Introduction

Mountain glaciers store a relevant portion of freshwater resources. In the densely populated Alpine region, glaciers are a source of freshwater for domestic, agricultural and industrial use, and a relevant economic component for tourism and hydro-electric power production. Modifications in the glacier storage capacity associated with climate change can have relevant impact, as glacier melt often supports the water supply during summer.

The front variations of Alpine glaciers show a general trend of glacier retreat over the past 150 years, with intermittent re-advances in the 1890s, 1920s and 1970–1980s (Patzelt 1985; Zemp et al. 2007; Calmanti et al. 2007; Diolaiuti et al. 2012). Often, the onset of the post-Little Ice Age retreat and the later periods of intermittent re-advances in the European Alps are attributed to changes in winter precipitation rather than temperature (Vincent et al. 2005; Zemp et al. 2007). In the last decades, glacier retreat in the northwestern Italian Alps has been extremely evident, owing also to the higher temperature rise in these mountains when compared to the global average (Ciccarelli et al. 2008).

Climate affects glaciers through changes in mass balance, determined by the difference between accumulation and ablation (e.g., Paterson 1994; Oerlemans 2001). Thus, a direct way to study glacier response to climate variability is to focus on the modifications of the glacier mass balance. However, mass balance data are available only for a limited number of glaciers. On the other hand, records of snout position variations are widely available, and they provide indirect information on glacier response to climate fluctuations.

In this work, we use snout position data and consider a regional viewpoint, trying to assess the overall response of glaciers in a given area to regional-scale climate variations. When using a regional perspective, an empirical stochastic modeling approach relating glacier snout fluctuations to climate variability can be of value (e.g., Calmanti et al. 2007), as a substitute for detailed dynamical models of individual glacier behavior (e.g., Jouvet et al. 2009). The study reported here follows this approach, with the aim of estimating the overall glacier response in the northwestern Italian Alps to different climate change scenarios in the coming decades.

The rest of this paper is organized as follows: In “Glaciers and climate data” section, we introduce the data, and in “Glacier response to climate variability”, section we explore the sensitivity of Alpine glaciers to fluctuations in precipitation and temperature. On the basis of these results, in “An empirical model for the average glacier response” section, we built an empirical statistical model that relates climatic variability to glacier snout fluctuations. “Impact of climate change on average glacier snout fluctuations” section is devoted to the study of the impact of the average response of Alpine glaciers in different climate change scenarios. Discussions and conclusions are reported in last section.

Glaciers and climate data

Snout position data

We analyze a set of glacier snout fluctuation data in the western sector of the Italian Alps, in the area of Piedmont and Valle d’Aosta. Figure 1 shows a schematic map of the study area, indicating the locations of the 14 large glaciers which have been monitored for several decades and have thus been considered in our analysis.

Fig. 1
figure 1

Schematic map of northwestern Italy. The location (arrows) and names of the 14 large glaciers considered in this work are indicated. The numbers following the glacier names indicate the average annual snout retreats in month/year. Thin solid lines indicate borders between different Italian regions (Piemonte, Valle d’Aosta) and/or countries (Switzerland, France, Italy)

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Glacier snout data are collected by the Comitato Glaciologico Italiano—Consiglio Nazionale delle Ricerche (GCI) and regularly published in the GCI Bulletin (http://www.glaciologia.it/). Following Calmanti et al. (2007), to which we refer for details, we consider the mean annual snout variation, \(\delta X_{i}^{j}\), which measures the change in the snout position of the j-th glacier from the previous to the current year i. To compare the behavior of different glaciers, we use standardized snout fluctuations, \(\delta x_{i}^{j}\)defined as:

$$\delta x_{i}^{j} = \frac{{\delta X_{i}^{j} - \overline{{\delta X^{j} }} }}{{\sigma^{j} }}$$
(1)

where \(\overline{{\delta X^{j} }}\)is the average snout fluctuation and \(\sigma^{j}\)is the standard deviation of snout fluctuations for the j-th glacier.

All snout fluctuation data considered here cover the period 1958–2009, while the starting date varies with the individual glaciers. Since glacier behavior can be rather different in the late stage of retreat, we considered only glaciers which are presently longer than 1,500 m. For these glaciers, even in the presence of a forecasted retreat of a few 100 m, we can assume that the glacier length remains large enough to justify the use of the empirical model derived below.

Table 1 reports the main characteristics of the selected glaciers. Almost all values of \(\overline{{\delta X^{j} }}\)are negative, confirming the overall retreat of Alpine glaciers in this area and the homogeneity of the regional behavior of large glaciers in the northwestern Italian Alps (Calmanti et al. 2007). This homogeneity allows for averaging the standardized time series of the individual glaciers, to obtain a signal describing the regional glacier behavior in the period 1958–2009. Before averaging the individual glacier data, we tested for the possible presence of outliers and we removed documented anomalies reported in the GCI bulletins, such as a residual snow layer at the time of measurement or glacier breakup.

Table 1 From left to right: names of the selected glaciers, glacier latitude and longitude from the National Snow and Ice Data Center, http://nsidc.org, length (from the 1958 CGI Inventory, Comitato Glaciologico Italiano, 1959, 1961), slope and orientation (aspect), starting date of snout fluctuation measurements (the study period considered here starts in 1958), number of sampled years from 1958 to 2009, average snout fluctuation and standard deviation of the snout fluctuations
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Figure 2 shows the average standardized glacier snout fluctuations after removal of the documented outliers, \(\overline{{\delta x_{i} }}\). A 5-year running mean has been applied to the averaged snout data to smooth out short-term fluctuations and highlight longer-term trends and cycles.

Fig. 2
figure 2

Average standardized annual snout variation for the dataset of 14 glaciers considered here. Error bars indicate maximum and minimum snout fluctuations on the ensemble of glaciers measured in a given year

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Climatic data

The local climate in the western sector of the Italian Alps is characterized by relatively mild winters and warm summers. The seasonal cycle is characterized by minimum temperatures in December/January and maximum temperatures in July/August. Precipitation has maxima in spring (May) and fall (November) and minima in July and January (Regione Piemonte 1998).

Here, we use temperature and precipitation as proxies of the whole set of climate parameters which can influence glacier dynamics. In past years, Arpa Piemonte developed a dataset of gridded temperature and precipitation records, with spatial resolution 0.125°, by means of a optimal interpolation (OI) technique (Kalnay 2003) which produces a spatial interpolation of the data provided by an ensemble of meteorological stations in Piedmont and Valle d’Aosta (Ronchi et al. 2008) for the period 1958–2009. The OI technique consists in the assimilation of arbitrarily displaced ground station data on a selected regular three-dimensional grid map based on a background field (BF). For temperatures, the background field is obtained by a linear tridimensional downscaling of the ERA-40 archive from 1958 to 2001 and of the ECMWF objective analysis from 2002 to 2009 (for precipitation the BF is obtained from the station raw data).

In this work, we opted for averaging the meteorological records provided by OI analysis over a large area (the whole Piedmont and Valle d’Aosta region) rather than considering only the single grid point closest to the glacier location itself (see Calmanti et al. 2007 for a detailed discussion about this issue) obtaining a mean regional climatic signal.

Considering standardized variables rather than raw values helps reducing the bias due to differences in the altitude and position of the grid points. Therefore, monthly averages for each grid point are standardized (by subtracting the mean and dividing by the standard deviation) and then averaged over all the grid points in the study region. The monthly data are then aggregated to provide seasonal values. We call the seasonally averaged precipitation and temperature data P(n − m) and T(n − m), where n and m indicate the months marking the beginning and the end of the aggregation period.

Glacier response to climate variability

Glacier snout variations respond to climatic fluctuations with a time delay from years to tens of years (Oerlemans 2001). We estimate the value of the time lag between climatic variables and glacier snout response by systematically examining the lagged cross-correlations between individual climatic variables and snout fluctuations. To this end, for each possible lag τ (in years), we generate systematically aggregated precipitation and temperature variables covering the whole year, considering increasing periods of aggregation (n − m) (from 3 to 6 months) and 12 possible starting months: Using precipitation as an example, we consider P τ (1-3), P τ (1-4), P τ (1-5), P τ (1-6), P τ (2-4), P τ (2-5), P τ (2-6), P τ (2-7), P τ (3-5) and so on.

By using this technique, we obtain 48 potential lagged predictors for each of the two main climatic variables (12 starting months times 4 possible durations of the aggregation period). Each time series is then passed through a 5-year running average. Finally, we compute the cross-correlation r between the aggregated climatic data and the snout fluctuations \(\overline{{\delta {\kern 1pt} x_{j} }}\)and estimate its significance. For each choice of starting month and aggregation period, the estimated lag between the specific climatic variable and the glacier response is determined by the value of τ associated with the maximum cross-correlation.

The confidence level for the lagged cross-correlation coefficient is estimated nonparametrically by employing a Monte Carlo randomization technique. In addition, an error bar, δr, associated at each value of r has been calculated by a jackknife procedure (Tong 1990). These nonparametric approaches do not require any assumption on the probability distribution of the data, and they can deal with time series passed through a running average such as those considered here.

Table 2 reports the values of the maximum cross-correlation of precipitation (upper panel) and temperature (lower panel) with snout fluctuations and the corresponding lags, for each starting month and duration of the aggregation period. All cross-correlations between snout fluctuations and meteorological variables have been computed in the whole period 1958–2009.

Table 2 Cross-correlations between climatic variables and snout fluctuations, as a function of the starting month on the horizontal and of the duration of the aggregation period on the vertical. Darker colors indicate stronger correlations, whereas light colors and white indicate progressively weaker correlations. The upper panel is for precipitation; the lower panel is for temperature. In each cell, the upper symbol indicates the climatic variable and the lower figure is the value of the cross-correlation (color figure online)
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For precipitation, all cross-correlations are significantly different from zero, except for the variables with starting month 4, P 0(4-7)–P 9(4-9). There are two main periods of the year during which precipitation shows strong positive correlation with snout fluctuations. The strongest signal comes from the positive correlation with winter precipitation ten years before the measured snout fluctuation, P 10(11-3), and it is consistent with the results of Calmanti et al. (2007). This effect is associated with the accumulation period, and the estimated lag is typical of Alpine glaciers with the size considered here. The second is the spring period (February–May) in the same year of the snout fluctuation, P 0(2-5). A direct effect of weather conditions on glacier snout measurements is excluded, as snout estimates are obtained in late August–early September. The strong positive correlation for spring precipitation in the current year (τ = 0) can be interpreted by noting that abundant spring precipitation can reduce the retreat of the glacier front, owing to the high snow albedo that protects the ablation region from solar radiation. On the other hand, measurements of the snout position could be affected by the presence of a residual snow layer which survived summer melt, leading to estimates of the glacier length which could be larger than the true one.

For temperature, all cross-correlations are significantly different from zero, and the correlation coefficients are larger than for precipitation. A period during which temperature is strongly correlated with snout fluctuations is August to January (T 7(8-1), with a lag τ = 7 years), and this is presumably associated with summer ablation. The period corresponding to the physical ablation period, approximately from May to October (T 7(5-10) in the table), also has a good correlation with snout fluctuations. However, there are two other sub-periods which have stronger correlations: T 4(5-8), with a shorter lag τ = 4 years and T 8(7-10) with a slightly larger lag.

The strongest negative correlation, however, is found for spring temperature in the current year, T 0(2-5). Higher spring temperature could indeed favor liquid precipitation, which tends to accelerate melting of the snow layer covering the ablation region of the glacier, while lower spring temperatures favor precipitation in solid form. Additionally, higher spring temperatures can accelerate snow melt, resulting in a lower protection of the glacier ablation zone.

An empirical model for the average glacier response

Model construction

We introduce a linear empirical model relating N temperature and M precipitation variables, defined in a given seasonal period in each year, to the glacier snout fluctuations. Since climatic data start in 1958 and some climatic variables determine glacier snout fluctuations with a lag up to 10 years, we use the empirical model to reconstruct glacier fluctuations from 1968 to 2009.

The model takes the form

$$\overline{{\delta x_{i} }} = F\left( {\left\{ {T_{{i - \tau_{T,n} }}^{n} ,P_{{i - \tau_{P,m} }}^{m} } \right\}} \right) + \sigma_{r} W_{i} = a_{0} + \sum\limits_{n = 1}^{N} {a_{T,n} } T_{{i - \tau_{T,n} }}^{n} + \sum\limits_{m = 1}^{M} {a_{P,m} } P_{{i - \tau_{P,m} }}^{m} + \sigma_{r} W_{i}$$
(2)

where \(\overline{{\delta x_{i} }}\)is the mean standardized snout fluctuation on year i, averaged over all glaciers in the sample, \(P_{{i - \tau_{P,m} }}^{m}\)and \(T_{{i - \tau_{T,n} }}^{n}\)are the standardized precipitation and temperature in a given seasonal period (symbolically represented by m and n) on years i − τ P,m and i − τ T,n , respectively, τ P,m and τ T,n are the chosen lags for precipitation and temperature, respectively, and for ease of notation, we have omitted the explicit indication of the months over which the climatic variables are averaged. The function F is the deterministic part of the model which depends on the ensemble of climatic variables, and here it is taken as a linear combination. The constant term a 0 measures the long-term trend and the coefficients a T,n and a P,m weight temperature and precipitation dependencies, respectively. The last term is a noise term where σ 2 r is the variance of the stochastic component, and W i is Gaussianly distributed white noise with zero mean and unit variance (Tong, 1990). This is a “maximally random” version of the stochastic term that ensures that no significant statistical structure is left out of the model. The model residuals are then estimated as \(r_{i} = \overline{{\delta x_{i} }} - F\left( {\left\{ {T_{{i - \tau_{T,n} }}^{n} ,P_{{i - \tau_{P,m} }}^{m} } \right\}} \right)\).

The first step in the construction of the empirical model is the choice of a proper subset of explanatory variables among the 96 reported in Table 2. To reduce the number of possible choices, for each starting month (i.e., for each column in Table 2), we selected the aggregation period which has the highest cross-correlation with snout fluctuations. In some cases, cross-correlation values were very similar, as in the case of T 8(7-10), T 7(7-11) and T 7(7-12) in the temperature table. In this instance, the underlying physical process is the same regardless of the cumulating period length. We thus got 24 variables, 12 for temperature and 12 for precipitation, respectively.

To further reduce the number of explanatory variables, we used a backward stepwise algorithm using the AIC statistics (Akaike Information Criterion) for selecting the “best” models in the sense of explained variance and parsimony. Applying the backward stepwise procedure to all 24 predictors, we obtained 12 possibly relevant predictor variables: T 0(1-6), T 8(7-10), T 3(12-5), T 5(6-8), T 4(5-8), T 0(2-5), P 0(2-5), P 6(5-8), P 0(4-8), P 0(1-5), P 0(3-5), P 10(10-3).

To obtain an even more parsimonious model, we focused on out-of-sample prediction. In this approach, the model parameters are determined by means of least-square fitting on a given portion of the data-set, and the model is then be used to forecast the snout fluctuations in the time period other than that used to estimate the model parameters.

To assess the combination of variables leading to the best out-of-sample prediction, we considered all possible combinations of the 12 significant variables identified by the backward stepwise regression. For each combination, we verified the ability of the linear model to perform out-of-sample predictions, by estimating the model parameters from the first 20 years of the available data (from 1968 to 1988, defined as the training period) and using the resulting model to forecast glacier snout fluctuations in the second portion of the time series, from 1989 to 2009 (verification period). The procedure was repeated 1,000 times for each parameter choice. We then computed the root mean squared error RMSE and correlation coefficient r between the average model prediction and the snout fluctuation data in the verification period. Since a good out-of-sample prediction should have low RMSE and high correlation r, the models with best out-of-sample prediction ability are those with the highest ratio between r and RMSE.

Model results

From the above analysis, the model with the best prediction ability and lowest AIC value has the following 4 predictors: P 10(10-3), P 0(3-5), T 0(2-5) and T 5(6-8). The parameter values for this model are reported in Table 3. The results of the randomization test indicate that the null hypothesis that the values of the model parameters obtained from the least-square fit are not significantly different from zero must always be rejected.

Table 3 Model parameter estimates obtained from the out-of-sample prediction procedure. The first column indicates the predictor, the second column reports the value of the parameter obtained by fitting the linear model to the whole data-set, the third column reports the 95 % prediction bounds and the fourth column reports the probability that the parameter estimate from a random reordering of the data is larger, in absolute value, than the original parameter. Probabilities have been estimated from a total of 1,000 random reorderings of the time series. In the bottom of the table, R 2, adjusted R 2, AIC and RMSE = σ r are shown. In the lower line, the values in square brackets refer to a model where only the delayed climatic variables P 10(10-3), T 5(6-8) are used
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The two explanatory variables that have the largest influence in the estimation of the standardized snout fluctuation are P 10(10-3) and T 5(6-8), followed by the less relevant T 0(2-5) and P 0(3-5).

The residuals were tested to verify whether they are normally distributed with zero mean (Tong 1990; Jacobson et al. 2004), have constant variance across the observation period (homoscedasticity, Breusch and Pagan 1979) and are uncorrelated in time (Durbin and Watson 1971). The results showed that the selected model was able to produce residuals which passed the normality and homoscedasticity tests. Only one time lag (at 5 years) shows a significant residual correlation.

The upper panel of Fig. 3 shows the standardized annual snout fluctuations and the in-sample deterministic reconstruction (hindcast) from the selected lagged-linear model. The lower panel of Fig. 3 shows the out-of-sample prediction using the training and verification periods discussed above.

Fig. 3
figure 3

Upper panel: in-sample deterministic reconstruction of the average annual snout variations with the selected empirical lagged-linear model, which parameters are reported in Table 3. Confidence bounds for the in-sample estimation are computed by including the white noise stochastic component (reported in Eq. 2). The procedure is repeated 1,000 times, and the uncertainty band is defined by the 5th and 95th percentiles. Lower panel: out-of-sample prediction of the average annual snout variation. The black solid line represents the data, the blue solid one the in-sample estimation in the training period, the red solid one the averages of 1,000 different out-of-sample predictions obtained by training the stochastic linear model on the first 21 years of data. The blue band represents the 5th and 95th percentiles of 1,000 different forecasts

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For completeness, we have tested the behavior of a model obtained considering only the same-year climatic variables, T 0(2-5) and P 0(3-5), and of a model obtained by considering only the delayed variables, P 10(10-3) and T 5(6-8). While the former does not satisfactorily reproduce glacier snout fluctuations (neither in-sample nor out-of-sample), the model based on the delayed variables only produces results that are substantially equivalent to those of the full model, albeit with slightly lower statistical performance. This further indicates that the main drivers of glacier snout fluctuations are the delayed variables and that same-year spring variables play a minor role.

Impact of climate change on average glacier snout fluctuations

EC-Earth and RCP scenarios

The global climate change scenarios considered here are produced by the EC-Earth model, a recent Earth system model developed by a consortium of European research institutions (Hazeleger 2012, see also http://ecearth.knmi.nl/).

For EC-Earth, climate change scenarios have been simulated using the recently developed representative concentration pathways (RCP), see Moss et al. (2010). For this work, two RCPs were selected: a no-climate policy high forcing scenario (RCP8.5, corresponding to 8.5 W/m2 total anthropogenic forcing in 2100) and a “stabilization without overshoot” scenario (RCP4.5, corresponding to 4.5 W/m2 total anthropogenic forcing in 2100).

One advantage of the EC-Earth consortium is that, in the framework of the international program CMIP5, several members of an ensemble of climatic simulations have been made available. All simulations were obtained with the same model and the same parameterizations, and differed simply by the choice of the initial conditions, which were randomly extracted, after the end of the transient phase, from a 700-year-long spin-up simulation with constant radiative forcing at the 1,850 level. For each scenario, the different members of the ensemble provide an estimate of the intrinsic climate variability of the model. The different simulations are publicly available on the “Climate Explorer” Web site (http://climexp.knmi.nl/). Here, we consider eight members of the ensemble of EC-Earth model projections for each RCP scenario.

In order to avoid potential biases deriving by a dissimilar representation of ground height between EC-Earth climate model and the OI analysis, first we upscaled the OI dataset to match the model horizontal resolution and then we calculated, for each model grid point, the average differences between the climate historical runs and the upscaled OI dataset during the period 1958–2009, both for monthly mean temperature and precipitation. The results show an average model bias of −1.8 °C and a mean model precipitation difference ranging from −7 % to +7 % during Autumn and Spring. On this basis, we removed the calculated bias from the temperatures produced by EC-Earth for the future scenarios. No bias correction was applied to the EC-Earth precipitation time series.

After removal of the temperature bias, the temperature and precipitation series for each EC-Earth grid point were standardized by using the mean and standard deviation from the upscaled OI data in the period 1958–2009, and used to drive the best-performing empirical glacier model for the period 2010–2100.

Multimodel SuperEnsemble and SRES A1B scenario

The Multimodel SuperEnsemble method (Krishnamurti et al. 1999) is based on averaging several model outputs, which are individually weighted by an adequate set of weights determined by comparison with observations during a suitably defined training period. The technique first interpolates the output of each model run on the OI grid by bilinear interpolation. For each grid point, the multimodel average S is defined as

$$S = \overline{O} + \sum\limits_{i = 1}^{N} {a_{i} \left( {F_{i} - \overline{{F_{i} }} } \right)}$$
(3)

where N is the number of models in the ensemble, a i are weights to be determined by the procedure, F i is the output of the individual i-th model, \(\overline{{F_{i} }}\) is the mean model value in the training period and \(\overline{O}\) is the mean of the observations in the training period. The weights a i are obtained with a Gauss--Jordan minimization on the difference between the average multimodel average S and the observations O over the whole set of grid points considered. Once the set of weights is determined, the same weights are used to average the model outputs in the forecast period (thus assuming stationarity of the weights). This technique has been applied to a large number of weather parameters in Piedmont, with a significant reduction in forecast error (Cane and Milelli 2006, 2010).

For our study, we used the outputs of several regional climate models from the ENSEMBLES project for the SRES A1B scenario runs. We use 1961–2009 as a training period and 2010–2088 for the future scenario period (this is the period over which the ENSEMBLES projections are available). For the time series of temperature and precipitation obtained with the Multimodel SuperEnsemble technique, bias correction is not necessary as the weights applied to the individual model outputs are determined explicitly to reproduce the observations in the training period and the model average is imposed to be equal to that of the observations.

The average temperature and precipitation A1B Multimodel SuperEnsemble projections were then standardized using the mean and standard deviation from the upscaled OI data in the training period, and used to drive the empirical glacier model identified in the previous section for the period 2010–2088.

Simulation of future snout fluctuations

For each scenario (and for each ensemble member in case of EC-Earth), we generated the time series of monthly precipitation and bias-corrected temperature, from which we obtained the seasonally averaged signals used to drive the glacier model. For the future projections, we adopted the best-performing model determined in the previous section. All the parameters of the glacier model were estimated from the training period 1958–2009.

Figure 4 shows the standardized annual snout fluctuations over the period 1958–2100, for the RCP4.5 and RCP8.5 scenarios with the EC-Earth global model, and for the A1B scenario with the Multimodel SuperEnsemble approach. When different members of the EC-Earth model ensemble are used, the uncertainty bands overlap with each other and the colored envelope provides an indication of the joint climatic and glacier model uncertainty for a given scenario.

Fig. 4
figure 4

Standardized annual snout fluctuations over the period 1958–2100. For each climate run, 1,000 realizations of the glacier models were generated to provide an estimate of the uncertainty bands, defined by the 5th and 95th percentiles of the distribution of values produced by the glacier model. The first portion with white uncertainty bounds refers to the training period 1958–2009 used to determine the parameters of the glacier model; the second portion with green uncertainty bounds refers to the simulation of the snout variations over the period 2010–2100 (or 2088) from the climate scenarios. Upper panel: EC-Earth for RCP8.5; middle panel: EC-Earth for RCP4.5; lower panel: Multimodel SuperEnsemble for A1B. The colored curves refer to the average of the glacier model runs; for each EC-Earth scenario, eight different climate simulations with the same model are used

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Figure 5 shows an estimate of the cumulated snout positions in the period 1958–2100 for the three scenarios. To obtain the estimated glacier snout position, we converted the non-dimensional standardized snout fluctuations into dimensional values. To this end, the standardized annual mean snout fluctuation \(\overline{{\delta x_{i} }}\) was multiplied by the square root of the average of the variances used to standardize the glacier fluctuations in the period 1958–2009 and added to the average of the mean fluctuation of each glacier. The snout position is always referred to the value measured in 1968, when we start simulating the glacier length fluctuations. For all scenarios, the results indicate a continuing retreat of the glacier snout position.

Fig. 5
figure 5

Snout positions in the period 1958–2100 for the three scenarios RCP8.5, RCP4.5 and A1B. Same details as in Fig. 4

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Discussion and conclusions

In this work, we have analyzed the 1958–2009 time series of snout fluctuations for an ensemble of fourteen large valley glaciers in the northwestern Italian Alps, searching for empirical correlations with the variability of temperature and precipitation recorded in this area. We have built a best-performing empirical regressive model which relates year-to-year snout fluctuations, averaged over the ensemble of glaciers, to temperature and precipitation variability as provided by a new gridded climatic data-set for the area under study, obtained by optimal interpolation methods from raw ground station data. The empirical model was then driven by the temperature and precipitation data provided by different scenarios for future climate conditions, obtaining a set of ensemble predictions of the possible average response of the large glaciers considered here to climate change in the coming decades. Such procedure can, of course, be applied to any other mountain area where snout fluctuation data and meteo-climatic conditions are available.

In our approach, temperature and precipitation must be considered as proxies for the much more complex set of meteorological and environmental conditions which affect glacier dynamics. Incoming solar radiation and cloudiness, for example, can play major roles, but their direct effect was ignored here, mainly because we were looking for a simplified empirical model that uses the most commonly available data such as temperature and precipitation. Despite these limitations, the best-performing empirical model derived here explains up to 93 % of explained variance.

From the analysis of the glacier and climatic data considered here, we detected a significant dependence of snout fluctuations on summer temperatures and winter precipitation, with a temporal delay of, respectively, 5 and 10 years. These delays are consistent with the earlier results of Calmanti et al. (2007) and represent the time taken by the climatic signal, through modifications of the glacier mass balance, to affect the snout position. Precipitation variations take a longer time to affect snout position than temperature fluctuations, presumably owing to the more rapid response to melting, which acts mainly on the lower portion of the glacier, than to an increase in ice mass, which happens mainly in the upper part of the glaciers and requires some time to propagate to the snout.

In addition to the delayed response, we also detected a strong and rapid (i.e., same year) response of the glacier snout position to temperature and precipitation in spring. Higher spring temperatures could accelerate melting of the snow layer covering the ablation region of the glacier, resulting in lower albedo and favoring snout retreat. Higher temperatures also favor precipitation in liquid form, which further contributes to the washing out of the snow layer protecting the glacier. Instead, spring precipitation has a positive effect, possibly due to the fact that abundant snow precipitation in spring can increase the albedo on the lower portion of the glacier and protect the ablation region from solar radiation. On the other hand, a spurious effect of spring climatic variables cannot be excluded, as snout position measurements in late summer could be biassed by the presence of a residual snow layer, leading to a larger estimate of the glacier length. In any case, a model including only the two delayed variables produces results which are very similar to those of the full four-variable model, indicating that spring conditions do not seem to play a crucial role.

The empirical snout fluctuation model obtained from the data analysis was then driven by the temperature and precipitation records provided by the global climate model EC-Earth for the RCP4.5 and RCP8.5 scenarios in the period 2010–2100, and by a Multimodel SuperEnsemble mean of regional climate models from the ENSEMBLES dataset, for the SRES A1B scenario in the period 2010–2088. Using several members of an ensemble of climate simulations for EC-Earth and an ensemble of 1,000 replicates of the empirical glacier models, we were able to provide a confidence interval for the average snout fluctuations and glacier retreat expected in the coming decades.

In all cases, the glacier model indicates a strong retreat of the glacier snouts in the northwestern Italian Alps, with retreats between about 300 and 400 m with respect to the current position by 2050. In general, the RCP8.5 and A1B multimodel scenarios indicate slightly larger retreats than RCP4.5, as could be expected from the different severity of these emission pathways.

In the estimation of future glacier behavior, we have used the best-performing model defined from the data analysis, which uses a total of four explanatory variables: summer temperature with a delay of 5 year, winter precipitation with a delay of 10 year, and same-year spring temperature and precipitation. To provide estimates of the maximal uncertainty associated with the imperfect knowledge of the proper proxies for glacier response to climate variability, we have run the whole set of empirical models with four explanatory variables considered in the analysis, thus using all possible groups of four randomly selected precipitation and temperature combinations. In this case (not shown for the sake of conciseness), the 95 % uncertainty bars become obviously much larger, but they nevertheless clearly indicate significant glacier retreat by the end of the century.

The modeling approach developed here is very simple, and its results confirm in a rigorous way the common view that summer temperatures and winter precipitation are good proxies, at least in a statistical sense, of the climatic drivers of glacier length fluctuations (see, e.g., Belloni et al. 1985 for an early study of Italian glaciers). The fact that a purely statistical approach quantitatively identifies the main climatic variables affecting glacier dynamics gives confidence in the analysis procedure and in its use for future projections of glacier snout fluctuations. The approach adopted here also allowed for determining the average time delays characterizing glacier response and indicated a possible role of same-year spring variables which should be further analyzed in the future.

The model discussed here is appropriate for describing the average response of (moderately) large Alpine glaciers and can be applied whenever glacier snout fluctuations, temperature and precipitation time series are available. When the glaciers shrink to smaller and smaller lengths, the model could cease to be applicable, and thus in principle, it cannot be used to predict when glaciers will disappear. For such cases, as well as for the description of smaller glaciers, a physically based minimal model (Oerlemans 2011) should probably be preferred.