Slope stability is a key topic, not only for engineers but also for politicians, due to the considerable monetary and human losses that landslides can cause every year. In fact, it is estimated that landslides have caused thousands of deaths and economic losses amounting to tens of billions of euros per year around the world. The geological stability of slopes is affected by several factors, such as climate, earthquakes, lithology and rock structures, among others. Climate is one of the main factors, especially when large amounts of rainwater are absorbed in short periods of time. Taking this issue into account, we developed an innovative analytical model using the limit equilibrium method supported by a geographic information system (GIS). This model is especially useful for predicting the risk of landslides in scenarios of heavy unpredictable rainfall. The model, hereafter named terrain stability (or TS) is a 2-D model, is programed in MATLAB and includes a steady-state hydrological term. Many variables measured in the field – topography, precipitation and type of soil – can be added, changed or updated using simple input parameters. To validate the model, we applied it to a real example – that of a landslide which resulted in human and material losses (collapse of a building) at Hundidero, La Viñuela (Málaga), Spain, in February 2010.

Landslides, one of the natural disasters, have resulted in significant injury and loss to human life and damaged property and infrastructure throughout the world (Varnes, 1996; Parise and Jibson, 2000; Dai et al., 2002; Guha-Sapir et al., 2004; Crozier and Glade, 2005; Khan, 2005; Toya and Skidmore, 2007; Raghuvanshi et al., 2015; Girma et al., 2015). Normally, heavy rainfall, high relative relief and complex fragile geology with increased manmade activities have resulted in increased landslide (Gutiérrez-Martín, 2015). It is essential to identify, evaluate and delineate landslide hazard-prone areas for proper strategic planning and mitigation (Bisson et al., 2014). Therefore, to delineate landslide susceptible slopes over large areas, landslide hazard zonation (LHZ) techniques can be employed (Anbalagan, 1992; Guzzetti et al., 1999; Casagli et al., 2004; Fall et al., 2006).

Landslides are a result of intrinsic and external triggering factors. The intrinsic factors are mainly geological factors or the geometry of the slope (Hoek and Bray, 1981; Ayalew et al., 2004; Wang and Niu, 2009).

The external factor which generally triggers landslides is rainfall (Anderson and Howes, 1985; Collison et al., 2000; Dai and Lee, 2001). Several LHZ techniques have been developed in the past, and these can be broadly classified into three categories: expert evaluation, statistical methods and deterministic approaches (Wu and Sidle, 1995; Leroi, 1997; Guzzetti et al., 1999; Iverson, 2000; Crosta and Frattini, 2003; Casagli et al., 2004; Fall et al., 2006; Lu and Godt, 2008; Rossi et al., 2013; Raia et al., 2014; Canili et al., 2018; Zhang et al., 2018). Within these categories, we want to highlight the empirical models that are based on rainfall thresholds (Wilson and Jayko, 1997; Aleotti, 2004; Guzzetti et al., 2007; Martelloni et al., 2011). Each of these LHZ techniques has its own advantage and disadvantage owing to certain uncertainties on account of factors considered or methods by which factor data are derived (Carrara et al., 1995). Limit equilibrium types of analyses for assessing the stability of earth slopes have been in use in geotechnical engineering for many decades. The idea of discretizing a potential sliding mass into vertical slices was introduced in the 20th century. During the following few decades, Fellenius (1936) introduced the ordinary method of slices (Fellenius, 1936). In the mid-1950s Janbu and Bishop developed advances in the method (Janbu, 1954; Bishop, 1955). The advent of electronic computers in the 1960s made it possible to more readily handle the iterative procedures inherent in the method, which led to mathematically more rigorous formulations such as those developed by Morgenstern and Price and by Spencer (Morgenstern and Price, 1965; Spencer, 1967).

Until the 1980s, most stability analyses were performed by graphical methods
or by using manual calculators. Nowadays, the quickest and most detailed
analyses can be performed using any ordinary computer (Wilkinson et al.,
2002). There are other types of software based on the modelling of the
probability of occurrence of shallow landslides LHZ, in more extensive areas
using geographic information system (GIS) technology and DEM (digital elevation model), as is the case of
deterministic models like the software TRIGRS, SINMAP, R-SHALSTAB,
GEOtop or GEOtop-FS, and r.slope.stability, among others (Montgomery and Dietrich,
1998; Pack et al., 2001; Rigon et al., 2006; Simoni et al., 2008; Baum et
al., 2008; Mergili et al., 2014a, b; Michel et al.,
2014; Reid et al., 2015; Alvioli and Baum, 2016; Tran et al., 2018). These
are widely used models for calculating the time and location of the
occurrence of shallow landslides caused by rainfall at the territorial
level, some even in three dimensions, in order to obtain a probabilistic
interpretation of the factor of safety. Currently other approaches and/or theoretical
studies for landslide prediction are used (for triggering and/or
propagation; Martelloni and Bagnoli, 2014; Martelloni et al., 2017). One
of the achievements of the presented study is to discretize the potential
slip mass in the critical profile of the slope, once unstable areas have
been detected through the LHZ (landslide hazard zonation) programs. The
terrain stability (TS) calculation tool is not limited to shallow landslides and debris flows but
allows analysis of deep and rotational landslides, which other models
often do not account for. We use the hydrological
variable

Limit equilibrium types of analyses for assessing the stability of earth slopes have been in use in geotechnical engineering for last years. Currently, the vast majority of stability analyses using this method of the equilibrium limit are performed with commercial software packages like SLIDE V5, SLOPE/W, Phase2, GEOSLOPE, GALENA, GSTABL7, GEO5 and GeoStudio, among others (González de Vallejo et al., 2002; Acharya et al., 2016a, b; Johari and Mousavi, 2018). Currently there are other slope stability models based on the theory of limit equilibrium that are still in analysis and testing, as is the case with the SSAP software package (Borselli, 2012), but in this case a general equilibrium method model is applied. Secondly, sometimes for commercial models, the introductions of parameters to perform calculations are not very interactive. For the stability analysis, different approaches can be used, such as the limit equilibrium methods (Cheng et al., 2007; Liu et al., 2015), the finite element method (Griffiths and Marquez, 2007; Tschuchnigg et al., 2015; Griffiths, 2015) and the dynamic method (Jia et al., 2008), among others. Limit equilibrium methods are well known, and their use is simple and quick. These methods allow us to analyse almost all types of landslides, such us translational, rotational, topple, creep and fall, among others (Zhou and Cheng, 2013). For the stability analysis, different approaches can be used, such as the limit equilibrium methods (Zhu et al., 2005; Cheng et al., 2007; Verruijt, 2010; Liu et al., 2015), the finite element method (Griffiths and Marquez, 2007; Tschuchnigg et al., 2015; Griffiths, 2015) and the dynamic method (Jia et al., 2008), among others (SSAP 2012, Slide V5, 2018). Also, limit equilibrium methods can be combined with probabilistic techniques (Stead et al., 2006) or with other models, like stability analysis of coastal erosion (Castedo et al., 2012). However, they are limited in general to 2-D planes and easy geometries. Numerical methods – finite element methods – give us the most detailed approach for analysing the stability conditions for the majority of evaluation cases, including complex geometries and 3-D cases. Nevertheless, they present some problems, such as their complexity, data introduction, the mesh size effect, and the time and resources they require (Ramos Vásquez, 2017).

The above-mentioned software packages provide useful tools for determining
the stability through the

Delft University of Technology has developed a well-known and free software program to
analyse landslides, the STB 2010 (Verruijt, 2010). This program is based
on a limit equilibrium technique, using a modified version of Bishop's
method to calculate the

In the model we developed, the TS model, we used the
limit equilibrium technique for its versatility, calculation speed and
accuracy. An analysis can be done studying the whole length of the breakage
(shearing) zone or just small slices. Starting with the original method of
slides developed by Fellenius (1936), some methods are more
accurate and complex (Spencer, 1967; Morgenstern and Price, 1965) than others
(Bishop, 1955; Janbú, 1954). Using Spencer's method (Spencer, 1967;
Duncan and Wright, 1980; Sharma and Moudud, 1992) here would mean dividing our slope into small slices that must
be computed together. This method is divided into two equations, one related
to the balance of forces and the other to momentum. Spencer's method imposes
equilibrium not only for the forces but also for the momentum on the surface
of the rupture. If the forces for the entire soil mass are in equilibrium,
the sum of the forces between each slice must also be equal to zero.
Therefore, the sum of the horizontal forces between slices must be zero as
well as the sum of the vertical ones (Eqs. 1 and 2):

Representation of the forces acting on a slice, considered in
Spencer's method (Spencer, 1967). W is the external vertical loads; Zn and
Zn

These equations must be solved to get the

The

As mentioned, the minimum

To analyse the slope using Spencer's method, a set of equations must be
solved to satisfy the forces and momentum equilibrium and to obtain
the

With that being said, if we assume that the forces between slices are parallel
(in other words, that

When solving the normal and parallel forces at the base of the slice of the
five acting forces, we obtain (

The pore pressure will be hydrostatic, defined by the following:

This equation is used in our proposed model for calculating the safety
factor (substituting the expression of

Figure 2 shows the results of applying the terrain stability model to an
irregular slope, including the initial and final points of the first failure
circle (shown in yellow). This circle corresponds with the initial value
introduced by the user into the FSOLVE function. The points of the slope are
extracted from a DEM model in ArcGIS 10 (Glennon et al., 2008). The slope
height is equal to 15 m, and the soil is considered to be uniform with the
following nominal properties:

Idealized cross section of a slope. In this example, the centre coordinates are equal to

The code works as follows: the initial circular failure curve is plotted
using the FPLOT tool, as shown in Fig. 2 (yellow line). In this example,
the centre coordinates are equal to

The next step is to apply Spencer's method to the different breakage
surfaces until the curve with the lowest

Results following the application of the software showing the slope
profile and surface damage. The

On the basis of this first curve (yellow line in Fig. 2), the program
enforces new restrictions:

The curve passes through the origin of slope

The centre of the possible circles of critical breakage is inside the
rectangular box defined as
(

To complete the second phase in the TS model operation, the effect of rain
infiltration must be introduced by the coefficient of the pore-pressure
factor

Outcome of the TS model after the introduction of the infiltration
factor, producing an unstable circular failure (

We can determine that if this infiltration factor value is small enough, taking into account the safety coefficients, the design may still be adequate, but critical information was missing to calculate this parameter.

To clarify the procedure employed in the suggested algorithm, the flow chart (block diagrams) presented in Fig. 5 demonstrates the calculation and iteration process as implemented in our software.

Sequential TS algorithm (block diagrams). Numbers in parentheses refer to numbers in the text.

Our algorithm (software) is more versatile compared to the STB 2010; the model developed here can analyse slope from right to left and vice versa, and the STB 2010 only allows the analysis from right to left. Other software programs, like the STB 2010, use a modified version of Bishop's method, a less accurate methodology than Spencer's method. A modified version of Bishop's method solves only the equilibrium in momentum, while the Spencer method also considers the equilibrium in forces.

Another improvement made by the TS code, in comparison with others, is that the use of Spencer's method allows us to analyse any type of slope and soil profile. In this procedure, we calculated the worst breaking curve by modifying the calculation points.

In the TS model, from the first slip rotational circle obtained in MATLAB, many circles were then calculated using the fmincon function, with some user restrictions. However, other models, like the STB 2010, require the definition of a quadrangular region (to look for the centres of rotational failures) and a point (namely five; see Fig. 9) to define the curve as where the failure must pass. Also, the number of circles that the STB 2010 model can analyse for their minimum value is limited to 100.

The TS model can detect relevant earth movements derived from rainfall
infiltration, both translational and rotational types (Stead et al., 2006),
such as those that usually occur in regions like India, the US,
South America and the UK, among other places. The programs
that do not contemplate this option will overestimate the

The TS model has an additional advantage: it also offers the opportunity to incorporate, in the same code, the stability analysis and the effect of the infiltration factor in the rainfall regime. This is a step forward from open-access programs, such as STB 2010, and also from alternative payment software, such as Slide.

In 2010, La Viñuela, Málaga, (Spain) experienced torrential
rainfall. The main consequence was a devastating landslide with serious
personal and material losses, as shown in Fig. 6. The coordinates where
this event occurred were in degrees (36.88371409801,

The study area is located in the county of La Viñuela, specifically in the Hundidero village, which is located immediately north of the swamp of La Viñuela (El Hundiero) and south of the Baetic System mountain ranges (southern Iberian Peninsula).

According to the Cruden and Varnes' (1996) classification, the slide corresponds to a rotational slide-like complex movement because it was generated in two sequences at different speeds. This type of mechanism is characteristic of homogeneous cohesive soils, as was the one analysed here (Cornforth, 2005; Rahardjo et al., 2007; Lu and Godt, 2008).

This event caused serious damage to different buildings. Regarding the damage caused, in the initial stretch of the slope (its head), a house was dragged and destroyed and another was seriously damaged. On the right bank of the mentioned house, another building was affected. In total, this event left a total of two buildings destroyed and one seriously compromised. Although 15 people lived in these houses, there were no fatalities. About 20 houses were to be constructed at the head of the slope; fortunately, the event happened before this construction. Figure 7 shows an aerial picture from 2006 before the disaster as well as the affected area and landslide in 2010.

For this example, we used data of IGN, the Spanish National Geographic
Institute (

From this map we obtained the topographic information to acquire all necessary profiles to study the landslide. Moreover, as our algorithm is a 2-D model, with this topographic map we study the critical curve of the slip in the most unfavourable profile of the landslide (Fig. 8).

It is well known that mass movements, such as landslides, are highly complex
morphodynamic processes. We selected The Hundidero as our study area because
it is prone to landslides. In order to analyse this case study using our
model, we first calculated the initial displaced volume of the study area.
According to the dimensions of the problem, the initial displaced volume was
calculated, equivalent to the volume of half an ellipsoid (Varnes, 1978;
Beyer, 1987; Cruden and Varnes, 1996) that has Vol

Characterization and longitudinal section of the rotational sliding (Geolen Engineering, 2010). The location of the dragged house is noted in red, which was analysed by the TS model.

The initial spit of land had an approximate size of 235 m in length by 70 m in width. Due to this initial displacement, there was a drag and a huge posterior planar displacement of about 550 m length, affecting a zone with several parcels of land and buildings. These sizes were confirmed using aerial photography and field data. The soil is basically composed of clays of variable thicknesses and of fine grain, with fluvial sediments and silty clay. The authors obtained these data by conducting a field survey as well as through the laboratory tests carried out by the laboratory Geolen S.A. (Geolen Engineering, 2010). From a geological and geotechnical point of view, according to a survey of those present as the laboratory extracted the materials, different lithological levels can be distinguished, as shown in Table 1.

Lithology of the area affected by the failure, according to the laboratory tests of Geolen S.A. No groundwater level was detected.

The laboratory tests included a sieve analysis (following UNE 103 101) in
three of the samples extracted from the field at depths of 1.80–2.00 m, of
which 70.3 % were composed of clay and silt; according to this, the sample
is classified as cohesive. The liquid limit and the plastic limit were
determined in two of the samples (following UNE 103 103 and UNE 103 104,
respectively), yielding liquid limit values of 57.5 % and 64.2 %, respectively, and a
plasticity index of 37 %. According to the lab results, the
material can be classified as high plasticity material with the potential of
having a high water content. The landslide analysed began in February 2010,
ending in March of that same year. However, based on the field inspection
and the analysis of the rainfall series in the La Viñuela region in 2010
(see Fig. 10), it can be inferred that the main causes of the event were the following:

the poor geomechanical parameters of the material that formed the affected hillside,

the hydrometeorological conditions in the days preceding and days after the event, according to the histogram.

Rainfall histogram at La Viñuela from August 2009 to April 2010. Rainfall data have been provided by the Spanish Meteorological Agency (station of Viñuela).

Most of the landslides observed during these days occurred as a consequence
of exceptionally intense rainfalls. The precipitation data were provided by
the meteorological station of La Viñuela (Fig. 10). It can be observed
that large amounts of precipitation fell during the months of December,
January, February and March of 2010, with peaks being, at the most,
60 L m

The rotational slide analysed had occurred between level 2 and level 3, when the water content reached that depth, as confirmed by the infiltration calculations in the terrain (see graphs in Fig. 9, reaching depths of up to 5 m). Two direct shear tests (consolidated and drained) were conducted in unaltered samples extracted from the boreholes at 3.00–3.60 and 4.00–4.60 m deep. The cut-off values of the soil are specified in Table 2. Those values were used in the developed software to obtain the safety coefficient and the theoretical failure curve.

Summary chart of the characteristics of the soil analysed at the
Geolen S.A. laboratory:

The dynamic and continuous tests were carried out by the Geolen S.A. laboratory
with an automatic penetrometer of the ROLATEC ML-60-A type. The data
obtained were transcribed by the number of strokes to advance the 20 cm
tip, which is called the “penetration number” (

This test is included in the ISO 22476-2:2005 standard as a dynamic probing super heavy and consists of penetrating the ground with a conical tip of standard dimensions. The depth of the failed mass can be estimated as well as the theoretical failure curve for an increase in the soil consistency (see data in Table 3).

Summary chart of the soil analysed at the Geolen S.A. laboratory. Bold values show, according to the data of the field penetrometers, the depth mobilized by the rotational sliding.

The change in the geomechanical response of the soil takes place at a depth
of 4–5 m, according to the results of

To analyse the topography of the critical section, we obtained the DEM data
from the ArcGIS 10 software program (Environmental System Research, 2017), with a scale of

To complete the input data, we plotted the hydraulic potential and the volumetric water content, as a function of depth in the ground for different time steps, using a previously developed infiltration model, as shown in Fig. 11 (Herrada et al., 2014). The figure shows the evolution of how the wetting front advances can be observed. These reached almost 5 m depth at the end of April 2010.

We applied the TS model using topographic data obtained from the ArcGIS 10
software program. We did so to obtain the degree of stability of the sliding
land based on the angle of internal friction, the cohesion, the density and
the angle of the slope we analysed. Figure 9 shows the analytical results
from the real slope by studying and analysing the most unfavourable profile
of the landslide studied. In addition we compared the results given by the
developed TS model and the results given by STB 2010 model, using free
surfaces in both cases. In our model the worst curve (shown in green) was
calculated automatically from the initial curve (shown in blue), resulting in

As can be noted, the failure curves are similar, and the safety
coefficients

A new calculation including the pore coefficient

This calculation and the theoretical failure curve provided by our model was
able to reproduce, in a realistic way, the landslide which occurred in
La Viñuela. Our model found that the critical surface area that
corresponded with the profile of the terrain was 12 927.45 m

As mentioned, the STB 2010 model does not allow stability calculations to
apply to rainfall infiltration on a hillside. Hence, it is not capable of
predicting a hillside's instability in a critical rainfall scenario, which
was critical in the slope analysed. The STB 2010 model found that the
hillside studied had an

If we compare the results of the penetrometric tests (Table 3) and the
laboratory tests (Geolen Engineering, 2010) summarized in the actual critical surface in
the most unfavourable profile of landslide (Fig. 9) with those offered by
our TS algorithm (Fig. 13) to which we apply the infiltration factor

A value of

Using the STB 2010 program, we would not have been able to previously detect the landslide of the case study of the paper, calculation that is not normally done in the stability calculations; with the calculation with our code we could have avoided the collapse of the building.

With these results, the terrain stability analysis performed using the developed model defines the slip-breaking curve that intuitively appears to be susceptible to failure fairly well, especially when heavy rains occur. For example, the landslides which occurred in the La Viñuela area could only have been predicted if the infiltration had been taken into account. Even then, it could not have been done with other available software programs, which were not able to consider it.

The terrain stability (TS) analysis defines the critical surface
to landslides in 2-D of each profile of the analysed slope and the safety slip
factor (

The TS model we developed uses Spencer's method, which is more precise
than the modified Bishop method; this model is used by other software such as the
case of the STB 2010, so it differs in the results it provides for
the

In the case study analysed, the slope was initially stable and was
determined by the analysis performed with the STB 2010 model. However, the
slope became unstable due to the heavy rains of that hydrological period,
which called for the application of the pore-pressure coefficient

The data that can be publicly accessed have been described within the text; however, raw data belong to the La Viñuela Municipality, which contracted the first author to conduct a field survey and analysis. Authors are not authorized to publicly share the data directly. However, some of them can be shared upon request.

Each author has made substantial contributions to the work. AGM contributed to the conception of the work, to the applied methodology, to acquisition, to formal analysis, to data elaboration and to writing the original draft of the paper. MAH contributed to the code development. JIY and RC contributed to the field survey, to formal analysis, to validation of the work and to writing the original draft of the paper. Each author has approved the submitted version and agrees to be personally accountable for their own contributions and for ensuring that questions related to the accuracy or integrity of any part of the work, even ones in which the author was not personally involved, are appropriately investigated, resolved and documented in the literature.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Advances in computational modelling of natural hazards and geohazards”. It is a result of the Geoprocesses, geohazards – CSDMS 2018, Boulder, USA, 22–24 May 2018.

This paper was edited by Albert J. Kettner and reviewed by three anonymous referees.