Introduction

The definition of hazard associated to debris-flow (DF) events is usually given by the combination of frequency of occurrence, magnitude, and intensity of a certain event (Jakob and Hungr 2005; Fell et al. 2008). For the DF triggered by rainfall, the frequency of an event may be related to the return period of the triggering precipitation. The magnitude refers to the total volume of material mobilized during the event. The intensity of a DF is defined as its ability to cause damage and is generally estimated through the impact energy of the flow against an obstacle. The energy of the impact depends on the characteristics of flow’s depth and velocity (Rickenmann 2005). The connection between the rainfall and its effects can be reconstructed by the simulation of the different processes, which take place during the spatial and temporal evolution of the flow, from its mobilization to its stop. In this sense, we can distinguish two distinct phases of assessment: the first aims at estimating the volume potentially mobilized by a given precipitation, with an assigned return period (initiation); the second has the objective to estimate the area of invasion; and the resulting intensity (propagation) (Bregoli et al. 2011).

Usually, two types of assessments can be distinguished for the evaluation of hazard: studies at regional scale and studies at local scale. The hazard assessment at regional scale generally implies a geographic information system (GIS), in combination with statistical analyses (e.g., Carrara et al. 1991; Mark and Ellen 1995; Ohlmacher and Davis 2003) or physical-based approaches (e.g., Montgomery and Dietrich 1994; Crosta and Frattini 2003). Detailed studies require numerical models and comprehensive field work to determine the hazard in the DF deposition areas (e.g., Medina et al. 2008).

The starting point of the DF hazard assessment at regional scale is the identification of prone areas where DF can initiate and propagate. This task is known as susceptibility assessment (Guzzetti et al. 2005), which is part of the hazard assessment that is usually produced for the management of the territories.

The zones of initiation are often placed in steep slope and unreachable territories, complicating direct observations. The rapid recovery of vegetation can hide scars and tracks created by the occurrence of the flow. Furthermore, DF being a catastrophic natural hazard, the associated susceptibility assessment at regional scale increases in interest as a fundamental task to spatially locate hazardous areas.

Different models have been evaluated and used to assess landslide susceptibility (Guzzetti et al. 2006). The estimation of model’s quality is a rising issue due to the needs to compare the performance of the different techniques used in literature (Frattini et al. 2010).

Recently, many authors have proposed models focusing on DF: for instance Carrara et al. (2008) uses physical and statistical approaches in the Alps, concluding that the latter are more performing; Chevalier et al. (2013) uses fluvio-morphological parameters and data mining techniques to assess the DF susceptibility in the Central-Eastern Pyrenees.

The initiation of DF is triggered by means of various mechanisms (Coussot and Meunier 1996; Hungr et al. 2001), but the mobilization from rainfall-triggered landslides seems to be the most common process (Iverson et al. 1997; Crosta and Frattini 2008).

For the present work, two physical models assessing the susceptibility have been selected. Both models are able to define the frequency of events. The first model concerns an infinite slope stability analysis coupled with a lateral groundwater flow mechanism (Montgomery and Dietrich 1994) implemented using a novel technique valid for a long duration of rainfall events. The second model applies an infinite slope stability analysis coupled to a vertical flow mechanism. Both methodologies are valid for granular soil material. Concerning the initial condition, the first methodology works only for dry soil, while the second one is also valid for initial partially-saturated soil.

The transition from shallow landslide sources to DF, here used, is not compulsory, but can be related on the basis of field observations’ database. When in the study area, debris flows are mainly triggered by failures of soil layers, sufficiently large compared to the thickness, models of shallow landslides initiation can be applied (Carrara et al. 2008; Papa et al. 2012).

The present work takes into account merely the initiation of DF due to shallow landslides induced by rainfall. Other types of triggering as, for instance, earthquakes, sediment entrainment, volcanic eruptions, and soil burnt or soil liquefaction are not contemplated.

Physical-based models, applied at regional scale, can entail a substantial uncertainty due to the high sensibility to input (Borga et al. 2002), but are useful in case of lack of data concerning historical events.

Trusting on the benefits improved by physical models, the main goal of this work is to present and validate the performance of the latter in evaluating the susceptibility due to DF triggered by shallow landslides at regional scale. Moreover, the selected models’ results are compared to the results obtained by a statistical model. The propagation of DF and thus the evaluation of depositional areas are neglected herein.

A second objective of the present paper consists in relating the susceptibility to the frequency of the event, which is thought to be an important, novel task.

The methodologies selected are applied and validated in the Boí Valley (North West of Catalonia, Spain), where a reconnaissance of DF tracks has been done (Chevalier et al. 2013). The validation is achieved through a quality analysis of the models’ output, previously aggregated in terrain units. The terrain units consist in headwater sub-catchments, which are commonly taken as resolution of susceptibility assessment output (Fell et al. 2008). A sub-catchment can be considered reactive when at least one DF tracks has been found within it. Finally, a comparison with a statistical model based on logistic regression is performed.

Models applied

Debris flows can be triggered during intense rainfall when high pore pressure is produced inside a loose sediment layer, reducing the ratio between resisting stresses and acting stresses. When the latter exceeds the former, a landslide, likely to be propagated as a DF, can be initiated. In case of shallow landslides, having a very small thickness when compared to the length and width of the slide, the behavior is described by the common Mohr-Coulomb failure approach to the infinite slope stability (Bromhead 1992).

Two different models are here adapted for the study of the initiation of DF. The first one, the Steady State Infinite Slope Method (SSISM), couples a Mohr-Coulomb failure mechanism with a steady state lateral flow. It takes into account the cumulated area and the local slope (as TOPMODEL, Beven and Kirkby 1979). To determine the probability in terms of return period of an event, a statistical analysis on extreme values has been done for large durations of events. A large duration is considered because the mechanism of lateral flow evolves to reach the steady state condition in a relative long time. The second one, the Transient Vertical Infinite Slope Method (TVISM), couples a Mohr-Coulomb failure mechanism to a vertical flow model (as Green and Ampt 1911). It is also known as transient piston flow of wetting front model (Iverson 2000; Crosta and Frattini 2003). The return period of events is estimated trough the Generalized Extreme Value (GEV) analysis (Jenkinson 1955).

In the selected approaches, rainfall is spatially distributed and represented by a constant-in-time value of intensity (see Eq. (8)). The meteorological scenarios analysed are synthetic and related to selected return periods. They do not reproduce measured events. Temporal and spatial rainfall uniformities are assumed, as they are one of the most important simplifications of the model (Montgomery and Dietrich 1994; Pack et al. 1998). The rainfall intensity is intrinsically related to duration, and the duration strongly influences the landslides’ trigger (Iverson 2000).

Steady state infinite slope method

A traditional approach to the slope stability calculates the water pore pressure, in a simplified way (Dietrich et al. 1992; Borga et al. 1998). In this study, it has been assumed that water table’s steady state conditions are reached after a rainfall of constant intensity and indefinite duration. For a given topographic element of the domain having b as width and a as upslope cumulative drainage area, the conservation of mass and the Darcy’s law in steady state condition at the given element can be expressed as:

$$ a\left( p\kern0.5em -\kern0.5em PE\right)\kern0.5em -\kern0.5em q\kern0.5em =\kern0.5em 0,\mathrm{for}\ h\kern0.5em \le \kern0.5em z $$
(1)
$$ q\kern0.5em =\kern0.5em bhK \sin \alpha \cos \alpha $$
(2)

where p is the rainfall rate, PE is the potential evapotranspiration, q is the groundwater outflow, h is the water table depth, K is the soil hydraulic conductivity and α is the slope.

Equations (1) and (2) are evaluated for every element of the domain. It has to be noted that a main assumption of this method is that the capacity of the soil infiltration is always considered in excess with respect to the net rainfall rate (p-PE). Thus, only the groundwater flow is taken into account, which consequently implies to neglect the over-land flow.

The ratio between the water table depth h and the thickness of the soil layer z may be derived by combining the Eqs. 1 and 2, by the following equation (Montgomery and Dietrich 1994),

$$ \frac{h}{z}\kern0.5em =\kern0.5em \frac{I}{Kz}\frac{a}{b}\frac{1}{ \sin \alpha \cos \alpha} $$
(3)

where I = (p-PE) is the net rainfall rate.

Using the classical approach of Skempton and DeLory (1957) for the stability of a completely saturated shallow soil layer, the factor of safety (FS) may be then computed as follows:

$$ FS\kern0.5em =\kern0.5em \frac{c^{\prime }+ z{\gamma}_w{ \cos}^2\alpha \tan \varphi}{z{\gamma}_s \sin \alpha \cos \alpha}\kern0.5em -\kern0.5em \frac{\gamma_w}{\gamma_s}\frac{ \tan \varphi}{ \tan \alpha}\left(\frac{h}{z}\right) $$
(4)

where γ s is the specific weight of saturated soil, γ w is the specific weight of water, c’ is the soil cohesion, and φ is the soil internal friction angle.

At limit equilibrium condition (FS = 1), the critical water table depth h c can be evaluated as:

$$ {h}_c\kern0.5em =\kern0.5em \frac{ \tan \alpha}{ \tan \varphi}\frac{c^{\prime }}{\gamma_w \sin \alpha \cos \alpha}\kern0.5em +\kern0.5em \frac{\gamma_s}{\gamma_w} z\left(1\kern0.5em -\kern0.5em \frac{ \tan \alpha}{ \tan \varphi}\right) $$
(5)

If h ≥ h c , the slope is recognized as unstable, while for h < h c the slope is recognized as stable.

Equation 4 shows two limit cases. The first one is the unconditionally unstable case (UUC), corresponding to an unstable slope even found in dry condition, which is given by h = 0 and FS ≤ 1, yielding to the relation (Eq. 6) on the slope.

$$ \tan \alpha \kern0.5em \ge \kern0.5em \frac{ \tan \varphi}{1\kern0.5em -\kern0.5em \frac{c^{\prime }}{z{\gamma}_s \sin \alpha \cos \alpha}} $$
(6)

The UUC’s nonsense is related to the steep slopes that are—in practice—rock cliffs.

The second limit case is the unconditionally stable cases (USC), corresponding to a stable slope even found in completely saturated condition, which is given by h = z and FS > 1, yielding to the relation (Eq. 7) on the slope.

$$ \tan \alpha <\frac{ \tan \varphi \left(1-{\gamma}_w/{\gamma}_s\right)}{1-\frac{c^{\prime }}{z{\gamma}_s \sin \alpha \cos \alpha}} $$
(7)

The USC’s nonsense is related to those flat zones that are always stable.

A slope that fulfills the Eqs. 6 or 7 is removed from the calculation. In general slopes higher than 45° are eliminated from the calculation, but that value has to be treated with care depending on the geomorphology of the area.

Coupling the Eqs. 3 and 5 at critical condition for h = h c , the critical rainfall rate I c , for a selected return period, may be evaluated as follow:

$$ {I}_c\kern0.5em =\kern0.5em \left(\frac{ \tan \alpha}{ \tan \varphi}\frac{c^{\prime }}{\gamma_w \sin \alpha \cos \alpha}\kern0.5em +\kern0.5em \frac{\gamma_s}{\gamma_w} z\left(1\kern0.5em -\kern0.5em \frac{ \tan \alpha}{ \tan \varphi}\right)\right) K\frac{b}{a} \sin \alpha \cos \alpha $$
(8)

Equation (8) is valid in the hypothesis of constant intensity and indefinite duration of rainfall. Consequently, the corresponding return period is not defined due to the undetermined duration. To overcome this difficulty, the duration of the rainfall event is fixed equal to the time necessary for the soil to reach the steady state condition of lateral groundwater flow. A simple relation to evaluate such interval time, as a mean value on a certain portion of basin formed by M cells, is proposed in Eq. 9 (Papa and Trentini 2010):

$$ {t}_s\kern0.5em =\kern0.5em \frac{1}{M}{\displaystyle \sum_{i\kern0.5em =\kern0.5em 1}^M\frac{a_i}{b_i}}\frac{\theta_{s_i}}{K_i \cos {\alpha}_i \sin {\alpha}_i} $$
(9)

where t s is the time to reach the steady state condition and θ s is the water content at saturation expressed as the ratio of volume of water over the total volume of soil, which is considered for convenience equal to the porosity. The use of θ s is justified under the simplified hypothesis of saturated soil even above the water table, due to the rainfall infiltration. In Eq. 9, it can be observed that the time to reach saturation only depends on morphological and physical parameters, and not on rainfall, being the formulation of steady state condition of groundwater flow.

In particular, in Eq. (9), t s depends on catchment’s area, slope, water content and hydraulic conductivity. Area and slope values’ uncertainty could be easily addressed, and the typical water content’s range of values is narrow. However, the uncertainty of the hydraulic conductivity follows a logarithmic logic; the real values could change by a few orders of magnitude (see as reference the Table 2). Observing that the hydraulic conductivity is linearly related to t s , it is immediately concluded that both have the same order of uncertainty. It suggests that the parameters’ uncertainty is higher than the effect of the simplification hypothesis done in Eq. (9).

Once the duration time of rainfall (t s ) is assessed, as a sort of characteristic time, the rainfall intensity for the different return periods can be derived by the intensity-duration-frequency (IDF) curves.

Due to the usual features of the soil and the value of the hydraulic conductivity, the magnitude of t s is of the order of days. It is arguable that the typical IDF curves, being used for urban drainage studies, are constructed—and thus valid—for duration ranging from few minutes to 24 h. According to this, the mathematical representation of the IDF curves provokes a resolution leak for long duration rainfalls: The confidence intervals completely overlap for long-term events and it becomes difficult to distinguish between distinct return periods.

Concerning groundwater flow, a longer duration is necessary to evaluate a long-term behavior. Thus, different IDF curves have to be established to overcome the aforementioned problem. A different relationship has to be established. A new methodology is proposed, built on a relationship that gives a return period of event (T r ), knowing the rainfall rate (I) and the duration (D = t s ). The first step of this new methodology is to organize the data coming from rain gauges as a typical extreme value analysis: (a) aggregate the daily rainfall rate for 1, 2, …, 10 days, which are the duration (D); (b) choose for every aggregation the highest rainfall for each year; (c) assign a T r to every data and for every aggregation. A general definition of return period is given in Eq. 17. For every long-term rainfall duration identified in (a), a different extreme values’ fitting is performed, giving a piecewise defined IDF. Following this methodology, the global influence of the duration parameter is not explicitly taken into account, but implicitly collected, every duration having its characteristic fitting coefficients. From a mathematical point of view, what is sought is the reduction of the confidence intervals for “long-term” range and uncertainty.

The IDF relationship here proposed (Wenzel 1982; Chow et al. 1988, p.459) has the following form:

$$ I\kern0.5em =\kern0.5em \frac{k{T_r}^m}{D^n\kern0.5em +\kern0.5em l} $$
(10)

where I is the intensity of rainfall (or rainfall rate), k, m, n > 0 and l are the parameters to be evaluated fitting the available rainfall data. In that application, it is assumed l = 0 for simplicity.

This methodology actually and ideally requires rain gauges’ time series, which are often scarce and difficult to obtain. However, it permits to deal with long-term durations associated to the lateral groundwater flow.

The critical return period T r,c is that return period of rainfall event whence a soil’s portion is recognized as unstable. Settling I = I c and D = t s and substituting Eqs. 8 and 9 in Eq. 10 after deriving T r , the critical return period is given by the following:

$$ {T}_{r, c}\kern0.5em =\kern0.5em {\left\{\left[\frac{ \tan \alpha}{ \tan \varphi}\frac{c^{\prime }}{\gamma_w sen\alpha \cos \alpha}\kern0.5em +\kern0.5em \frac{\gamma_s}{\gamma_w} z\left(1\kern0.5em -\kern0.5em \frac{ \tan \alpha}{ \tan \varphi}\right)\right] K\frac{b}{a} \sin \alpha {\left(\frac{1}{M}{\displaystyle \sum_{i\kern0.5em =\kern0.5em 1}^M\frac{a_i}{b_i}}\frac{\theta_{s_i}}{K_i \cos {\alpha}_i \sin {\alpha}_i}\right)}^n{k}^{-1}\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$ m$}\right.} $$
(11)

It has to be noted that T r,c is evaluated for every elementary cell of the domain of study even if t s is averaged on a certain sub-domain defined by the expert.

Transient vertical infinite slope method

Iverson (2000) gives the following analysis of the time scale of the infiltration process, handling the Richard’s equations: (1) the quasi-steady (analog to SSISM) groundwater response in time is in the range from 100 days to 102 years being proportional to a/D 0 , where D 0 is the reference diffusivity of the soil; (2) the transient groundwater response in time is in the range from 100 minutes to 100 years being proportional to z 2/D 0 . The time scale of the studied processes seems to be more congruent with the latter. This suggests the interest in introducing a model of vertical infiltration. A typical transient piston flow of wetting front is selected (Green and Ampt 1911). The infinite slope stability analysis is coupled to the vertical flow mechanism. This method is valid for initial condition of partial saturated soil.

First of all, the critical water table depth h c is calculated thanks to Eq. 5. Then, the infiltrated volume (F), the infiltration rate (f), and time (t c ) of Green and Ampt theory are calculated, respectively, in Eqs. 12, 13, and 14.

$$ F\kern0.5em =\kern0.5em {z}_f\varDelta \theta $$
(12)
$$ f\kern0.5em =\kern0.5em \frac{dF}{dt}\kern0.5em =\kern0.5em K\left(1\kern0.5em +\kern0.5em \frac{\varDelta \theta}{F}{\psi}_f\right) $$
(13)
$$ K{t}_c\kern0.5em =\kern0.5em F\kern0.5em -\kern0.5em {\psi}_f\varDelta \theta \ln \left(1\kern0.5em +\kern0.5em \frac{F}{\varDelta \theta {\psi}_f}\right) $$
(14)

where z f is the water table depth, Δθ = (θ s  − θ i ) where θ s is the saturated soil moisture content and θ i the initial soil moisture content, f is the infiltration rate and ψ f the suction (negative). Generally, θ i  = 0.0 for initial dry condition and θ s  = 0.4 as a maximum value.

Assuming that h = h c  = z f at critical condition, the Eqs. 12, 13, and 14 are solved with known values at critical condition.

Expecting a short time of response, the total daily critical rainfall volume P d can be back calculated using the infiltration rate f and the time t c in the IDF curves. Those curves are usually regionalized. Here, disclosing the further application of this work, a form of IDF curves valid for the Catalonian region is given in Eq. 15 (Témez 1978).

$$ {P}_d=24 f{(11)}^{-\left(\frac{28^{0.1}-{t_c}^{0.1}}{28^{0.1}-1}\right)} $$
(15)

It is worth observing that any IDF curves can be easily implemented in the algorithm.

The calculation of return period associated to P d , is achieved through a generalized extreme value (GEV) function.

The GEV cumulated distributed function is shown in Eq. 16:

$$ \mathbb{F}\left( X;\mu, \sigma, \xi \right)= \exp \left\{-{\left[1+\xi \left(\frac{X-\mu}{\sigma}\right)\right]}^{-1/\xi}\right\} $$
(16)

where X is the rainfall data, μ ∈ ℝ is the location parameter, σ > 0 is the scale parameter, and ξ ∈ ℝ is the shape parameter.

The definition of return period T r (in years) is in Eq. 17, which shows which probability P of occurrence has a determinate rainfall P d  = X.

$$ \mathbb{F}(X)\kern0.5em =\kern0.5em \underset{\bar{\mkern6mu}}{P}\left( x\kern0.5em \le \kern0.5em X\right)\kern0.5em =\kern0.5em \frac{T_r\kern0.5em -\kern0.5em 1}{T_r} $$
(17)

Combining Eqs. 16 and 17, the return period of an assigned rainfall is achieved as follow:

$$ {T}_r=\frac{1}{1- \exp \left\{-{\left[1+\xi \left(\frac{X-\mu}{\sigma}\right)\right]}^{-1/\xi}\right\}} $$
(18)

Data requirement

The digital elevation model (DEM) is the main data used by the presented physical approaches. Concerning the susceptibility assessment of landslides and DF initiations, many authors have discussed the role of DEM resolution in applying geomorphological and physical-based approaches. Montgomery and Foufoula-Georgiou (1993) recommend a resolution of less than 30 × 30 m; Tarolli and Tarboton (2006) suggest a grid resolution not larger than 10 × 10 m; Dai and Lee (2002) uses a 20 × 20 m cell’s resolution in evaluating landslides in Hong Kong; Borga et al. (1998), Crosta and Frattini (2003), and Carrara et al. (2008) use a grid resolution of 10 × 10 m in assess landslides and DF in northern Italy; Borga et al. (2002) use a grid resolution of 5 × 5 m for small shallow landslides in the Dolomites; and Vanacker et al (2003) use a DEM resolution of 5 × 5 m in assessing slope stability in the Ecuadorian Andes.

The rainfall dataset is used to calibrate the long duration IDF and the GEV parameters: daily rainfall record is required.

The necessary characteristics of the soil are the internal friction angle (φ, in radiant), the cohesion (c’, in Pa, including the root cohesion), the thickness of the soil (z, in m), and the hydraulic conductivity (K, in mm/day or mm/h or m/s, but must have the same unit as the rainfall intensity).

When in the calculation of FS (Eq. 4), the effect of vegetation is included, the total cohesion c’ is split into the soil cohesion c s , and the root cohesion c r . The influence of weight of the tree species (W) is also taken into account, yielding to the following:

$$ FS\kern0.5em =\kern0.5em \frac{c_s\kern0.5em +\kern0.5em {c}_r\kern0.5em +\kern0.5em \left(\left({\gamma}_s\kern0.5em -\kern0.5em {\gamma}_w h/ z\right) z{ \cos}^2\alpha \kern0.5em +\kern0.5em W \cos \alpha \right) \tan \varphi}{z{\gamma}_s \sin \alpha \cos \alpha \kern0.5em +\kern0.5em W \sin \alpha} $$
(19)

The sensibility analysis of Borga et al. (2002) suggests that the ranking of the parameters’ influence, considering a proper range of variability, is: α, φ, h, c r , z, c s , W. For instance, an increase of 25 % in slope produces a decrease of nearly 25 % in FS while a decrease of 25 % of slope yield to an increase of more than 25 % in FS. This particularly high sensibility to the terrain morphology can be partially enhanced using high resolution DEM. Equation 19 is also especially sensible to the internal friction angle which encompasses also a certain drawback in its estimation. The sensibility of FS due to the other parameters’ variation is lower. In particular, the influence of W can be considered as low as to be neglected, yielding to the simplified following equation:

$$ FS\kern0.5em =\kern0.5em \frac{c_s\kern0.5em +\kern0.5em {c}_r\kern0.5em +\kern0.5em z{\gamma}_w{ \cos}^2\alpha \tan \varphi}{z{\gamma}_s \sin \alpha \cos \alpha}\kern0.5em -\kern0.5em \frac{\gamma_w}{\gamma_s}\frac{ \tan \varphi}{ \tan \alpha}\left(\frac{h}{z}\right) $$
(20)

The slope is assessed through the DEM, the water table through hydrological models while information on the internal friction angle, soil depth, and soil cohesion is only sometimes available. On the other hand, root cohesion is related to the plants’ species and age, and local climate. Past studies (Sidle 1992; Wu and Sidle 1995; Mattia et al. 2005) show a great variability in root cohesion and can only give an order of magnitude, mainly between 1 and 15 kPa (Schmidt et al. 2001; Bathurst et al. 2006).

An analysis of slope stability at large scale deals with the spatial variability of parameters. The values can strongly vary in a given small area. Unique physical and mechanical properties of soil can be fixed in the study area, as well as different values for every homogeneous region.

It is important to point out that the correct procedure to distinguish between regions with different physical-mechanical characteristics is based on soil maps. Nowadays, such maps are sometimes available at local scale (Papa et al. 2012) but not usually available at regional scale. A common way to define soil properties is to extrapolate them from geological and land coverage maps that are more ordinary. Homogeneous regions can be reclassified as portion of territories with the same geology formation and/or the same land coverage feature.

To define properly the physical-mechanical soil’s properties of a given homogeneous region, laboratory tests should be carried out. This tests concern the evaluation of soil’s parameters as for instance cohesion, internal friction angle, or hydraulic conductivity. However, the lab tests are not always feasible when dealing with a wide territory. When that condition occurs, expert criteria help in the task of assigning soil’s parameter values at homogenous portion of territory. However, this approach is a rather delicate task and should be carried out by a geologist or engineer, familiar with the geological, geomorphological, and geotechnical characteristics of the study area. The approaches presented use the geological maps and coverage maps as explained, applying the method’s validation.

Algorithms implementation and operational details

The proposed methodologies and algorithms are implemented using Java environment and declared as GNU/GPL open source code GITS-UPC (2010). The algorithms require common GIS commands (i.e., slope, curvature, accumulated flow…). These tools are provided through the SEXTANTE GIS library (Gimenez and Olaya 2008). Other new libraries have been implemented in the present framework by the authors to carry out the presented tasks. Algorithms are included inside SEXTANTE to be available to extern applications. It is to be noted that SEXTANTE is not a GIS, but a library that could be accessed from different open source as well as commercial GIS. A command line application to use SEXTANTE library without GIS is developed in that study. The data exchange formats of information are ESRI ASCII for raster and ESRI shapefile for vectorial.

Resolution of dataset input has to be the same for each raster used and it has to be congruent with the resolution of the DEM.

Nowadays, local authorities provide more and more free, high resolution DEMs. These models have a grid sizes equal or more than 10 × 10 m.

Usually, for a regional survey, a study’s area of the order of 102 km2 is common. For instance, such a wide area implies a file dimension of the order of 10 or 102 Mb when the raster resolution is, respectively, 30 × 30 m or 5 × 5 m. Due to the combination of different raster layers operated by the algorithms, an ordinary personal computer with 4 Gb of RAM cannot achieve the calculation, limiting the use of high resolution raster. The choice of a 30 × 30 m is constrained if the user does not want to face a laborious partition of the studied domain.

To overcome this issue for the presented applications, a Linux system is selected and used over a workstation having 14 GB of RAM. This has been done due to the limited control of Java’s heap size in Windows® systems unlike to what happens in Linux systems. However, the algorithms can also be used in a Windows® environment.

Application to the Boì Valley

The selected area, where the presented tools are applied, is the Boí Valley. It is located in the Alta Ribagorçana region (North West of Catalonia, Spain; Fig. 1) and is delimited by the catchment of the river Noguera de Tor. The basin has an area of 264 km2; a perimeter of 122 km; a maximum, mean, and minimum elevation of, respectively, 3028, 1962, and 822 m above the sea level (m asl); and a mean slope of 29°.

Fig. 1
figure 1

Location of the Boí Valley, with the studied catchment delimited in black. Based on the 1:500,000 map of the Institut Cartogràfic de Catalunya (ICC 2011)

Full size image

The valley lies in the central Pyrenees (Axial Pyrenees) where the presence of U-shape valleys and cirques shows the effects of past glacial erosion’s activity. The result is that tills and bedrocks are the most common outcrops. Presence of colluvium is nonetheless frequent as deglaciation induced many slopes destabilizations in the region. Many landslides and debris flows (see examples in Fig. 2) are documented by the previous studies (Clotet and Gallart 1984; Baeza 1994; Portilla et al. 2010; Corominas and Moya 2010). The study focuses on shallow landslides that developed into debris flows.

Fig. 2
figure 2

An example of erosional activity in the Boi Valley: a scar located on an ancient glacial deposit favored a debris flow in Erill la Vall. It is shown an oblique photo on the left (2011) and an aerial photo on the right (ICC 2011). The white polygon represents the estimated area of invasion

Full size image

Inventory of past events

Because reconnaissance of debris flows in this part of the Pyrenees is not systematic, many debris-flows events are not reported and an inventory was needed. The inventory elaborated for this study is based on the interpretations of aerial pictures. The inventory also takes into account the real events observed in situ (Chevalier et al. 2013).

GoogleTMEarth readily offers the display of aerial viewing, spanning from 2003 to 2008. It was used to determine where in this area unreported debris flows could have occurred. The report of erosional activity was assessed through the consideration of a series of criterions such as vegetation’s change in the landscape, presence of landslide’s scar(s), clear visibility of a torrent/gully/stream where roughness could be observed, or presence of potential deposition fans. The same series of criterion were used when the aerial pictures from 1956 to 1957 shared by the Catalan Cartographic Institute (ICC) through their Internet application OrtoXpres 1.0 (www.ortoxpres.cat/client/icc/). Eventually, two sets of aerial pictures (1975/1976 and 1982/1984) were also consulted (Chevalier et al. 2013). More details on the inventory methodology can also be found in Hürlimann et al. (2012).

Guzzetti et al. (2006) noted that clustering the landslides events in time intervals and studying the susceptibility within these intervals had played a role in theirs results. Their conclusion was that a unique multitemporal landslides inventory gave better results. For this reason, the debris-flow tracks were all gathered in a unique inventory (see Fig. 3), regardless the origin of the aerial pictures considered to identify it.

Fig. 3
figure 3

DEM (IGN 2008), inventory of DF tracks, first-order sub-catchments, and reactive sub-catchments of the Boí Valley

Full size image

In the selected basin, 83 DF tracks triggered by shallow landslide have been recognized.

Digital elevation model and soil properties evaluation

A DEM of 5 × 5 m resolution, available from the National Geographic Institute of Spain (IGN 2008; Fig. 3), has been chosen for this study. The DEM has been built based on a stereo correlation of orthophotos, with a resolution ranging from 25 to 50 cm/pixel.

Past studies show that the infinite slope stability method has a high sensibility to the input soil parameters. Furthermore, the local soil heterogeneity may be the main factor responsible for the timing and location of landslides. For instance, locally, an error on internal friction angle is propagating lineally on the safety factor calculation (Borga et al. 2002). For the present study, a soil map is not available, so it is decided, in order to evaluate soil properties, to operate as follows.

The available land coverage map has a scale of 1:5,000 based on orthophotos with a resolution of 2.5 m and taken from 2005 to 2007 (CREAF 2008). The land coverage map provides different soil classes as in Fig. 4. In a way, geological maps integrate this information (Fig. 5). The geological map is obtained from ICC (2002), with a scale of 1:250,000.

Fig. 4
figure 4

Land coverage map for the Boi Valley. Extracted from the map MSCS-3, CREAF (2008)

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Fig. 5
figure 5

Hydraulic conductivity map of Boi Valley after the reclassification of the geology classes. Legend codes are shown in Table 2

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With the purpose of calculating shallow landslide and DF, the areas that are not playing a role in the process should be eliminated. The previous step is the elimination (“no data” in GIS) of lakes and water basins. The elimination turns those zones into a detention basin stopping the area accumulation and thus the groundwater flow’s propagation. Glaciers and permanent snows are also eliminated because considered as long-term stable slopes. Urban zones and roads are also of particular interest, where the stability is high when compared to mountain slopes. Here, the treatment of water bodies is not appropriate because urban areas and roads are involved in the computation. For instance, in the calculation of cumulated area or flow direction, lakes are working as “black hole”, while roads and building are not able to stop the flow. A response to that is an artificial increase in stability parameters—such as cohesion and internal friction angle—where urban areas and transportation infrastructures are recognized. Lakes, glaciers, permanent snows, urban areas, and roads have been identified in the land coverage map.

Tills and colluviums have been identified from the geological map.

Once the different homogeneous soil’s classes have been identified from land coverage and geology (see description of classes in Table 1 and Table 2), the next step is to assign values of soils’ physical-mechanical properties. The reclassification of cohesion, internal friction angle, porosity, and suction is based on the land coverage map and is reported, for each identified class, in Table 1. The hydraulic conductivity’s values are based on the reclassification of geological map and are reported for each class in Table 2. The soil thickness is based on land coverage classes but with the superposition of tills and colluviums classes, identified from the geological map. The importance in locating tills and colluvium resides in the assumption that these are generally the main reserve of soil in the studied area. It should be noted that the assigned values find their basis in literature (Bathurst et al. 2006; Liu et al 2011) and other studies carried out in surrounding areas (Hürlimann et al. 2010; GITS-UPC 2010).

Table 1 Reclassification of land coverage map and assignation of thickness, cohesion, internal friction angle, porosity, and suction values. ND No data
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Table 2 Hydraulic conductivity values per geological classes (ND No data)
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The water content at saturation θs is generally set at 40 %. The initial soil water content θ i , necessary for the TVISM calculation, should be set at 0 % for dry soil. In the present calculation, it is set at 20 % in order to take into account an initial partially saturated soil, simulating the presence of light, long-term antecedent rainfall. The latter is likely to be an important factor influencing DF triggering (Wieczorek and Glade 2005). The saturated density γ s is arbitrary settled at 2,200 kg/m3.

The hydrology

The rainfall’s analysis has been carried out thanks to rain gauges’ data available in the area. Data from the Vilaller’s rain gauge in the Barravés Valley (Ebro River Basin Authority) have been selected. The rain gauge has gathered 50 years series of daily rainfall, from 1941 to 1991.

Data has been reorganized (see Fig. 6a) in order to obtain IDF curves of large rainfall duration (see Fig. 6b and Table 3). A proof of the organized dataset of precipitation and the mentioned IDF curves can be found in Fig. 6c. The same dataset has been used to define the parameters of the GEV function (see Table 3).

Fig. 6
figure 6

Vilaller rain gauge data: a values of rainfall intensity aggregated in terms of duration of days and the corresponding return period; b long duration IDF family of curves at selected return periods, defined fitting the data in Fig. 6a and following the form of Eq.  10 ; c proof of the similitude between the organized dataset of rain gauge (in solid lines) and the fitting curves (in dashed lines)

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Table 3 Calibrated parameters of the used function to estimate the rainfall (data from Vilaller rain gauge)
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For a long duration event, the Potential Evapotraspiration (PE) has to be accounted. Following the indicative values of PE in Brower and Heibloem (1986), PE is fixed to 6 mm/day.

SSISM being a simulation of a long duration event response, the run is achieved using a net rainfall, detracting PE from the total.

The TVISM runs are achieved using a total rainfall, having a short time response of infiltration.

Resulting susceptibility maps

Three simulations were run: SSISM, TVISM, and TVISM without outcrop (TVISMwo).

It has to be stated that the TVISM and TVISMwo follow the same physical behavior as transient vertical flow, but the latter used a slightly different soil properties dataset when compared to the former. Observing the land coverage map an inaccurate differentiation between scree and rocks has been detected. Rocks are often classified as scree and thus frequently recognized by the model as unstable. Due to this issue, it has been decided to run and compare the following two simulations: in the first the outcrops (called “rocks” in class 5 of Table 1) participate in the calculation; while in the second the outcrops have been removed from the calculation, a priori assumed as always stable slope. In Table 4 and Fig. 7d, the sketches of the simulations’ results, relative to the entire catchment, are reported. In Fig. 7a, b, c, the results’ maps of the three simulations are reported.

Table 4 Results of the runs over the whole studied catchment, where A is the susceptible area over the total basin’s area A tot
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Fig. 7
figure 7

Performed simulations over the whole studied catchment: a result of the SSISM; b result of the TVISM; c result of the TVISMwo simulation, after removing outcrops zones; d results of the different models in terms of return period and the susceptible area (A) over the total basin’s area (A tot )

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A definition cell-by-cell of the results (Fig. 7a, b, c) is congruent with the physical behavior of the process, but not congruent with the resolution of input-output of the presented methods. Moreover, the relationship between points of the DF track’s inventory and susceptible cells is a major drawback, as points are subjectively placed along the DF paths.

Following the work of Chevalier et al. (2013), first-order sub-catchments have been created in order to aggregate the results in significant terrain units. The sub-catchments have been derived from DEM and have been defined with a minimum contributing area of 1 km2 (Fig. 3). This last feature led to exclude from the analysis of 19 DF tracks which are placed in not represented, smaller sub-catchments.

A sub-catchment is considered reactive when at least one DF tracks has been found within it. It is to be noted that in a single sub-catchment more than one DF tracks can be placed.

The created set of terrain units consists in 79 sub-catchments, 26 of them being reactive (Fig. 3).

The aggregation of results has been executed through the zonal statistic, yielding to an up-scaling of models’ output at the mentioned terrain unit scale. This operation is essential to distribute the susceptibility at sub-catchment scale.

Two analyses have been carried out.

The first one concerns the evaluation of the relationship between the minimum number of DF and the susceptible area in a terrain unit. For each model used, the previous relationship is monotonically increasing (see Fig. 8a, b, c). This relation assesses that the more DF tracks have been found, the more susceptible the area has been recognized by the models.

Fig. 8
figure 8

Relationships between the minimum number of DF tracks per sub-catchment and the average susceptible area (A sc ) per sub-catchment for each return period for the different models outputs: a SSISM; b TVISM; c TVISMwo. Linear regressions equation and their coefficients of determination are in square boxes having borders congruent with the line style

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The second analysis has been done to define thresholds to assess the reactivity of a sub-catchment. The selected reference value of return period is 10 years. The motivation of this choice is strictly linked on the observation’s drawback: the reconnaissance of a DF track is influenced by the temporal resolution of aerial photos and the vegetation recovery. With this basis, the time scale of observed DF has an expected return period of the order of 10 years.

Table 5 shows the derived thresholds, associating the dataset of DF and the models’ results. The thresholds are in terms of the average susceptible area (A sc ) contained in a sub-catchment, whether a DF is present or not both in absence or presence of DF.

Table 5 Derivation of thresholds of presence or absence of DF tracks in a sub-catchment for the return period of 10 years. A sc is the average susceptible area contained in a sub-catchment, A sc,tot is the sub-catchment’s total area
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At first glance the TVISM model’ thresholds are clearer, in respect to the SSISM: the averages of susceptible area offer more differences in case of absence/presence of DF.

Once the thresholds have been defined, the maps of susceptibility aggregated in first-order sub-catchment can be drawn. Figure 9a, b, c reported the susceptibility maps created with the A sc /A sc,tot threshold for the different models.

Fig. 9
figure 9

Susceptibility maps with frequency of event aggregated in first-order sub-catchment for: a SSISM; b TVISM; c TVISMwo. The threshold value used is relative to A sc /A sc,tot

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Test of the models and comparison to a statistical model

In the practice of DF susceptibility evaluation using different methods, it is important to assess their reliability through the use of objective techniques (Carrara et al. 2008; Frattini et al. 2010). Five selected indices of quality (IQ) are based on the matrix’s elements of Table 6. These (IQ) are derived from the receiver operating characteristics (ROC) analysis (Fawcett 2006).

Table 6 Matrix model definition after Fawcett (2006)
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$$ \mathrm{Precision}\kern0.5em =\kern0.5em {(IQ)}_{\Pr}\kern0.5em =\kern0.5em \frac{TP}{TP\kern0.5em +\kern0.5em FP} $$
(21)
$$ \mathrm{Recall}\kern0.5em ={(IQ)}_{\mathrm{Re}}\kern0.5em =\frac{TP}{TP\kern0.5em +\kern0.5em FN} $$
(22)
$$ \mathrm{F}\hbox{-} \mathrm{measure}\kern0.5em ={(IQ)}_{\mathrm{Fm}}\kern0.5em =\kern0.5em \frac{2 TP}{2 TP\kern0.5em +\kern0.5em FP\kern0.5em +\kern0.5em FN} $$
(23)
$$ \mathrm{Success}\ \mathrm{rate}\kern0.5em =\kern0.5em {(IQ)}_{\mathrm{Sr}}\kern0.5em =\kern0.5em \frac{TP\kern0.5em +\kern0.5em TN}{TP\kern0.5em +\kern0.5em FP\kern0.5em +\kern0.5em TN\kern0.5em +\kern0.5em FN} $$
(24)
$$ \mathrm{Weighted}\ \mathrm{success}\ \mathrm{rate}\kern0.5em =\kern0.5em {(IQ)}_{\mathrm{Wsr}}\kern0.5em =\kern0.5em \frac{W_{TP} TP\kern0.5em +\kern0.5em {W}_{TN} TN}{W_{TP} TP\kern0.5em +{W}_{FP} FP\kern0.5em +\kern0.5em {W}_{TN} TN\kern0.5em +\kern0.5em {W}_{FN} FN} $$
(25)

The weights of Eq. 25 are based on the dataset, proportional to the ratio of the number of sub-catchments exhibiting DF, over the total sub-catchment’s number, yielding to the values of W TP  = W FN  = 2 and W FP  = W TN  = 1.

The SSISM and TVISM models are then compared to a statistical model: Chevalier et al. (2013) has run a logistic regression (LR) in an area of Pyrenees, including the Boí Valley. LR was chosen because the dataset, which consisted of 1,005 first-order catchments, exhibited nonlinear behavior (Landwehr et al. 2005; Witten et al. 2011). The LR handles 14 fluvio-morphological parameters aggregated per terrain units. The model has been calibrated using the aforementioned dataset, whence the present study’s dataset has been extracted. With comparing intentions, the LR results have been clipped for the Vall de Boí.

The LR model identifies a unique frequency of susceptibility, while physical models identify different return periods. Thus, for a congruent comparison, the 10 years return period has been selected, justified by the time scale of observed DF previously discussed.

Matrices and indices of quality of physical and statistical models are reported in Table 7. To better compare the different models, a differential index of quality Δ(IQ) (as percentage) has been defined as in Eq. 26. (IQ) x,y are indices of quality associated to the two models to be compared.

Table 7 Matrix models and indices of quality for the physical models’ results and the statistical model result
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$$ \varDelta (IQ)\kern0.5em =\kern0.5em \frac{(IQ)_x\kern0.5em -\kern0.5em {(IQ)}_y}{\left({(IQ)}_x\kern0.5em +\kern0.5em {(IQ)}_y\right)/2}100 $$
(26)

Table 8 shows the differences in reliability, using the same models, but applying the two different thresholds. It is possible to observe that the threshold based on A sc /A sc,tot enhances the result in respect to the threshold based on A sc , for all the physical models. The improvement is particularly marked for the SSISM model (17 %) and for the recall index (up to 37 %), which means that the model better avoids false negative responses.

Table 8 Evaluation of the best threshold type for physical models, comparing IQ (following Eq.  26 )
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Investigating the best physical model, Table 9 reveals that the transient vertical flow-based model works better in respect to a steady state lateral flow-based model (average improvement of 18 %), having a remarkable improvement in precision (28 %) and F-measure (19 %). The latter explains that the TVISM better avoids false responses. The spatial agreement between SSISM and TVISM (Fig. 10a) shows that the former overestimates the susceptibility, especially in the SE part of the basin, while both agree in the upper part of the basin. Both models are not identifying the susceptibility in the central part of the basin, where reactive sub-catchments have been found.

Table 9 Evaluation of the best physical model, comparing IQ (following Eq.  26 )
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Fig. 10
figure 10

Comparison a, b, and c, and union d between the used models: a spatial agreement between observation, SSISM, and TVISM based on A sc /A sc,tot threshold; b spatial agreement between observation, TVISM based on A sc /A sc,tot threshold, and LR model; c spatial agreement between observation, TVISM based on A sc /A sc,tot threshold, and LR model; d union between TVISM based on A sc /A sc,tot threshold and the LR model. 0 means absence of DF or no reactivity predicted by models and 1 means presence of DF or reactivity predicted by models

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Manually removing the outcrop’s zones in TVISM results enhances the average quality by 1 %, improves by 8 % the precision but worsens the recall by 7 %. Observing these limited improvements suggests that the post-process practice of removing outcrop is probably not justified to achieve a better result.

The comparison with the LR statistical model (Table 10) points out that the latter considerably improves the performance in respect to the SSISM model (18 %), having a better response for every index. When contrasting the LR model with the TVISM model, it essentially leads to the same IQ average. With respect to TVISM, LR considerably enhances the recall (29 %) and substantially worsens the precision (15 %) and the success rate (12 %). The worse recall and better precision show that TVISM is more effective in restraining false positives but overestimates false negatives. Although the last matter could be read as a dangerous result, the success rate and the weighted success rate show that, for the TVISM, false rates are restrained.

Table 10 Comparison between physical models and the statistical model, based on IQ (following Eq.  26 )
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In general, LR consistently overestimates reactive sub-catchments, while TVISM slightly underestimates the reactivity.

Comparing TVISMwo with LR leads to similar conclusions, highlighting the difference in recall in favor to LR, and the differences of the other indices in favor to TVISMwo. On average, TVISMwo wins over the LR by only 2 %.

The spatial agreement between physical and statistical models (Fig. 10b, c) shows a considerable mismatch. In the upper part of the basin, the two models agree, while in the central and lower part they do not match. The TVISM, as opposed to LR, is able to better predict the reactive sub-catchments on the watershed boundaries, while it is not able to predict the reactivity in the center of the basin. This observation leads to the conclusion that the two models are partially complementary. The resulting union of TVISM and LR models (TVISM∪LR) shows that all the reactive catchments are matched at the expense of a pronounced overestimation (Fig. 10d). In the same vain, the test analysis in Table 11 points out the poor precision of the union, together with the absence of false negative. The comparison between the TVISM∪LR and its generating models entails that the union generally improves by 4 % when compared to the TVISM and by 5 % when compared to the LR (Table 12).

Table 11 Matrix models and indices of quality for the union of TVISM and LR results (TVISM∪LR). Best values for each index in bold
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Table 12 Comparison between the union of physical and the statistical model (TVISM∪LR) and the originating TVISM and LR model, based on IQ (following Eq.  26 )
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Discussion and conclusions

Two different physical models of shallow landslides initiation of DF have been implemented in a toolbox and presented. The first one is based on a steady state model of lateral flow coupled with a Mohr-Coulomb mechanism of infinite slope failure (SSISM); the second is based on a transient vertical flow coupled with the same Mohr-Coulomb mechanism (TVISM). Both models are able to define the susceptibility at regional scale in terms of frequency of events (return periods).

As discussed by previous authors, the lateral flow of groundwater schema has a large time of response (order of days or years). As a consequence, it is not always representative of the DF’s time scale. An IDF family of curves for a large duration of rainfall has been defined in order to link the hydrology with the physical model. The new IDF curves need rain-gauges’ data to be calibrated.

The vertical flow, as opposed to the horizontal one, has a short time of response (order of hours or days). It is more appropriated to the DF’s time scale of event. Like the SSISM model, the TVISM model needs rain-gauges’ data to be performed.

The models are applied in the watersheds of Boí Valley, in the Central Pyrenees.

The inputs are to be carefully handled in order to perform the simulations in the best possible environment. A digital elevation model of 5 × 5 m grid size is used. The hydrology has been defined through local rain-gauges’ data, including evapotranspiration for long-time response’s model. The weak point of physical models of shallow landslide consists in the evaluation of soil parameters, which is a compulsory, often complicated task. Here, an exhaustive reclassification of land coverage and geological maps has been performed to define physical and mechanical soil parameters, including roots’ associated cohesion. In the case of TVISM, a post-process has been performed in order to eliminate the outcrop areas that are not properly recognized. Thus, a separate methodology (TVISMwo) is introduced with the aim of improving the quality of results.

To mount the models presented in this article, a reconnaissance of DF events has been accomplished in the selected basin, finding 83 DF tracks.

After running the models, the resulting spatial outputs are then aggregated in first-order sub-catchments, which are used as basic terrain units of study. The aggregated results show a monotonic relationship with the number of DF tracks found in each sub-catchment and the relative predicted susceptible area. Successively, different thresholds of reactivity have been defined for the two physical models, based on predicted susceptible area over the terrain unit.

The reapplication of the defined thresholds permits to evaluate the predicted reactivity at terrain unit’s scale.

These last results of physical models are related to the presence of DF for each sub-catchment, using a test analysis, which permit to evaluate the performance of the used methodologies.

The test of the models shows that the performance of TVISM globally enhances the SSISM by approximately 18 %. This considerable achievement shows that a model based on vertical groundwater flow is more representative than a model based on horizontal flow.

The TVISMwo only improve TVISM by 1 %, which does not probably justify the post-process of eliminating outcrops.

The physical models are then compared to a statistical model (logistic regression). The result of the comparison shows that the SSISM is less accurate and probably cannot compete against LR. On the contrary, TVISM’s performance is almost equivalent to the LR: precision and success rates are better, but the number of reactive sub-catchments is underestimated. In general, LR overestimates the reactivity but limits the false negative cases. In other words, this work states that physical models based on transient vertical flow that have a certain quality in input can compete with statistical models.

The spatial mismatch between physical and statistical model outputs has suggested the evaluation of the union of results. The comparison between the TVISM∪LR and its generating models implies that the union generally improves by more than 4 %, which also creates a certain overestimation of susceptibility. The mentioned improvement of TVISM∪LR with respect to its generating models’ results might not justify the employ of the union: running TVISM and LR are time consuming. However, every improvement in this topic should be received as an achievement and it was included in the presented work.

A suitable preliminary procedure can consider the union of the two model results as a first susceptibility assessment. Successively, field observations or photo interpretation can reduce the overestimation previously commented. Local detailed study can be carried out in order to generate detailed hazard and risk maps.

As it has been mentioned, the presented methodologies entail certain limitations that have to be considered carefully when applying the models. Moreover, some simplifications have been introduced in order to reduce the degree of complexity.

SSISM and TVISM can predict DF only triggered by shallow landslides induced by rainfall. The rainfall is taken as constant in time. Other triggering behaviors are not taken into account; therefore, when other mechanism is recognized in the study area, the user should consider other methodologies.

The reapplication of the presented methodologies in areas of the same region with the same geomorphological features and where a database of real event is handled is immediate. However, a great concern still exists about the applicability of the presented models in other regions having different geomorphological settings. Regarding the physical models (SSISM and TVISM), the methodologies being deterministic only need the redefinition of thresholds (as in Table. 5) through the use of a new database of real events. This redefinition is possible using GIS techniques.

The statistical model LR entails a certain amount of morphological parameters strictly regionalized as, for instance, the ones related to the elevation of the terrain units. Thus, the logistic regression formulation has to be redefined with the database of past events within the selected area. This operation is quite complex as involving the data mining which is a not widespread technique for engineering and geologist practitioners.

However, in general, statistical models are suitable for evaluating DF susceptibility. They do not use soil’s parameters, which often have uncertainty. With the correct procedure, they yield to satisfactory results. In some cases, physical models are more complex than statistical models. Nevertheless, the present work shows that physical-based models, and in particular, the TVISM model, entail a good performance, and they cannot be discarded a priori for a regional DF susceptibility assessment. Moreover, physical models are also able to perform the susceptibility in terms of return period.