The Schilthorn massif site (SCH) is situated at 2970 m a.s.l. (above sea level) in the north-central part of the Swiss Alps. The lithology of this non-vegetated site is dominated by deeply weathered dark limestone schists forming a surface layer of mainly sandy and gravelly debris up to several metres in thickness over presumably strongly jointed bedrock. Within the framework of the European PACE project (Harris et al., 2003), the site was chosen for long-term permafrost observation and was consequently integrated into the Swiss permafrost monitoring network PERMOS as one of its reference sites (PERMOS, 2016). The monitoring station at 2910 m a.s.l. is located on a small plateau on the north-facing slope, and comprises a meteorological station (shortwave and long-wave radiation, air temperature, humidity, snow height, wind speed and direction) and three boreholes (14 m vertical, 100 m vertical and 100 m inclined) with continuous ground temperature measurements from 1999 onwards (Vonder Mühll et al., 2000; Hoelzle and Gruber, 2008; Noetzli et al., 2008; Harris et al., 2009; PERMOS, 2016). Borehole data indicate permafrost of at least 100 m thickness, which is characterized by ice-poor conditions close to the melting point. Maximum active-layer depths recorded since the start of measurements in 1999 were generally around 4–5 m until the year 2008 but have increased to 6–7 m since 2009. During the superposition of the very warm winter of 2002/2003 with the summer heatwave of 2003 (Schär et al., 2004), the active-layer depth increased exceptionally to 8.6 m, reflecting the potential for degradation of permafrost at this site (Hilbich et al., 2008).

The monitoring station is complemented by soil moisture measurements from 2007 onwards and geophysical (mainly geoelectrical) monitoring from 1999 onwards (Hauck, 2002; Hilbich et al., 2011; Pellet et al., 2016). The snow cover at Schilthorn can reach maximum depths of about 2–3 m and usually lasts from October through to June/July.

The rock glacier Murtèl-Corvatsch (COR) is situated in the Upper Engadine, eastern Swiss Alps, and ranges from 2750 to 2600 m a.s.l., facing north–northwest. The surface consists of large blocks of up to several metres high, which are composed of granodiorite and metamorphosed basalt (Schneider et al., 2013). Below this coarse blocky surface layer of approximately 3–3.5 m in thickness, a massive ice core (up to 90 %, Haeberli, 1990; Haeberli et al., 1998; Vonder Mühll and Haeberli, 1990) is present down to 28 m, with a frozen blocky layer below reaching from 28 to 50 m, probably adjacent to the bedrock (Arenson et al., 2002).

The main monitoring station is situated on a flat ridge at 2670 m a.s.l. and comprises a meteorological station (short- and long-wave radiation, air temperature, surface temperature, humidity, snow height, wind speed and direction) established in 1997 (Mittaz et al., 2000; Hoelzle et al., 2002; Hoelzle and Gruber, 2008) and two boreholes drilled in 1987 and 2000 (PERMOS, 2016), which show significant small-scale heterogeneities in the rock glacier (Vonder Mühll et al., 2001; Arenson et al., 2010). Permafrost temperatures are around -2 ∘C at 10 m depth, and the active layer has a thickness of 3.2 m on average. Annual precipitation at the site is about 900 mm (982 mm St Moritz 1951–1980; 856 mm Piz Corvatsch 1984–1997), with a typical snow cover thickness of 1–2 m. Mean annual air temperature (MAAT) is -1.7 ∘C for the observation period of March 1997 to March 2008 (Scherler et al., 2014). Geophysical monitoring (mainly ERT) has been conducted since 2005 (Hilbich et al., 2009).

The Lapires (LAP) talus slope is located on the western slope of Val de Nendaz in Valais (46∘06′ N, 7∘17′ E) in the western Swiss Alps, ranging from 2350 to 2700 m a.s.l. with a north–northeast orientation. Its surface consists of gneiss schists, and the talus shows a thickness of more than 40 m at the locations of the boreholes described below. Snow avalanches and minor rockfalls with variable frequencies from one year to another affect the slope (Delaloye, 2004; Delaloye and Lambiel, 2005; Lambiel, 2006). The Lapires talus slope shows an active layer of about 4–5.5 m thickness situated on top of an ice-rich (30–60 %) permafrost layer of around 15 m thickness, with temperatures very close to the melting point (Scapozza et al., 2015; Staub et al., 2015).

The monitoring station consists of a meteorological station (air temperature and shortwave radiation since 1998, wind speed and direction and snow depths since 2009) installed in 1998 and three further boreholes installed in 2008 along a longitudinal profile (Scapozza et al., 2015). MAAT was +0.5 ∘C at 2500 m a.s.l.

Compared to the strong microtopography of Murtèl rock glacier, the Lapires talus slope is comparatively homogeneous regarding slope and microtopography. The permafrost distribution within the talus slope is discontinuous (mainly related to heterogeneous substrate dominated by fine-grained material in the western part and coarse-blocky material in the eastern part) and linked to a complex system of internal air circulation, also called the “chimney effect” (Delaloye and Lambiel, 2005). This air circulation is responsible for ground cooling at the bottom of the talus slope, where cold air is sucked up in winter. These 2-D (or potentially 3-D) processes cannot be explicitly simulated by the CoupModel; however, their effect on the thermal regime has been indirectly confirmed by specific 1-D distributed CoupModel simulations at this site (Staub et al., 2015).

The active rock glacier Ritigraben (RIT) is located in the area Grächen-Seetalhorn (46∘11′ N, 7∘51′ E), Valais, western Swiss Alps, and covers an area between elevations of 2260 m and 2800 m a.s.l. Block sizes at the surface range from 0.5 up to several cubic metres. Active layer depth is almost constant at 4 m.

A 30 m borehole was drilled in 2002 in the lower part of the rock glacier at an altitude of 2615 m a.s.l., which is gradually being sheared off from the base upwards due to the movement of the rock glacier. As a result, temperature is currently only measured to a depth of 13 m. Borehole temperatures indicate the formation of a seasonal talik between 11 and 13 m depth, which appears to be directly linked to snow meltwater and rainfall infiltration (Zenklusen Mutter and Phillips, 2012). The effect of these processes on the thermal regime has recently been analysed by explicit process modelling using the model SNOWPACK (Luethi et al., 2016).

The monitoring station is complemented by an automated weather station (net radiation, air temperature and relative humidity, surface temperature, snow depth, precipitation and wind speed and direction) installed in 2002 (Herz et al., 2003).

The Muot da Barba Peider (MBP) talus slope is located near the top of the NW-oriented flank of the Muot da Barba Peider ridge at 2960 m a.s.l. above the village of Pontresina, Upper Engadine, eastern Swiss Alps. The slope is 38∘ steep and is covered with coarse blocks (Zenklusen Mutter et al., 2010). The bedrock consists of gneiss from the upper Austroalpine nappe. Two adjacent (50 m apart) 18 m deep boreholes were drilled in 1996.

The drilling stratigraphy shows ground ice occurrences inside the talus, which reach a depth of about 4 m, with frozen bedrock below (Rist et al., 2006). Active layer depth varies between 1 and 2 m (Zenklusen Mutter et al., 2010). Due to the presence of experimental snow avalanche defence structures near borehole 1, the snow cover persists longer there in spring/summer, and thus influences the ground thermal regime (Phillips, 2006).

An automatic weather station was installed in 2003, showing MAAT of -3 ∘C. Regional values for mean annual precipitation are around 1500 mm at this elevation (Zenklusen Mutter and Phillips, 2012). Maximum snow depths have ranged between 0.5 and 3 m since 2003.

The study site of the Stockhorn (STO) plateau is situated on an east–west-oriented mountain crest around 3410 m a.s.l., to the west of the Stockhorn summit (3532 m a.s.l.) above Zermatt (45∘59′ N, 7∘49′ E), western Swiss Alps. The lithology consists of Albit-Muskowit schists, and the surface is characterized by patterned ground that has developed in a thin debris cover. Significant amounts of ground ice could be observed in large ice-filled cracks during construction works of a new ski lift in summer 2007 (Hilbich, 2009). Two boreholes only 30 m apart were drilled in 2000 as part of the PACE project (Harris et al., 2003). The recorded borehole temperatures show that the Stockhorn plateau is strongly affected by 3-D topography effects (Gruber et al., 2004) because the 100 m deep borehole close to the north face exhibits significantly colder temperatures than the borehole 17 m deep located close to the southern edge of the plateau. A meteorological station (measuring shortwave and long-wave radiation, air temperature, humidity, snow height, wind speed and direction) was installed in 2002. A soil moisture station was added in 2014.

The MAAT at this site is -6.4 ∘C for 2002–2012, and the annual precipitation is around 1500 mm (Gruber et al., 2004; based on King, 1990, and Begert et al., 2003). This site is characterized by low precipitation and high solar radiation (mean shortwave incoming radiation from 2002 to 2013: 209.3 W m-2) due to particular conditions created by surrounding mountain ranges exceeding 4000 m a.s.l. (Gruber et al., 2004).

Site-specific climate scenarios have been developed for eight meteorological variables at daily resolution for the period from 1951 to 2099 (Rajczak et al., 2016). The scenarios are based on an ensemble of 14 regional climate model (RCM) projections from the EU ENSEMBLES project (van der Linden and Mitchell, 2009). It should be noted that some variables have fewer GCM/RCM chains available: 7 for mean wind speed and maximum wind gusts and 13 for global radiation. Only the 13 chains with global radiation were used in the present study.

The ensemble accounts for a comprehensive range of model uncertainty, and is forced by the IPCC SRES A1B emission scenario (Nakicenovic and Swart, 2000). Due to their limited spatial resolution, site-specific features are typically not resolved by climate models, and even on resolved scales, models are subject to biases (e.g. Kotlarski et al., 2014). Statistical downscaling (SD) and bias correction (BC) techniques serve to attain representative conditions for the site scale and to remove model biases. SD/BC applications derive an empirical relationship between observations and model output. The established relationships are in turn used to translate long-term climate simulations to the site scale. Calibrating SD/BC techniques, however, requires long-term observations (e.g. 30 years and more), a prerequisite not met by the monitoring sites of the present study.

To obtain robust and reliable climate scenarios at the six considered sites, a newly implemented SD/BC method was used that specifically targets locations that lack long-term data. A detailed description and comprehensive validation of the approach is given by Rajczak et al. (2016). It is designed as a two-step procedure sketched in Fig. 1. In the first step, climate model simulations are downscaled to match long-term observational measurements at a most representative site (MRS) within a surrounding measurement network (e.g. MeteoSwiss weather stations). In the second step, the downscaled and bias-corrected time series from the MRS are spatially transferred to the site of interest (e.g. a permafrost monitoring site). Both steps rely on the quantile mapping (QM) method, a well-established statistical downscaling and bias correction technique (e.g. Themessl et al., 2011). The concept behind QM is to correct the distribution of a given predictor (e.g. climate model output) in such a way that it matches the distribution of a predictand (e.g. observations of the same variable at a monitoring site). Values outside the range of calibrated values are treated using the correction for the 1st (99th) quantile. Within this study, the spatial transfer is performed from an objectively selected MRS within the MeteoSwiss monitoring network. Consequently, Rajczak et al. (2016) show that the MRS is, in many cases, not the closest station but rather one at a similar altitude.

For calibration, we used series of borehole temperature data for each site with a minimum length of 10 years (Table 1). Borehole data are often considered as “ground truth data”, but potential measurement errors are possible due to several reasons (such as sensor or logger drift, logger failure and infiltration of water inside the borehole casing, to name a few). Borehole data in mountainous terrain are also influenced by 3-D thermal and hydrological processes (cf. Gruber et al., 2004; Lüthi et al., 2016), which are a source of additional uncertainty in 1-D model studies, especially in areas with large topographic variability. Further, an unequal repartition of data gaps may introduce a bias in the calibration. The gaps within the borehole temperature series have not been filled in order to avoid the introduction of inconsistency and additional errors in the data used for calibration. Periods with gaps are consequently ignored in the calibration process.

The model used for this study is the CoupModel, a 1-D numerical model combining soil, snow and atmospheric processes (Jansson and Karlberg, 2004; Jansson, 2012). This model has already shown that it is well suited to simulate mountain permafrost processes at Schilthorn (Engelhardt et al., 2010, Scherler et al., 2010, 2013; Marmy et al., 2013) and Murtèl rock glacier (Scherler et al., 2013, 2014). It also includes an optional procedure for semi-automatic calibration based on statistical indicators (see Sect. 4).

The model couples the water and heat transfer of the soil using the general heat flow equation: δ(CT)δt-LfρδΘiδt=δδzkδTdz-CwTδqwδz-Lvδqvδz, where C (J K-1) is the heat capacity of soil, Cw (J K-1) is the heat capacity of water, T(z, t) (K) is the soil temperature, Lf and Lv (J kg-1) are the latent heat of freezing and vapour, ρ (kg m-3) is the density, Θi(z, t) is the volumetric ice content, k (W m-1 K-1) is the thermal conductivity, t is the time, z is the depth and qw(z, t) and qv(z, t) (kg m-2 s-1) are the water and vapour fluxes.

The lower boundary condition is derived from the sine variation of the temperature at the soil surface and a damping factor with depth. The maximum model depth is different for the various sites due to the varying maximum depth of the available boreholes, but it is at least 30 m for all sites and well below the depth of zero annual amplitude (see Fig. 3). The prescribed heat flux at the lower boundary condition is therefore negligible. This enables comparatively stable conditions at the lower boundary, and accounts for the often isothermal conditions found in Alpine permafrost at this depth (Scherler et al., 2013; PERMOS, 2016). However, the long-term variability of permafrost conditions at the lower boundary cannot be simulated using this approach. The hydraulic boundary condition is given by gravity-driven percolation if the lowest compartment is unsaturated.

The upper boundary condition is calculated using the complete energy balance at the soil surface (or snow surface, if present). The convective heat inflow of water is given by precipitation and snowmelt multiplied by the surface temperature and the heat capacity of liquid water (Cw): qh(0)=Ts-T1Δz/2+CwTa-ΔTPaqw(0)+Lvqv(0) where qh(0) (J m-2 day-1) is the soil surface heat flow, Ts is the soil surface temperature, T1 is the temperature in the uppermost soil layer, ΔTPa is a parameter representing the temperature difference between air and precipitation, qv(0) and qw(0) are the vapour and water fluxes at the surface and Lv is the latent heat of vapour. For periods with snow cover, the upper boundary condition is calculated assuming a steady-state heat flow between the soil and a homogeneous snowpack using the thermal conductivity of snow. Temporally changing insulation conditions of the snow cover can be simulated by a critical snow height that corresponds to the snow height that completely covers the soil. It mainly depends on the surface roughness and reflects the fact that 50 cm of snow induces different insulation properties for a surface consisting of 1–2 m high boulders (e.g. for a rock glacier, COR) compared to a rather homogenous surface covered by sandy soil (e.g. at SCH; cf. also the discussion in Staub and Delaloye, 2016). The fraction of bare soil is then calculated by a ratio between 0 and this threshold (see Table 2) and further used to estimate the average soil surface temperature and surface albedo. This critical snow height is one of the parameters with the largest influence in our calibration procedure.

Snow is simulated by partitioning precipitation into rain and snow depending on temperature threshold parameters. The snow cover is assumed to be horizontally and vertically homogenous. Snowmelt is estimated as part of the heat balance of the snowpack, including net radiation, sensible and latent heat flux to the atmosphere, heat flux in precipitation, snow temperature change and heat flux to the soil. Further important processes in CoupModel are listed in Table 2 together with the respective equations. Symbols and units are listed in Appendix A.

The soil structure consists of 18 to 25 compartments (depending on the site) with increasing thickness with depth, ranging from 0.1 m in the upper layers to 4 m in the lower layers (Fig. 3). Initial conditions are estimated by the model using the first values of the meteorological data series. To avoid imprecise initial conditions, the model is run from 1981 onwards, although observational time series usually begin around the year 2000. No additional spin up is needed as the model usually reaches stable conditions (i.e. not influenced by initial conditions) after 10 to 15 years. Model tests with longer spin-up times only showed negligible differences with respect to the procedure described above, which may also be due to low ice contents in the bedrock layers at larger depths, which exert no large cooling effect on the surface from below during thawing. However, this approach clearly neglects all long-term effects of past climatic conditions on the ground thermal regime at larger depths. Therefore, simulation results at larger depths should not be interpreted in a climate context.

The GLUE method was used to test a large number of parameters at each site and to statistically assess their relative importance in the model. The relative importance of each parameter in the model is calculated based on the standardized covariance matrix of the tested parameters and related model performances using the LGM (Lindeman, Gold and Merenda) method (Lindeman et al., 1980) that averages the sequential sums of squares over all orderings of regressors. We group the parameters into six categories: (1) snow parameters (Train, Tsnow, ρsnowmin, Sk, Meltrad, Melttemp, ΔScrit), (2) albedo parameters (αdry, αwet), (3) hydraulic conductivity (kw, gm), (4) porosity (Φ), (5) thermal conductivity (Ksoil) and (6) evaporation (ψeg), and evaluate the influence of each parameter group on the statistical performance indicators r2 and ME at three different depths (near-surface, around 10 m, and the maximal depth of each borehole). The r2 accounts for variance, whereas ME accounts for absolute errors. This joint analysis of correlation and mean error is needed, as small temperature biases near the freezing point may result in large errors when latent heat processes are not adequately represented. While minimizing the ME ensures that the absolute values are near the observed ones, the correct simulation of the timing of freeze/thaw events can be improved by maximizing the correlation coefficient. It is clear that in the case of long-lasting freeze/thaw events, a good correlation will always be difficult to achieve, but a reasonable good match was achieved at least for the near-surface layers by optimizing the correlation. In a future step, other quantities such as the energy content of the ground (Jafarov et al., 2012) could be used for calibration, but variables directly related to the amount of water present (see Sect. 7) can also be used to enhance the calibration.

The results of the calibration are shown in Fig. 5 (left panels). Hereby, the relative importance of the six groups of parameters are shown for the three different depths as well as the absolute importance of the varying parameters on the simulations results (in %). A large relative importance identifies a parameter or process as being dominant with respect to the other parameter groups; however, it can still have a low overall importance on the simulation results, if the absolute importance is low.

As already noted in many previous studies (e.g. Lütschg et al., 2008; Gubler et al., 2013; Scherler et al., 2013; Atchley et al., 2016), Fig. 5 shows that the snow parameters have the greatest importance on the calibration performance for all sites. This importance is obviously pronounced at the surface, as the snow conditions represent a large part of the upper boundary condition by influencing the ground surface temperature during the snow-covered period. The variation of r2 at the surface is explained by snow with a relative importance usually above 50 %, ranging from 34 % at RIT up to 72 % at COR and 90 % at LAP. The differences between the sites can be explained by different snow conditions: there is a mean of about 280 days with snow cover per year at RIT, whereas COR only has about 200 days of snow cover days per year, indicating that the relative importance of snow for model calibration decreases with increasing snow cover. Hence a site with long and thick snow cover is less sensitive to variations in the snow parameters, as the snow persists anyway during a long period, than sites with less snow and a faster transition between snow-covered and snow-free ground. At LAP, snow cover conditions are additionally influenced by the presence of ski tracks and frequent occurrence of avalanches (cf. Staub et al., 2015).

In comparison with r2, ME is less influenced by snow parameters, as snow cover is more important for seasonal temperature variability (i.e. by accurately reproducing the transition between snow-covered and snow-free ground) than for absolute temperatures values. Interestingly, the relative influence of snow on ground temperatures is still large at greater depths: snow explains 12 % of the r2 and 10 % of the ME at MBP at 17.5ṁ; 65 % of the r2 and 43 % of the ME at RIT at 30 m, 19 % of the r2 and 54 % of the ME at SCH at 13.7 m; 8 % of the r2 and 8 % of the ME at STO at 98.3 m. At 57.95 m at COR, the snow shows a very limited influence as it explains only 0.1 % of the r2 and 0 % of the ME at this depth. This is probably related to the thick model layer with high porosity (cf. Fig. 3), where massive ice is permanently present, which decouples the lowest layers from processes at the upper boundary.

The albedo parameters have a significant influence on the calibration results at all sites, with relative importance for the ME ranging from 17 % at SCH to 74 % at LAP, reflecting the calibration of the surface temperature amplitudes. The r2 (reflecting the inter-seasonal variation) is less or not influenced by the albedo. In some cases at greater depths (r2 and ME at 10 m at COR, 8 m at LAP), albedo appears to have a high relative influence, sometimes higher than at the surface. This is most likely not related to realistic physical processes: intermediate depths, which are located between the well-calibrated upper and lower boundary conditions, are difficult to calibrate with any of the parameters tested (see the low percentages of absolute importance in Fig. 5). Therefore, those values are interpreted as statistical artefacts.

The sum of the influence of snow, albedo and evaporation parameters ranges from 58 (SCH) to 100 % (RIT) near the surface, from 26 % (COR) to 97 % (RIT) at medium depth and from 7 (STO) to 96 % (RIT) at larger depths for r2. This highlights the major role played by the upper boundary condition in the calibration. LAP and COR are exceptions as the importance of the upper boundary parameters is high at the surface (90 % for r2 and 97 % for ME at LAP and 78 % for r2 and 86 % of the ME at COR) but negligible at larger depths, where variation of r2 and ME is mainly due to variation of the thermal conductivity. The model needs to broadly tune the thermal conductivities (between 0.3 and 2.5 W m-1 K-1) of certain layers (10–15 m) to correct the temperature where a missing process or an incorrect soil structure parameterization need to be corrected. LAP and COR are two ice-rich sites (as seen e.g. in the geophysical results by Hilbich, 2009; Hilbich et al., 2009), with large blocks at the surface and high porosity. The combination of these effects decouples the intermediate layers from the upper boundary conditions to a larger extent than at MBP and RIT, which are also sites with coarse-grained material, but with a smaller estimated porosity by the model (Fig. 3).

Not surprisingly, the thermal conductivity plays a large role at depth where the relative importance (ME) ranges from 34 % at SCH to 89 % at STO and even 100 % at LAP (an exception is COR with 11 %). At MBP, thermal conductivity plays a large role even at the surface (67 % of the r2). As mentioned above, a potential radiation bias could be present in the input data of MBP due to the absence of on-site measured global radiation. A compensation of a potential bias would be expected either in the near-surface thermal conductivities or in the albedo values. Although the critical snow height parameter ΔScrit for MBP is very low, indicating a potential model compensation as it affects the albedo calculation, the albedo values themselves were calibrated with average values (αdry = 24.1 %, αwet = 19.1 %), which rather points to the absence of a large radiation bias. Similarly, the calibrated thermal conductivity values for the near-surface layer are about average (around 2–4 W m-1 K-1) compared to the other sites and do not indicate a large bias towards radiation-based surface temperatures that are too warm.

Among all sites, only RIT is insensitive to changes in the thermal conductivity (3 % of ME at 30 m depth). On the other hand, evaporation has a strong influence on calibration even at larger depths (44 %), which is in strong contrast to all other sites, where this parameter shows only little influence (between 0 and 10 %). When analysing the specific values obtained for the different calibration parameters, the parameter related to evaporation (water tension Ψeg, cf. Table 2) did not show specifically high or low values for RIT, but the parameterized values for Tsnow (minimum temperature at which precipitation only falls as snow) and ΔScrit (critical snow depth, at which the whole surface is considered to be covered by snow) were very low (Tsnow = -4.86 ∘C) and high (ΔScrit = 1.9 m), respectively. Whereas the former leads to comparatively large precipitation input as rain, the latter leads to an almost never completely snow-covered surface. In addition, the wet soil albedo for RIT is calibrated with the lowest value of all sites (αwet = 7.0), whereas its dry albedo is comparatively high (αdry = 34.6). In total, this parameter combination enables additional energy input by liquid water into the subsurface, which of course also explains the high sensitivity to evaporation. Even though this parameter combination may lead to an unrealistic process representation in the model, it is still in good accordance with observations, as at RIT the effect of 3-D advective water flow from the melting snow cover has been observed in borehole temperatures (Zenklusen Mutter and Phillips, 2012; Luethi et al., 2016), which explains this specific calibration outcome. Of course, the real 3-D process of meltwater infiltration cannot be explicitly included in our model.

Porosity and hydraulic conductivity of different horizons show little or no influence on calibration performance. For porosity this is not surprising as the parameter ranges are narrow to keep porosity close to reality. The only site showing sensitivity of changes in porosity is MBP (20 % of importance for the r2 at 10 m and 19 % at 17.5 m), which is specifically sensitive to changes of the porosity of the second soil layer (1.6 to 3.6 m depth). This points to an imprecise initial soil structure set-up that the model needed to correct, in this case the thickness of the surface blocky layer with high porosity.

When considering the absolute importance (% in Fig. 5, left panels), we notice that deep boreholes (COR, RIT and STO) have low percentages, which is not surprising as the temperatures at those depths vary on much longer timescales and depend primarily on the structural set-up of the model. As their future evolution is influenced by past climates, which are not included in the present study, simulated temperature changes at large depths will not be discussed within this study. However, their correct representation for present-day climate is important as the lower boundary condition for shallower levels. Contrary to these deep levels, the surface at all sites shows the highest sensitivity to the tested parameters, due mainly, as explained above, to the high importance of snow and albedo parameters.

After the LGM analysis, the most sensitive parameters for each site were identified to be used in the second iteration of the GLUE calibration procedure (cf. Fig. 4) to refine the calibration. The parameters listed in Fig. 5 (right panels) are the four to six most important parameters in the variation of statistical indicators; their relative importance for the variation of r2 and ME at three different depths is represented by the pie charts. One parameter that shows high sensitivity is ΔScrit, which allows the model to correct for the imprecise snow conditions and systematic biases in the building of the snow cover. The biases regarding the disappearance of the snow cover in early summer are corrected by the parameter Meltrad (coefficient for the importance of global radiation in the melt function of the snow). The thermal conductivity (ksoil) is important to adjust temperatures at middle and lower depth (COR, LAP, SCH and STO) but also at the surface (MBP). It can also be seen that snow parameters (blue colours in the pie diagrams) have stronger influence at the two bedrock sites (SCH, STO) compared to talus slopes and rock glaciers (COR, MBP, RIT, LAP), where other processes such as advection, convection and latent heat processes (due to the higher ice content) play a major role at depth.

To identify the most accurate runs among the 20 000 runs of the second iteration, we apply a selection based on two balanced criteria: (i) selecting the runs with the highest r2 (i.e. seasonal and interannual variability) in layers close to the surface and (ii) reducing the ME as much as possible (i.e. model temperature bias, leading to a model that is too warm or too cold globally) at greater depth. This option has been preferred over a globally best r2 or ME averaged over all depths because the latter would put the weight equally to all depths, whereas the surface is more important (and more accurate) regarding decadal changes.

Figure 6 shows the performance of the calibration at each site for three different depths, indicating the obtained value of r2 and ME. It has to be pointed out that a low r2 or high ME value does not mean that a better result at a certain depth cannot be obtained by GLUE because the selection process is a compromise between r2 and ME at several depths. Most calibration runs produce either well-calibrated temperatures near the surface or at greater depths, but not both for the same set of calibration parameter.

Calibration at the surface is very good at LAP and STO (r2 > 0.8), indicating a good representation of the upper boundary condition, especially regarding snow timing and duration. At the four other sites, r2 at the surface ranges between 0.65 (COR) and 0.77 (MBP). The comparatively low values at COR are not surprising due to the presence of very coarse blocks (> 2 m) at the surface inducing additional processes in the active layer that influence the near-surface sensors in the borehole (Scherler et al., 2014). The general variation and absolute values of near-surface ground temperature are satisfactory. Some systematic mismatches exist, such as insufficient cooling during winter at SCH and LAP, or excessive cooling in winter at MBP. At MBP, this is compensated by an equally high excessive warming during summer. At STO, the general behaviour of the near-surface temperature is accurately reproduced by the calibration, but with a reduced amplitude (warmer in winter and cooler in summer). At COR, there is an insufficient warming in summer, leading to a negative bias at the surface.

Temperatures at or around 3 m are the most challenging to calibrate as the influences of the upper and the lower boundary conditions have to be balanced. Moreover, this depth is usually within the active layer, and a small error in temperature (and/or soil water content) will lead to a mismatch in active layer thickness (e.g. at STO). Without putting a specific focus on matching the active layer thickness, the transition between frozen and unfrozen conditions is difficult to reproduce, especially given that subsurface structure and composition is generally unknown. The selection process showed that the selection of the best r2 at this depth led to the introduction of a strong positive bias in the absolute value (leading to disappearance of permafrost) and to poor calibration results at lower depth. A reduction of the ME at this depth led to a better representation of the permafrost conditions at all sites, but as a consequence the seasonal variations at this depth could not always be reproduced.

Seasonal variations at this depth are only reproduced correctly at MBP (low ME and high r2) and, to a certain extent, at COR and SCH (cf. Fig. 6). At SCH a warm bias is introduced in the model at 3 m depth, which can be explained by insufficient cooling during winter at the surface which propagates to larger depths. At COR, the warm bias at the surface is not reproduced at 3.55 m due to the permafrost conditions at this depth in the model. At RIT and STO, the model shows a constant temperature at the freezing point, leading to a large positive bias (1.74 K at RIT and 1.32 K at STO). At LAP, the model also shows temperatures at the freezing point at 3.6 m, and it is only able to reproduce some seasonal variations at the end of the calibration period. The bias at LAP is slightly positive (0.25 K).

Calibration of the lowermost layer is always satisfactory even though the model shows a small positive bias at STO (0.56 K), RIT (0.29 K) (probably originating from the propagation of the warm bias at 3 m) and MBP (0.28 K), and a negative bias at SCH (-0.25 K). Even if the calibration resulting from the GLUE procedure is not always satisfactory, it represents the optimal set-up for the given initial model for each site under the constraints of the semi-automated calibration approach presented in this study.

The GLUE calibration method is not meant to determine the physical value of a parameter. The model is physically based regarding its underlying equations, but has to rely on parameterizations for many of the complex processes in the subsurface and at the soil–snow–atmosphere boundary. The values for all model parameters at all depths cannot be known exactly, especially as almost no direct measurements of these properties are available. The GLUE method enables the value that gives the best fit with observations within the number of tested runs to be found. However, as the system is complex, with sometimes highly uncertain initial and boundary conditions, non-linear processes and simplifications of the model structure make an optimum calibration impossible (Beven, 2002). It is therefore more meaningful to analyse the residuals and the sensitivity to parameters than the values of the parameter themselves.

The calibration with GLUE depends on several subjective initial assumptions: (a) the choice of tested parameters and their range; this choice has to be made by the modeller prior to the calibration, and is a result of previous tests to identify relevant and sensitive parameters, and (b) the choice of criteria of acceptance. For the former, we tried to include a representative set of parameters for surface processes (snow, albedo, evaporation), subsurface processes (thermal and hydraulic conductivity) and properties that are characteristic of the specific geomorphological sites (porosity) in order to provide enough degrees of freedom for a satisfactory calibration. In addition we used our prior experience with CoupModel (cf. Engelhardt et al., 2010; Scherler et al., 2010, 2013, 2014; Marmy et al., 2014; Staub et al., 2015) to identify the most sensitive parameters. We tried to fix the allowed parameter range to physically plausible ranges, and verified that the obtained values during calibration were not distributed at the limits of these ranges. Regarding the choice of criteria of acceptance, we gave priority to good correlation coefficients near the surface and at intermediate levels while making sure that mean errors were acceptable at all depths. Here, different simulation results would have been obtained by e.g. giving more weight to intermediate levels; however, due to the uncertainties regarding the influence of past climates at the lower boundary, and regarding the exact representation of temperature evolution near the freezing point, the results would be less certain than in the case of a well-calibrated model at the upper boundary. Finally, uncertainties of the calibration add themselves to the uncertainties of observation and climate models when considering the long-term simulations.

One challenge of the calibration with GLUE is that there are many parameters to calibrate that are often underdetermined with respect to the available data. Therefore, the optimum is sometimes poorly defined, especially for sites that include processes like 2-D air circulation, which is not taken into account in the present model formulation. According to Beven (2002), an increased physical realism of the model structure does not aid in obtaining a better calibration. The perfect model would include an extremely large number of parameters and be unique to each site, and this is of course unrealistic.

In comparison with other permafrost modelling studies (e.g. Scherler et al., 2013; Westermann et al., 2013; Fiddes et al., 2015), the calibration method reaches a satisfactory calibration level for most of the sites. The obtained biases in the calibration may originate from several phenomena (which are very likely linked): (a) neglect of a sensitive model parameter in the calibration process, (b) parameter ranges that are too narrow, which do not allow the global optimum to be reached, (c) insufficient number of runs to find the optimum for each site, (d) errors regarding the initial model structure (soil type, horizons, etc.), (e) biases introduced in the reconstruction of the input meteorological data or (f) errors or imprecision in temperature measurements.

In addition, several potentially relevant processes such as convective flow of air in the coarse blocky layer or 2-D air or water circulation are not included explicitly in the CoupModel. In a previous study, this was solved by artificially creating a heat source/sink to reproduce convection within the coarse blocky layer of rock glacier Murtèl (Scherler et al., 2013). A similar parameterization for advective water flow within the SNOWPACK model has been published by Luethi et al. (2016) for Ritigraben.

Other processes that were not taken into account in the model concern the snow redistribution by avalanches or by wind that often takes place in high mountain environments (Hoelzle et al., 2001; Lehning et al., 2008; Mott et al., 2010; Gisnås et al., 2016). However, we could quantify the influence of several snow parameters. Snow has an especially strong influence at sites with shorter snow cover duration: there it is the most important parameter for the variations at the surface, but it also has a strong influence at deeper layers. The sites with a long-lasting snow cover (RIT and MBP) showed a reduced sensitivity to snow parameters as the snow is present most of the time, and the transition between snow-covered and snow-free conditions is less difficult to simulate. In general, the definition of the upper boundary conditions (snow, albedo, evaporation) appears to be a crucial issue as they influence the performance of the calibration of the whole soil column.

Facing the scarcity of measured data, it is difficult to check whether the calibration obtained by the semi-automated procedure is robust for outputs other than temperature. Possibilities exist to validate the calibration with electrical resistivity data (related to water/ice content) or direct soil moisture data but a thorough analysis of the quality of the present calibration or a calibration improvement by including these data in the calibration routine would be beyond the scope of this paper, especially as data do not exist for all sites. First tests have been made in this direction at STO, with promising results of a joint calibration using temperature and electrical resistivity data (Python, 2015). Efforts are also currently being made towards the installation of a soil moisture network in mountain environments (SNF project SOMOMOUNT, http://p3.snf.ch/project-143325). Soil moisture and geophysical monitoring data could then serve as additional validation of the thermal calibration (Pellet et al., 2016) as shown for the example of SCH (Fig. 9). Figure 9a and b show the soil moisture output of the model set-up giving the best fit with observed temperatures in comparison with on-site measured data that stem from soil moisture sensors adjacent to the borehole (see Hilbich et al., 2011). Although some biases are present, like the absolute value of the maximal peak in early summer (about 10 % mismatch at 12 cm), the absolute minimum during winter (about 7 % mismatch at 12 cm), or the stable summer maximum at 60 cm, the general behaviour is well reproduced: the mean values and the timing of freezing–thawing is satisfying. In a second step, we manually calibrated the soil physical parameter used in the water retention curve to define the minimal residual water, which also has a notable influence on the freezing-point depression. By this, the agreement with measured soil moisture was substantially improved (Fig. 9c and d), showing that model calibration can easily be improved if additional data sets are available. Figure 10 shows the resulting temperature difference at 10 m depth in the long-term simulations between the improved calibration and the reference run, indicating colder temperatures (∼ 0.3 K) and later permafrost degradation at 10 m depth compared to the reference run.

Given the various sources of uncertainty mentioned above and the choice of only one emission scenario (A1B) in the climate simulations, the results of the long-term simulation should not be considered as a prediction but rather as a projection of the range of the possible evolution of permafrost in the Swiss Alps under a given emission scenario. Our long-term simulations showed that the permafrost evolution is strongly influenced by the specific regional climate scenario applied (i.e. the specific GCM/RCM chain) but also by differently calibrated CoupModel set-ups. Climate scenario uncertainty appears to be the dominant component of uncertainty in this study.

A similar climate impact study has been carried out by Scherler et al. (2013), but with a different calibration procedure of the CoupModel and a different RCM downscaling technique for SCH and COR. In comparison to their results for SCH, the timing of permafrost degradation at 10 m around 2020–2030 and the moment when the entire seasonal thaw layer cannot refreeze anymore in winter are modelled similarly, but the consecutive warming after the start of degradation is smaller in the present study. Similarly, the 20 m layer shows a rapid degradation in Scherler et al. (2013), whereas it remains below the freezing point for most of the GCM/RCM chains in the present study. The discrepancies are mainly explained by a slightly different soil structure, which was part of the calibration approach in the present study. At COR, the results of Scherler et al. (2013) show slow warming at 10 and 20 m. In the present study, the warming is also slow, but once the 10 m layer is thawed, the warming propagates faster to deeper layers than in the results of Scherler et al. (2013); see Fig. 11. This difference is not surprising as Scherler et al. (2013) manually introduced a site-specific seasonal heat sink/source to compensate for the effect of air convection in the coarse blocky surface layer. By this, permafrost was conserved longer in the model than in a model set-up without parameterized convection. In addition, higher ice contents within the rock glacier ice core were simulated in Scherler et al. (2013) than in the present study (85 % vs. 62 %, cf. Fig. 3), which decelerates warming as well. In contrast, the calibrated porosity values near the surface are higher in the present study (49 %) than the manually calibrated values of the previous study (10 %). Porosity values in heterogeneous rock glaciers are of course always highly uncertain, but it has to be noted that the best results of the GLUE procedure were not obtained with the highest porosities for the deeper layers: during the selection process, the consideration of the r2 tended towards high porosities, but the best performances were obtained with lower porosities when considering the ME (cf. Fig. 3).

In contrast to Scherler et al. (2013), the cooling effect of convection in the coarse blocky surface layer was not hard-coded by an explicit source/sink term, but rather represented indirectly through automatic adaption of site-specific subsurface parameters during calibration, e.g. a comparatively high albedo (∼ 25 %), low critical snow depth parameter and particularly a larger porosity (see above). Nevertheless, the absence of an explicit convection parameterization for coarse blocky subsurfaces is still the major shortcoming of the CoupModel regarding mountain permafrost applications (cf. also Staub et al., 2015), and it leads to a probable overestimation of the warming at this site (Fig. 11). However, it is not yet clear how the cooling by convection would evolve in a context of climate change and permafrost degradation, which is why an explicit treatment of this process would be favourable compared to the static, hard-coded energy source/sink approach used in Scherler et al. (2013).

At all six sites, significant permafrost degradation is projected, driven mostly by the projected increase in air temperature during snow-free periods and the prolongation of these periods due to snow cover decrease. This is in good agreement with earlier sensitivity studies using the same model (Marmy et al., 2013) and similar studies from other regions (Etzelmüller et al., 2011; Hipp et al., 2012). In general, the sites with blocky material and higher porosity (COR, LAP, MBP) show a lower sensitivity to climate change, whereas the bedrock sites (SCH and STO) tend to have a more rapid degradation. At most places, a high porosity is coupled with higher interstitial ice contents, hence requiring more energy to melt the ice and warm the ground.

Changes simulated in the snow cover duration are mostly influenced by the increasing air temperature and much less by the change in mean annual precipitation sum. This is in agreement with both Wang et al. (2014), who stated that the increase in atmospheric freezing level is responsible for most cryospheric changes in the future, and Steger et al. (2013), who found that Alpine snow cover changes in the ENSEMBLES GCM/RCM chains are mostly driven by temperature increases. Our CoupModel simulations showed a decrease of snow cover duration of about -20 to -37 %, which is on the same order of magnitude as the results by Bavay et al. (2009), who projected a mean reduction of snow cover duration of ∼ 30–35 % for two Alpine catchments (run under the B2 and A2 scenarios), and by Schmucki et al. (2014), who projected a decrease of snow cover of 32–35 % for high-elevation sites. These numbers are furthermore consistent with Steger et al. (2013), who analysed Alpine snow cover changes in the ENSEMBLES climate models themselves. During the next 10–20 years, this reduction of snow cover may have an opposite effect to ground warming in summer: a decrease of the snow cover in autumn and early winter can lead to a cooling of the ground because the cool winter temperature can better penetrate the ground with no snow cover or reduced snow cover. However, sensitivity studies for a whole range of air temperature and precipitation changes suggest that until the end of the century, the effect of warming will dominate over the potential cooling effect in late autumn/early winter (Marmy et al., 2013). In spring and late summer, the decrease of snow cover has always had a warming feedback because the snow is no longer present to isolate the ground from the positive summer temperatures.

The results of the long-term simulations have to be considered with caution as uncertainty may arise at several steps of the model chains: errors in the measurements used for calibration, structural errors of the model, choice of parameters and choice of their tested ranges, biases introduced during the calibration, emission scenario uncertainty or GCM/RCM chains' uncertainty.