Introduction

Among the commonly used definitions of landslide hazard, that of Varnes and IAEG Commission on Landslides and other Mass-Movements (1984) is most commonly evoked. By this definition, landslide is considered as the probability of a potentially damaging event (landslide) occurring in a specified period of time and in a given area. Due to an upsurge in landslide events around the world (Gutierrez et al. 2010), landslide investigators are becoming increasingly determined to develop integrated methods to better tackle the problem. This is done by addressing the two fundamental questions raised by Varnes’s definition which includes “where” landslides can occur and “when” they are most likely to occur. However, much attention has been given to the spatial dimension of the hazard “where” than the timing aspect “when”. Gokceoglu and Sezer (2009) on their review of publish articles on landslides noted that most publications, especially those in international journals, were focused on spatial probability assessments rather than temporal probability, although landslide risk is typically done because of triggers (Tiranti and Rabuffetti 2010; Malamud et al. 2004; Havenith et al. 2003) which addresses the temporal problem. Rainfall and earthquakes have been identified as the main triggers of landslides in many parts of the world (e.g. Bai et al. 2014; Giannecchini et al. 2012; Marques et al. 2008; Chleborad et al. 2006; Aleotti 2004; Crosta and Frattini 2003; Glade et al. 2000; Canuti et al. 1985; Caine 1980; Govi and Sorzana 1980). Although the Bamenda Mountain is located along the Cameroon Volcanic Line (CVL) where earthquakes due occur, rainfall is known to have triggered more landslides in the area (Guedjeo et al. 2013; Gountié et al. 2012; Eze and Ndenencho 2004; Lambi 2004) than seismic activities (Afungang 2015). The role of rainfall is reflected in its effects on surface (top soil) and subsurface saturation which has been responsible for the  slope movements. The impact of earthquakes is only felt on the southern portion of the CVL close to the active Cameroon Mountain volcano. For instance, in 1999 and 2001, earthquakes triggered hundreds of catastrophic landslides along the slopes of the Mt Cameroon and these events are well described in Ayonghe et al. (2004, 1999). However, the characteristics of the main trigger of landslides in the area (rainfall) and its temporal probability have not yet been well established.

The objective of this paper is to determine the temporal probability of landslide events based on empirical rainfall and landslide inventory analysis. This involves defining threshold levels, estimating the probability of the threshold being exceeded and the probability of landslides occurring after the threshold is exceeded. This is based on the assumption that the temporal probability of rainfall that caused past landslides corresponds to the temporal probability of landslides. This probability has been determined using direct and indirect methods. This includes the use of physical process-based models (Crosta and Frattini 2003; Wilson and Wieczorek 1995; Montgomery and Dietrich 1994) or dynamic hydrological models (Terlien 1998), and statistical models that are based on the definition of empirical thresholds (Giannecchini et al. 2012; Bui et al. 2012; Saito et al. 2010; Guzzetti et al. 2007; Aleotti 2004). The physically based approach uses physical terrain properties and requires characterisation of materials involved in the slide. This approach is mostly adapted to small areas because the model requires geotechnical data on the characteristics of the area (e.g. soil properties, changes in groundwater level, discharge conditions and shear properties). Given the spatial extent of the study area (regional scale), detail soil characteristics could not be easily measured and thus, the physical models could not be applied. On the other hand, the statistical approach required historical records of rainfall-induced landslide events and daily rainfall records which were feasible available. Landslide inventory was possible through review of the literature, image interpretation and field work, and time-series rainfall data was available from the Bamenda weather station. Given the scale of the study area and the available information, the empirical approach was most suitable for use in the assessment. Among the many methods used in determining rainfall thresholds for landslide initiating such as frequentist method (Brunetti et al. 2010), quantile regression (Saito et al. 2010) and Bayesian inference (Guzzetti et al. 2007), the rainfall intensity–duration and cumulative amount–duration methods that have been used in some parts of the study area (Afungang 2015) were chosen for use in the study.

Study area characteristics

The Bamenda Mountain Region is situated along the Cameroon section of the West African Volcanic Line (WAVL) which is found around the Gulf of Guinea. The general massif lies between longitudes 10°3′54′′ E and 10°21′57′′ E and latitudes 6°4′18′′ N and 5°46′15′′ N (Fig. 1) and covers approximately 913.7 km2. The study area can be divided into four broad relief units including the volcanic mountains, the intermediate highlands, the eastern valleys and the north-western lowlands. Volcanic activities or the uplift movements that took place in the late Tertiary and Quaternary created high mountain ranges along the northern section of the line. The landscape is marked by a multitude of landforms with sharp uplifted features and separated from each other by narrow-like or V-shaped valleys. The area has wide altitudinal differences with elevations ranging from 2600 m ALS in the volcanic mountains to 1010 m ALS in the valleys and plains.

Fig. 1
figure 1

Location and physiography of the study area

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The Bamenda Mountain is geologically diverse. It is composed of igneous rocks including those from volcanic formations (basalts, trachyte, rhyolite, tuffs and ignimbrite), plutonic formations (Pan African granite, dolerite mafic rock and diorite coarse-grained rock) and metamorphic rocks (migmatite, gneiss, amphibolite and quartzite). There are also small concentrations of sedimentary rocks in the area mainly sandstones, but which were not possible to represent on the lithology map (Fig. 1) extracted from 1:500.000 geology map. Most outcrops with a size less than 500 m or a resolution of less than 250 m (<250 m) could not be detected and were merged into larger rock formations. Superficial formations here are from geological deposits composed of unconsolidated sediments derived from weathering processes and fluvial actions linked to the Quaternary. The main types of superficial formations here include: talus deposits or scree, gravel, colluvium, sand, silt, clay and volcanic ash. Most recent types are anthrosols, which are deposits formed from human activities such as crop cultivation, cattle rearing and infrastructural construction. This category is weak and unconsolidated and are easily be moved by denudational activities. The area is made up of two principal drainage basins. The northern part has a dendritic drainage system, and the southern portion has a radial pattern network that enables spatial distribution of water into unstable zones.

The climatic condition of the area depends on the periodic circulation of the south-west Monsoon winds and the north-east Trade winds known as the Harmattan winds in the area. Changes in the inter-tropical convergence zone (ITCZ) between these two wind systems determine the start and the end of the rainy season in the area. Most often, the Monsoon winds blow across the area from March to October bringing in a low-pressure system with heavy rainfall. It is during this period that most landslides are initiated. The study area has a mean annual precipitation (MAP) of about 2375 mm/year (Fig. 2) with temperatures ranging between 13 and 22 °C (Ndenecho 2011). However, the area is generally cold when compared with average temperatures (20–25 °C) in the southern part of the country.

Fig. 2
figure 2

Annual precipitation distribution at Bamenda rain gauge from 1970–1971 to 2012–2013 hydrological year. The horizontal line is mean annual precipitation (MAP). The red histograms indicate years with two or more registered landslides

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The mean annual precipitation (MAP) for the study period as shown in Fig. 2 is 2435.4 mm/year, which is slightly higher than the known MAP (2375 mm/year) for the area. MAP is at times used to evaluate the occurrence of landslides (Giannecchini et al. 2012) with landslides most probable to occur when daily rainfall exceeds the MAP. Rainfall amounts ≥ MAP were registered ten (10) times in 43 years, and within this period, landslides event was registered seven times and in 3 occasions, landslides were not initiated. Contrarily to the view that landslides will be initiated when the MAP is exceeded, landslides were initiated in 5 occasions when rainfall was below the MAP. This suggests that landslides events in the study area are determined not only by cumulative rainfall, but also by the relationship between rainfall intensity and duration (R ID) and may be other factors as well (e.g. land use).

Materials and methods

Information on past landslide events and rainfall data was the two types of information used in the assessment. Landslide data were obtained from a number of sources, and data on rainfall were taken from reference weather station in the area. Both data were analysed using different models to determine thresholds for rainfall-induced landslides and temporal probability of landslides for the study area as shown below.

Landslide inventory

Shallow translational and deep rotational landslides are the two main types of landslides affecting the study area and were thus considered in the study. Landslides were registered based on the movement mechanism as indicated by the scar and morphology of the slide. Both translational and rotation slides did occur very close to each other (i.e. in the same area), and the mapping of these events was done in three phases. In the initial phase, landslides were compiled from scientific reviews (e.g. Afungang 2013; Guedjeo et al. 2013; Ayonghe et al. 1999) and newspapers (e.g. Vanguard news July 2009). The description in these reviews on the location, type, size and time of occurrence of landslides guided the second and third stage of the inventory which included photo-interpretation and field work. Aerial photographs with a resolution of 1000 dpi and satellite images (mainly Google Earth images) with a resolution of <5 m covering the area were exploited in the inventory process using remote sensing and GIS techniques. Field work was carried out in 2009, 2010, 2013 and 2014. About 72 % of landslides registered through photo-interpretation were confirmed in the field. About 66.6 % of the landslides were registered during field surveys. From the different inventories methods used, a total of 335 landslides were registered including 144 rotational slides and 181 shallow translational slides. The rotational slides covered an estimated area of 12,740 m2 which is 52.6 % of unstable area, and the translational slides occupied about 11,940 m2 representing 48.4 % of unstable area. Rotational slides were fewer than the translational type but accounts for greater portion of unstable areas due to their large scars or source area. Most landslides events before 1994 had no precise dates and were obtained from literature reviews. These events were those that caused huge material damage and human casualties, and some of the scars had completely been eroded by denudation and human activities. Consequently, the total landslides registered for the entire 43 years could not be used. The final inventory for the study was limited to 19 years (1994–2013) considering only slides whose dates were known. Within the 19-year study period, the landslides considered were believed to have been triggered by 15 rainfall episodes among which ten of the rainfall episodes triggered two or more slides. This puts the registered landslides reoccurrence event rate at 0.53 % or 1.9 events per year. About 80 % of the landslides measured between 10 and 90 m2. The most destructive were those that measured ≥100 m2. Some examples include the 4 August 2009 landslide at Akum village (Fig. 3a, b) which cut off the main highway leading into Bamenda town and that which occurred 3 days later (7 August 2009) at “S” bend up station that placed the regional governor’s residence at high risk (Fig. 3c, d). Another major landslide occurred on the same 7 August 2009 at Siesia neighbourhood killing one person, wounded four others and destroyed eight houses and farmlands. Results from the inventory showed that, from 1994 to 2013, there were 46 deaths, hundreds of injuries, major damages to road infrastructure and buildings in the Bamenda Mountain Region linked to moving debris from landslides.

Fig. 3
figure 3

Source: Vangardcameroon.com news (2009)

Landslides in the study area photograph a mile 1 Akum translational landslide 04 August 2009, b aftermath effects, c up station “S” bend rotational landslide 7 August 2009 and d aftermath effects.

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Rainfall data

Time-series daily rainfall records from the Bamenda rain gauge from 1994 to 2013 were used in defining the rainfall threshold for the study area. The Bamenda rain gauge is located on the highest peak in the study area (2400 m) and is the only rain gauge among the four rain gauges located in and around the study area with daily rainfall records needed for threshold analysis. About 80.9 % (271) of the 335 registered landslide events occurred along the escarpment slope within a perimeter of up to 3 km2 from the Bamenda weather station. The remaining 11.1 % of the slides were registered within a 4-km2 proximity to each of the weather stations located in the lowland (Santa, Bambui and Bamessing rain gauges). Since topographic location of weather stations is an important factor in the recording of climatic conditions (Petrucci and Polemio 2009), and the fact that the majority of slides occured at close proximity to the Bamenda rain gauge, it was chosen as the reference rain gauge for the study area.

Determination of rainfall thresholds

Rainfall is one of the most recognized triggers of landslides, and investigators in many parts of the world have for many years attempted to determine the amount of precipitation needed to trigger slope failure. This has been achieved by establishing rainfall thresholds levels using physically based models (Baum et al. 2010; Godt et al. 2008; Salciarini et al. 2008; Montgomery and Dietrich 1994; Crosta and Frattini 2003; Wilson and Wieczorek 1995), empiric historical or statistical approach (Giannecchini et al. 2012; Cannon et al. 2011; Jaiswal and van Westen 2009; Guzzetti et al. 2008, 2007; Chleborad et al. 2006; Coe et al. 2004; Godt et al. 2005; Caine 1980), or using a combination of the two methods (Crozier 1999). Determining rainfall conditions that did cause and did not trigger landslides has been a central point in threshold definition. A number of methods to define thresholds have been used in different parts of the globe, but none of these methods are universally applied because of differences in physiographic setting (Bui et al. 2012). Each model performs adequately mostly in areas for which they were developed (Jakob et al. 2006). Among the approaches mentioned above, the empirical approach that defines rainfall threshold based on past landslide events is most commonly used (Coe et al. 2004; Gabet et al. 2004; Glade et al. 2000; Canuti et al. 1985; Ellen and Wieczorek 1988; Govi and Sorzana 1980). Empirical-based thresholds have been established using methods such as the daily rainfall method (Dahal and Hasegawa 2008), cumulative rainfall method (Polemio and Sdao 1999), intensity–duration method (Giannecchini et al. 2012; Zêzere et al. 2005; Corominas and Moya 2008), antecedent rainfall method (Bui et al. 2012; Jaiswal and van Westen 2009; Marques et al. 2008; Zêzere et al. 2005) and normalized critical rainfall method (Giannecchini et al. 2012; Aleotti 2004). Each of these methods has its advantages and disadvantages (Guzzetti et al. 2007), and the type of method to be used depends on landslide typology in question (Martelloni et al. 2011). The intensity–duration (R ID) and the cumulative antecedent rainfall method (R DA) were chosen for this research with the assumption that translational landslides are initiated by rainfall of high intensity and low duration while contrary rainfall conditions activate rotational landslides.

Among the two methods chosen for the threshold definition, the intensity–duration (R ID) rainfall threshold was the first method that was tested. One of the basic tasks in assessing the relationship between rainfall and landslide occurrence using this method was to reconstruct the antecedent rainfall that triggered past landslides. This involved the reconstruction of the event dates, rainfall for short and long duration prior to the landslide event and rainfall on the day of the event. Daily rainfall for the 19-year period (1994–2013) from the Bamenda weather station (reference gauge) was used to reconstruct the antecedent rainfall for 1, 3, 5, 10, 15, 30, 60 and 90 days prior to the date of registered landslide events. The antecedent rainfall was calculated as:

$$Px_{n} = Px_{1} + Px_{2} \ldots P_{n}$$
(1)

where Px n is maximum accumulated rainfall for nth day, Px 1 is accumulated rainfall for day 1 and P n is accumulated rainfall for “nth” day.

Only maximum daily rainfall for each year was used, considering that such maximums were the most likely rainfall that triggered the past landslide events. The threshold was established by defining the critical combination of rainfall intensity–duration for past landslides and their return periods. It was assumed that the best critical rainfall combinations (rainfall intensity–duration) under which past landslides occurred were those with the highest return periods. Determining rainfall return periods is important in the prediction of landslide hazards (Bhandari et al. 2010) since rainfall conditions that triggered past landslides will trigger the same type of landslides and will affect a similar area each time they reoccur (Zêzere et al. 2005). The return period for each registered event was therefore calculated. The distribution of rainfall to determine the return periods was done using the Gumbel (1958) law that was defined as:

$$F \left( x \right) = { \exp }\left[ { - { \exp }\left\{ { - \frac{x - u}{\alpha }} \right\}} \right]$$
(2)

The reduced variable was denoted as: \(\frac{x - u}{\alpha }\) for which the Gumbel distribution was expressed as: \(Fm = \left( {Fy_{(m)} } \right) = { \exp }\left\{ { - { \exp }\left( { - y_{(m)} } \right)} \right\}\) and hence \(y_{\left( m \right)} = F^{ - 1 } \left( {F_{m} } \right) = - { \ln }\left\{ { - { \ln }\left( {F_{m} } \right)} \right\}\), where U = location parameter and α = scale parameter.

The critical pair calculation above assumes only critical values for extreme combinations. It is only statistically based and do not consider the relationship between rainfall and the characteristics of the soil. However, it provides the best means to separate rainfall episodes that were characterized by landslide activity and those rainfalls that did not trigger landslides (Giannecchini et al. 2012; Zêzere et al. 2005).The critical combination of rainfall intensity–duration (R ID) was based on the extreme probability for each landslide event to occur, and this was calculated through logistic regression analysis (Ascenso 2011). This was expressed as:

$${\text{EP}} = 1 - { \exp }( - { \exp }( - ( P_{xn} - \beta )/\alpha ))$$
(3)

where Ep is the extreme probability, P xn is maximum accumulated rainfall for a landslide event.

The alpha (α) and beta (β) values for the Gumbel distribution were derived from the mean (m) and standard deviation (sd) for each rainfall episode that initiated landslides. These were stated as:

$$\alpha = \left( {\frac{{P_{x1:xn} }}{2}} \right) - m\left( { - { \ln }_{x1:xn} } \right)\beta$$
(4)
$$\beta = \left( {\frac{{P_{x1:xn} }}{2}} \right) - {\text{sd}}\left( { - { \ln }_{x1:xn} } \right)$$
(5)

is maximum accumulated rainfall for day 1 to n-day. The impact of rainfall on landslide activity was analysed by combining rainfall intensity–duration (R ID) with the highest return period on a bi-logarithm graph. The bi-logarithmic scale had rainfall duration (R D) on the x-axis and intensity (R I) including registered landslide events on the y-axis (Fig. 5).

Daily antecedent rainfall threshold (R DA) was the second method that was used in determining threshold for rainfall-induced landslides in the area. Antecedent rainfall plays an important role in landslide initiation, especially in less impermeable soil where much soil suction and pore water pressure are needed to trigger soil movement (Aleotti 2004). However, their role is negligible in shallow movements like debris flows (Corominas and Moya 2008). Determining the number of days believed to have caused soil saturation prior to the day of the event itself is very strategic in assessing the role played by antecedent rainfall. Different durations ranging from 2 to 125 days have been proposed. Some examples include: 3 days (Dahal and Hasegawa 2008); 10 days (Bai et al. 2014; Crozier 1999; Glade et al. 2000); 15 days (Bui et al. 2012; Jaiswal and van Westen 2009; Zêzere et al. 2005; Aleotti 2004; Wieczorek et al. 2000); and 25 days (Terlien 1998). To determine the number of antecedent rainfall days, daily rainfall for reported landslides whose dates were known was plotted with corresponding antecedent rainfall together with yearly maximum rainfall for 1 day for years without reported slides over the 19-year study period. This was done for different intervals (3, 5, 7, 10, 15 and 30 days) following the method used by Zêzere et al. (2005). Then the thresholds for the area were determined by constructing a scattered plot between daily rainfalls that initiated landslides (y-axis) against the X-day antecedent rainfall (x-axis). The threshold function for the envelope curve was obtained using the lower-bound precipitation (Chleborad et al. 2006; Bui et al. 2012) following the plotted points in the scattered graph.

The temporal probability of landslides was assessed after the (R ID and R DA) thresholds were established. The temporal probability of landslide occurrence was considered as the probability (P) of rainfall (R) exceeding the threshold (R TH), and the probability of landslides occurring after the threshold must have been exceeded \(P\left\{ {L/R > R_{\text{TH}} } \right\}\). These two probabilities were calculated separately and later joined to estimate the temporal probability if there was a slope failure in the area. In order to estimate the first probability \(P\left\{ {R > R_{\text{TH}} } \right\}\), it was assumed that: (1) for landslides [L] to occur, rainfall in a given period (e.g. daily and antecedent rainfall) must exceed the threshold; (2) landslide will not or rarely occur below the threshold; and (3) the probability of occurrence will remain constant for all time intervals of a fixed length. This assertion can be expressed by the function: \(P[L/R > R_{\text{T}} ] = 1\) and \(P [L/R \ge R_{\text{T}} ] = 0\). The probability of landslides to occur \(P\left\{ L \right\}\) depends on the exceedance probability of \(P\left[ {R > RT} \right]\), which is \(P\left[ L \right] = P\left[ {R > RT} \right]\). Therefore, to estimate the exceedance probability, the future mean recurrence interval (\(\mu\)) which is the average time interval between successive slides had to be calculated. This was done using the historical landslide inventory, considering that the rate at which past landslides occurred will remain the same in the future. After obtaining this value, the Poisson distribution was then used to calculate the probability of n landslides occurring during time t (Crovelli 2000) as shown by the function:

$$P\left\{ {N\left( t \right) = n} \right\} = {\text{e}}^{ - \lambda t} \frac{{\left( {\lambda t} \right)n}}{n!}\quad \quad n = 0, 1, 2, 3, 4 \ldots .$$
(6)

where N(t) is the number of landslides to occur during a given time and λ is the rate of occurrence. Thus, the probability of experiencing one or more landslides events during time t also known as the “exceedance probability” was expressed as:

$$P\left\{ {N\left( t \right) \ge 1} \right\} = 1 - {\text{e}}^{ - \lambda t} = 1 - {\text{e}}^{ - t/\mu }$$
(7)

where μ is the future mean recurrence interval between successive slides and t is a period of time in the future for which the exceedance probability is calculated.

To estimate the second probability, that is the probability of landslides occurring after the threshold must have been exceeded \(P\left\{ {L/R > R_{\text{TH}} } \right\}\), the frequency of landslide occurrence after the threshold had been exceeded was calculated. The first and second probability was combined in a single function and was mathematically expressed as:

$$P\left\{ {L\left| {R > RT} \right.} \right\} \times P\left\{ {L\left| {R > RT} \right.} \right\}.$$
(8)

Using this function (Eq. 8), the temporal probability for the study area was thus estimated. This probability was based on the intensity–duration (R ID) threshold and daily antecedent rainfall (R DA) threshold estimations. Due to data deficiency (incomplete landslide inventory) and the risk of gross underestimation, only registered landslides from 1994 to 2013 were used because <80 % of the inventory was complete.

Results and discussion

Rainfall distribution

The highest annual rainfall in the 43-year study period (1970–2013) was recorded in 1999/2000 hydrological year with 2811.8 mm and the lowest was recorded during the 1973/1974 hydrological year (Fig. 2) with 1930 mm. Within this time, the hydrological year 2009/2010 was the rainiest with a total of 205 rainy days with the 1988/1989 hydrological year being the driest year with only 195 rainy days. About 82 % of the annual rainfall was received within 4 months (June–September) raising the spectral of landslides in this area. About 99.75 % of all registered landslides occurred between June and September which have 65.85 % of the annual rainfall, and the average rainfall for the four rainiest months (June, July, August and September) was 391.1 mm (Fig. 4). The maximum daily rainfall ranged from 60.3 to 129.3 mm recorded on 9 July 2013 and 30 June 2006, respectively.

Fig. 4
figure 4

Average monthly rainfall and percentage (%) of landslides events from 1994 to 2013: red bars (Bambili rain gauge), blue bars (Bamenda, reference rain gauge)

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Intensity–duration (R ID) rainfall threshold

The plotting of critical rainfall duration (x-axis) and intensity and landslide events (y-axis) on a bi-logarithmic scale (Fig. 5) gave a rainfall intensity–duration function \(R_{\text{TH}} = 94.063 D^{ - 0.517}\) which represents the rainfall threshold for the occurrence of translational and rotational landslides in the study area. The rainfall episodes that were used in the critical combination of rainfall intensity–duration were those episodes that triggered up to ten landslides. It was observed that rainfall that did not trigger any landslides was below the regression line meaning a good separation of the events. The fitted function (R 2 = 0.878) shows a good correlation with the plotted points.

Fig. 5
figure 5

Critical rainfall intensity–duration (R ID): (1) is rainfall episodes associated with shallow translational landslides, (2) is rainfall values associated with deep-seated rotational landslides

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The graph (Fig. 5) also shows that shallow translational landslides are triggered by high-intensity rainfall over short duration (group 1) and deep-seated rotational slides are initiated by low-intensity and long-duration rainfall (group 2). The timing of the two types of movement may be caused by differences in slip surface depths, soil characteristics, infiltration levels and effective hydraulic conductivity that control the soil’s pore water pressure (Iverson 2000). In order for rotational slides to be activated, rainfall for a longer duration is required so that water can penetrate into greater depths, especially in clay-rich soil, but this less water is needed to activate translational slides. Rainfall of high intensity (94 mm in <1 day) was sufficient enough to infiltrate into the upper layers of the soils causing the soil layers to move. Most areas affected by translation slides had soft soil composed largely of silt (82 %), clay (15 %) with a natural mixture content of 67 % and a plasticity index between 32 and 49 % (Afungang 2013). The depth of the affected soils affected by translational slides ranged from 0.5 to 4 m, and areas affected by rotational slide had higher percentages of clay soils with depths ranging between 3 and 8 m.

Most landslides were found to have been initiated by rainfall durations of up to 15 days. About 74 % of landslides were triggered within 2 days and up to 47 % initiated just in a single day. About 27 % of events were triggered by rainfall lasting between 4 and 15 days and were mostly deep-seated rotational landslides. The return period for rainfall that activated landslides varied between 7.5 and 69.9 years for recorded events. Events with long return periods (<20 years) correspond to extreme rainfall episodes that initiated more than ten landslides in the area (Afungang 2015). The model rightfully forecasted 89 % of landslides to occur above the threshold, and the failure to predict probability was 1/9 or 0.11.

Daily antecedent rainfall threshold (R DA)

Although some authors consider the rainfall intensity–duration threshold as a typically type of threshold (Godt et al. 2005), it has been criticized because of its highly statistically based nature and because it does not take into consideration the response of the soil to rainfall, especially the draining capacity of the soil (Simoni et al. 2007). Considering the fact that 93.5 % of landslides in the area occur between July and September which is already 5 months after the start of the rainy season, it is assumed that considerable amount of soil saturated had been attained before the landslides are triggered. This means that rainfall intensity–duration threshold R ID function depends on a certain level of antecedent soil moisture. The definition of the antecedent rainfall threshold R Ad was therefore important in the assessment. The central aspect in the R Ad threshold model was to determine the number of antecedent rainfall days prior to the actual day of the incident. After analysing the cumulative rainfall for different periods and landslide events as described in “Methodology”, the best correlation was noticed with 10-day antecedent rainfall (Fig. 6). This meant that soil saturation was attained after 10th day of antecedent rainfall after which additional rainfall on the day of the event is very likely to trigger slope movements.

Fig. 6
figure 6

Relationship between daily rainfall and 10-day antecedent rainfall (1994–2013): blue diamond indicates landslide events

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The rainfall threshold for landslides occurrence based on 10-day antecedent rainfall was estimated using the lower end of the envelop curve and was mathematically estimated as R TH = 96.2 −0.478R 10Ad. Any precipitation above the lower bound is considered to trigger ≥1 landslide, and below the bound, landslides activity is not expected to occur, or may rarely occur under certain conditions (e.g. unprecedented land-use change). The high intercept (96.2) and the small slope (−0.478) mean that landslides are generally triggered by daily precipitation of high intensity when the soil must have been sufficiently soaked by 10 days of antecedent rainfall. About 66.6 % of landslides occurred between 82 and 100 mm of rainfall and 140–270 mm of antecedent rainfall.

Validation of rainfall threshold model

Validation of rainfall threshold model is an important step in landslide assessment, although they remain imperfect indicators of landslide occurrences (Chleborad et al. 2006). The validation of prediction models is an essential step that improves the prediction power of the model, and this step makes the model scientifically acceptable. The R ID threshold was validated using historic landslides and daily rainfall data for the year 2009 which was one of the deadliest years in landslide history in the area. This information was not used in building the threshold model. The validation model (Fig. 7) was constructed by comparing the daily rainfall (real rainfall) with the critical rainfall for the entire study period to confirm rainfall conditions that triggered past landslides. The relationship between critical rainfall and real or daily rainfall on the day of landslide events determines the efficiency of the model. When this relationship is negative (i.e. when the difference of the minimum value between daily rainfall and critical rainfall is <0), landslides are more likely to occur, but in a contrary situation where this relationship is positive, landslides are not expected to occur.

Fig. 7
figure 7

Validation of R ID model using the 2008–2009 hydrological year and corresponding landslide events (The red line is the threshold; the blue curve is the minimum value difference between critical rainfall and daily rainfall; and the dashed vertical lines are landslide event dates)

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The result of the model validation (Fig. 7) shows that two landslide events (i.e. 4 and 7 September 2009) occur when the relationship between daily rainfall and critical rainfall was negative. This means that landslides were initiated when daily rainfall was above the critical rainfall. All periods (days) with negative rainfall values to the threshold are likely to have landslides, and the higher the negativity of this value to the threshold, the higher the probability of landslides. The critical rainfall that initiated the 7 September rotational slide was slightly higher than that which triggered the 4 September translational landslide. This therefore supports or confirms past research findings that shallow translational movements are mostly triggered by high-intensity rainfall over short periods and deep-seated rotational slides are activated by prolong rainfall with nearly constant rate or low intensity in the area (Afungang 2015, Guedjeo et al. 2013; Eze and Ndenencho 2004).

Temporal probability of landslides occurrence

From the analysis, the R ID threshold was exceeded 24 times by rainfall episodes in 19 years putting the future mean recurrence interval (μ) between successive threshold exceedance at 0.792. On the other hand, the R DA rainfall threshold was exceeded 22 times for the same number of years giving a future mean recurrence rate of 0.864. The annual probability for rainfall to exceed the R ID threshold \(P\left\{ {R > R_{\text{TH}} } \right\}\) was 0.717 and the \(P\left\{ {R > R_{\text{TH}} } \right\}\) for the R DA threshold was 0.686 calculated using Eq. 7. In some studies cited in the paper (e.g. Bui et al. 2012; Jaiswal and van Westen 2009), the frequency of the threshold being exceeded was estimated per unit area delimited based on physiographic characteristics, land use and rain gauge location. In this study, this type of delimitation was not possible since only one weather station (Bamenda rain gauge) had daily rainfall records. The frequency of landslides in the study area was accurate to the 80.9 % (271) of the 335 landslide events registered around the reference rain gauge (Bamenda escarpment and environs) and may diminish with increase distance from the reference rain gauge.

In the second probability, rainfall exceeded the R ID threshold 24 times, and during those episodes, more than one landslide occurred in 09 occasions giving an estimated probability of \(P\left\{ {L\left| {R > RT} \right.} \right\}\;0.375\). On the other hand, the R DA rainfall threshold was breached 22 times, and in those rainfall episodes, landslides were triggered on 09 occasions corresponding to an estimated probability of \(P\left\{ {L\left| {R > RT} \right.} \right\}\;0.409\). The temporal probability of landslide occurrence was obtained by multiplying the two probabilities \(P\left\{ {R > R_{\text{TH}} } \right\} \times P\left\{ {L/R > R_{\text{TH}} } \right\}\). From this expression (Eq. 8), the annual probability of having one or more rainfall episodes that can initiate landslide in any given year stood at 0.269 when the R ID threshold is considered and 0.281 when the R AD threshold is considered (Table 1). Since it is not always evident that landslides will occur every time that the threshold is exceeded and based on the fact that the thresholds were defined based solely on registered lopes failures, the use of the join probability helped to close the gap between the probability of the threshold being exceeded and the likelihood that landslides will be actually be initiated after a threshold breach. The join probability also helps to reduce forecast errors in cases where unforeseen circumstances such as changes in local conditions (e.g. land-use changes) have a greater role to play in slope failure. This might have an impact on the definition of the threshold since the environmental reaction to rainfall is affected through land use. The temporal probability was estimated for 1-, 5-, 10- and 25-year return period (Fig. 8).

Table 1 Exceedance probability estimate of rainfall threshold and landslide occurrences
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Fig. 8
figure 8

Temporal probability of landslides occurrence for different return periods: threshold 1 is R ID and threshold 2 is R AD

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The graph (Fig. 8) shows that the temporal probability of landslides occurrences increased from year 1–5 and then remains almost stable till the 25th year. Due to heavy rainfall each year, landslides event is likely to occur on a yearly basis, especially during the peak of the rainy season. The validation of this model using the frequency of landslide across the area shows that shallow translational landslides occur almost yearly while rotational slides occur after two or more years.

Conclusion

The peak of the rainy season in the area (August) directly coincides with the peak of landslide occurrences in the area, indicating that there exists a strong correlation between rainfall and landslide initiation. The computing of rainfall episodes and landslides whose dates were known forms the basis of the prediction model. Translational and rotational landslides were registered within the same area, suggesting that the differences in slope movements may be highly dependent on rainfall conditions (e.g. intensity, duration and amount) rather than the geophysical setting of the area. Experience from the study shows that temporal assessment of landslide hazard using statistical analysis highly depends on the completeness of the landslide inventory and quality of the rainfall data. A number of landslide events were left out because of lack of adequate dating information. It was realised that some errors and exaggeration in predicting future events emanating from incomplete landslide inventory can be minimised by computing the temporal probability using only events whose dates were known. The use of two methods to determine rainfall thresholds that initiate landslides in the area helped to check the applicability of each of the threshold methods in the area. The slight difference between the two threshold functions shows that both methods can be used in the area and that the rainfall data and landslide inventory used in the analysis were good. However, the thresholds seem to be higher than it should be considering that some landslides occurred far below the threshold. This therefore suggests that a better inventory is needed to improve the definition of the threshold and more details about each slide are required to differentiate human and rainfall-triggered events.

The frequency of landslides is an important component in estimating the temporal probability of rainfall-induced landslides, and this aspect may be affected when the inventory is not incomplete. The use of rainfall data from a single rain gauge may have reduced the ability of the model to predict landslides in far-off distances (>4 km). The model performs well with well-distributed network of rainfall data collected from many rain gauges and landslide inventory where slide dates are known. The closer the landslides to the rain gauge, the better the efficiency of the threshold model to predict future slides. Contrarily to some assessments that consider temporal probability of landslides to depend solely on the probability of rainfall exceeding the threshold, we consider it as a join probability of rainfall exceeding the threshold and the probability of landslides occurring after the threshold has been exceeded. The validation of the models shows that it can effectively be used in the area to predict translational and rotational landslides and it may work differently when other landslides types (e.g. debris flows and mudslides) also common in the area are concern. This research shows that temporal probability of landslides can be achieved in data deficient areas and the combination of the two probability models improves the prediction of future landslides. This research provides information that can be used by land-use managers to limit the exposure of people and property to future landslides which most often take many by surprise.