Zone 1 of the PGIM was derived from modelled ground temperatures. The ground temperatures were calculated based on a multiple linear regression analysis using the explanatory variables PISR and elevation (as a proxy for mean annual air temperature). These are the two most important parameters for the surface energy balance (Hoelzle et al., 2001) and are used in almost all permafrost distribution models. Ground temperatures measured in 15 reference boreholes were used as predictor variables. These boreholes were chosen from areas without ice-rich permafrost in Switzerland and Italy, close to the Swiss border (uppermost 15 sites in Table 1). Temperature is measured by thermistors in the boreholes at multiple depths between 15 and 100 m with a sub-daily temporal resolution. The thermistors commonly have a measurement accuracy of around 0.1 ∘C or better, and the types of thermistor and data loggers are specified in PERMOS (2016a).

The basic concept was to attribute a PISR value, an elevation value and a mean annual ground temperature (MAGT) to each of the 212 thermistors. Based on this dataset, the regression parameters a, b and c in Eq. (1) were determined and later used in Eq. (4) to calculate the ground temperatures in zone 1. The four detailed work steps involved are explained below. 1MAGT=a+b⋅R+c⋅E, where MAGT is the mean annual ground temperature measured by each individual borehole thermistor, R is the solar radiation value for each individual thermistor and E is the elevation value for each individual thermistor.

The regression result depends on the following parameters: PISR, elevation and reference MAGT. Changes in these parameters will influence the regression result. Elevation is independent from external influences and therefore uncritical for the result. Reference MAGT can be influenced by environmental conditions as well as by measurement errors, which are not considered here. Our small- to medium-sized statistical sample of measured ground temperatures might be distorted in comparison to the total statistical population. To test the sensitivity of our result to changes in the statistical sample, we carried out a 10-fold cross-validation by randomly splitting the reference MAGT into 10 samples, 9 of which were used as training data for our regression model and 1 as test data. The validation was carried out 10 times with each of the 10 samples as a test sample; subsequently the resulting MAGT deviations of all 10 runs were averaged. The calculation of PISR values, especially in steep terrain, included several other parameters such as the distance threshold, a slope threshold, an aspect-dependent weighting factor and assumptions on the timing of snow coverage. Indeed, the model was optimized by applying these parameters. The PISR values are however not an independent statistical unit of a sample of observations but are all based on the same calculation. This means that they can introduce systematic errors to the model, e.g. due to simplified assumptions of the snow cover timing, but they are not the origin of random changes in the regression result.

Zone 2 includes all forms of ice-rich permafrost such as rock glaciers or ice-rich talus slopes. Therefore, we defined areas in which the burial of ice or snow by rockfall can lead to the development of ground ice or in which epigenetic ground ice may have been relocated due to ice creep. We carried out nine work steps, as shown in Fig. S2:

Avalanche snow and rockfall deposits were assumed to accumulate at the foot of slopes steeper than 40∘. Potential locations of deposits were modelled by calculating runoff tracks from such slopes using ESRI ArcGIS with a 25 m digital elevation model (DEM; Fig. S2a) This was done in areas above 2000 m a.s.l., as only few, azonal permafrost sites exist below in the Alps (e.g. Cremonese et al., 2011).

These areas were then reduced stepwise by excluding spatial intersections with other datasets through the following:

All areas steeper than 30∘ (Fig. S2c), which barely contain ice-rich permafrost (Kenner and Magnusson, 2017), were removed. Snow avalanches seldom form deposits in such steep slopes, and epigenetic segregation ice in talus slopes would creep downslope.

In a final step, the resulting polygons were checked and manually edited if necessary. Some still contained areas in which surface bedrock excludes the development of ice-rich permafrost. In a few cases, parts of rock glaciers were missing due to errors in the reproduction of creep paths or due to small terrain steps steeper than 30∘. Manual editing included two tasks: all areas showing a bedrock surface, infrastructure or >50 % vegetation cover (for some reason not captured by the SAVI) were removed from zone 2. Missing parts of rock glaciers were added to zone 2 if at least parts of them were already captured by the automatic mapping approach. An exemplary editing task is shown in Fig. S3. The human polygon editor was not aware of the positions of the validation points during this process.

Using the same validation dataset, we validated the PGIM and two other permafrost maps of Switzerland in addition to compare the results: the Alpine Permafrost Index Map (APIM) created by Boeckli et al. (2012) and the Potential Permafrost Distribution Map (PPDM) (Gruber et al., 2006), available online on the swisstopo web map service (swisstopo, 2019a). A more detailed methodical background to the PPDM can be found in Haeberli (1975), Keller (1992) and Gruber et al. (2004). The permafrost maps were validated using a set of 98 evidence points of permafrost occurrence or absence, of which 10 represent the Swiss reference boreholes used to set up the regression model for PGIM zone 1 (Table 2). The reference boreholes are distinguished from the other records in the PGIM validation results. A more detailed verification, e.g. of modelled temperatures, was not possible due to the lack of data. The validation dataset partly consists of records collected by Cremonese et al. (2011), of which we only used direct evidence of permafrost occurrence or absence having exact coordinates. Further validation points were provided by continuous near-surface ground temperature data (GST) measured at 38 automatic weather stations in the Swiss Intercantonal Measurement and Information System (IMIS) (Russi et al., 2003). To balance the number of permafrost and permafrost-free validation points, only IMIS stations above 2400 m a.s.l. were used, which mostly lie within the critical elevation belt of discontinuous permafrost. The IMIS stations measure near-surface ground temperature at 10 cm depth with a Campbell 107 temperature probe. Of these 38 IMIS stations, 33 record a constant zero curtain during winter and are therefore expected to be on permafrost-free ground (Hoelzle, 1992). The remaining 5 stations show fairly constant winter GST between -3 ∘C and -4 ∘C and are located on active rock glaciers. They were therefore classified as permafrost sites. A few additional validation sites were added from different sources (Table 2).

All classes of the PGIM were attributed with the number of validation records lying within them indicating permafrost occurrence or permafrost absence. Additionally, zone 2 of the PGIM was validated against an inventory of 124 rock glaciers in the Albula Alps created by Kenner and Magnusson (2017).

Predicting the ground temperatures of the ice-poor reference boreholes on the basis of elevation and PISR yields a correlation coefficient of 0.94 and a standard deviation of 0.58 ∘C (Table 3, Fig. 1). This highlights the strong dependence of ice-poor permafrost on these two factors and its relatively high predictability. Including ice-rich permafrost in this regression analysis causes a drastic drop of the correlation coefficient and thus in the predictability of permafrost (Table 3 and Fig. 2).

Although thermistors of individual boreholes show clear deviations from the regression line, the cross-validation indicates a high robustness of the regression analysis result. The standard deviation between the modelled and measured ground temperatures stayed constant at 0.58 ∘C. The MAGT calculated during the cross-validation differed from the MAGT calculated based on the entire set of reference temperatures by a mean value of -0.003 ∘C and a standard deviation of 0.025 ∘C. The highest deviation found for a single thermistor was 0.14 ∘C. Explanations for the deviations of single boreholes or thermistors are presented in Sect. 4.1.

An example section of the PGIM is shown in Fig. 3. The entire map is available online in a web map service at https://www.slf.ch/pgim (last access: 10 July 2019) or as a shapefile at 10.5281/zenodo.1470165 (Kenner, 2018b). Together, zones 1 and 2 indicate a potential permafrost area (area considered by the map to potentially contain permafrost) of 2000 km2 in the Swiss Alps, which is considerably less than that indicated by the APIM (3710 km2; Böckli, 2013) and also less than on the PPDM (2550 km2; Gruber et al., 2006). To estimate the actual permafrost area (area effectively containing permafrost), Böckli (2013) considered all areas of the APIM with an index value >0.5. This results in an area of 2160 km2 for the APIM. The PGIM includes 830 km2 in the core area of zone 1 and 600 km2 in zone 2, of which a maximum of 90 % is expected to include permafrost according to the validation output. This results in an actual permafrost area of <1400 km2 in the Swiss Alps, which corresponds to <3.4 % of the area of Switzerland. For comparison, Keller et al. (1998) gave a value of 4 %–6 %.

The permafrost distribution over elevation is shown in Fig. 4 for different aspects. In very shady, north-facing slopes, ice-poor permafrost occurs down to around 2550 m a.s.l. In south-facing slopes, ice-poor permafrost terrain generally starts about 350 m higher. Ice-rich permafrost can occur in all aspects and has no sharp lower boundary. While the map indicates the highest frequency of ice-rich permafrost slightly above 2500 m a.s.l. for slopes facing north-west to north-east, it is at around 2600 m a.s.l. for slopes facing south-east to south-west.

The validation of the PGIM (Fig. 5) confirms the high accuracy of zone 1. Only two validation sites representing ice-poor permafrost are located outside the core area of zone 1, labelled “permafrost” (Fig. 5). In turn, no permafrost-free sites were located in the core area of zone 1. Zone 2 (potential ice-rich permafrost) includes 21 sites indicating permafrost and 2 indicating permafrost absence. Zone 2 furthermore includes 95.5 % of the rock glacier area recorded in the Albula Alps inventory (Kenner and Magnusson, 2017). This value applies to the automatically created version of zone 2 before it was manually edited.

The validation of the APIM (Boeckli et al., 2012) is shown in Fig. 6. The zones with a permafrost index of 0 (no permafrost) or 1 (definite permafrost) have a similar error rate to the corresponding classes in the PGIM but contain fewer validation records. The indices between 0 and 1 contain a rather homogeneous ratio of permafrost and no-permafrost sites; an increase in permafrost frequency is only visible for the very highly indexed areas (>0.8).

The validation result of the PPDM (Gruber et al., 2006) is shown in Fig. 7. The different probability ranges reflect the actual permafrost frequency quite well for the high probability classes but show larger deviations for the lower classes. Several permafrost evidence points exist outside the permafrost zonation of this map.

While the permafrost modelling based on the regression analysis was successful for ice-poor permafrost, it is not applicable for ice-rich permafrost (Table 3). This makes ice-poor permafrost much better predicted than ice-rich permafrost. The high correlation coefficient achieved by the regression analysis is remarkable because the borehole temperatures represent different landforms with strong differences in substrate and snow coverage. These factors, which influence ground temperatures (Haberkorn et al., 2015b; Hoelzle and Gruber, 2008; Schneider et al., 2012; Zhang, 2005), are represented in the regression result by rather small deviations of less than 1 K (Fig. 8).

Nevertheless, deviations exist due to advective cooling (Flüelapass; Fig. 8a and b; Phillips et al., 2009), substrate characteristics (relatively warm glacial polish at the lower Grépillon borehole; Fig. 8a and c) or temperature disturbances due to former glaciation (upper Grépillon borehole; Fig. 8a and c). Additional deviations might arise from the climate warming signal in the borehole temperatures. While near-surface temperatures might be in accordance with the current climatic conditions, temperatures at greater depth are still influenced by previous decades with colder climate conditions. As temperatures at several depths are included in our reference dataset, depth-dependent deviations can occur. Our model for ice-poor permafrost does thus not represent a permafrost distribution which is in equilibrium with the current climate conditions but a snapshot of the current distribution of ice-poor permafrost, which is currently adapting to warmer climate conditions.

Ice-rich permafrost cannot be satisfactorily predicted based on surface energy fluxes and requires the consideration of mass wasting processes such as rockfall and avalanche activity, as well as creep rates and varying glaciation during the Holocene. As these processes are often not known in detail, the accuracy of the cartographic representation of ice-rich permafrost is limited, as discussed in Sect. 4.3.

The uncertainty of a map can be quantified by the validation points, which are clearly mapped as being permafrost or not. In the PGIM, definitive permafrost is indicated by the core area of zone 1. In the APIM definitive permafrost is indicated by a permafrost index of 1 (for validation, values higher than 0.994 were rounded up to 1). The PPDM does not have a zone of definitive permafrost. Definitive permafrost absence is indicated on all three maps for areas outside the permafrost zonation. The PGIM could attribute 69 % of the validation points to a definitive class, while the APIM reached 33 % and the PPDM 23 % (Figs. 5–7).

Accuracy can be measured by the number of validation points wrongly attributed to a definitive class or by the plausibility of the description of a class. In the PPDM, seven permafrost sites occur outside the permafrost zonation. The definitive permafrost classes of the APIM and the PGIM predict all validation points correctly – with the exception of one site (Emshorn-Oberems), which is wrongly attributed on both maps. A weakness of our accuracy analysis is that the landforms and geographical locations of the validation sites do not represent the natural variability. Terrain- or region-related errors of the permafrost zones, which are not captured in this accuracy analysis, are therefore possible.

The APIM includes almost all areas in Switzerland in which permafrost will occur and is therefore a useful tool to exclude permafrost at a certain location. However, similar to the PPDM, it shows weaknesses in the reproduction of permafrost-free areas, while the PGIM performs better here. This might be caused by the elevational permafrost gap phenomenon introduced in Sect. 1. Figure 9a shows the example of the research site Flüelapass (Kenner et al., 2017), with a permafrost-free belt between the ice-poor and ice-rich zones.

Mapping solely based on thermal influences is not able to reproduce the permafrost gap and either neglects the ice-rich permafrost at the base of talus slopes (Fig. 9b) or overestimates the permafrost further upslope (Fig. 9b, 9c). This problem leads to a high number of permafrost-free validation points in the zones of medium permafrost probability on the comparison maps: for example in the 60 %–70 % probability zone on the APIM or the zone “local permafrost possible, patchy to extensive” on the PPDM (Figs. 6, 7). This may also cause the rather random distribution of permafrost-free validation points over the remaining probability classes of the APIM. In the PGIM the permafrost gap becomes visible when plotting the mapped permafrost area against elevation as shown in Fig. 4. A more accurate identification of this permafrost gap is an important step because it enables a better planning of ice-sensitive infrastructure in alpine terrain.

Zone 2 of the PGIM has a relatively high uncertainty. The low number of permafrost-free validation points wrongly attributed to this zone (2 out of 33; see Fig. 5) might rather overestimate the accuracy of the zone due to a general lack of permafrost-free validation points in talus slopes. However, there is very little ice-rich permafrost outside this zone, as indicated by the 95 % representation of the Albula rock glacier inventory within the automatically created raw version of zone 2. Accordingly, zone 2 should not be interpreted as a reliable representation of ice-rich permafrost but rather as a best guess including most of the ice-rich permafrost in Switzerland, with some by-catch of permafrost-free ground. The greatest challenges in mapping ice-rich permafrost are the correct representation of rock glaciers and the differentiation between loose rock sediments, which can contain ice-rich permafrost, and bedrock, which cannot. Merging existing, manually created rock glacier inventories in Switzerland to a nationwide inventory would improve zone 2 as the model approach could be focussed on ice-rich talus slopes.

Kenner and Magnusson (2017) highlighted the influence of the combined effect of lithology and precipitation on the occurrence of ice-rich permafrost: ice-rich permafrost is less frequent in sedimentary rock areas with high precipitation rates and relatively abundant in drier areas with crystalline or metamorphic lithology. These regional climate- and lithology-induced differences are difficult to implement in a map and must be carefully interpreted by the user.

The PGIM is the first large-scale permafrost map indicating permafrost temperature and ground ice content. The ice-rich permafrost in Zone 2, located in lower elevations than zone 1, typically has temperatures at or slightly below 0 ∘C (PERMOS, 2016a). Knowledge of the distribution of ice-rich and/or warm permafrost is particularly important for engineering purposes, as ice affects the ground stability and bearing capacity strongest and should therefore be avoided during the infrastructure planning phase (Bommer et al., 2010). The construction of infrastructure on ice-rich permafrost can lead to destabilization of infrastructure through subsidence or creep induced by ice warming or even melting beneath the infrastructure. Construction activity and the subsequent use of infrastructure can lead to rapid changes in hydrology and ice content (Duvillard et al., 2019). The hydration heat of concrete and heat from machinery in buildings are particularly problematic if the permafrost contains ice (Phillips et al., 2007). Permafrost in rock walls is very sensitive to climate fluctuations (Noetzli and Gruber, 2009), and rock temperatures influence rock slope instability (Davies et al., 2001; Gruber and Haeberli, 2007; Krautblatter et al., 2013). In general, substrates with negative ground temperatures require specially adapted construction materials to prolong the service life of infrastructure (Bommer et al., 2008).

The mapping approach of zone 1 can probably be adapted to other mountain regions or future climate scenarios without requiring any local ground temperature reference data. Equation (4) in Sect. 2.1 defines the distribution of ice-poor permafrost solely by the parameters PISR and elevation as a proxy for MAAT. PISR can be calculated globally based on digital elevation models. By setting radiation to 0, Eq. (4) represents a direct conversion between elevation and MAAT. If the elevational MAAT distribution is known, elevation models for other climate regions can be adapted using this conversion and can then be used in the regression formula defined in this study. As for different climate regions, the elevation models can also be adapted to future scenarios of air temperatures. However, when adapting our results to a different climate, the transient effects have to be considered. As explained in Sect. 4.1, our model represents the actual current permafrost distribution in the Swiss Alps and is therefore not in equilibrium with the current climate. This disequilibrium might be smaller or larger in future or in different world regions. The performance of our regression formula for permafrost temperatures in other world regions should be tested in a future study using the validation dataset of worldwide borehole temperatures. This validation dataset is currently in preparation (ESA, 2018). This universal application of our method would only be feasible for mapping ice-poor permafrost. Our approach for modelling ice-rich permafrost can only be used for regions very similar to the Swiss Alps, as it is designed for non-arid, vegetated areas and requires special datasets such as information on past glaciation.