The SSB was drilled in early July 2010, using refrigerated compressed-air-flush drilling technology. The stratigraphy was obtained using analyses of the cuttings (sampled every 10 m) and, for the first 100 m, through analysis of video logging. Since September 2010, the thermal regime of the SSB was monitored with thermometers placed according to the PACE protocol (Harris et al., 2001). The accuracy of the thermometers is 0.1 ∘C and the resolution is 0.01 ∘C. The thermistors recorded the daily ground temperature (minimum, maximum and average) at 20, 25, 35, 40, 60, 85, 105, 125, 145, 165, 205, 215 and 235 m of depth. Since 1998, the main climatic parameters at the site (air temperature, snow cover, incoming radiation) have been monitored. Below the 20 m depth, no significant seasonal variations in temperature are recorded.

The thermal properties of the three main facies observed in the stratigraphy were measured in the laboratory at three different temperatures (0, -1, -3 ∘C). Thermal diffusivity and specific heat were measured by NETZSCH-Gerätebau GmbH (Selb, Germany) using a NETZSCH model 457 MicroFlashTM laser flash diffusivity apparatus. Thermal diffusivity measurements were conducted in a dynamic helium atmosphere at a flow rate of ca. 100 mLmin-1 between -3 and 0 ∘C. Specific heat capacity was measured using the ratio method of ASTM E1461 (ASTM, 2003) with an accuracy of more than 5 %. Density of the rock at room temperature was determined using the buoyancy flotation method with an accuracy of better than 5 %. Thermal conductivity was calculated following Carlsaw and Jaeger (1959): λ=ρ⋅cp⋅κ, where λ is the thermal conductivity (Wm-1K-1), ρ is the bulk density (gcm-3), cp is the specific heat capacity (Jg-1K-1), and κ is the thermal diffusivity (m2s-1).

The temperature anomaly in the borehole at time t at depth z is modelled by the solution of the heat equation ∂A∂t-∂∂zκ∂A∂z=0 for the domain t,z∈-tmax,0×0,zmax. Note that Eq. (1) can be derived from the classical formulation of Carlsaw and Jaeger (1959) under the hypothesis that the density and the specific heat capacity are constant with respect to the depth z (see also Liu and Zhang, 2014), which is a good approximation for the SSB (see Sect. 4.1 and Appendix). Further, we have indicated with the earliest time tmax, for which we will reconstruct the GST and with the depth of the borehole zmax. Equation (1) can be solved to compute the temperature anomaly at any given past time t and depth z from the boundary values A(t;0), which represent the GST history. If the GST data A(t,0) are piecewise constant, the solution of the direct problem for Eq. (1) can be found explicitly (see Carlsaw and Jaeger, 1959). In our case, we need to solve the inverse problem of finding the GST from the borehole data, which provide the anomaly measured at present (t=0) or past times (t>0) at some depth z in the borehole.

In order to exploit the abovementioned explicit solution, it is customary to approximate the GST with a piecewise constant function (see Fig. 3): GSTt=τk,t∈-tk,-tk-1τ∞,t<tN, where tk, for k=1,…,N, is the sequence of times in the past at which we want to compute the value of the GST, and the τk's are the unknown values to be computed. The diffusive nature of the heat equation has the effect that fine details of GST signals are averaged away as time progresses. Therefore, in the field data, one can find signals coming only from long-wavelength GST variations that occurred in the distant past, whereas short-wavelength signals are observable only if produced in the more recent history. In order to take into account long- and short-wavelength variations in GST for which each of them makes sense, contrary to the common use of choosing uniformly spaced time points, we choose tk=1+0.2k2 so that the reconstruction points are closer to each other in the recent past and more separated for distant ages. The choice of the parameter 0.2 is such that the reconstructed GST can contain signals of wavelength of at least 33 years from 1600 onwards, 23 years from 1800 onwards, 16 years from 1915 onwards and 9 years from 1985 onwards.

Once the sequence tk is chosen, the relation between the borehole temperature at depth zj predicted by the model and the unknown values τk of the GST anomaly is linear. When comparing the anomaly A(z,t) described by the above equation with the measured data in the borehole, one has to take into account that measured data represent the superposition of the anomaly with a background signal (linearly increasing with depth) coming from the heat flow and past climatic changes since the Last Glacial Maximum as found for deeper boreholes by Safanda and Rajver (2001) or by Rath et al. (2012). This linear trend can be identified by linearly fitting the data from the deepest part of the borehole (below 60 m in our case). Following Eq. (3), imposing that the borehole temperature anomalies predicted by Eq. (1) for tk years ago at depth zj agree with the measured data leads to the linear system Lτ=m, where the column vector τ=τ1,τ2,…,τN,τ∞ collects the unknown GST values, m is the column vector of detrended measured data and L is a matrix with M×(J+1) entries (see the Appendix). Each row in L (and entry in the vector m), corresponds to a measured temperature in the well at present or at some time in the past. In this fashion, the GST reconstruction can be based not only on a single temperature profile but also on the variation in the temperature profile between the present and some years ago. To the best of our knowledge, this possibility, which enhances the robustness of the reconstruction, has never been exploited before in the literature. Given the detrended measures m, we must compute the vector τ solving the linear system Eq. (4). However it is well known that the inverse problem for the heat Eq. (1) is severely ill-posed and thus directly solving the linear system Eq. (4) would lead to a computed GST that would be highly oscillating and very far from the true physical values for τ. It is then necessary to introduce a regularization process by modifying the original problem (Eq. 4), in order to obtain an approximation that is well posed and less sensitive to errors in the right-hand side of Eq. (4). Classical regularization techniques include the truncated singular value decomposition (TSVD) and the Tikhonov regularization in standard form (Hansen, 1998), applied in Beltrami and Bourlon (2004) and Liu and Zhang (2014), respectively. In this paper, we propose the use of the generalized Tikhonov regularization, where the damping term is measured by a proper seminorm. In practice, instead of dealing with the linear system Eq. (4), we solve the minimization problem Lτ-m+αRτ, where α>0 is the regularization parameter and R is the regularization matrix. The use of a regularization matrix R for this application is a novelty, although several other regularization smoothing parameters were already used (i.e. Shen et al., 1992; Rath and Mottaghy, 2007). If R is simply the identity matrix, then the problem (Eq. 5) reduces to the standard Tikhonov method used in Liu and Zhang (2014). When α is large the restored GST is very smooth but the differences between the measured data and the temperatures in the well that would be computed by Eq. (4) from the recovered GST are large. On the contrary, when α is too small the data fitting is good but the GST becomes highly oscillating due to the ill-posedness. A good trade-off is not trivial and several strategies can be explored for estimating an optimal value of α: as an example, the generalized cross validation (Golub et al., 1979) often provides good results.

A common choice for R is a finite difference discretization of a differential operator (Hansen, 1998). In this paper, we consider a standard discretization of the Laplacian so that the constant and linear components of the solution are not damped in the Tikhonov regularization, Eq. (5), while we have a penalization of high oscillations. The details of the chosen regularization and of the GST inversion employed are described in the Appendix.

In order to validate our GST inversion method we have generated a synthetic data set as follows. An ideal GST was chosen (dashed curve in Fig. 4) and Eq. (1) was solved using a finite difference method with a spatial grid spacing of 1 m. Homogeneous Neumann boundary conditions were imposed at the well bottom and the ideal GST as Dirichlet data at z=0, thus obtaining synthetic data for the measurements of temperature in the well. The computed temperatures were saved for the depth at which the real thermometers in SSB are located (see Sect. 3.1), for the present time, as well as for 1, 2 and 3 years before present. We then used the generated data as input to the inversion algorithm described in the previous section and compared the reconstructed GST with the ideal one used to generate the synthetic data.

In the first experiment we fed our inversion algorithms only with the synthetic data for the present time. The value of α that best fits the exact GST is α = 0.15, but in Fig. 4 one can see that also varying this value by 33 % the reconstructed GST does not vary significantly.

Next we also fed the inversion algorithm with the synthetic data for the past years. First, the inversion is expected to be more accurate since the algorithm can average not only the temperature at a given depth but also the variation in the temperature in the last years at that depth. Moreover, the algorithm should also be more robust since it relies on a larger data set. Both these effects can be appreciated in Fig. 5, where it can be seen that the inversion in the last 50 years is more accurate than the inversion of Fig. 4 and that a wider variation in the value of α is possible without affecting the quality of the reconstruction very much.

The SSB stratigraphy is characterized by four different facies of dolostone (Fig. 6): a massive dolostone (from grey to pinky grey) comprises more than 90 % of the profile; three other facies (white dolostone, black stratified limestone, brownish dolostone) are thin intercalations (maximum 3.5 m of thickness and located mainly in the first 42 m). In particular facies d was not analysed for thermal analyses because it is very limited and does not have any lateral continuity.

The mean annual thermal profiles of the last three years (2013, 2014 and 2015) show a negative gradient between 20 m (a depth corresponding approximately to the depth of zero annual amplitude, ZAA) and 60 m that does not vary (-0.8 ∘C/100 m in all three years). At greater depth, the gradient is positive with slightly different slopes between 60–105; 105–125; 125–205; 205–215 and 215–235 (Fig. 7 and Table 1).

Table 2 shows the thermal properties of the three main stratigraphic facies encountered in the borehole. Facies a and c show similar density and thermal properties while facies b has higher density and higher conductivity. All facies have heat capacity values that increase with a decrease in temperature. In facies a, this behaviour also occurs for thermal conductivity and diffusivity values. In contrast, facies b and c show a reversed bell shape behaviour, with the minimum value recorded at -1 ∘C and an absolute maximum at -3 ∘C. Therefore, from a thermal point of view, only facies b is different. Moreover, at depths below the level of zero annual amplitude, this facies occurs only at depths of 34.5 and 90 m with a negligible thickness (2 and 1 m respectively) and at 142.5 and 205 m where it reaches 3–3.5 m in thickness. Clearly, the thermal influence of this facies is negligible; indeed, the gradient between 60 and 235 m is approximately the same as that between 60 and 105 m and between 125 and 205 m. The effects of the different thermal diffusivities measured in the different facies of the SSB are also illustrated in Fig. 8 where is possible to notice that the difference of temperature a posteriori between a model with a constant diffusivity equal to the average value of facies a between 0 and -1 ∘C (red dots) and the model with the different diffusivities for each different facies layer (blue dots) is absolutely negligible (< 0.02 ∘C) at all the depths, with the exception of the uppermost 20 m, where the difference is higher but still very low (0.06 ∘C).

According to the model proposed in Sect. 3, we found the best fitting with the thermal profiles (Fig. 7) using a heat flow of 70 mWm-2 (Della Vedova et al., 1995), a thermal diffusivity value equal to the mean between the value obtained for 0 and -1 ∘C for facies a, which is more widespread in the borehole, and an α value of 0.95 as shown in Fig. 9.

The linear system (4) was assembled including the detrended data measured at SSB in 2015 (Tj=0), in 2014 (Tj=1) and 2013 (Tj=2), at the 13 depths listed in Sect. 3.1, resulting in 39 equations. The anomalies of the GST reconstruction obtained with respect to the reference period between 1880 and 1960 has been computed using the value of α=0.95 for the regularization parameter (Fig. 10).

In permafrost environments, snow cover can influence GST variability in both space and time (e.g. Zhang, 2005; Schmidt et al., 2009; Morse et al., 2012; Rodder and Kneisel, 2012; Schmid et al., 2012; Guglielmin et al., 2014). This is especially the case for alpine areas where topography influences both the re-distribution of the snow by wind drift and actual snow cover evolution (e.g. melting date and duration). Nevertheless, GST and air temperature are well correlated (R2=0.8027) and present a very similar pattern over the last 15 years with only a slight warming (Fig. 11). This relatively slight effect of snow at this site is probably due to the high wind velocities during winter that, on average, prevent build-up of a thick snowpack. Figure 12 illustrates the temporal variability in snow cover on the GST. In general, the highest (> ± 5 ∘C) differences between mean daily GST and mean daily air temperature occur when there are large drops of air temperature during the winter. Sometimes, large differences also occur when there are large drops of air temperature during the summer where there is little or no snow cover because of high solar radiation that heats the ground surface. Correlation is even better between monthly mean air temperature, mean annual air temperature (MAAT) and mean annual GST (MAGST) (R2=0.8712 for this latter). This agrees with the results of Zhang and Stamnes (1998), who found that in a flat area in northern Alaska, changes in seasonal snow cover had a smaller effect than MAAT on the ground thermal regime.

Our reconstruction after the cold GST anomaly, between AD 1906 and 1941, shows a slightly positive peak (ca. 0.1 ∘C) in 1930 and afterwards a very unstable period with a first sharp decrease in temperature until 1989 (between -0.2 and -0.6 ∘C) and a second even sharper increase, reaching the uppermost GST anomaly value of the last 500 years (around 1 ∘C) in 2011.

On a regional scale, the Stelvio data can be compared with the MAAT obtained for the Alps by Christiansen and Ljungqvist (2011) (Fig. 10) and Trachsel et al. (2010). The maximum of the slight temperature increase during the first half of the 20th century in the Stelvio data (1930) falls exactly in the middle of the relative warming period between 1925 and 1935 in the Alps found by Trachsel et al. (2010) and is in good agreement with the date (1928) indicated by Christiansen and Ljungqvist (2011). Later, the sharp GST anomaly decrease was delayed in the Stelvio data (1989) with respect to the 1950–1965 period found by Trachsel et al. (2010) and 1965–1975 period found by Christiansen and Ljungqvist (2011). Finally, the most recent increase in temperature culminated in the Alps in 1994 (Christiansen and Ljungqvist, 2011), while in the Stelvio data it culminated in 2011.

The Stelvio reconstruction shows a long period of negative anomaly between AD 1560 and 1860 with colder conditions (< -2 ⋅ SD) between 1683 and AD 1784 and with a peak at -1.5 ∘C around AD 1730. This period of negative anomaly falls within this well-known cooling period (LIA). It is recognized in several kinds of proxy data although there are differences in both magnitude and timing across the world. According to Neukom et al. (2014), synchronous cold temperature anomalies occurred at a decadal scale in both hemispheres between AD 1594 and 1677. They also found two phases of extreme cold temperature in the Northern Hemisphere with the first between AD 1570 and 1720 and the second between 1810 and 1855. Syntheses of the LIA in the European Alps have been presented by Trachsel et al. (2012) and Christiansen and Ljungqvist (2011). Considering the common colder periods in these two Alpine syntheses, the LIA has three main negative peaks at AD 1570–1600, 1685–1700 and 1790–1820.

The LIA period has also been characterized by a widespread worldwide glacier advance, although the comparison between glacial evidences and temperature fluctuations is problematic because glaciers respond with different timescales (mainly depending on their size) and also reflect the precipitation regime, which is even more variable in space and time. According to Holzhauser et al. (2005), the LIA advance of the main Swiss glaciers has three peaks around AD 1350, 1640 and 1820–1850 respectively with the two later phases almost synchronous, also in the Eastern Alps (Nicolussi and Patzelt, 2000).

Close to the location of the SSB, the maximum LIA advance was diachronous. Nearby glaciers show a maximum LIA advance in AD 1580 (Trafoi Valley glacier; Cardassi, 1995), around AD 1770 (Solda glacier; Arzuffi and Pelfini, 2001) and in AD 1600 (La Mare glacier; Carturan et al., 2014).

The borehole area was presumably over-capped by the Vedretta Piana glacier until 1868. Due to the geomorphological position (on a watershed divide) the possible glacier should have been very thin and possibly cold based, as already stressed by Guglielmin et al. (2001). However, considering Fig. 10, the glacier should have been present in the borehole area with a buffering effect only between AD 1711 and 1834, with a peak at 1760, when the difference between the GST anomaly and the MAAT anomaly was maximum. This peak is pretty similar to the peak of the LIA in the Solda glacier (AD 1770) but not to the peak in the Trafoi glacier (AD 1580); this could be related to Vedretta Piana having a more similar glacier size and aspect (NE-N) to the Solda glacier than to the Trafoi glacier, although this latter is the closest to the Vedretta Piana.

Several deep Alaskan boreholes have been used to demonstrate the 20th century warming (e.g. Lachenbruch and Marshall, 1986; Lachenbruch et al., 1988) but only a few studies in Europe illustrate GST reconstructions that span a time period greater than 100–150 years (e.g. Isaksen et al., 2001; Guglielmin, 2004). In North America, only Chouinard et al. (2007) show a GST pattern of the last 300 years in the context of the permafrost of northern Québec. There, after the LIA (AD 1500–1800), an almost constant and marked warming of ca. 1.4 ∘C until 1940 was found, followed by a cooling episode (≈ 0.4 ∘C) which lasted 40–50 years, and finally a sharp ≈ 1.7 ∘C warming over the past 15 years.

There is a some similarity between the Stelvio reconstruction and the pattern of Canadian permafrost GST reported by Chouinard et al. (2013) after the LIA. Indeed, at our site there was also an almost simultaneous but greater cooling (0.9 ∘C) in the period between 1941 and 1989, followed by a sharp warming of ca. 1.7 ∘C. Conversely, GST reconstructions can be obtained with different models and it is interesting to compare our data with, for example, the PMIP3–CMIP5 simulations that include the effect of aerosol forcing by García-García et al. (2016): there, in the last 500 years, the GST shows a cold anomaly (LIA) between 1582 and 1840, with the most negative peaks between 1798 and 1840, slightly delayed with respect to our data.