The landslide database used in this study includes the DISASTER database and has detailed information about the date and location of landslide occurrence. The DISASTER database was carried out by exploring several daily and weekly newspapers, published in Portugal between 1865 and 2010, which include all the landslides that caused fatalities, injuries, missing people, evacuated and homeless people. The method used to construct the DISASTER database has been widely described and can be found in Zêzere et al. (2014). Additionally, using the same newspaper sources, landslides that did not cause any human damage during the same time period were identified and included in the database that supported this study. It should be pointed out that falling walls and instabilities directly resulting from engineering works were rejected. Similarly, the landslides in active coastal cliffs were not included in the database. The database structure is divided into two sections: landslide features and landslide damages. The first section includes information on landslide type, temporal and spatial location, triggering factors and newspaper metadata. The second section refers to the human consequences of landslides (fatalities, injuries, missing people, evacuated and homeless people), and direct and indirect damage to buildings, structures, roads and railroads.

Our analysis is focused on the dates of landslide occurrences. The newspapers are a reliable data source, despite the existing uncertainty concerning the spatial location of many reported landslide events, as well as on their type. Only landslides with at least 1 day of accuracy were included in the database. The spatial accuracy of landslide cases was divided into five classes, following Zêzere et al. (2014): (i) locations with the exact coordinates (accuracy associated with scale 1:1000), (ii) locations based on local toponymy (accuracy associated with scale 1:10000), (iii) locations based on local geomorphology (accuracy associated with scale 1:25000 scale), (iv) locations in the centre of the parish and (v) locations in the centre of the council. A total of 400 landslide cases were inventoried, with the majority (83 %) located with accuracy corresponding to classes (i) to (iii). These landslides affected clay (40.24 %), sandstone and conglomerate (22.52 %), limestone (16.52 %), volcanic (11.11 %), marly and marly limestone (9.01 %) and granite (0.60 %). The landslide type was classified following the Cruden and Varnes (1996) classification scheme. Slides are the dominant landslide type in the database (53.8 %), followed by falls (14.4 %). Flows and complex slope movements are less representative (2.4 and 1.5 %, respectively). The landslide type is unknown in 27.9 % of the cases. In this study the analysis was performed for all landslide types, following the approach of similar studies (e.g. Brunetti et al., 2010; Rosi et al., 2012; Peruccacci et al., 2017).

In this study the following definition was adopted for landslide events: an individual landslide or a set of landslides that occurred on a precise date (day). In those cases where the activity period of a landslide was reported as lasting several days, the first day of the period was considered for the landslide event.

The selection of the reference rain gauge took into account the available time series, the data quality and resolution and the climatic representativeness. The daily rainfall data were collected at the Lisboa-Geofísico rain gauge (latitude 38.72∘ N, longitude 9.15∘ W, elevation 77 m), located within the city of Lisbon. The rainfall daily measurements at Lisboa-Geofísico started in 1864 and is one of the few rain gauges with centennial-long daily records in Portugal. A long time series of rainfall data is an important condition for creating comprehensive thresholds based on the analysis of the rainfall return period. In addition, this rain gauge presents reliable data, the quality and completeness of which was already tested and confirmed by Kutiel and Trigo (2014). The rainfall measurements have been taken without interruption and always in the same place since 1864. Furthermore, the rain gauge is climatically representative of the Lisbon region, with a rainfall regime influenced mainly by the atmospheric general circulation and the oceanic proximity.

The daily rainfall refers to the period between 09:00 UTC on the previous day and 09:00 UTC on the day of measurement, whereas the landslide dates are ascribed to a period from 00:00 to 23:59 UTC. Due to this difference, the date of each landslide event reported by the newspaper was compared with the daily rainfall registered in 3 days (from the day before up to the day after), and the day registering the highest rainfall amount was selected as the day of the landslide event.

The reconstruction of cumulated rainfall follows the method proposed by Zêzere et al. (2005). In a first step, the daily rainfall data registered at the Lisboa-Geofísico rain gauge during the period 1864/1865–2009/2010 were organized by climatological year (September to August). The decision to use the climatological year instead of the hydrological year (October–September) is justified by the rainfall regime of the study area. Starting the analysis in September, after the month with the low values of rainfall (August), we capture the complete transition period towards the wet season in each year. Afterwards, for each day, from 1864 to 2010 the cumulated antecedent rainfall was calculated for the durations of 1, 2, 3, 4, 5, 10, 15, 20, 30, 40, 50, 60, 75 and 90 days.

The maximum annual records of daily rainfall and cumulated rainfall for each duration were extracted and analysed using the theoretical distribution described by Gumbel (1958). This distribution is also known as the distribution of Fisher–Tippett and is applied to the extreme values. With the Gumbel law it is possible to obtain the probability of occurrence of each rainfall value within the series with N values. The reduced Gumbel distribution (y) is calculated with Eq. (1): y=-ln-lnmN+1, where m is the position number of the respective observations and N is the total number of observations. Considering this distribution, the theoretical frequencies can be calculated by the average and standard deviation for the reduced Gumbel distribution (My and Sy) and for the rainfall values (Mx and Sx). Eq. (2) expresses the theoretical trend: y=α(x-μ), where y is the reduced variable and x the rainfall value. The parameters α and μ are calculated as follows: 1/α=Sx/Syμ=Mx-My/α. Finally, the probability of exceedance of any rainfall value is given by the Eq. (5): P(x)=1-e-e-y. For each landslide event the cumulated antecedent rainfall was assessed for the durations of 1, 2, 3, 4, 5, 10, 15, 20, 30, 40, 50, 60, 75 and 90 days. For each antecedent rainfall the return period (RP) was calculated with Eq. (6): RP=11-e-e-y. The pair (cumulated rainfall duration) with the highest return period was considered to be the critical rainfall combination, responsible for triggering the landslide event. This assumption is not physically based, but has been applied in previous work (e.g. Marques et al., 2008; Zêzere et al., 2008, 2015) and provides the best discrimination of the rainfall events related to landslide activity (Zêzere et al., 2005). Moreover, this approach agglomerates the rainfall that triggered the landslide event and the antecedent rainfall that contributed as a landslide preparatory factor.

As was previously mentioned, our landslide database was collected from newspaper sources and in some cases the rainfall triggering is not clear. Therefore to calculate the threshold we decided to use only the landslide events which have a critical rainfall combination with a return period exceeding 3 years. The boundary is arbitrary, but this criterion reduces the possibility of considering landslide events with a triggering factor other than rainfall (e.g. human action). The landslide events associated with critical rainfall combinations with return period less than 3 years were assumed to not be triggered by rainfall.

Finally, the climatological years without landslide records in the database were selected and the maximum yearly cumulated rainfall was identified for durations lasting from 1 to 90 consecutive days. These data were further used as rainfall events that did not generate landslide events and are crucial for the thresholds definition and calibration.

The critical distance where the rain gauge is regionally representative was evaluated by drawing several buffers up to 60 km from the rain gauge (5, 10, 15, 20, 30, 40, 50 and 60 km). The ratio between the non-rainfall-triggered landslide events and the rainfall-triggered landslide events within each buffer was used to identify the area where the rain gauge is representative.

During the analysed time period (1864/1865–2009/2010) landslides in the study area were mostly triggered by rainfall and the earthquake trigger can be neglected (Vaz and Zêzere, 2016). The human action was an additional landslide-triggering factor, in particular through artificial cuts and drainage constraints associated with the progressive enlargement of urban areas. As it was already mentioned, the reference rain gauge is located in the city of Lisbon, where the landslides induced by human action are expected to be higher in number when compared with the outside of the urban area. Following this assumption, the ratio between the non-rainfall-triggered landslide events and the rainfall-triggered landslide events should decrease as the distance from the gauge increases. If this relation does not occur we assume that the rain gauge is no longer representative for the corresponding buffer. Therefore, the lowest ratio between non-rainfall-triggered landslide events and rainfall-triggered landslide events was considered to define the critical distance where the rain gauge is regionally representative to assess rainfall thresholds for landslide occurrence.

Landslide events registered within the critical distance from the rain gauge were considered and rainfall thresholds were established using linear and potential regression, based on cumulated rainfall duration with the highest return period. The lower-limit and the upper-limit rainfall thresholds were also defined following the suggestion by Glade et al. (2000). The lower-limit and the upper-limit rainfall thresholds were defined by linear regression based on two pairs. The lower limit was established by iteratively selecting two landslide events associated with different durations with the lowest values of cumulated critical rainfall and ensuring that the complete set of landslide events fall above the threshold. The upper limit was established by iteratively selecting the two highest pairs (cumulated rainfall/duration) that did not trigger landslides and ensuring that the complete set of non-landslide events fall below the threshold.

When representing thresholds we avoid using logarithm scales, and thresholds were established as linear relationships instead of using a power law, with a single exception (the potential regression threshold). These options maximize the zone between the lower-limit and upper-limit thresholds, thus allowing the distinction between rainfall events that generated (did not generate) landslide events.

The performance of rainfall thresholds was evaluated using ROC metrics. ROC analyses are commonly used to validate susceptibility landslide models (Beguería, 2006; Kappes et al., 2011) and it is based on confusion matrices. The principles used in these analyses can also be applied to calibrate the rainfall thresholds (e.g. Staley et al., 2013; Gariano et al., 2015a; Zêzere et al., 2015). The confusion matrix is used to assess the correct and incorrect predicted observations, for positive and negative cases (Beguería, 2006). Therefore, the analysis is based on the evaluation of true positive (TP), false negative (FN), true negative (TN) and false positive (FP) cases. When applied to rainfall thresholds the TP corresponds to the landslide events in which the rainfall combination (cumulated rainfall duration) is above the threshold. The FN are landslide events for which the rainfall combination (cumulated rainfall duration) is below the threshold. The rainfall combinations that did not resulted in landslide events are classified as TN if they are below the threshold or FP if they are above the threshold.

Also, four ROC metrics functions described by Staley et al. (2013) were used in this study (Table 1). The true positive rate (TPr) is the proportion of landslide events that were correctly predicted by the threshold (Table 1). The false positive rate (FPr) is the proportion of rainfall events above the threshold for which there is no information on landslide occurrence. The false alarm rate (FAr) is the ratio between false predictions and the complete set of rainfall events above the threshold. The threat score (TS) is used to evaluate the threshold to maximize the number of correct predictions while minimizing the rate of FP and FN. A TS = 1 represents a perfect model but is reduced by incorrect predictions.

The probability of a rainfall event above the rainfall threshold resulting in a landslide event was measured by the positive predictive rate (PPr), which was previously described by Bradley (1997) and Fawcett (2006). The PPr measures the relationship between the rainfall events above the threshold that resulted in landslide events and the complete set of rainfall events located above the threshold, as follows: PPr=TPTP+FP. Therefore, the PPr is the opposite of the FAr and can also be calculated by the expression: PPr=(1-FAr). Using this approach, several linear rainfall thresholds were plotted in the zone between the lower-limit and the upper-limit rainfall thresholds, and the corresponding PPr were calculated in order to compute the probability of landslide events associated with each threshold. In addition, the probability of each rainfall threshold was computed based on the return period of the corresponding cumulated rainfall duration.

Lastly, the performance of the lower-limit threshold was assessed beyond the critical distance of the rain gauge. For each buffer referred to in Sect. 3.2 the ratio between the FN and the total set of landslide events (TP + FN) was systematically evaluated. We assume the lower-limit threshold can only be applied to those buffer distances where this ratio remains stable.

Within the area located up to 60 km from the reference rain gauge 223 landslide events were identified dating from 1865 to 2010 (Fig. 4). However, the return period computed for the cumulated rainfall does not exceed 3 years in 92 landslide events. Therefore, according to the criterion defined in Sect. 3.2, these landslide events were assumed not to have been triggered by rainfall.

The ratio between the number of non-rainfall-triggered landslide events and the number of rainfall-triggered landslide events was calculated for each buffer zone shown in Fig. 4. The results are summarized in Table 2 and were used to define the critical distance at which the rain gauge is regionally representative, and to select the landslide events considered to compute the rainfall thresholds. We acknowledge the ratio differences that occurred only to the second decimal place (Table 2), but these differences can be interpreted considering the characteristics of the study area. Within the 5 km buffer the calculated ratio is relatively high (0.65). The first buffer zone includes Lisbon city centre, which explains the high number of landslides triggered by factors other than rainfall, mainly due to human actions. In the following buffer zone (10 km) the ratio decreases to 0.63. This decrease was expected as the urban area extension decreases in the second buffer, thus justifying the lower number of non-rainfall-triggered landslides. The ratio between the non-rainfall-triggered and the rainfall-triggered landslide events increases to 0.66 within the 15 km buffer zone, and the ratio ranges between 0.66 and 0.70 in the next buffer zones up to 60 km from the rain gauge. The increasing ratio in distance exceeding 10 km from the rain gauge cannot be attributed to the occurrence of an unexpectedly high number of non-rainfall-triggered landslide events, but can only be explained by a decrease in spatial representativeness of the rain gauge data in areas beyond 10 km. Therefore, we consider 10 km the critical distance at which the rain gauge is representative, and the rainfall thresholds were computed considering only the landslide events registered within this zone.

In the area located up to 10 km from the reference rain gauge of Lisboa-Geofísico 60 landslide events, with return periods below 3 years, were assumed to be non-rainfall-triggered landslides, and therefore were not considered for the threshold calculation and analysis. Moreover, 96 rainfall-triggered landslide events were identified, which include 187 individual landslides. The yearly and monthly distributions of these landslide events are shown in Figs. 2 and 3, respectively. The rainfall-triggered landslide events occurred mainly in wet years: 89 % of landslide events were registered in years with rainfall above the MAR. The climatological years 1876/1877, 1946/1947 and 1968/1969 are in the top regarding the number of landslide events (six events in each year). In these three climatological years the annual rainfall was above 933 mm at the reference rain gauge, which exceeds the MAR by more than 30 %. However, there is not a direct relationship between the MAR and landslide events because landslide occurrence is usually related to rainfall events over a few days or weeks, which are not expressed by the mean annual rainfall. Indeed, landslide events were also registered in 10 years with annual rainfall below MAR, as was the case for 1909/1910, which registered two landslide events.

The monthly distribution of landslide events follows the rainfall distribution over the year in a Mediterranean climate, with dry summers and wet winters. The landslide events essentially coincide with most rainy months, as 92 % of events occurred from November to March. Within this period, January and February stand out with the highest concentration of landslide events (24 and 22.9 %, respectively). Besides the monthly rainfall percentile, Fig. 3 represents the 30-day cumulated antecedent rainfall for each landslide event and shows that 96 % of landslide events are above the 70th percentile. If we consider the 90th percentile this value decreases to 79 %, but it continues to highlight the exceptionality of rainfall during the 30 days before the landslides are triggered.

For each landslide event the critical cumulated rainfall duration was obtained following the method described in Sect. 3.2. The obtained critical durations associated with landslide events range from 1 to 90 consecutive days. The monthly distribution of critical durations is shown in Fig. 5 for the rainfall-triggered landslide events. The shorter rainfall events (less than 20 consecutive days) occurred mainly from September to December (56 %) at the beginning of the rainy period. On the contrary, when associated with longer rainfall periods (more than 20 consecutive days) the landslide events were more frequent from January to May (86 %).

Figure 6 illustrates the cumulated rainfall duration combinations that resulted in landslide events and the typical return periods established for 3, 5, 10, 25, 50, 100, 150 and 200 years. Around 64 % of the cumulated rainfall duration that resulted in landslide events have a return period below 10 years. However, four landslide events had a rainfall amount and duration with a very high return period, above 150 years. Figure 6 also identifies the landslide events that include multiple landslides and the landslide events that are constituted by a single landslide. The distribution of both groups is inconclusive, as the landslide events containing multiple landslides are not always directly related to the exceptionality of the rainfall event, i.e. the critical cumulated rainfall duration combination with a higher return period.

The rainfall conditions (cumulated rainfall duration) associated with each landslide event were considered to define rainfall thresholds using linear and potential regression (Fig. 7). The linear regression follows the equation R=5.5D+124.6, where D is the duration in days, whereas the potential regression follows the equation R=67.8D0.46 (Table 3). The coefficient of determination is very high in both cases (R2=0.8 and 0.9, respectively). Both rules can be used as rainfall thresholds for landslide occurrence in the study area; however none of them ensure a low number of false negative occurrences (i.e. landslide events below the threshold).

To calibrate the thresholds, the maximum yearly rainfall for each duration (1 to 90 consecutive days) was calculated for those climatological years without records of landslide events in the analysed period (1865–2010). These records represent rainfall events not associated with landslides and are symbolized by grey dots in Fig. 7 (1428 dots). The majority of these rainfall events (96.6 %) drop below the threshold obtained with the potential regression. However, there are 57 false negatives occurrences (i.e. events that occurred without being predicted), as well as 48 false positives (i.e. rainfall events lying above the threshold, without any landslide reported).

In the next step, the lower-limit and the upper-limit rainfall thresholds were determined. The former establish the threshold below which there are no true positives (landslide events), whereas the latter establish the threshold above which there are no false positives (rainfall events without landslides). The lower-limit threshold follows the equation R=4.4D+56.5, and the upper-limit threshold follows the equation R=7.3D+235.8, where D is the duration in days (Table 3).

Table 3 also summarizes the ROC metrics for the regression thresholds (linear and potential) and the lower-limit and the upper-limit thresholds. The TPr measure the proportion of landslide events that occurred when the combinations of rainfall duration are exceeded and show the efficiency of a threshold to predict a landslide event. On the other hand, the FPr measures the proportion of combinations of rainfall duration that are above the threshold but did not result in any known landslide event. For the potential regression threshold, the TPr is not very high (0.41, best value is 1) but the FPr is a good result (0.03, the best value is 0), which means that the thresholds have a low probability of a false detection. The TPr is equal to 1 for the lower-limit threshold, considering that it was drawn to avoid FN occurrences. However, the FPr and the FAr are very high (0.37 and 0.85, respectively) as a consequence of the typical low values of the threshold. The lower limit is a conservative threshold, and its main advantage is predicting all the landslide events, but it also includes a very high number of false positive events. On the contrary, the upper-limit threshold is only surpassed by true positive occurrences, so the FPr and FAr have the best result (0 value). However, the TPr is very low (0.03) reflecting the high number of false negative events. The threat score (TS) provides a better understating of each threshold performance as it relates to the TP, FN and FP occurrences. The linear regression threshold has the best result with 0.29 of TS when compared with the potential regression threshold (0.27), the lower-limit (0.15) and the upper-limit (0.03) thresholds (Table 3). The false alarm rate (FAr) also gives a better result for the linear regression threshold in comparison with the potential regression threshold (0.47 and 0.55, respectively).

Although only the FPr returns very good values, the regression thresholds, linear or potential, can be used as acceptable thresholds to predict landslide events in the study area. However, the lower-limit and the upper-limit thresholds should not be excluded, as the zone between these rainfall thresholds defines the boundary conditions where any rainfall event may (or may not) cause a landslide event.

The PPr summarized in Table 3 gives the probability of a rainfall event resulting in a landslide event when the threshold is exceeded. The value ranges from 0 to 1 where 1 indicates 100 % probability of landslide occurrence. Accordingly, when the lower-limit threshold is exceeded, the probability of occurrence of a landslide event is relatively low (0.15). On the contrary, when the upper-limit threshold is reached the occurrence of a landslide event is certain (PPr=1). The PPr associated with the regression thresholds is close to 0.5, but higher for the linear trend in comparison with the potential trend (0.53 and 0.45, respectively).

The systematic comparison between true positives and false positives and the PPr calculation were taken into consideration to draw five intermediate rainfall thresholds in the zone between the lower-limit and the upper-limit rainfall thresholds, representing the 20, 30, 40, 50 and 60 % probabilities of the occurrence of a landslide event (Table 3 and Fig. 8). Within this chart, any rainfall event exceeding the PPrx% threshold has the x % probability of generating a landslide event in the study area. Further probabilities could not be computed due to a lack of data.

To analyse the performance of the PPr rainfall thresholds presented in Fig. 8, the ROC metrics were calculated and are summarized in Table 3. As expected, the false alarm rate decreases as the PPr increases, and the same occurs with the true positive rate and the false positive rate. According to the threat score (TS), the PPr 40 % and the PPr 50 % are the rainfall thresholds with the best performance (TS = 0.34 in both cases).

The return period of the rainfall associated with each calculated threshold presents a wide variation according to the considered number of consecutive days of cumulated rainfall (Fig. 9). As a rule, shorter durations (below 10 days) present a high return period in comparison with longer durations, independently of the type of rainfall threshold. In the case of the upper-limit threshold, the PPr 60 % and the PPr 50 % thresholds, the obtained return periods for the shorter durations are less realistic and the corresponding rainfall values were never registered in the rainfall data series of the Lisboa-Geofísico rain gauge.

Figure 9 also shows that the rainfall threshold is easier to reach for periods ranging from 15 to 45 consecutive days, namely for the regression threshold (linear), the lower-limit threshold, and the PPr 20 %, PPr 30 % and PPr 40 % thresholds. For the mentioned durations these thresholds will be exceeded by rainfall events with return periods less than 10 years. However, for durations longer than 45 consecutive days, the return period of the corresponding rainfall denotes an increasing trend for all the thresholds, although the return period remains lower when compared with periods less than 10 consecutive days.

Data summarized in Figs. 8 and 9 can be combined to better characterize any rainfall threshold. Taking as an example the PPr 60 % threshold, we can state that the highest yearly probability that this threshold is exceeded is 5 % (20-year return period) associated with 30 to 60 consecutive days. The probability of landslide occurrence is 60 % given rainfall exceeding the threshold. Therefore, the maximum yearly combined probability of a landslide event associated with the PPr 60 % threshold is 3 %.

Although the rainfall thresholds for landslide occurrence were defined taking into consideration the landslide events registered up to 10 km from the reference rain gauge of Lisboa-Geofísico, we admit that the obtained thresholds may be valid for distances greater than 10 km. In accordance, the performance of the lower-limit threshold was evaluated for each buffer zone represented in Fig. 4. The ratio between the FN and the total set of landslide events, FN / (TP + FN), for the different buffer zones is summarized in Table 4. As expected, the lowest ratio (0.167) corresponds to the buffer zone of 10–15 km. The ratio remains relatively stable within buffer zones up to 50 km from the rain gauge (ratio ranging from 0.2 to 0.297), and increases significantly in the buffer zone of 50–60 km (0.5). Therefore, taking into account the ratio FN / (TP + FN) we can conclude that the prediction model remains efficient up to 50 km from the rain gauge. Consequently, although established with landslide data registered up to 10 km, the thresholds identified for the Lisboa-Geofísico rain gauge may be applied with reasonable confidence for the area within 50 km.

The concept of a landslide event is not straightforward as has been applied in the literature to describe a landslide or a set of landslides usually related to a specific triggering factor, such as an intense rainstorm (Crozier and Glade, 1999; Zêzere et al., 2014). When the landslide event is a single landslide, generally there is no problem with identifying the date of the event that will be related to the daily rainfall data for the rainfall threshold assessment. However, when several landslides are triggered over consecutive days in a study area, this may be a source of bias for the rainfall threshold definition. Usually, a date between the start and the end of the rainfall event is selected (e.g. Gullà et al., 2012; Gariano et al., 2015b), and therefore, a unique combination of cumulated rainfall duration is calculated. The selection of the landslide event date is critical for this method as it can lead to an overestimation of the threshold, particularly if the end date of a long-lasting rainfall event is chosen. In these cases, the chosen cumulated rainfall duration may be not representative of the triggering conditions of landslides that occurred at the beginning of the event. To address this problem, in this work a landslide event was considered to be an individual landslide or a set of landslides that occurred on a precise date (day). Therefore, in those cases in which different landslides occurred on consecutive days, each day was considered a landslide event and the corresponding antecedent rainfall was used for the rainfall threshold assessment. In addition, when the activity period of a landslide was reported as lasting several days, the first day of the period was considered for the landslide event.

Several benefits and drawbacks can be outlined regarding the use of a single rain gauge or multiple rain gauges to assess rainfall thresholds for landslide initiation. The use of multiple rain gauges is a typical option for assessing rainfall thresholds (e.g. Caine, 1980; Gariano et al., 2015b; Peruccacci et al., 2017). The main advantage lies in the proximity of the rain gauge from the landslides, which provides a better relationship between rainfall and landslide triggering. However, the rainfall thresholds obtained in different rain gauges may be biased due to the different topographic and physiographic contexts characterizing each point of rainfall measurement. In these circumstances, the obtained rainfall thresholds will be biased by the differences regarding the rainfall regime of each location. Therefore, the merging and comparison of several rainfall data sets obtained in different places should be preceded by the normalization of rainfall data.

In addition, this type of analysis demands a high density of rain gauges, which is only available for recent years. In Europe the number of stations increased after 1960 and peaked between 1980 and 1990 (Haylock et al., 2008). In Portugal, a reliable rain gauge network has only existed since 1980, when the mean distance between neighbouring rain gauges was about 7.9 km (Belo-Pereira et al., 2011). Therefore, the use of multiple rain gauges to assess rainfall thresholds in the Lisbon region before 1980 would increase the threshold uncertainty due to the very low density of the available rain gauges. The restriction of the analysis to the period 1980–2010 was a possible way of overcoming this limitation. However, this option was not considered because the number of landslide events would be drastically reduced (from 96 to 15 landslide events, Fig. 2), and the same would happen concerning the number of available rainfall events that did not generate landslides. The reduction of data representativeness would decrease the reliability of obtained rainfall thresholds.

Our landslide database covers a 145-year period (from 1865 to 2010) and we decided to analyse the complete period; thus, the selection of a single rain gauge was inevitable. The Lisboa-Geofísico rain gauge has taken uninterrupted rainfall measurements since 1864 and it is one of the few rain gauges in Portugal with long-term rainfall data set. The comparable rain gauges (Évora, Porto, Guarda and Coimbra) are located more than 100 km from Lisbon. A long time series is important for applying the return period and the Gumbel probability to the rainfall data. Moreover, it provides a more reliable relationship between the rainfall conditions and landslides, strengthening the reliability of the obtained thresholds. This is particularly important for a climate with great variability, such as the one existing in the study area.

Identifying the rainfall responsible for the landslide occurrence is the basis for any empirical rainfall threshold calculation. A range of procedures that define the rainfall critical period associated with landslide events have been proposed in the literature (e.g. Guzzetti et al. 2007; Segoni et al., 2014). Moreover, even the definition of critical rainfall is not straightforward. Aleotti (2004) defined the critical rainfall as the rainfall period starting when a sharp increase in rainfall intensity is identified and ending when the first landslide is triggered. Therefore, in such circumstances, the cumulated rainfall before the rainfall increase is considered antecedent rainfall and is not included in the critical rainfall. Brunetti et al. (2010) and Peruccacci et al. (2012, 2017) use the concept of “rainfall event”, as a period of continuous rainfall separated by a dry period, with a seasonal variability concerning the length of the dry period (48 h in the dry season and 96 h in the wet season). In our study, the critical rainfall joins together the antecedent rainfall (acting as a landslide preparatory factor) and the rainfall that triggered the landslide event. Our procedure for defining the critical rainfall combination, which is responsible for preparing/triggering the landslide event, is based on the return period calculation and selects the cumulated rainfall with the highest return period.

This approach has the advantage of being objective and easily reproducible for other areas, and provides rainfall thresholds with the most optimistic results concerning the ROC metrics. However, the use of the return period implies a “rigid” statistical relationship between the rainfall and landslides, which does not always occur. Moreover, in some cases, the cumulated rainfall associated with the landslide triggering is not very high. However, the use of other empirical procedures to define the critical rainfall period, as previously mentioned, is based on subjective observations like the duration of the dry period that bounds the rainfall events.

Probably, the identification of the critical rainfall period for a specific landslide event can only be precise using coupled geotechnical and transient hydrological physical models. However, each slope is unique and the rainfall is not uniform both in time and space, which explains the difficulty of establishing rainfall thresholds based on physically based models at the regional scale.

The uncertainty related to the triggering factor is particularly high when newspapers are the main source of information and, additionally, when a long time series is being analysed. As a rule, only newsworthy content is reported by newspapers, which certainly create bias in the landslide database. For instance, landslides that generated human damage or occurred in an urban environment are usually highlighted, which increases the probability that landslides triggered by human action are included in the database, despite their exclusion whenever a description suggests an anthropic intervention. On the contrary, landslides triggered by rainfall that did not generate any social or economic damage were probably unreported by the newspapers. In addition, the long time that elapsed since the occurrence of some landslides inhibits the use of recent methods and techniques to confirm the rainfall triggering. For example, the confirmation of landslide events using aerial photo interpretation is only possible in Portugal for the period after 1947.

Using field-based landslide inventories in the Lisbon region, Zêzere et al. (2015) considered a rainfall-triggered landslide event any date on which at least five individual landslides are known to have occurred on natural slopes. This criterion reduces the possibility of including landslides triggered by human action. However, this criterion cannot be used in the present study, because landslides reported in newspapers are a small sample of the total number of triggered landslides. Therefore, any reported date, even those reporting a single landslide, should be admitted as a “landslide event candidate”. Addressing the issue of the triggering factor, those landslide events associated with rainfall combinations (cumulated rainfall duration) with a return period below 3 years were rejected as rainfall triggered. Using field-based landslide inventories in the Lisbon region, Zêzere et al. (2015) showed that only 12 % of landslide events triggered by rainfall have a cumulated rainfall return period below 3 years and landslide events were not registered with rainfall conditions with return periods below 2 years. Given our data source feature (based in newspaper) and our study area (integrated in an urban area) a more conservative boundary was preferred, and therefore, the selection of a 3-year return period.

This criterion can eventually eliminate some (a few) landslide events triggered by rainfall in the study area. However, the possibility of including non-rainfall-triggered landslide events would increase by not applying this criterion. The inclusion of non-rainfall-triggered landslides in the analysis would bias the rainfall thresholds as well as the ROC metrics, generating a higher number of undesirable false alarms.

This criterion proved to be suitable for distinguishing between rainfall events that triggered and did not trigger landslide events in the study area. However, further investigations should be carried out on this topic, namely in other study areas. An alternative method to the 3-year return period criterion could be the calculation of the thresholds in the range 5–10 km, thus excluding the current urban area. However, the landslide database used in this analysis covers a very long time period (145 years) and the urban area extension did change considerably. For example, at the end of the 19th century extensive rural zones were present within the 5 km buffer. Moreover, this option would reduce the number of landslide events considered in the analysis from 96 to 37, which would reduce the reliability of the obtained rainfall thresholds.

The discussion on the spatial representativeness of a rain gauge data series used to assess rainfall thresholds for landslide activity is scarce in the literature, which is surprising, taking into consideration the large number of papers dealing with empirical rainfall thresholds published in recent years. In previous work using multiple rain gauges, the distance between the gauge and the landslides is the criterion used to select the rain gauge, along with the setting features (e.g. elevation difference and morphological settings) (e.g. Brunetti et al., 2010; Peruccacci et al., 2017). However discussion on the topic is scarce and different distances have been proposed for the same region. For instance, for the Calabria region (Italy) Vennari et al. (2014) used 12 km as a limit, whereas Gariano et al. (2015b) used 5 km.

To the best of our knowledge the spatial representativeness of a single rain gauge used to assess rainfall thresholds was never addressed before. In this work, we applied a method to compute the critical distance based on the ratio of non-rainfall-triggered landslide events and rainfall-triggered landslide events tested along several buffer zones starting from the rain gauge at 5, 10, 15, 20, 30, 40, 50 and 60 km. Our method takes into account both the source of landslide data (newspapers) and the location of the rain gauge in the urban area. We acknowledge that this method is valid in urban areas, as is the case of the Lisbon region, and can be applied in other zones with a similar context. However, the method cannot be directly applied in non-urban areas, which is a drawback.

In addition, effort was made to evaluate the regional performance of the lower-limit threshold, which was proved to be applied with reasonable results up to 50 km from the rain gauge. It should be pointed out that the climatic and topographic features of the study area allow for a spatial enlargement of the threshold. The rainfall regime of the region is spatially consistent and it is mainly influenced by the general circulation of the atmosphere and by proximity to the ocean, with the same weather types associated with high rainfall (Trigo and DaCamara, 2000; Ramos et al., 2014). In addition, the orographic effect on the rainfall distribution is low in the region, which enlarges the spatial representativeness of the reference rain gauge. However, the distance at which the thresholds can be applied will always be connected with high levels of uncertainty associated with the rainfall discontinuity both in space and time. Therefore, the consideration of the lower-limit threshold up to 50 km should be used only if no other threshold is available.