The present study uses historical rockfall data that are extracted from the landslide database of Germany see. Scientific publications, governmental reports, police reports, civil protection reports, newspapers, field data collections, and GIS and web analyses were the information sources for the landslide database, which currently contains about 6000 mass movement events of different types. The database mainly covers the last 200 years, with the oldest event dated to 1137. Information on 670 rockfall events (Fig. ) is included in the rockfall dataset. The focus of the present study is on a number of geomorphological processes (e.g. rockfall, rock topple, debris fall, debris topple) that are characterised by the rapid gravitational downslope fall of debris or rocks . Due to the different particle sizes and volumes of the detached masses, the mentioned processes are subsumed under the generic term rockfall in this study . In addition to an identification number, the location (i.e. coordinates) and the date of occurrence for each rockfall event are stored in the dataset. For a total of 343 (642) rockfalls the day (year) of occurrence is known, while the remaining are undated. The time span of rockfall occurrences ranges between 1480 and 2018, with the majority of them (n=621) recorded from 1873 onwards. It is important to note that the rockfall database is not comprehensive. The increase in the number of recorded events with time (Fig. ) is not due to climatic conditions but reflects the fact that data on rockfall events were more readily available in recent years.

The dense spatial clustering of rockfall events and high temporal data homogeneity guide the selection of three study areas (Fig. ; ES is the German part of the Elbe Sandstone mountains, HL is northern Hesse and southern Lower Saxony, and HR is western Hesse and Rhineland-Palatinate).

For this study, datasets with a long record and high horizontal resolution were used in order to identify meteorological and hydrological conditions for as many rockfall events as possible with sufficient accuracy. It was therefore decided to use the gridded REGNIE dataset for daily precipitation amounts. The dataset is compiled from spatially interpolated gauge measurements of the quality-controlled German weather service (Deutscher Wetterdienst, DWD) stations. REGNIE is available since 1931 for western Germany. For the new (post-reunification) federal states, the time series starts in 1951. The grid boxes have a size of 1 km2.

In order to study precipitation intensities, the gridded radar-based climatology RADKLIM was used. The dataset includes hourly precipitation from radar measurements adjusted to station observations and has a horizontal resolution of 1 km × 1 km. For the present study, the daily maxima were extracted. The time series is comparatively short, as it only starts in 2001.

For temperature, it was decided to use the gridded E-OBS dataset as it goes back to the year 1950. The horizontal resolution of the grid is 0.1∘ × 0.1∘, which corresponds to approximately 7 km × 11 km in Germany. For the analysis of freeze–thaw cycles, the ensemble mean of near-surface atmospheric daily minimum and daily maximum temperatures provided in the v21.0e version of the E-OBS dataset was used. A freeze–thaw cycle was defined as the transition from a daily minimum temperature below - 1 ∘C to a daily maximum temperature higher than 0 ∘C.

The subsurface water content (e.g. soil moisture, cleft water, water in matrix pores) is measured generally only at very few sites. Spatially consistent soil moisture monitoring in Germany, for example, relies on modelled soil moisture . In this study we attempt to utilise modelled soil moisture as a representative for all types of subsurface water. We analyse the results of a simulation with the state-of-the-art, grid-based hydrological model mHM , which was calibrated at hydrological stations using daily time series of observed discharge. The model has a daily time step, a horizontal resolution of 5 km × 5 km and six vertical levels from the surface to a depth of approximately 1.8 m. The hydrological model mHM considers different soil types. Each soil type has different soil layers and thus site-specific soil characteristics such as substrate distribution and hydraulic conductivity. The infiltration from the surface into the ground depends on these soil characteristics. The set-up is based on European datasets as described in and . We analysed the relative moisture content (i.e. degree of saturation) for the entire column from the surface to a depth of approximately 1.8 m. It is common practice for this model to further normalise these values using percentiles as the variability of the modelled values is too low.

With respect to our aim to develop a statistical model that can be used to analyse the rockfall probability under climate change conditions, a challenging point of using simulated soil moisture is that it is stored only for some climate scenario simulations. Additionally, the moisture variables and the depth levels they represent differ between climate models. Therefore, the usage of a pore-water proxy (D) as an alternative to simulated soil moisture was tested as a predictor for the logistic regression model. D is defined as the difference between precipitation accumulated over a period of time (Precacc) and the potential evapotranspiration (PET) during this period: 1D=Precacc-PET.

The term D is also the basis for the calculation of the standardised precipitation evapotranspiration index SPEI;, which includes a standardisation of D in order to allow comparisons between different climatic regions, which is not necessary here. Different empirical methods exist to determine potential evaporation. In this study, the method first proposed by in the version modified by was applied. As input parameters it needs extraterrestrial radiation (which depends on latitude and day of the year), the period mean of maximum and minimum daily temperatures, and mean precipitation over the period of interest (which is used as a proxy for cloudiness). D does not depend on the material of the ground and is an indicator of general water availability, thus accounting for water in rock discontinuities as well as in matrix pores. Therefore, further mentions of pore water include water in discontinuities if not specified otherwise.

A relationship between the triggers and events can only be established for the sites and periods for which both elements are known. Thus, the analyses carried out in this paper include only data from grid boxes that contain the site (es) of at least one rockfall event occurring within the observational period of the respective record. Percentiles for soil moisture and D are determined using the observations at these sites rather than all grid points within Germany. We refer to them as “across-site” percentiles.

Weight of evidence (WOE) can be used to describe the relationship between an independent and a dependent variable and to rank the predictive power of different independent variables e.g.. It is defined as the logarithm of the Bayes factor : 2WOE=ln⁡f(XY¯=1)f(XY¯=0), where X is the independent variable and Y is the binary information of whether the event occurred or not. f(XY¯) denotes the conditional probability density function for X if Y is true (=1) or false (=0).

In practice, a continuous independent variable (e.g. precipitation amount) is split into bins containing an equal number of observations. The WOE for each bin (b) is then calculated separately. It depends on the fraction of days with an event (here rockfall) to that of uneventful days. For categorical variables the WOE is determined for each category. 3WOEb=ln⁡%NoRockfallb%Rockfallb

An integral measure for the strength of the relationship between the dependent and independent variable is the information value IV;. It is calculated as 4IV=∑b=1nbins%NoRockfallb-%Rockfallb×WOEb, with nbins being the number of bins. According to , a predictor is not useful for statistical modelling if the IV value is less than 0.02. The IV is also used to rank the variables according to their influence.

Logistic regression is used to model the relationship between predictor variables and the probability of a binary response variable. For logistic regression, a generalised linear model with a logit link function is fitted . The probability p of the binary event (e.g. rockfall yes/no) can be expressed as 5p=11+exp⁡(-b0-b1x1-…-bixi), where x1,…,xi are the predictors (i.e. meteorological and hydrological variables) and b0,b1…,bi are the regression coefficients. The regression coefficients are determined by maximising the log likelihood. For this study, the glm function of the statistical software R was used for this task.

The classical score to compare logistic regression models of different complexity is the Brier skill score, which, however, becomes unstable for rare events such as ours . We therefore use the logarithmic skill score (LSS) instead, which behaves similarly to the Brier skill score but performs better for extreme probabilities . The logarithmic skill score quantifies the percentage gain of using the statistical model over just predicting the climatological probability and is calculated as follows: 6LSS=1-LSLSclim⋅100%, where LS =1n∑k=1nLSk is the logarithmic score, with n being the number of forecasts and k indicating an individual forecast.

The value of LSk is determined using the forecasted probability pk calculated by the logistic regression model. LSk=-ln⁡(pk),if rockfall event occurs-ln⁡(1-pk),if rockfall event does not occur

LSclim is calculated analogously using the climatological probabilities eventsn (number of events per number of forecasts).

When comparing two statistical models predicting the same n situations, a higher logarithmic skill score indicates better predictions.

Another option for comparing statistical models that were fitted based on the same observations is the Akaike information criterion AIC;, which estimates the prediction error: 7AIC=2i-2ln⁡(L), where L is the likelihood and i is the number of predictors. Here, a lower AIC value is associated with the better model. The risk of overfitting is considered by penalising a high number of predictors.

Ensuring that no overfitting takes place can also be achieved by cross validation, which tests the statistical model on a sample of independent data. For this study, the full event catalogue was divided into five approximately equally sized groups, with events from the different clusters equally distributed between the groups. The statistical model was then trained using only four of the groups and afterwards applied to predict event probabilities in the remaining group. The logarithmic skill score for that group was calculated. The process was repeated for all groups, and a mean cross-validated logarithmic skill score was determined (LSScv).

The weight of evidence analysis is used to analyse the potential of different predictors to influence rockfall probability. All variables were screened individually. Figure  shows the most robust estimate possible for each variable. It is based on all rockfall events that occurred during the observation period of the respective dataset and all unique observational time series at the location of these events. A graphical inspection of the result already reveals that a relationship between the independent variable and the probability of rockfall exists for all variables. Moreover, the IV value is higher than 0.02 for all variables.

For a consistent comparison of the IV values, the analysis was repeated with the number of grid boxes, time steps and events reduced to the subset covered by all datasets (see the Supplement). This slightly increases the IV values for daily precipitation and soil moisture. The highest IV in the short common period (2001–2013) is obtained for daily precipitation (IV = 0.85). Soil moisture and precipitation intensity have similar IV values of 0.25 and 0.23, respectively, followed by freeze–thaw cycles (IV = 0.05). To take into account that the thawing process might take several days, a time span preceding the event was evaluated. Comparing different time spans, it turned out that the IV value associated with a freeze–thaw cycle immediately before the rockfall event (i.e. the preceding 2 d) was too low to be considered useful for statistical modelling (see Fig. S2a in the Supplement). Extending the analysis period backward in time increased the IV value, with a peak reached after 9 d (Fig. ). The WOE analysis also confirmed that, in accordance with the findings of , thawing increases rockfall probability while freezing decreases it (see Fig. S2).

Logistic regression is a well-established statistical method to determine probabilities for a binary event (e.g. rockfall vs. no rockfall) based on the conditions of independent variables. Here, a logistic regression model using precipitation, soil moisture or the pore-water proxy D, and freeze–thaw cycles as independent meteorological and hydrological variables is fitted. The consideration of individual rockfall clusters (Sect. ) provides information on possible regional differences of the results.

The logistic regression models were fitted using n=es⋅ts data points, with ts being the number of days for which meteorological and hydrological data are jointly available among all variables used as independent parameters in the model. The number of event sites (es) at which a rockfall event was recorded within the period covered by the meteorological and hydrological observations depends on ts. Other than for the WOE analysis, we neglected the fact that some grid boxes enclose the site of more than one event and did not merge these sites (thus, es is events). The background is that the logarithmic skill score and the Akaike information criterion can only be used to compare statistical model alternatives if they are based on the same data points n. Due to the different spatial resolutions of the individual meteorological datasets, merging would change n for each new combination of input variables. n is only reduced for evaluations involving precipitation intensity that are carried out based on a much shorter period and a lower number of sites than all other evaluations. This will be considered when comparing the results.

To find the best performing statistical model, numerous combinations of the potential predictors were compared. Table  lists the results for a selection of these tests. Evaluated predictors include daily precipitation (precip_1day), the local percentile of daily precipitation calculated using wet days (precip_1day_lperc), across-site percentile of simulated total column soil moisture (sm_perc), across-site percentile of parameterised pore water (D_perc), hourly precipitation (precip_1hr), the local percentile of hourly precipitation calculated using wet hours (precip_1hr_lperc), a categorical predictor denoting to which cluster an event belongs (cluster) and a binary predictor indicating if a freeze–thaw cycle occurred at the site during the previous 9 d (ftc). As the full notation of the model equations is space consuming, we use the compact symbolic form used in the R programming language . The operator + denotes adding another predictor term, : marks the product between two predictors and ∗ indicates that all possible combinations of interactions between the predictors are considered. Thus, the term (precip_1day_lperc∗D_perc) + ftc in row 16 of Table  translates to 8pk=1/[1+exp⁡(-β0-β1precip_1day_lperck-β2D_perck-β3ftck︸if ftck=TRUE-β4precip_1day_lperckD_perck)].

The performance of the statistical models listed in Table  is compared with the help of the cross-validated logarithmic skill score (LSScv) and the Akaike information criterion (AIC). A scientifically sound comparison is possible between models with identical es. Comparing models 1–3 shows that daily precipitation is more important than soil moisture and performs best if included in the form of its local percentile (denoted by the suffix “_lperc”). For hourly precipitation, LSScv and AIC indicate that absolute values lead to better results than local percentiles (models 4 and 5). The ranking between all three predictors suggested by models 1–5 is confirmed if the analysis is repeated using only the 139 events that took place during the period covered by all predictor datasets. The cross-validated logarithmic skill score for this short period is 4.38, 1.33 and 0.76 for precip_1day_lp, sm_perc and precip_1hr, respectively (not included in the table). Models 6–8 reveal that considering soil moisture in addition to the local percentile of daily precipitation improves the statistical model, with the best result obtained by using both variables individually as well as their interaction term (model 8). LSScv can be further increased by adding the binary information of the occurrence of a freeze–thaw cycle in the previous 9 d to the set of predictors (model 11). Adding the binary cluster information (models 9, 10, 12) has the effect of fitting individual bi coefficients (Eq. ) for each cluster. One would expect a better model performance if the different geological regions represented by the clusters would respond differently to the meteorological or hydrological triggers. Comparing the LSScv for models 9 and 10 to that for model 8 and the LSScv of model 12 to that of model 11 shows that this is not the case here. Model 10 demonstrates the importance of cross validation. This model exhibits the highest number of regression coefficients resulting in an LSS higher than for model 8. The LSScv is, however, lower than for model 8, indicating that the LSS improvement is achieved by overfitting. At first sight it seems that including hourly precipitation considerably improves the statistical model as the LSScv in model 14 is higher and the AIC lower than in model 11. It must be kept in mind, though, that the radar climatology is still comparatively short, and fits including hourly precipitation are based on a small subset of rockfall events, making a direct comparison of LSScv and AIC with model 11 impossible. Therefore, the result of model 11 applied to the data subset used to fit model 14 is summarised in line 15 of Table . It shows that the increase in LSScv and decrease in AIC seen for model 14 has to be attributed to the shorter time series, and the inclusion of hourly precipitation does not improve the statistical model.

An encouraging result, with respect to facilitating the analysis of climate scenario simulations, was obtained when substituting the across-site percentiles of modelled relative soil moisture used in the logistic regression model 11 with across-site percentiles of D (D_perc, Eq. ). In order to use D as a proxy for pore water, it was accumulated over a period of time. The optimal number of days for the accumulation period was determined by successively reducing the length of the period starting at 2 weeks. The logarithmic skill score of the logistic regression model increased with decreasing duration and reached a plateau at 5 d (Fig. ). With this accumulation period, we obtained the results shown on line 16 of Table . The cross-validated logarithmic skill score of that model is 4.06; thus model 16 outperforms model 11.

In addition to the combinations shown in Table , it was also investigated whether the regression coefficients depend on the slope angle at the event site. For this we downloaded the Copernicus digital elevation model (DEM) at 25 m horizontal resolution and calculated the slope angle at the rockfall locations using the methodology proposed by . The slopes at the event sites calculated using the DEM data appear plausible at many of the sites. There are, however, also locations for which we determined a slope angle of only a few degrees, which is inconsistent with the occurrence of rockfall events. Possible explanations could be an insufficient spatial resolution of the DEM or the possibility that the slope was altered by the event and is therefore no longer captured in the DEM dataset representative of the year 2011. In addition to this, for large-scale rockfall events it is difficult to determine the exact location at which the slope needs to be estimated. Overall, including the slope angle calculated using the DEM as an additional parameter in the logistic regression model did not improve the results.

In summary, Table  shows that the best results are obtained from the logistic regression model 16, which is expressed in Eq. (). The corresponding regression coefficients are β0=-10.48, β1=-2.969×10-3, β2=-1.413×10-2, β3=0.435 and β4=4.053×10-4.

Model 16 (Eq. ) can now be used to predict changes in rockfall probability valid on average for specified changes of the meteorological conditions and the pore-water preconditions. The response of the rockfall probability to variations in the local daily percentile of precipitation and the percentile of the pore-water proxy D is depicted in Fig. . The probability esn used as a reference to calculate the logarithmic skill score is marked with a horizontal line. As the rockfall database is not comprehensive, this value should not be interpreted in absolute terms. With D and local daily precipitation set to median values (Fig. a), rockfall events can be expected to appear with approximately climatological probability. Less (more) precipitation leads to a probability below (above) climatological average. Increasing the local precipitation from the median to its 90th percentile approximately doubles the probability of a rockfall event. The amount of precipitation associated with the 50th percentile varies between 1.2 and 6.8 mm and between 6.5 and 31.2 mm for the 90th percentile, depending on the site. The occurrence of a freeze–thaw cycle in the previous days increases the probability of an event by about 50 %. Precipitation becomes more effective when the pore-water amount is high (Fig. b). When D is at the 95th percentile, increasing precipitation from its median to its 90th percentile makes rockfall events almost 4 times more likely. This dependence of the slope of the probability density function for precipitation on D (and for D on precipitation) is made possible by the inclusion of the interaction term between precip_1day_lperc and D_perc in Eq. (). The logistic regression model suggests that the influence of pore water on rockfall probability is on average less pronounced than the influence of daily precipitation. At most sites an increase in pore-water amount in the absence of strong precipitation has hardly any effect (Fig. c).

In terms of event numbers, the combination of D_perc ≤ 50 and precip_1day_lperc ≤ 50 includes 42 % of all days but only 25 % of all events, while the combination D_perc ≥ 90 and precip_1day_lperc ≥ 90 includes only 2 % of all days but 19 % of all events (Table ). Combinations of high (low) precipitation percentiles with low (high) D percentiles rarely occur. The climatological frequency of rockfalls in the Elbe Sandstone cluster ES for the combination D_perc ≤ 50 and precip_1day_lperc ≤ 50 indicates that this cluster includes a higher number of events not associated with a meteorological trigger than the other clusters.