A rockfall dataset for Germany is analysed with the objective of identifying the meteorological and hydrological (pre-)conditions that change the probability for such events in central Europe. The factors investigated in the analysis are precipitation amount and intensity, freeze–thaw cycles, and subsurface moisture. As there is no suitable observational dataset for all relevant subsurface moisture types (e.g. water in rock pores and cleft water) available, simulated soil moisture and a proxy for pore water are tested as substitutes. The potential triggering factors were analysed both for the day of the event and for the days leading up to it.

A logistic regression model was built, which considers individual potential triggering factors and their interactions. It is found that the most important factor influencing rockfall probability in the research area is the precipitation amount at the day of the event, but the water content of the ground on that day and freeze–thaw cycles in the days prior to the event also influence the hazard probability. Comparing simulated soil moisture and the pore-water proxy as predictors for rockfall reveals that the proxy, calculated as accumulated precipitation minus potential evaporation, performs slightly better in the statistical model.

Using the statistical model, the effects of meteorological conditions on rockfall probability in German low mountain ranges can be quantified. The model suggests that precipitation is most efficient when the pore-water content of the ground is high. An increase in daily precipitation from its local 50th percentile to its 90th percentile approximately doubles the probability for a rockfall event under median pore-water conditions. When the pore-water proxy is at its 95th percentile, the same increase in precipitation leads to a 4-fold increase in rockfall probability. The occurrence of a freeze–thaw cycle in the preceding days increases the rockfall hazard by about 50 %. The most critical combination can therefore be expected in winter and at the beginning of spring after a freeze–thaw transition, which is followed by a day with high precipitation amounts and takes place in a region preconditioned by a high level of subsurface moisture.

Landslides are geomorphological hazards associated with damage and fatalities
to people and their connected structures

Against this background, the present study focuses on multiple rockfall
clusters spanning all of Germany. Rockfall is the removal of superficial
and individual rocks from a rock cut slope

In this context, the question arises as to whether a statistical model focused on
meteorological parameters can accurately predict rockfall occurrence. A first
investigation conducted on a monthly basis by

The present study uses historical rockfall data that are extracted from the landslide database of Germany

The dense spatial clustering of rockfall events and high temporal data homogeneity guide the selection of three study areas (Fig.

The ES cluster mainly includes the German parts of the Elbe Sandstone Mountains, which are located on both sides of the upper reach of the river Elbe between the Czech city Děčín and the Saxon city Pirna. Geologically, the area is dominated by compact Cretaceous sandstones. Fracturing and formation of cracks and fissures came about by extensive uplift processes and long-term tectonic stresses. Fluvial incision accounted for a heavily dissected relief with numerous horizontal cracks, vertical joints and clefts, and small gorges

The HL cluster embeds large parts of the northern German Central Uplands, i.e. the Hesse Highlands and Lower Saxon Hills. Predominantly, the geological conditions are characterised by Middle Lower Triassic Bunter Sandstone. Pronounced dissections were caused by tectonic stresses

The HR cluster comprises large parts of the Hunsrück Hills in Rhineland-Palatinate and a small part of the Taunus Hills in Hesse. Geologically, Devonian bedrock, namely slate and quartzite, is predominantly present in this area. Distinct plateaus alternate with ridges and incised valleys

Location of rockfall events analysed in this study. Three distinct clusters – ES (Elbe Sandstone), HL (Hesse and Lower Saxony) and HR (Hesse and Rhineland-Palatinate) – are marked in red, blue and orange, respectively. All other events are coloured in light blue.

Time series of the number of rockfall events per year included in the database. The years at which the meteorological and hydrological observations start are indicated.

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

In order to study precipitation intensities, the gridded radar-based climatology RADKLIM

For temperature, it was decided to use the gridded E-OBS dataset

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

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 (

The term

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

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

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.

An integral measure for the strength of the relationship between the dependent and independent variable is the information value

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 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

The value of LS

LS

When comparing two statistical models predicting the same

Another option for comparing statistical models that were fitted based on the same observations is the Akaike information criterion

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 (LSS

The weight of evidence analysis is used to analyse the potential of different predictors to influence rockfall probability. All variables were screened individually. Figure

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

Weight of evidence (WOE) for

Dependence of the IV for freeze–thaw cycles on the period used for the analysis.

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

The logistic regression models were fitted using

To find the best performing statistical model, numerous combinations of the potential predictors were compared.
Table

Symbolic formulae for a list of logistic regression models tested in this study and main characteristics associated with these models. The characteristics include the number of coefficients that needed to be determined, the number of event sites (es) that were used for fitting, the logarithmic skill score (LSS), the logarithmic skill score determined by cross validation (LSS

The performance of the statistical models listed in Table

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

In addition to the combinations shown in Table

In summary, Table

Model 16 (Eq.

In terms of event numbers, the combination of

Dependence of the cross-validated logarithmic skill score on the accumulation period of

Probability for rockfall predicted by the logistic regression model. A dashed (solid) curve denotes the result for situations with (without) the occurrence of a freezing episode in the previous 3 weeks. The horizontal line marks the climatological probability.

Percentage of days with combinations of precip_1day_lperc and

In this study, a statistical model was developed that is able to describe changes in the probability of rockfall events in Germany that can be expected under different meteorological and hydrological conditions. It is important to keep in mind that a statistical relationship is not proof of a cause-and-effect relationship. As rockfall occurrence in Germany exhibits a seasonal cycle with a maximum in January

The logarithmic skill score used to evaluate the fit of the statistical model describes the percentage improvement over a model that always predicts a climatological probability for rockfall events. The skill score of our model is just over 4 % and improves to more than 5 % if only the last 20 years are used for model fitting. A value of 4 % appears to be not much, but it has to be interpreted keeping the physics of rockfall events in mind. A rockfall event can only be triggered if the slope is predisposed, after many years of weathering. Because of this, most of the time strong rainfall in an area with high soil moisture or pore-water preconditions remains without consequences (i.e. false alarms). Prediction errors (i.e. missed alarms) may also stem from events triggered by non-meteorological mechanisms or processes not captured by the chosen predictors. This seems to be the case for some events in the Elbe Sandstone cluster ES. The model skill obtained using the selected meteorological–hydrological parameters as predictors, however, suggests that non-meteorological influence and missing predictors seem to be subordinate factors in the rockfall process for the selected study regions.

As this model was developed for the purpose of detecting changes in rockfall probability in climate scenario simulations, the low skill score on account of the overall low probability for rockfall does not pose any problems. For a warning system, the number of false alarms would be too high. This limitation could only be overcome by including information on the predisposition in the statistical model. Unfortunately, this is not feasible as it would be far too expensive to monitor every slope operationally. Nevertheless, the concept of using a logistic regression model instead of fixed thresholds would also have advantages for warning systems. The probability for rockfall relative to a baseline climatology could be determined with Eq. (

We found that daily precipitation is the most important factor to trigger rockfall events in Germany. The best fit for the statistical model was obtained when using local percentiles rather than across-site percentiles (not shown) or absolute values. A possible interpretation could be that most rock slopes are balanced under normal climate conditions but can become unstable in the presence of above-normal precipitation amounts. The presence of freeze–thaw cycles increases the probability by approximately 50 %. Pore water on its own is unlikely to trigger a rockfall event. It can weaken porous material, making it more susceptible to a trigger like precipitation. The fact that both simulated soil moisture and

Quantitatively, our findings are in contrast to those of

Using a rockfall dataset for Germany, it was possible to build a statistical model that is able to quantify changes in rockfall probability in response to changes in pore water and meteorological factors identified in geophysical studies as potential triggers for rockfall events. The model can be regarded as representative for the low mountain ranges in Germany. It can also be used in other central European low mountain regions with similar climatological, hydrological, geological and topographical characteristics for which no customised modelling approach exists.

The model was developed in order to be applied to climate change simulations, with the aim of determining if the probability of rockfall events can be expected to change in response to global warming. Applying the statistical model to climate simulation output is facilitated by the fact that the model works with percentiles for most predictors. Thus, only temperature for the evaluation of freeze–thaw cycles needs to be bias corrected. In addition, the complex simulation of soil moisture can be substituted by a pore-water proxy (i.e. accumulated precipitation minus potential evaporation), which can be easily calculated from climate model output.

For application in climate change studies, it is important that the statistical model considers the interaction between the triggering factors as these are expected to show opposing trends. While heavy precipitation is likely to increase in the future

The meteorological data used in this study are freely available. After registration the E-OBS dataset can be downloaded from

The supplement related to this article is available online at:

KMN conducted the statistical analyses and prepared the draft manuscript. SR and BD collected and analysed the rockfall data. SR prepared Fig. 1 and provided the rockfall data description. TMK wrote the introduction and BG conducted the soil moisture simulations. BD and UU supervised the project and provided advice and feedback in the process.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work used resources of the Deutsches Klimarechenzentrum (DKRZ) granted by its Scientific Steering Committee (WLA) under project IDs b1152 and bm1159.
We acknowledge the E-OBS dataset from the EU-FP6 project UERRA (

This research has been supported by the Bundesministerium für Bildung und Forschung (grant nos. 01LP1903A, 01LP1903K and 01LP1903E).We acknowledge support from the Open Access Publication Initiative of Freie Universität Berlin.

This paper was edited by Paola Reichenbach and reviewed by two anonymous referees.