, m.

r(t)

rainfall rate, m h−1.

S

water sorptivity, m h−1/2.

SM

hydraulic gradient, m/m.

t

time, h.

tC

time of ponding according to time compression approximation, h.

tP

time when infiltration rate equals rainfall rate, h.

v

flow velocity if surface runoff, m/h.

V

soil water volume, m3.

Vcrit

maximal water volume without free water, m3.

VLS

volume of a landslide, m3.

Vmin

minimum landslide volume for validity of power law distribution, m3.

Vpond

ponding water of a soil column, m3.

W

weight per area acting on soil-bedrock interface, kg/(m s2).

zbr

bedrock elevation, m.

zc

elevation of center of soil column, m.

zi

elevation of neighboring soil column, m.

zn

elevation of soil column, m.

zsu

surface elevation, m.

α

inverse of characteristic capillary head, m−1.

β

slope angle, °.

γ

friction angle, °.

Δt

computing time step, h.

δNLS

number of landslides in certain range of landslide size.

δVLS

range of landslide size, m3.

ɛ

volume balance, m3.

θ

volumetric water content.

θ0

initial volumetric water content.

θr

residual volumetric water content.

θS

saturated volumetric water content.

Θ

effective water saturation of soil column.

ρr

density of soil minerals, kg m−3.

ρw

density of water, kg m−3.

σN

normal stress related to weight of soil column, kg/(m s2).

σS

suction stress related to soil matric suction, kg/(m s2).

τC

compressive strength, kg/(m s2).

τr

tensile shear strength provided by tree roots, kg/(m s2).

τS

shear strength, kg/(m s2).

τsat

basal shear strength reduction in presence of free water, kg/(m s2).

τt

total tensile strength related to capillary and root effects, kg/(m s2).

ϕ

soil porosity.

ζ

exponent of power law of landslide volume-frequency relationship.

Acknowledgments

[75] This work is part of the projects Local and Regional Hydrologic and Geomorphic Factors Determining Landslide Patterns, funded by Swiss National Science Foundation (SNSF), and Triggering of Rapid Mass Movements in Steep Terrain (TRAMM), funded by the Competence Centre Environment and Sustainability (CCES) of the ETH Domain. We are grateful to Christian Rickli (Swiss Federal Institute for Forest, Snow and Landscape Research, WSL) for providing access and allowing to use his data set of the two event-based landslide inventories Napf 2002 and 2005. Special gratitude goes to Stuart Stothoff for providing the soil depth model. We greatly appreciated the insightful comments of three anonymous reviewers on the previous form of the manuscript.

Ancillary