Similar to exceedance-frequency curves which are a standard method in natural hazard risk analysis for plotting losses against a specific exceedance-frequency for historic events , the damage-frequency curve plots the calculated damage against the return period. We use these curves to show synthetically generated avalanche risk scenarios and to draw conclusions between calculated damages. Therefore, we show possible outcomes at other return periods that have not been specifically calculated but rather estimated through interpolation. These curves thus provide a more complete picture and give decision makers a wider range of available interpretations of the results.

CLIMADA is an open-source and open-access Python package for probabilistic risk assessment . It allows the impact computation of natural hazards using intensity maps, considering exposure and its vulnerability. It is possible to compute current and future risk, including climate change, that modifies the hazard and socio-economic development affecting the exposures and vulnerability. Finally, one can compute the reduction in risk and cost from adaptation options to perform a cost-benefit analysis . For all the model outputs, CLIMADA provides a module to perform uncertainty and sensitivity analyses using global (quasi-)Monte Carlo sampling.

In this project, we use CLIMADA to compute the risk of avalanches to more than 13 000 individual buildings from each of the 40 000 simulated avalanches using the hazard intensity maps described in Sect. , the exposures distribution described in Sect.  and the impact functions introduced in Sect. . As defined in the CLIMADA tool , the impact is expressed by the following risk quantities.

The expected annual impact for each exposure (affected building) is expressed in monetary terms per year (Swiss Francs (CHF) per year). It combines the hazard intensity with its expectation (return period) and the damage degree from the impact functions.

To cover a large range of potential avalanche events, we consider three different avalanche scenarios (see Table ). One frequent scenario [A] corresponding to a snowfall event with a 30-year return period, one intermediate one [B] (100-year return period) and one extreme [C] scenario (300-year return period). The definition of the scenarios is implemented according to the maximum 3 d snow depth increase and directly determines the mean fracture depths and thus the release volume and the modeled avalanche run-out lengths. The higher the return period, the larger the mean avalanche volume. The term return period (rp) describes the average number of years between two comparable events of the same intensity at the same location. By hazard frequency we denote the probability of occurrence calculated as a reciprocal value of the return period (1/rp). This definition is common practice in avalanche hazard mapping . To look at the overall avalanche risk of all scenarios later in the risk analysis, we adjusted the three avalanche scenarios for the combination with regard to the return period by p30=p30-p100=1/30-1/100=0.0233 and p100=p100-p300=1/100-1/300=0.00667 and p300=1/300=0.0033. This allows us to combine the hazard scenarios and recalculate the overall risk, such as the expected annual impact for single objects over all hazard scenarios in the entire study region.

The derivation of the 3 d snow depth increase was based on a long-term snow measurement series at the meteorological station named “ME2”, representative of the chosen area. As shown in Fig. , two extreme value statistical methods were applied: the generalized extreme value distribution with maximum likelihood estimations (GEV-MLEs) and the Gumbel distribution with maximum likelihood estimations (GUM-MLEs). Table  shows that both methods produce similar values, but slightly higher values are produced with Gumbel. This depends on the choice of the fitted correlation through the value distribution. Since the Gumbel distribution rather represents a “worst case” for the respective return period, and confirms the application of this method, these values were chosen for avalanche fracture depth determination. To account for elevation of the release area, a correction factor has to be applied. estimated this factor to be about ±2 cm per 100 elevation meters, whereas Swiss practitioners usually use a gradient of 5 cm per 100 elevation meters . In consideration of the large avalanche scenarios, we applied 5 cm per 100 elevation meters to each release polygon, corresponding to existing studies in the Swiss Alps . This elevation correction was applied based on the reference elevation of the measuring station ME2 at 1320 m a.s.l. This correction was calculated for each individual release area based on the difference between the respective mean elevation of an avalanche release area and the elevation of the measuring station ME2. Since the 3 d snow depth increase (see Fig. ) at the meteorological station Meien is measured in flat terrain, a standard slope inclination correction was conducted for all avalanche release polygons to consider an adapted fracture depth d0. As is standard in avalanche practice, we assume that less snow accumulates in steep terrain than in flat areas. Depending on the average inclination of the individual release area, the slope correction factor of the snow height for each potential release area is individually calculated and corrected. This is applied automatically in the ArcGIS Python script for fracture depth d0 calculation by using a slope factor ψ generally depending on the snow strength (increasing with snow depth) ψ=0.291/sin⁡(α)-0.202×cos⁡(α), with α being the slope angle and the fracture depth d0 being a function of the 3 d snow depth increase d0* and the slope factor ψ, expressed as d0=d0*f(ψ) . To correct for the influence of blowing snow, a snow drift factor was added, depending on the scale of the scenario . In practice, this factor strongly depends on local conditions in the release zones . For the 30-year return period scenario we added 20 cm and for the 100-year return period scenario 30 cm and the 300-year return period scenario 50 cm of drifting snow correction according to . With the method explained above, we are now able to use the fracture depth of different return period scenarios for further analysis.

Forest influences the snowpack structure by interception and changes in the micro-climate. An increased topographic ground surface roughness in forests and irregular layers in the snowpack prevent avalanche formation. In some cases dense forests are even able to stop movement of small avalanches due to higher friction in the flow and the avalanche run-out zone . To take forest cover into account for hazard mapping, we used the algorithm developed by and introduced by to identify protective forest in the study area (Fig. ). The algorithm developed by Bebi et al. is based on a database of 150 forest avalanches and yields a logistic regression model, taking into account the parameter's slope gradient, degree of forest cover and widths of gaps between trees. Together with a vegetation height model and a high-resolution elevation model, a general logistic regression model was used to calculate forest conditions to define if avalanches can occur or not. The thus derived protective forest layer was calculated for frequent (return periods < 100 years) and extreme scenarios (return period > 100 years). The protection forest model is improved and extended by a “shrub forest” layer and a ground roughness layer. Since shrub forest also has a protective function against avalanche release, the existing protective forest layer was expanded with a shrub forest component. Compared to normal protection forest, shrub forest has a reduced protection capacity, which is taken into account. This is a significant improvement over previous protective forest layers in which shrub forest was not taken into account. In this study, in a frequent scenario with a 30-year return period, shrub forest prevents avalanche release, whereas in the large avalanche scenarios with 100 or 300-year return periods, shrub forest no longer has any influence. The ground roughness layer classifies rough ground to mitigate avalanche release to a certain extent. The implemented ground roughness is taken into account for the protective forest layer calculation in frequent avalanche scenarios. This procedure results in a binary protection forest layer (green forest layer in Fig. ) that divides the area of investigation into areas with and without protection forest. With a forest layer generated this way, it is assumed for the large-scale avalanche mapping process that an avalanche release in the forest is not possible . Further details on the algorithm are delivered by .

Terrain models used for the selection of the release areas no longer contain buildings or infrastructure. Thus, the algorithm can designate release areas in places where there is actually infrastructure. Of course the formation of avalanches is also altered by infrastructures, which can completely prevent avalanche formation or subdivide release areas into smaller ones, e.g., a road cutting through a steep slope. Therefore, infrastructure such as roads, railway lines and buildings are also added to the binary layer as being objects that prevent avalanche release.

Identifying the potential avalanche release areas is a crucial hazard assessment. Individual expert analysis and manual release area selection is no longer efficient for a large-scale application. We therefore use an automated approach introduced by and further developed by , and , applied to large scales by .

This approach is based on a terrain analysis in a GIS and the “object based image classification” (OBIA) method to delineate the individual release polygons, which serve as an input for the large-scale avalanche hazard indication modeling . The algorithm analyses the slope geometry and identifies all possible potential avalanche release zones automatically for a given set of input data. With the fracture depth derived from the snow depths (Sect. ) and the area of the release zones, as well as applying the corresponding correction factors, the volume for each release area could be determined. With the automated processing, the single-potential avalanche release areas are mapped and classified into four release area size classes: large (release volume >60000 m3), medium (25 000–60 000 m3), small (5000–25 000 m3) and tiny (<5000 m3). Depending on these release volumes, different friction parameters are assigned to the associated avalanche for the subsequent simulation. These friction parameters μ (basal friction) and ξ (turbulent friction) are assigned according to the RAMMS standard modeling procedure , which is standard for hazard mapping in Switzerland.

We used RAMMS::LSHIM (Rapid Mass Movement Simulation::Large-Scale Hazard Indication Mapping), which is a special version of the base module of RAMMS::AVALANCHE, a numerical avalanche simulation model capable of simulating avalanches in complex topography . RAMMS::AVALANCHE is based on an efficient second-order numerical solution of the depth-averaged avalanche dynamics equations and the two-parameter Voellmy model and has been calibrated with numerous observed avalanches, such as those at the SLF test site in Vallée la Sionne (Arbaz, Valais, western Swiss Alps). For the simulations, a three-dimensional digital terrain model, with a spatial resolution of 10 m; the generated potential release areas; and the avalanche fracture depth (Table ) served as inputs to calculate avalanche volumes, flow velocities, flow heights and impact pressures for 40 000 individual avalanches. This was done for all three scenarios. Each avalanche was assigned to a release zone and registered individually, which allowed us to allocate each avalanche with its pressure to a return period, to be used subsequently in a risk analysis.

For avalanche release volumes <25000 m3, the binary forest layer is considered for the flow simulations with the assigned friction parameters μ (mu: basal friction) and ξ (xi: turbulent friction). Values of μ=0.020 and ξ=400 were assigned for the simulations in RAMMS::LSHIM because protective forest has a strong influence on friction during avalanche runout in frequent scenarios with smaller avalanche volumes. For avalanche release areas with volumes >25000 m3, the influence of the protective forest is neglected, since the energy generated during the flow process mostly destroys the protective forest, which thus hardly has any effect on the size of the affected area in the run-out zone . In the extreme scenarios, the potential release zones with avalanche volumes <25000 m3 are simulated with forest influence. In the frequent scenarios, on the other hand, the potential release areas with avalanche volumes >25000 m3 are simulated without forest influence. This is part of an extended set of algorithms for automatic release area identification and the standard friction parameter set, which is commonly used in practice . For a detailed description of the RAMMS::AVALANCHE application in RAMMS:LSHIM, see .

Figure  shows the resulting large-scale hazard indication map and, in detail, how the individual scenarios were simulated, taking into account protection forest.

EconoMe is an online tool for the evaluation of the effectiveness and economic efficiency of mitigation measures against natural hazards . It provides a sophisticated methodology for risk analysis and the evaluation of the cost–benefit ratio of protective measures and is used by private companies and cantonal and federal authorities for decision support regarding the distribution of subsidies. All values used for this study were generated using the EconoMe methodology and are described in Sect.  and listed in Table .

Four main datasets were used to determine monetary values: (1) STATPOP, statistical population data which provide information on the number of building inhabitants; (2) STATENT, statistical data specifying which buildings are used for economic purposes; (3) construction zones Switzerland, providing information about settlement areas or agricultural zones; and (4) a 3D dataset of all Swiss buildings, including the volume of individual objects. To allow a high performance in risk calculation and a broad applicability, buildings in the exposure data frame are considered as point objects located in the center of a building polygon. Using these four datasets, an exposure point layer was created according to the guidelines for creating risk overviews as described in the following.

The GIS layer with assigned monetary values was created for 13 304 individual objects. To generate this monetary building layer, objects needed to be classified using the above-mentioned datasets, also listed in Table . The classification was carried out by plotting the statistical data STATPOP and STATENT on a 2D building layer. The monetary values of buildings were assigned according to the building types also listed in Table . According to EconoMe standards, the average number of persons in one housing unit (= HU) was assigned as 2.24 persons per unit . By using STATPOP, we identified the number of persons living in each building and determined the number of housing units rounded to the next smaller number. When the number of persons per object was below 2.5, the object was classified as a single residential building; when the number of persons per object was higher than 2.5, it was classified as a multi-residential building. Based on the number of HU, monetary values could be assigned to the objects as listed in Table .

Since there is no comprehensive data available for evaluating the value of companies and buildings, the STATENT dataset was used. These data provide building coordinates for each registered company in Switzerland. If one or more STATENT data point plots on the location of a building, this building was classified as a business building. The value corresponds to the volume of the building , which was taken from the Swissbuildings3D dataset, yielding a value of CHF 280 per cubic meter volume. For objects with the combined usage “business and living”, on which both the STATENT and STATPOP data points apply, the number of residential units was taken for the assignment of the monetary value, as these have higher value.

Buildings located within the Swiss construction zone plan with no inhabitants and a volume of less than 100 m3 were classified as uninhabited outbuildings, with a value of CHF 60 000 per unit (garages and parking lots including vehicles). Uninhabited outbuildings with a volume greater than 100 m3 were classified as economically used buildings, and their value was determined by the volume.

Buildings located outside the Swiss construction zone plan with a volume less than 100 m3 were classified as agricultural outbuildings (barns), with a value of CHF 80 per cubic meter. If their volume is greater than 100 m3, they were classified as agricultural main buildings with a value of CHF 180 per cubic meter . This method allowed the creation of an exposure data layer, shown in Fig. , with a monetary value for each individual building.

The extent to which an object is damaged by a hazard is referred to as the “hazard impact” or “impact”. We determined the impact by avalanches using so called “impact functions”, which are equivalent to “vulnerability functions” or “damage functions”, as defined in the CLIMADA methodology , which express the damage an object suffers at a certain avalanche intensity (pressure in kPa). These functions are based on values in EconoMe . These step functions originate from the standardized hazard mapping procedure in Switzerland, where continuous avalanche impact pressures p are divided into three pressure classes (0<p≤3 kPa, 3<p≤30 kPa and p>30 kPa). These step functions follow the Swiss standard and are based on an evaluation of building damages and expert judgment. According to construction type, objects show varying damage susceptibility to avalanche impact pressure. We defined impact functions with three components for each object type and combined them into a impact function set to describe the following avalanche impacts:

the mean damage degree (MDD), expressing the mean percentage of damage an object suffers under a certain avalanche impact pressure, expressed in kPa;

The mean damage ratio is a standard vulnerability unit. It is described as the ratio between the damage of affected objects and the total value of all objects exposed to the hazard. The application of these standard vulnerability components to avalanches is a special case. We assume in this framework that all objects exposed to a certain avalanche impact pressure also suffer damage according to the mean damage degree thresholds. Therefore, we set the mean damage ratio as equal to the mean damage degree.

Since the values of the exposure were determined with the generalized value assignment procedure introduced by , it is clear that these can deviate from real life building values. The FOEN (Federal Office for the Environment) method provides an estimation of a building value at a large scale but cannot resolve the differences in value of similar building types. To account for the potentially large fluctuation in asset value, an uncertainty of ±20 % of the total exposures value is assumed. In other words, for each run of the uncertainty sampling, all the assets' values are multiplied with a value “ET”, uniformly sampled from [0.8,1.2].

Industrial and residential buildings are the most costly building types. The steps of the impact functions for residential buildings and industrial differ by 40 %. To account for a large degree of uncertainties, this value of the step jump in the function was taken as the possible uncertainty range. For agricultural buildings or outbuildings, this value can be even higher, as they often have a relatively simple construction. But due to their lower value they play a less important role. To define a “standard case” in our analysis, we consider the function steps of industrial or residential buildings as the variability range. Therefore, for the mean damage degree of the impact function a value range of ±40 % was taken for the uncertainty analyses (cf. Table ).

To define a range for the hazard intensity uncertainty analysis, we refer to data recorded in experiments at the Vallée de la Sionne test site. measured the impact pressure of dry, wet or mixed avalanches with different flow regimes on obstacles. In our simulations, we only simulated “dry avalanches”, which mostly have velocities higher than 10 ms-1. Pressure values of avalanches measured by Sovilla et al. show avalanche impact pressures ranging between 200 and 700 kPa . An extrapolation of the dataset would yield maximum pressures of about 1000 kPa, which corresponds to the data obtained in our simulations shown in Fig. . The average pressure of the measurements would be around 600 kPa. Therefore, we assumed an uncertainty range of ±50 % for avalanche impact pressure (cf. Table ). The obtained samples are illustrated in the Appendix, Sect. , in Fig. .

The sensitivity analysis determines which input parameter's uncertainty has the strongest influence on the model output uncertainties . In this study, we used the Sobol sensitivity indices as implemented in CLIMADA , which can be computed from the same Sobol sequence as used for the uncertainty analysis (cf. Fig. , Appendix Sect. ). The Sobol sensitivity analysis method is a popular, variance-based method that quantifies the contribution of the variance of one or two parameters to the unconditional variance of a model output . It is expressed as the Sobol sensitivity index. After the distribution of the impact metrics has been computed for all samples (see Sect. ), the sensitivity index for the each metric was computed. We considered the first-order Sobol index S1 that describes the contribution of a single model input to the output variance. This index therefore provides information on the most influential parameter, with reference to its system (model).

The first-order sensitivity index S1 identifies which parameter impacts the aggregated average annual impact most. Figure a–c show that the mean damage degree plays the most important role in this model for all scenarios. For the scenario with a return period of 30 years and the scenario with a return period of 100 years, the exposure and the hazard intensity seem to be almost equal. In the extreme scenario with a 300-year return period, substantially more objects are affected (see Fig. ), and the exposure plays a slightly more important role here. The sensitivity index of the hazard intensity in the 30- and 100-year return period scenarios (Fig. a and b) is slightly higher than the S1 of the hazard intensity in the 300-year return period scenario in Fig. c. It is thus scenario-dependent whether the hazard intensity or the exposure have more influence. However, in all scenarios the mean damage degree defined by the impact functions is the most relevant parameter. For the combined hazard scenario (Fig. d) it is similar to the 300-year return period scenario. Both the exposure and the hazard intensity have almost the same high sensitivity index, but again the mean damage degree is more decisive.

The proposed approach has a fast and powerful applicability, which allows us to map potential release areas over very large areas, where only the computational capacity and the computation time define the limits of its application. This method is applicable worldwide and requires a small number of input data, making it one of the most unique and powerful tools for large-scale hazard mapping. We consider that the advantages of the large-scale applicability and the preciseness of the results meet the goals of our study.

Compared to a detailed hazard assessment, there are some limitations . In such a multi-step process, avalanche release areas are manually identified by experts performing a detailed terrain assessment, including an evaluation of protection forests, which is used as input for a numerical modeling of avalanche run out . In our approach, potential release areas are only identified on the basis of terrain and scenario characteristics. This is an approximation, as actual avalanche release zones may vary in shape and size in nature. However, a validation of this algorithm carried out by Bühler and colleagues showed that results are in good agreement with avalanche observed in the terrain .

The definition of the protection forest was based on datasets compiled at a specific point in time in the past (2020) using remote sensing methods. However, with the effects of extreme windstorms, bark beetle infestations or consequences of climate change (e.g., dry periods), the forest structure can change rapidly . A detailed consideration of these effects and an expert evaluation of the protective function of a particular forest is not applicable for large-scale applications and cannot be imperative for the preparation of risk overviews.

Using 3 d snow depth from a single weather station results in uncertainties for both the scenario definition and fracture depth. Both have a direct effect on the avalanche volume and subsequently on the extent of avalanche runout. The longer the time series, the more extreme snowfall events would have been recorded. However, the time series of 66 years used in our study is common practice for avalanche hazard assessment. The time series available are comparatively long, and longer series were not available either. However, this involves bigger uncertainties for estimation of fracture depths at high return periods. As can be seen in Fig. , the GEV method results in ±30 cm for a 100-year return period and ±50 cm for a 300-year return period, and the Gumbel method results in ±20 cm for the 100-year return period and ±25 cm uncertainties for the 300-year return period. Due to the 66-year record period, it can be assumed that the estimate of 3 d snow depth increase for a 30-year scenario contains lower uncertainties (±15 cm). This consideration of uncertainties does not take into account local wind effects that can arise because of topography changes at ridges and the back of mountains that significantly influence snow deposition in the release areas. These effects can lead to a high variability of the snow pack in the release zones. However, these local conditions cannot be taken into account in such large-scale applications. Only a wind load factor (30–50 cm) depending on the scenario size was added. Without detailed wind and snow deposition studies, this uncertainty in the modeling process is hard to quantify and has to be accepted.

Further, the topography and the assigned friction parameters play the most important role in the run-out flow simulation of an avalanche. For single slope assessments, different parameters (μ and ξ) would be assigned for each change in the flow cross-section (gully, channel or open slope), and thus, more precise results could be obtained. This leads to a less differentiated hazard pattern compared to a single slope approach in the avalanche run-out zone. Another effect that includes uncertainties is that all theoretically possible potential release areas are simulated with our model. In reality, a more differentiated picture of avalanches would result for each catchment zone, in which some release areas produce an avalanche and others do not. Over an entire catchment area, this leads to an overestimation of avalanche risk because too many objects are affected. This effect is not uniform for each slope aspect or catchment area. As far as we know, there are no studies on the release probability of avalanches that can be applied to large scales to represent an actual natural avalanche release scenario in a model. The advantage of our approach is, however, that all potential avalanche release areas that have the required geometrical properties for avalanche formation are actually covered in the simulation and taken into account for risk assessment. Here, the need for further research is identified to design probabilistic avalanche sets to cover real life avalanche release scenarios. In our risk analysis, each individual avalanche is treated as an individual, independent event, and the resulting damage is averaged out over all events impacting an object.

In this paper, we identify the monetary avalanche risk for buildings at a large scale. To do this, some generalizations have to be made, which leads to certain limitations in the level of detail.

The impact functions define the damage rate in steps and pressure classes. Actual damage does not always depend on these classes. If an avalanche hits an object, it is important at what angle the building is oriented in relation to the flow direction of the avalanche, what the specific flow regime is as well as how high the avalanche flows , or whether the building has structural weaknesses (e.g., windows or doors) exposed to the avalanche flow. Even though the impact step function was derived from expert assessments and damage surveys of avalanche incidents, it is unlikely that damage occurs according to these steps or is linear to the avalanche impact pressure – depending strongly on the individual situation. Even minor damage to the foundations of a house can lead to total damage and reconstruction. In our method, all these details cannot be taken into account. To ensure the performance of our approach, we generalize. We consider buildings as point objects with a generalized vulnerability. When using central points in building polygons, we naturally underestimate the influence of the actual three-dimensional building size in the avalanche flow. Small buildings are therefore equally considered as large buildings. Nevertheless, this approach allows a simple and performative applicability and an efficient large-scale approach but introduces inaccuracies in detail. However, we try to address the framework of these uncertainties by means of an overall uncertainty analysis.

Further, our method of the avalanche impact calculation does not take into account the temporal effects of the hazard. In reality, the first avalanche that destroys an object completely “prevents” its destruction by further events. After many minor avalanches have occurred in the same catchment area, the probability of a major event at the same location decreases. To cover these effects in a risk tool, a detailed probabilistic study would be needed for each individual catchment area. Local meteorological weather events and time-dependent interactions of individual avalanches would have to be investigated in detail. This would increase the level of detail but significantly weakens the independence of the place of application as well as the large-scale applicability for the identification of avalanche risks.