We applied the operational setup of the SNOWPACK model used for avalanche forecasting in Switzerland. The simulations were driven with meteorological data from AWSs located in flat terrain at the elevation of potential avalanche starting zones throughout the Swiss Alps . An overview of the spatial distribution of these AWSs is provided in Fig. . The measured meteorological data were pre-processed with MeteoIO to filter out potential measurement errors and fill measurement gaps using temporal interpolation or spatial interpolation with neighbouring stations. To reduce errors related to the meteorological input data in the validation (Sect. ) of the models developed in this study, we also produced SNOWPACK simulations using a quality-checked data set of meteorological measurements from the AWS Weissfluhjoch (2536 m a.s.l.) .

In addition to the simulations on flat terrain, forced with measured snow depths, simulations were also performed for four “virtual” slope orientations (north, east, south and west) with a slope angle of 38∘, including snow redistribution from windward to leeward slopes as described in and . Model output was available for up to 124 AWSs. We used SNOWPACK simulations for the four “virtual” slope orientations from the 21 winters 2001/2002 to 2021/2022.

To assess snow instability from simulated snow stratigraphy, we applied the instability model to the simulated snow profile at 12:00 LT on the day of interest, as described in . The instability model requires six input features describing the simulated snow layer of interest and the overlying slab. The output probability Punstable that a snow layer is unstable is determined by the fraction of trees in the ensemble of 400 classification trees that classify the layer as potentially unstable. Applying the instability model to every snow layer of a given snow profile allows computing the following properties (see also Fig. ):

Critical weak layer properties: the critical weak layer relevant for natural avalanche release is defined as the layer with the highest probability of instability, i.e. the layer where Punstable=max(Punstable). In case of ties, we selected the layer deepest in the snowpack. For each snow profile, we then extracted the following three layer properties: max(Punstable), which we refer to as Pcrit; the depth z below the snow surface in centimetres (zcrit); and the grain type (gtcrit). We grouped grain types into three classes considering the primary grain type: (i) persistent grain types (pg), including depth hoar, buried surface hoar, facets and rounding facets; (ii) precipitation particles (pp), including decomposing and fragmented precipitation particles; and (iii) other grain types (other), including rounded grain types, melt forms, melt-freeze crusts and ice layers. (See also for the grain type classification.)

The rate of snowfall and the amount of new snow are known to be important indicators of natural dry-snow avalanche activity, also called direct-action avalanches e.g., and also for the potential size of avalanches e.g.. Therefore, we also calculated:

Height of new snow in 24 h (HN1d).

Conventionally, the height of new snow is measured in the flat field. Consistent with this definition, the new snow amounts provided by SNOWPACK are therefore for the flat field as well, regardless of whether it is a simulation in the flat or on a virtual slope. However, we also considered the thickness of precipitation particle layers, zpp, which should capture the amount of recently fallen snow including snow transport by wind, since we used the SNOWPACK version including snow redistribution by wind. All other parameters depend on aspect since they were selected from the virtual slope simulations where, for instance, energy input and snow accumulation vary depending on aspect.

Lastly, we also extracted the minimum of the natural stability index sn38, which is implemented in SNOWPACK. Sn38 describes for each snow layer the ratio of the shear strength of the layer to the shear stress exerted by the overlying slab .

To validate model predictions, we used a data set of re-analysed regional danger levels. This data set is a subset of the forecast regional avalanche danger levels published by the national avalanche warning service in Switzerland. The data set only contains cases for which the forecast danger level was either validated or corrected (about 5 % of the cases) after considering multiple pieces of evidence, as described by . An updated version of this data set is publicly available . The data set consists of 36 582 re-analysed regional danger levels for specified warning regions, the smallest spatial units used in the Swiss avalanche forecast, for the forecast seasons from winter 2001/2002 to 2021/2022. In addition, the critical aspects and elevation range where the danger level applies, and the validity date of the forecast, are indicated. The danger level is assigned according to the five-level European Avalanche Danger Scale . The frequency of the danger levels in this data set is: 1 (low) 35 %, 2 (moderate) 29 %, 3 (considerable) 29 %, 4 (high) 7 % and 5 (very high) 0.3 %. In this re-analysed subset, the proportions for 4 (high) and 5 (very high) are slightly larger than in the original forecasts.

To estimate avalanche size for a given failure layer depth, we used data set AV2 (Sect. ) to relate avalanche size to observed failure depth (zobs) using logistic regression functions of the form 4P(S≥s)zobs=11+e-β0+β1⋅zobs, where P(S≥s) is the probability that avalanches greater than or equal to size s (s∈[2,3,4,5]; Table ) were observed given the observed failure depth (zobs).

The P(S≥s) functions were derived using observed data only, as SNOWPACK simulations were not available for the locations of the avalanche release areas. To analyse the performance of the size indicators combined with the depth parameters extracted from SNOWPACK, we estimated probabilities for different avalanche sizes on the AvDs from the training data set (Sect. ) using the simulated depth parameters z (HN1d, HN3d, zpp, zcrit and zdeep, described in Sect. ) as proxies for the potential failure depth. The resulting estimated probabilities for different avalanche sizes (s∈[3,4]) were then compared with the observed median and maximum avalanche sizes on the respective AvD using the Brier score (Eq. ) with the probabilities pi=P(S≥s)(zi) and the observed outcome Yi equal to 1 if an avalanche of size ≥s was observed and 0 otherwise. To evaluate how well the avalanche size estimator captures rare events, we also calculated the Brier score BS+ on the subset of positive observed outcomes, i.e. the data points that had an observed avalanche of size ≥ s.

To evaluate the performance of the avalanche day predictors (P(AvD)) and the avalanche size estimators (P(S≥s)), we used two independent data sets:

We used the observations of natural dry-snow avalanches in the region of Davos (data set AV3; Sect. ) to determine AvDs and nAvDs as described in Sect. . We labelled a day as a nAvD if there were no dry-snow avalanches in the region of Davos and the two surrounding regions (1000 and 5000 km2), which is in line with the definition described in Sect. . A day was labelled as an aspect-specific AvD if the AAI in the region of Davos was larger than 0.01 for the respective aspect and within an elevation band of ±250 m around the AWS Weissfluhjoch (2536 m a.s.l.) and if at least one natural dry-snow avalanche was observed within each of the two surrounding regions (1000 and 5000 km2), regardless of aspect and elevation. The definition of an AvD was thus slightly adapted compared with Eq. (1) due to the lack of consistent information on aspect and elevation of the observed avalanches within the two larger surrounding regions. Thus, AvD labels in the validation data set are somewhat less reliable compared with the original definition. The resulting data set consisted of 273 avalanche days and 984 non-avalanche days during the five winter seasons 2014/2015 to 2018/2019. For each of these 1257 d, we calculated aspect-specific values of P(AvD) and P(S≥s) using SNOWPACK virtual slope simulations driven with quality-checked data from the AWS Weissfluhjoch (see Sect. ). With the adapted definition of AvD (no consideration of aspect for the two surrounding areas not covered by the avalanche observations in the region of Davos), we obtained some more AvDs than in the other data sets. This follows from the fact that an east-facing avalanche in the region of Davos will count towards an AvD if there are other avalanches in the surrounding areas with unknown aspect.

Avalanche days were generally associated with new snow (HN1d=25 cm, HN3d=59 cm, p<0.001; see row = All in Table ). In contrast, nAvDs were typically characterized by no new snow (HN1d=0, HN3d=0, median values and p<0.001). Consequently, the median thickness of the layers including precipitation particles varied in a similar way (AvD: zpp=73 cm; nAvD: zpp=0 cm). The simulated critical weak layer was at a median depth of 75 cm on AvDs and 22 cm on nAvDs. The simulated critical weak layer had a significantly higher probability of instability on AvDs compared to nAvDs (Pcrit=0.92 vs. Pcrit=0.33, respectively; p≤0.001) and it was more often composed of persistent grain types (77 % vs. 47 % of cases, respectively; p≤0.001). As indicated in Table , these values varied between subsets. For instance, the median depth of the most critical weak layer was 44 cm on AvDs in 2020 and 91 cm in 2021, while on nAvDs the values were 4 and 65 cm, respectively. Similarly, on AvDs, the depth of the weak layer was 88 cm when the critical weak layer consisted of persistent grains (pg) compared with 47 cm for precipitation particles (pp).

At least one potentially unstable layer was detected in 84 % of the AvDs, and in only 2 % of the nAvDs. Moreover, in 7 % of the profiles, there was at least one other potentially unstable layer below the critical weak layer. These cases were rare on nAvDs (1 % of the profiles), but quite frequent on AvDs (66 %). The median difference in the depth between the critical and the deepest potentially unstable layer (zdeep-zcrit) was 14 cm (IQR: 4–44 cm). On AvDs, these layers were 15 cm deeper (IQR: 5–49 cm) compared with only 4 cm (IQR: 2–26 cm) on nAvDs. If such a deeper weak layer existed, it primarily consisted of persistent grains (90 %).

All explored variables (HN1d, HN3d, zpp and Pcrit) showed highly significant differences between avalanche days and non-avalanche days as demonstrated in the previous section. In the following, we will first explore their potential for a binary classification of AvDs and nAvDs, and then derive continuous functions describing the probability for an AvD.

The optimal thresholds (“thr”) to distinguish between nAvD and AvD for the seven subsets varied when cross-validating the model. For instance, threshold values ranged from 9 to 17 cm for the 24 h amount of new snow HN1d (median 12 cm) or from 22 to 47 cm for zpp (median 32 cm) (Table ). Applying these thresholds to the test sets, i.e. the data not used for training, showed that all four variables performed similarly well in correctly predicting nAvDs (TNR∈[0.96,1]; Fig. ). In contrast, larger variations were observed in the TPR, i.e. the proportion of correctly predicted AvDs. TPR was highest for HN3d (TPR=0.81) and Pcrit (TPR=0.79). The precision, i.e. the proportion of predicted AvDs that also were observed as AvDs, was highest for the two new snow parameters (PPV(HN1d)=0.84, PPV(HN3d)=0.83). However, these two parameters also showed a greater variation in PPV between subsets compared with Pcrit, which had a more consistent performance though a slightly lower PPV of 0.80. Overall, in terms of a balanced performance maximizing the F1 score, both HN3d and Pcrit had similar values (median F1 score of 0.80 in cross-validation). All approaches by far outperformed the natural stability index sn38 (median cross-validated F1 score of 0.24). Due to the limited discriminatory power of sn38, this variable was not considered further in the subsequent development of continuous models.

Analysing differences between the subsets in more detail also provided interesting insights. For instance, the optimal balanced zpp threshold to differentiate AvD from nAvD was 40 cm when the critical weak layer consisted of precipitation particles (pp) compared with 22 cm for persistent grains (pg); it was 47 cm in the region “elsewhere” and 22 cm in the inner-alpine region of Davos–Zuoz, where persistent weak layers are more frequently observed e.g.see Table 3. Similar results were also obtained for the two new snow variables, thus confirming what is known from a process-based point of view: when persistent weak layers are present, less new snow is needed to trigger natural avalanches .

The coefficients for the best-fitting sigmoidal functions f yielding the avalanche day estimators P(AvD)(x)=f(x) with x given by HN1d, HN3d, zpp or Pcrit are shown in Table . The Brier score (BS) was lowest for HN3d (BS=0.021 and BS+=0.156; Table ). Exemplarily, Fig.  shows the P(AvD) functions for Pcrit and HN3d. The values of P(AvD) predicted with these two variables correlated strongly (Pearson correlation coefficient r=0.82). The thresholds where the functions reached P(AvD)=0.5 (Table ) were slightly higher compared with the thresholds of the binary classification models described above (Table ). The F1 scores resulting from these thresholds deviated from the optimal F1 scores obtained with the simple classifiers by less than 1 %. Therefore, we only evaluated the performance of the continuous avalanche day predictor functions in the validation (Sect. ).

Finally, we explored the performance when averaging the P(AvD) predictions based on HN3d and Pcrit. Taking the mean of both models resulted in slightly better performance compared with the best-performing approach P(AvD)(HN3d): the BS decreased from 0.021 to 0.019, while the BS on the subset of AvDs, BS+, decreased from 0.156 to 0.144. Translating the mean probability into a binary classification resulted in a TPR of 0.81, a TNR of 0.99 and a high PPV of 0.95. Thus, the combined model detected more than 80 % of the avalanche days correctly and had the overall highest F1 score of 0.87.

In data set AV3, containing 5912 observed avalanches (Sect. ), the recorded failure depth zobs correlated with avalanche size (rs=0.45 and p<0.001; Fig. a). The median failure depth increased from 30 cm for size 1 avalanches to 100 cm for size 5 avalanches. While there is considerable overlap, the distributions of zobs were significantly different between pairs of consecutive avalanche size classes (Wilcoxon rank-sum test: p<0.001). Based on this data set, we derived logistic functions P(S≥s)(zobs) to estimate avalanche size from zobs (Fig. b; the respective coefficients are provided in Table ).

Comparing P(S≥3) and P(S≥4) with the observations from the data set AV1 on AvDs with at least two recorded avalanches, we obtained the lowest BS if the median avalanche size was estimated with HN1d as a proxy for the failure depth. For the largest recorded avalanche, on the other hand, zdeep was the best predictor (Table ). Considering only the subsets of data points where the avalanche size of interest was indeed observed (BS+ in Table ), i.e. for the 175 cases S≥3 when an avalanche of size 3 or larger was observed, zdeep had the lowest Brier scores for both the median and the largest avalanche size. Thus, zdeep outperforms the variables related to the amount of new snow in terms of capturing minority events, even for the median avalanche size, but has a tendency to predict avalanches larger than observed.

To develop the avalanche day predictor, we created a robust binary target variable (AvD versus nAvD) imposing restrictions on the observed avalanche activity in the vicinity of the AWS (Eq. ), ensuring a high reliability of the labelling. With this approach, the target variable included rather extreme cases of widespread activity versus no activity at all, which should be taken into account in model interpretation. Due to the necessary adaptation of the AvD / nAvD definition in the Davos validation data set, described in Sect. , the reliability of the avalanche day labels is lower as the definition is relaxed. This also shows in the ratio AvD / nAvD, which ranged from 6 % to 15 % in the training data set, while for the Davos validation set it was 27 %. In conclusion, the adapted definition leads to a higher proportion of AvDs, which seems responsible for the lower performance.

As the exact timing of avalanche release was not included in the data sets of observed avalanches, the explanatory variables were extracted from the snowpack and instability model simulations at fixed time steps (12:00 LT). This introduced uncertainty in the explanatory variables of both the training and validation data sets. With avalanche data sets from remote detection systems, providing the exact release time, this uncertainty would be removed. However, so far such data sets only cover short time periods and are very local in scope e.g., whereas the training data set used in this study included avalanches from the entire Swiss Alps observed over three winter seasons.

In a first step, we analysed the predictive power of the explanatory variables to distinguish between AvDs and nAvDs using different subsets of the training data set (AV1). An optimized threshold-based classification resulted in a reasonably high performance (cross-validated F1 score: 0.80) of Pcrit and clearly outperformed the conventional natural stability index sn38 (cross-validated F1 score: 0.24). While sn38 seems well suited to model natural avalanche activity from a physical point of view, the parameterization of this simple criterion within SNOWPACK has some weaknesses. Indeed, the shear stress can be simply calculated from the load and thus only inherits the errors from estimating precipitation mass based on measured snow depths, yet shear strength is a rather complex microstructural parameter. The current SNOWPACK parameterization of shear strength is based on density and grain type , which may not be sufficient to capture the influence of microstructure as also pointed out by . In particular, the evolution of the SNOWPACK shear strength over time only depends on density if grain type does not change. The poor performance of sn38 is in line with other studies . For instance, analysed sn38 based on field measurements and concluded that critical values of stability indices are less useful than their trends, a result confirmed by who showed that using time derivatives of sn38 has a higher predictive power. In contrast, the 3 d sum of new snow (HN3d), recognized as an important indicator of avalanche activity in past studies , yielded a classification performance (cross-validated F1 score: 0.80) similar to that obtained with Pcrit. Interestingly, the thickness of layers including precipitation particles, zpp, resulted in a slightly lower classification performance (cross-validated F1 score: 0.76), although it presumably captures the complete snowfall event, in contrast to HN3d. Potentially, using the mass of recent slab layers, which is more directly related to the load on the weak layer, may lead to better results than the depth of these layers.

Evaluating continuous one-dimensional sigmoidal P(AvD) functions for the four considered input variables (HN1d, HN3d, zpp and Pcrit) on the training data (AV1) resulted in negligible differences in F1 scores (≤1 %) compared with the F1-optimized threshold-based classification. The best performance in terms of F1 and Brier scores was obtained by taking the average probability from P(AvD)(Pcrit) and P(AvD)(HN3d), which was also confirmed by the validation on the independent data set from the region of Davos (data set AV3; F1=0.75). With this data set (AV3), the performance of the P(AvD)(HN3d) model in terms of predicting AvDs was rather low (TPR=0.55). A possible explanation is the more frequent formation of persistent weak layers in the region of Davos due to its relatively dry, inner-alpine snow climate, compared with the mean snow climate in the Swiss Alps. If weak layers are present within or at the snow surface, avalanches can release with smaller amounts of new snow e.g., which was also illustrated by the differences in optimal thresholds for the subsets from the training data (Sect. ). The combination of P(AvD)(HN3d) with P(AvD)(Pcrit) presents an alternative to using snow-climate-specific thresholds, as the Pcrit variable captures the presence of weak layers.

Most of the recently developed snow instability models are based on statistical methods which account for non-linear, complex relationships between target and explanatory variables. Here, we chose a rather simple approach based on one-dimensional sigmoidal functions which cannot account for interactions between explanatory variables but do allow for a simple interpretation of model output. Nevertheless, it should be noted that Pcrit itself is based on the output of a random forest model, which renders the interpretation of P(AvD)(Pcrit) difficult with respect to the original input parameters of the instability model. For a discussion of the influence of these input parameters on the direct output of the instability model, the layer-specific probability of instability, Punstable, see .

For a model to be considered useful, it has to provide more information than can be obtained from basic prior information , for instance, when simply assuming the base rate of avalanche days as the constant probability for an avalanche day. Thus, the potential benefits of a threshold-based classification can also be explored using the concept of information gain . Applied to our context, information gain is defined as the difference between the base rate probability of avalanche days and the posterior probability or the positive predictive value or precision;. As shown in Table  for the avalanche observations in the region of Davos (data set AV3), particularly combining the models P(AvD)(Pcrit) and P(AvD)(HN3d) provided a clear information gain (PPV≥0.76) compared with the base rate of avalanche days in this data set (BR=0.22). While the new snow model P(AvD)(HN3d) had a similar PPV, the combination of the two approaches resulted in a comparably balanced proportion of correctly detected AvDs and nAvDs.

Avalanche size is classified according to the destructive potential of the avalanche e.g., which is strongly influenced by the volume and mass of the snow in motion. Thus, avalanche size depends on the failure depth and the extent of the slab which released, the snow entrained in the avalanche path and also on the terrain itself e.g.. Of these factors, the one-dimensional snowpack simulations in combination with the instability model only provide information on the failure depth.

We estimated avalanche size as a function of various proxies of failure depth (HN1d, HN3d, zpp and zdeep). The correlation between size and failure depth of observed avalanches was demonstrated here (Fig. ) and in previous studies . The overall best indicator of the largest avalanche size in terms of Brier scores was obtained with the avalanche size estimator based on the simulated depth of the deepest weak layer, zdeep. This suggests that information on snow stratigraphy provides important additional information on avalanche size compared with using only indicators related to the amount of new snow. The size estimator based on zdeep, however, overestimated the occurrence of large avalanches. This might result from the above-noted oversimplification of the size estimator with one single input parameter, but also from the quality of the observed avalanche size distributions which represent a single, and often rather small, sample (median of two avalanches on avalanche days) from the potential avalanches on a given day. Moreover, there may be a reporting bias towards reporting larger avalanches . In addition, and as recently shown by , considerable uncertainty related to individual avalanche size estimates may exist. We cannot account for any of these factors in our analysis. Interestingly, also used a functional fit on the observed relation between avalanche size and failure depth to estimate avalanche size from a simulated proxy of failure depth. However, a comparison between our avalanche size estimator and this previous approach is not yet possible, as only indirectly validated simulated avalanche size as part of a classification tree for the prediction of danger levels.

The three key factors that characterize the avalanche danger levels are snowpack stability, the frequency distribution of snowpack stability and avalanche size . The models developed allow, for the first time, the use of a fully data- and model-driven approach to estimate these key factors of regional avalanche danger. We demonstrated that with increasing danger level, the probability for natural avalanches estimated by the avalanche day predictor P(AvD)(Pcrit) also increased for the stations and virtual slopes that matched the elevation and the critical aspects of the respective danger level (Sect. ; Fig. ). Interestingly, the benchmark model P(AvD)(HN3d) separated danger levels 1 (low) and 2 (moderate) from the upper danger levels high (4) and very high (5) even more strictly. This simple model P(AvD)(HN3d) could thus be of particular use for operational forecasting, especially when only meteorological variables and no detailed snow stratigraphy simulations are available. In line with these results, the proportion of AWSs predicting natural avalanche activity in regions with the same danger level increased from 1 (low) to 4 (high). Of particular note is that the avalanche day predictors indicated a wide range of conditions for danger level 3 (considerable), suggesting that splitting this danger level into several sub-levels as proposed by may allow for a better differentiation of avalanche situations. With respect to avalanche size, the predictions of the estimator based on zdeep showed a reasonable increase in the probability for large avalanches with increasing danger level, which is consistent with the definition of the danger levels.

Comparing the model-driven predictions of instability related to human-triggered avalanches with other studies exploring the relationship between the indicators of instability characterizing regional avalanche danger and the danger levels showed similar patterns (Fig. a). For instance, studies exploring Rutschblock stability test results, a stability test indicative of human triggering of avalanches , showed that the proportion of test results classified as very poor or poor increased with increasing danger level . Similarly, the proportion that at least one human-triggered avalanche was recorded in the area of observation or the proportion of observations indicating human-triggered whumpfs or shooting cracks increased in a similar manner. A similarly good agreement between model predictions and studies describing the (expected or observed) occurrence of natural avalanches is visualized in Fig. b. For instance, the proportion of P(AvD) predictions and the proportion of days on which natural avalanches were mentioned in the danger description in the Swiss avalanche forecast , or on which avalanches were observed in the region of Davos;, showed low values at 1 (low) and 2 (moderate), and high values at 4 (high) and 5 (very high).

We relied on simulated snowpack stratigraphy for virtual slopes with a 38∘ incline as used for operational avalanche forecasting in Switzerland. This simulation setup has been used operationally for several years, for instance, to assess snowpack wetting e.g. using the approach by or to monitor the presence of persistent weak layers. More recently, the Swiss avalanche warning service has been testing machine-learning models predicting wet-snow avalanche activity , snowpack instability and the avalanche danger level in a real-time setting. These models use as input snowpack simulations made for the four virtual slope aspects. As shown by for the instability model and the avalanche danger model, and by for the wet-snow avalanche model, predictions based on these simulations correlated with the aspects considered the most avalanche-prone or with signs of instability or avalanche activity.

Overall, we conclude that fully data- and model-driven aspect-specific predictions describing the probability of human-triggered avalanches and the occurrence of natural avalanches are clearly related to observational data and may therefore be suitable for estimating snowpack stability at the regional scale.