The main topographic data used in this study are a 1 m resolution digital elevation model (DEM) derived from two WorldView-2 (WV2) satellite stereo images acquired on 31 July 2010, covering the lower glacier, and 18 September 2010, covering the upper glacier. The DEM was mosaicked with tiles from the Canadian Digital Surface Model (CDSM) (20 m resolution) to include all adjacent topography that could cast shadows on the glacier. The merged DEM was resampled to 100 m resolution to allow for faster calculation with the mass balance model. The firn area was delimited by a mean snowline delineated from Landsat satellite images from the year 1986 to 2013 (Fig. 1c). A total of 18 cloud-free images were chosen near the end of the hydrological season (30 September) and used to map the mean transient snowline position at the end of summer, which was used as a proxy for the equilibrium line altitude (ELA). Image dates ranged between 22 August and 2 October, necessary to find cloud-free images capturing the transient snow line near the end of the ablation period.

To take into account historical glacier contraction in mass balance simulations, multitemporal DEMs and glacier boundaries from Tennant and Menounos (2013) (hereafter “TM2013”) were used to update the glacier geometry over time in the mass balance model. TM2013 derived DEMs and glacier extents from aerial stereo photographs from 1979, 1986 and 1993. For 1999, they used the Shuttle Radar Topography Mission (SRTM) DEM of February 2000, which they attributed to best represent the glacier surface at the end of the 1999 summer ablation season, due to the penetration of the radar wave in the following year's winter snowpack. The glacier extent in 1999 was derived from the closest cloud-free, 30 m resolution Landsat 5 Thematic Mapper (TM) image in September 2001. The 2009 DEM and glacier extent from TM2013 were derived from Satellite Pour l'Observation de la Terre 5 (SPOT 5) stereo images with a resolution of 2.5 m. Points matched on stereoscopic image pairs were gridded to a 100 m resolution in the ablation area and to 200 m in the accumulation area, where low contrasts resulted in a smaller number of elevation points and varying amounts of data gaps. We re-interpolated all TM2013 DEMs to continuous 100 m resolution using shape-preserving linear interpolation. The 2010 WV2 DEM was used instead of the 2009 DEM from TM2013, which particularly suffered from extensive gaps in the accumulation zone, but the glacier extent of August 2009 was conserved as a boundary for the 2010 WV2 DEM. The slope, aspect and sky-view factors were derived from all DEMs to be used as inputs for the mass balance model. A more recent, 2 m resolution DEM was built from a stereo pair of Pléiades satellite panchromatic images acquired in September 2016 and using the NASA Ames Stereo Pipeline (ASP) (Shean et al., 2016). This DEM was used to update the geodetic mass balance from TM2013 (in the Supplement). Since the 2010 WV2 DEM has the highest resolution and few gaps, it was considered the most reliable and used for model calibration and climate sensitivity experiments.

Two static balance simulations were performed, one using the 1979 DEM as the initial boundary condition and the other with the 2010 DEM. These were compared with a dynamical simulation in which the glacier geometry was adjusted with the multitemporal DEMs to consider the impact of glacier recession on mass balance. The TM2013 glacier boundaries were used, but two ice masses, disconnected from Saskatchewan Glacier since 1979, were excluded from the original TM2013 outlines (see Fig. 1c). The lateral, debris-covered moraines were also excluded from the glacier outlines (see Fig. 1c). The term “reference mass balance” (Bar) is used hereafter to refer to glacier-wide mass balance simulated with a fixed reference geometry, while the term “conventional mass balance” (Bac) is used for the simulation with adjusted glacier geometries (Huss et al., 2012). The effect of dynamical adjustment on Bar was obtained by subtracting the reference balance using the 1979 geometry (Bar1979) from Bac.

The physically based, distributed glacier mass balance model DEBAM (Hock and Holmgren, 2005) was used to simulate the mass balance of Saskatchewan Glacier over the period 1979–2016. The surface mass balance is expressed as 1bt=Pst-Mt-S(t), where b(t) is the point surface mass balance at time t, Ps is snow precipitation, M is melt and S is sublimation. The model calculates the distributed mass and energy balance on each 100 m × 100 m grid cell from the hourly downscaled NARR meteorological forcing data including air temperature, relative humidity, precipitation, wind speed and incoming shortwave global radiation. The energy at the surface available for melt on the glacier QM (W m-2) was calculated according to Eq. (2) and converted into meltwater equivalent M (m w.e. h-1) using the latent heat of fusion: 2QM+QG+QR+QL+QS+LWS↓+LWT↓+LW↑+1-αI+DS+DT=0, where I is the direct (beam) incoming shortwave solar radiation; DS and DT are the diffuse sky and terrain shortwave radiation, respectively; α is the albedo; LWS↓ and LWT↓ are the longwave sky and terrain irradiance, respectively; LW↑ is longwave outgoing radiation; QS is the sensible-heat flux; QL is the latent-heat flux; and QR is the energy supplied by rain (Hock and Holmgren, 2005). The ground heat flux in the ice or snow QG is often small for temperate glaciers and was neglected (e.g. Hock, 2005; Yang et al., 2021). Fluxes are positive towards the glacier surface and measured or calculated in watts per square meter. The model allows for different parameterisations for calculating energy balance components, depending on the availability of forcing data. The parameterisations used in this work are detailed in the next sections.

The simulated mass balance has been validated at the point scale against available seasonal and annual glaciological mass balance observations since 2012 and was at the glacier scale using the reconstructed geodetic mass balance from 1979 to 2016. These data are described in detail in the Supplement. The sensitivity of the reconstructed mass balance was tested with respect to (i) the NARR interpolation method, (ii) the NARR bias correction method, (iii) replacing NARR forcings with their AWS counterpart and (iv) uncertain model parameters. For (i), the model was run with NARR forcings, respectively, interpolated with the nearest-neighbour and bilinear methods. For (ii), the model forced with the bias-corrected NARR forcings was compared with a model forced with the raw NARR but correcting for the elevation difference between the NARR and the reference stations using the mean measured temperature and precipitation lapse rates (e.g. Fiddes and Gruber, 2014). For (iii), the NARR forcings (GNARR,Ta,NARR,WSNARR,RHNARR) were replaced one at a time by the AWS observations and the simulated point mass balances compared with stake observations for 2015, the only year with continuous AWS data and concurrent glaciological observations. For (iv), while the physical nature of the model did not require formal calibration, four uncertain model parameters were subjected to a sensitivity analysis to characterise their impact on the modelled mass balance. The precipitation lapse rate was varied within ± 4 % per 100 m, which corresponds to the standard deviation of the precipitation lapse rate calculated from the permanent weather network (see Sect. 3.2.5 and the Supplement). The ice albedo (αice) was varied within ± 0.03, which corresponds to the spatial standard deviation of ice albedo observed from satellite images (see Sect. 3.3.2). The aerodynamic roughness lengths for ice (z0_ice) and snow (z0_snow) were varied within ± 1 mm, which covers the range of values by Munro (1989) on nearby Peyto Glacier (see Sect. 3.3.4). The sensitivity to roughness lengths was also extended to ±1 order of magnitude, as an extreme case.

The validated DEBAM model was used to perform a climate sensitivity analysis of the reference mass balance (with respect to the 2010 glacier hypsometry: Br2010) to potential changes in air temperature (ΔTa) ranging between 0 and 7 ∘C (1 ∘C interval) and precipitation (ΔP) ranging between -20 % and +20 % (5 % interval). These warming and precipitation change scenarios encompass mean annual changes projected by ensemble general circulation model (GCM) simulations for the mid (2041–2070) and late (2071–2100) 21st century relative to the most recent 30-year climatological period (1981–2010) and under different Representative Concentration Pathway (RCP) scenarios (IPCC, 2013). The ensemble climate projections from the Coupled Model Intercomparison Project Phase 5 (CMIP5) were obtained from the KNMI (Koninklijk Nederlands Meteorologisch Instituut, Royal Netherlands Meteorological Institute) Climate Change Atlas (Trouet and Van Oldenborgh, 2013) for RCP2.6 (n=32), RCP4.5 (n=42), RCP6.0 (n=25) and RCP8.5 (n=39) for the grid point closest to the ELA of Saskatchewan Glacier (Fig. 1). The number of simulations (n) depended on the availability of the CMIP5 models for each scenario (IPCC, 2013). The IPCC AR5 Atlas (Intergovernmental Panel on Climate Change Fifth Assessment Report) subset was used, which uses only a single realisation of each model and weights all models equally, where model realisations differing only in model parameter settings are treated as different models (IPCC, 2013). The DEBAM model was run 63 times for every combination of ΔTa and ΔP perturbation imposed on the Ta,NARR and PNARR records over the 30-year reference period 1981–2010. Changes in mass balance for each sensitivity run were plotted as response surfaces, which provides a simple way to assess climate sensitivity across a range of possible climate change scenarios (e.g. Aygün et al., 2020b; Prudhomme et al., 2010). Mean temperature and precipitation changes along with their 95 % confidence intervals were overlaid onto the response surfaces to show the most likely future climate trajectories given by the GCM projections.

Daily and monthly averages of air temperature (Ta), relative humidity (RH), incoming solar radiation (G) and wind speed (WS) measured at the glacier AWS show notable differences between the 2 years of observation (Fig. 2). The winter of 2014–2015 was, overall, colder than 2015–2016, with frequent cold excursions below -15 ∘C and a winter absolute minimum of -27 ∘C vs. -17 ∘C in 2015–2016, although conditions were warmer in December. Relative humidity was generally high throughout the year (mean = 79 %), illustrating the predominantly humid climate of the Columbia Icefield but decreases noticeably in summer. The variability in daily RH is similar between the 2 years of measurements. The incoming solar radiation shows pronounced seasonality, varying between ∼50 W m-2 in winter and ∼300 W m-2 in summer, with daily variations between 50 W m-2 in winter and 150 W m-2 in summer caused by variable cloud cover. A gentle breeze blows on average on the glacier (mean wind speed = 4.46 m s-1), but wind speed shows significant day-to-day variations as well as higher values in winter. A gradual increase in wind speed is notably observed from the lowest monthly mean value in May 2015 (1.76 m s-1) to a maximum in February (7.13 m s-1). The historical precipitation records from the Columbia Icefield and Parker Ridge stations contain several gaps but still portray the seasonal and interannual variability in precipitation near the glacier (Fig. 2e). The mean annual accumulated precipitation throughout the historical period with complete data was 874 mm a-1 but varied between 276 and 1704 mm a-1. Precipitation data are more abundant in winter, with 58 % of precipitation falling between October to March, mostly as snow, and 42 % falling during April–September, mostly as rain.

The NARR meteorological variables used to drive the glacier mass balance model were compared with data from the glacier AWS (2014–2016) and the 29-year-long merged daily precipitation record (Fig. 3). Even prior to applying bias correction, Ta,NARR, RHNARR and GNARR show a good correlation with AWS observations on a daily scale, for both NARR spatial interpolation methods. As expected, the correlation is poorer for WSNARR, likely because the local glacier katabatic wind recorded by the AWS is not well represented in the NARR due to its coarse grid resolution. The NARR precipitation is also rather poorly correlated with observations (r=0.30). Biases in raw NARR variables are relatively small compared to the mean and range of values recorded (blue dots in Fig. 3), except for GNARR (30.4 W m-2) and PNARR (0.55 mm d-1), which represent 15 % and 25 % of their mean measured values over their period of observation, respectively. The cold bias (-1.26 ∘C) observed for Ta,NARR from the closest grid cell is consistent with the elevation difference between the AWS (2193 m) and the NARR grid cell (2430 m) (ΔZ= 237 m), which results in an expected temperature difference of -1.19 ∘C using the mean observed lapse rate of -0.5 ∘C per 100 m (see Sect. 4.3). Neither the scaling nor the EQM correction methods improved the Pearson correlation coefficient (r) – primarily since it is a relative measure of the synchronicity between two time series and is unaffected by the mean values. The EQM method was found to improve Ta,NARR best, closely followed by the scaling method, while the scaling method was slightly superior for RHNARR, WSNARR and PNARR. However, scaling only slightly reduced the errors for WSNARR and PNARR and had no effect on RHNARR, which had an initial low error. The diurnal scaling correction applied to GNARR also reduced its errors. Overall, the scaling bias correction method was the more efficient approach across all variables and both NARR spatial interpolation methods and was thus applied to all variables for consistency except for relative humidity, which was left uncorrected. Similar results, although with expectedly higher errors, were found for the interpolated NARR hourly data (Table S3 in the Supplement).

The bilinear NARR interpolation method resulted in slightly lower RMSE and bias values for the raw variables, i.e. before bias correction (blue bars and dots in Fig. 3), except for the slightly higher bias for RH. However, after applying the station-based bias correction, the bias and RMSE values were very similar among the two methods. As such, the NARR forcings downscaled from the nearest NARR grid cell were used as primary model forcings for the mass balance reconstruction and climate sensitivity experiments, and the sensitivity of the reconstructed mass balance to the NARR spatial interpolation method was further investigated in Sect. 4.6.

Monthly and annual averages of the downscaled NARR variables from the nearest NARR grid cell are displayed in Fig. 4. There is no visible trend in mean annual Ta,NARR over the 30-year period except since 2010, but there is a noticeable increase in minimum temperatures, with e.g. only 2 years with a monthly mean colder than -15 ∘C in 2000–2015 compared to 7 years prior to 2000. The positive trend seen in mean annual RHNARR is driven by increasing annual minima, while annual maxima show no trend, and so the seasonal amplitude decreases over time. The monthly RHNARR averages decrease in July and August (mean = 72 %), while winter months have higher values (mean = 80 %–82 %) (Fig. 4b). No noticeable trends occur in atmospheric emissivity (εNARR) and GNARR, despite the observed trend in RHNARR (Fig. 4d, f). A progressive decline in WSNARR occurs from 1984 onward, reaching the lowest annual value of the period in 1995 (∼4.3 m s-1) (Fig. 4c). A more subdued increase in WSNARR occurs afterward until 2010, followed by a decline. Finally, mean monthly precipitation shows no long-term trend but significant seasonal and interannual variability (Fig. 4e). A slight increasing trend in PNARR is noted in the last part of the record, since ∼2000.

On-glacier diurnal air temperature lapse rates were found to vary from -0.55 ∘C per 100 m at night and to a maximum of -0.34 ∘C per 100 m at midday following an increase during the day (Fig. 5a). The strength of the linear relationship between air temperature and elevation, as measured by the correlation coefficient (r), is generally high (r>0.95) but decreases slightly during daytime hours (r=0.92). While wind speed increased during the day, downglacier winds prevailed, with little deviation of the wind direction within the day (Fig. 5a). The wind blows dominantly downglacier, with the relative wind direction showing a mixed contribution of the main accumulation area upwind of the AWS and the glacierised plateau north of the AWS. Stronger daytime downglacier winds, possibly driven by a larger thermal gradient between the lower ice-free valley and the glacier, could result in downglacier cooling and correspondingly shallower near-surface lapse rates or even inverted lapse rates, as shown on the neighbouring Athabasca Glacier (Conway et al., 2021). Closer inspection of hourly lapse rates revealed that inversions only occurred 1.7 % of the time between May and August on Saskatchewan Glacier. On a monthly scale, the lapse rate, calculated from seven stations from the permanent network, varied between -0.58 and -0.42 ∘C per 100 m without any systematic seasonal pattern (Fig. 5b). The correlation for the monthly lapse rates is also more variable than for the diurnal lapse rates, varying between low values (r=0.6) in winter and higher values (r=0.94) in summer. The mean on-glacier summer (May–August) lapse rate (-0.46 ∘C per 100 m) was very close to that calculated from the permanent weather station network for the same period (-0.49 ∘C per 100 m), which gives confidence in extrapolating the monthly lapse rates from the network to the glacier surface. Superimposing the on-glacier anomalies of diurnal lapse rates onto the mean monthly lapse rates allowed for a better representation of the diurnal changes associated with the glacier wind.

The simulated annual specific (glacier-wide) conventional mass balance (Bac) was overall negative throughout the period (mean = -0.72 m w.e. a-1) with pronounced interannual variability (SD = 0.57 m w.e. a-1) (Fig. 7a). The cumulative conventional mass balance simulated with the multitemporal DEMs agrees well with the geodetic estimates (Fig. 7b). The simulated and geodetic cumulative mass balance were -26.79 and -25.59 ± 8.44 m w.e., respectively, for 1979–2016. The cumulative error in the geodetic estimates increases in 1999 due to the large error in the SRTM DEM, even though it was coregistered to the high-quality WV2 2010 DEM (in the Supplement).

The simulation with the 1979 reference DEM (Bar1979), when the glacier was thicker and larger, results in a larger cumulative mass loss (∼-3 m w.e. over 37 years) than when using the 2010 DEM (Bar2010) with the smallest historical extent (Fig. 7b). The difference essentially arises from the larger extent in 1979 which provides more area available for melting at lower elevations. The conventional mass balance simulation remains between the limits of the two endmember reference simulations, with a difference in cumulative mass loss of ∼ ± 1.4 m w.e., at the end of the period. The effect of dynamical adjustment was overall small from 1986 (first DEM update) onward (mean = 0.06 m w.e. a-1) but accelerated over the last 15 years (Fig. 7a).

Monthly energy balance shows that the sensible-heat flux (QS) dominates energy gains throughout most of the year (Fig. 9). The contribution of QS is fairly constant throughout the year, increasing only slightly in July–August and decreasing slightly in spring (March–May). The contribution of the net solar radiation flux (SW*) increases systematically from low values in winter (November–February) when the sun angle is low and the glacier is covered by highly reflective snow to peak values in July–August when the sun angle is high and low-albedo ice is exposed in the ablation area. Only in July and August does the net solar radiation (SW*) become the dominant energy source. The latent-heat flux (QL) is small over Saskatchewan Glacier, due to the generally high relative humidity (see Fig. 2). QL is positive on average and highest in summer, reflecting the predominance of deposition and condensation processes over sublimation. QL represents a small but non-negligible (7 %) heat gain throughout the year, which reaches 11.5 % in July–August. Energy loss occurs mainly by radiative cooling, i.e. through a negative net longwave radiation flux (LW*). Lower air and surface temperature, respectively, reduce the incoming atmospheric longwave radiation and outgoing longwave emissions from the glacier surface, thereby reducing LW* in winter. LW* increases somewhat in summer (June–August), mainly because the glacier surface is near its melting point, limiting longwave radiation losses. The energy supplied by rain (QR) has a negligible influence on the energy balance. Melting (QM) predominantly occurs between May and October and peaks in July–August, due to the elevated SW*, QS and QL fluxes and radiative cooling (LW↑) limited by the melting surface.

Four processes influence mass balance during the year (Fig. 10a). Snowfall and snow accumulation dominate during the accumulation season (October–April). Melt mainly occurs from May to October and peaks in July–August in response to the positive surface energy balance (Fig. 9). Deposition/condensation and sublimation fluxes are small. Net deposition predominates, while net sublimation occurs in the spring (April–June), when there is high incoming radiation and the upper reaches of the glacier have not yet reached the melting point (Fig. 10a). Although the QL heat flux was found to be non-negligible during summer (Fig. 9), the resulting mass loss is itself negligible compared to melting because the latent heat of sublimation/deposition is 7 times larger than that for melting. Moreover, the latent-heat flux has a pronounced diurnal cycle, switching from deposition at night when cooling of moist air causes the vapour pressure to increase relative to the melting glacier surface, while daytime heating reverses the vapour gradient between the glacier surface and the atmosphere, causing sublimation (Fig. 10b). Hence the two regimes tend to compensate each other, but nighttime deposition slightly dominates daytime sublimation, leading to a net positive deposition/condensation flux on average to the glacier surface.

Spatial patterns of the simulated reference mass balance (Bar2010) (Fig. 11) show an annual average snowfall of 1.54 m w.e., over the glacier with a minimum of 0.30 m w.e., near the toe, to ∼3 m w.e., over the upper reaches. Annual melt can reach 7.86 m w.e. a-1 at the glacier margin and 0.54 m w.e. a-1 in the upper accumulation zone. Net deposition/condensation predominates on average over the glacier, but fluxes are small (<0.03 m w.e. a-1), while net sublimation only occurs on the upper reaches of the glacier, mostly in the spring (Figs. 11c, 10a), corresponding to areas with high incoming solar radiation (Fig. S4 in the Supplement). On average, melting losses (mean =-2.22 m w.e. a-1) exceed snow precipitation gains (1.54 m w.e. a-1) and the small condensation gain (mean = 0.01 m w.e. a-1), yielding a mean negative reference annual balance (Bar2010) of -0.67 m w.e. a-1.

The static sensitivity of mean mass balance (Bar2010) components to climate perturbations (ΔTa=0 to +7 ∘C and ΔP=-20 % to +20 %) is shown in Fig. 12. The reference scenario (1981–2010) yields an average annual mass loss of -0.68 m w.e. a-1 (Fig. 12c). The response surface for Bar2010 shows that the glacier-wide mass balance is sensitive to changes in air temperature and much less sensitive to changes in precipitation (Fig. 12c). The ΔBa contours also become steeper and narrower with increased warming, which indicates a reduced sensitivity to precipitation and increased sensitivity to temperature, respectively. The seasonal mass balance response surfaces help to understand the Bar2010 sensitivities (Fig. 12a, b). The Bwr2010 response surface shows that a precipitation increase of +20 % can buffer the negative impact of warming on Bw up to +3 ∘C of warming but only up to +0.5 ∘C for Bar2010. Moreover, a warming of more than +6 ∘C with no change in precipitation would suppress net accumulation in winter, given the current glacier extent (2010) (Fig. 12a). The sensitivity of winter mass balance to temperature changes also increases markedly with warming, as seen by the progressive tightening of the contours in Fig. 12a. This is interpreted to result from decreasing accumulation due to the increasing shift from snowfall to rainfall and increased ablation during winter (October–April) due to the earlier disappearance of the snow cover under more pronounced warming. Conversely, the temperature sensitivity of summer mass balance (Bsr2010) increases only slightly with the warming scenario, and the steep contours in Fig. 12b suggest a small sensitivity to precipitation changes. The increased temperature sensitivity of Bar2010 with warming indicated in Fig. 12c is therefore mainly attributed to decreasing accumulation from reduced snowfall fraction and increased winter ablation as the climate warms and the snow cover retreats upglacier earlier in the spring (Fig. 12a).

The IPCC RCP scenarios for the mid (2041–2070) and late (2071–2100) 21st century were overlaid onto the response surfaces to show the most likely future climate trajectories. The RCP projection have significant uncertainties, as shown by their wide confidence intervals, and the annual mass balance change can vary by as much as ± 3 m w.e. a-1 within a single scenario. This illustrates the usefulness of scenario-free response surfaces to assess glacier mass balance sensitivity to climate as a background to evolving climate projections (Aygün et al., 2020b; Prudhomme et al., 2010). Nonetheless, given the current ensemble climate scenarios, the reference mass balance could decrease by -0.5 to -2.0 m w.e. a-1 by the mid century and by -0.5 to -4 m w.e. a-1 by the end of the century, relative to baseline conditions (Bar2010=-0.68 m w.e. a-1) and depending on the RCP scenario considered.

Since mass balance displays a large sensitivity to temperature and because glacier melt is the outcome of complex glacier–atmosphere energy exchanges, the sensitivity of energy and mass fluxes to warming alone was further investigated in Fig. 13. The increasingly more negative mass balance in response to warming is dominated by increased melting (∼93 %), while increasing condensation/deposition accounts for ∼-3 % (mass gain) of the net annual mass changes in response to warming (Fig. 13a, b). Warming alters the precipitation phase, with the snowfall ratio decreasing non-linearly from 0.80 under present climate to 0.47 at ΔTa+7 ∘C (Fig. 13c). This progressive conversion of snowfall to rainfall accounts for ∼10 % of the mass changes in response to warming (Fig. 13b).

The total energy input to the glacier surface increases with warming temperatures, and this energy surplus is predominantly used for melting (QM), which shows a non-linear increase with respect to warming (Fig. 13d, e). Interestingly, the increase in energy supply with warming is mainly driven by an increase in net solar radiation (SW*) and latent-heat flux (QL), with more subdued increases in the temperature-dependent sensible heat (QS) and net longwave radiation fluxes (LW*) (Fig. 13e). Since cloud cover remained unchanged in the sensitivity experiments, the increase in SW* with warming is entirely driven by the decreasing albedo, as snow cover duration on the glacier decreases (Fig. 13c). Since the relative humidity also remained constant in our sensitivity analyses, warming leads to higher atmospheric vapour pressures, since the saturated vapour pressure of the air increases with warming. Since the glacier surface is constrained to the melting temperature (0 ∘C) during a large part of the year, the increase in surface saturated vapour pressure in response to warming will, on average, be less than that of the atmosphere, causing the vapour pressure gradient to increase and boost QL fluxes (condensation/deposition) to the surface. Similar reasoning applies to QS; i.e. the near-surface temperature gradient will increase in response to atmospheric warming. While the rainfall ratio increases with warming, its influence on the energy balance is insignificant (Fig. 13e), but the reduced snowfall greatly impacts winter accumulation (Fig. 12a). Increasing net solar radiation (SW*) contributes from 51 % to 42 % of the increase in QM(ΔQM), with this contribution decreasing with warming. The contribution of QL to ΔQM increases from 27 % to 29 % in response to warming, while that of LW* increases from 5 % to 9 %. The contributions of QS (∼19 %) and QR (∼1 %) are more constant across the warming spectrum (Fig. 13f).

The results in Fig. 13 allow for apportioning the mass balance sensitivity to warming to four different processes (Table 1): (i) atmospheric warming, which causes an increase in the temperature-dependent fluxes (ΔLW∗+ΔQS) and contributes on average 24.3 % to the mass balance sensitivity to warming; (ii) a precipitation phase change feedback, which contributes 10.3 %; (iii) an albedo feedback, which contributes 44 %; and (iv) a humidity feedback, which contributes 22.3 %. While the contributions from atmospheric warming and the humidity feedback increase with the level of warming, the precipitation phase feedback remains constant, while the albedo feedback decreases over time (Table 1).

This study focused on reconstructing the mass balance of a glacier using a physically based model constrained by a sparse set of glacio-meteorological data without calibration. This situation is common to many mountain glaciers around the world where logistical and financial constrains preclude continuous monitoring programs. In this context, the outputs of reanalysis products represent a useful alternative for driving glaciological models. Several previous studies have used reanalyses to force hydrological and glaciological models in mountainous regions using statistical downscaling. Among the downscaling strategies used, some did not rely on in situ observations, such as such as the linear theory of orographic precipitation used by Jarosch et al. (2012) and Clarke et al. (2015) and the extrapolation to the glacier surface of the vertical structure of air temperature in reanalysis products (Fiddes and Gruber, 2014; Jarosch et al., 2012). Hofer et al. (2010, 2012, 2015), on the other hand, used station-based downscaling and found that combining different types of reanalysis variables and the spatial averaging of reanalyses grid cells led to superior downscaling performance. Earlier work by Radić and Hock (2006) used temperature and precipitation from ERA-40 reanalyses to force a mass balance model for Storglaciären, Sweden. They used bilinear interpolation of the nine grid cells centered on the glacier and obtained good mass balance simulation results after correcting the ERA-40 temperature bias with a lapse rate tuned to optimise the mass balance simulation but no correction on ERA-40 precipitation. Koppes et al. (2011) also used a simple approach by regressing temperature and precipitation from the closest NCEP–NCAR reanalysis grid cell against station data in Patagonia. The station-based scaling approach used in this study to correct biases in the NARR is simple compared to station-free (e.g. Jarosch et al., 2012) or multivariate regression (Hofer et al., 2010) approaches but is similar to the station-based methods used by Radić and Hock (2006) and Koppes et al. (2011). The comparison between the NARR and station observations was reasonably good (r= 0.31–0.98) given the short AWS record used for comparison. Three variables (Ta, RH, G) showed strong correlations (r= 0.85–0.98) between the NARR and AWS observations, and the simple scaling bias correction removed much of the bias present (Fig. 3). Moreover, the cold bias in NARR air temperature was consistent with the elevation difference between the AWS and the NARR grid cell and the local temperature lapse rate. The low bias and high correlation for NARR air temperature and relative humidity, as well as solar radiation to a lesser extent (Fig. 3), are consistent with previous findings from Trubilowicz et al. (2016), who showed that these variables agreed well with measured values at high-elevation stations in the southern Coast Mountains of British Columbia, Canada. Wind speed (WS) on Saskatchewan Glacier was however poorly represented (r= 0.37–0.38), most probably because thermal winds (katabatic and valley winds) are not represented at the coarse 32 km spatial resolution of the NARR. Trubilowicz et al. (2016) also reported lower and site-dependent accuracy for NARR wind speeds. More sophisticated wind downscaling (e.g. Vionnet et al., 2021; Wagenbrenner et al., 2016) could help improve further modelling at this site and other upland icefield-outlet valley glacier settings.

The positive bias in NARR precipitation was consistent with the higher elevation of the NARR grid point relative to the merged precipitation record (Fig. 3). However, once the effect of the elevation difference was corrected using the calibrated precipitation lapse rate (15.6 % per 100 m), the NARR was found to underestimate observations by 10 %. This is consistent with the recent study by Hunter et al. (2020), who showed that the NARR underestimates precipitation in the mountain regions of British Columbia, Canada. The NARR precipitation also correlated rather poorly with the off-glacier daily historical precipitation record (r= 0.30–0.31), showing that the daily variability in NARR precipitation is not well represented. Precipitation is notoriously more difficult to represent in reanalysis products, especially in complex terrain with steep orographic gradients and localised convective activity (Hofer et al., 2010; Mesinger et al., 2006). The station-free, linear orographic model for precipitation (LOP) method used by Jarosch et al. (2012) might perhaps be better suited than station-based downscaling in steep topography. The authors reported an improvement in the median relative error (M=-3.1 % to -20.9 %) with respect to monthly precipitation totals in the Canadian Rockies, compared to the raw NARR which underestimated station precipitation (M=-9.5 % to -42.6 %). However, the median absolute error (MAD) of the relative error did not change much and even increased in some instances, i.e. from 13.5 %–31.3 % for the raw NARR compared with 19 %–29.5 % for LOP (see Table 3 in Jarosch et al., 2012). The station-based scaling used in this study resulted in M= 3.8 % and MAD = 33 %, compared to M=27 % and MAD = 41 % for the raw monthly NARR precipitation. Hence the improvement seen is greater than that reported for the station-free LOP model by Jarosch et al. (2012) in the Rockies.

Ebrahimi and Marshall (2016) reported that the NARR precipitation for Haig Glacier, also in the Canadian Rocky Mountains, poorly represented the observed winter-accumulation totals. Nevertheless, NARR precipitation has been found to be reliable at the monthly scale and to represent a useful input for hydrological modelling in North America generally (Chen and Brissette, 2017). Our results suggest this finding also applies for glaciological modelling, given that bias correction is applied. The underestimation of precipitation in the NARR combined with the positive bias in the raw NARR global radiation mostly explains the exaggerated mass loss simulated by the mass balance model when forced with the lapsed NARR, i.e. when only precipitation and temperature are corrected to the elevation of the reference stations (Fig. 8a). More elaborate topographic corrections of solar radiation (Fiddes and Gruber, 2014) could improve the downscaling of NARR solar radiation in the absence of ground observations, but precipitation biases remain difficult to correct in this situation.

The choice of the NARR spatial interpolation method for downscaling to stations had an overall small effect on the comparison with station data (Fig. 3). The bias and RMSE were slightly reduced when using the bilinear interpolation of nine grid cells, compared with the nearest-grid-cell method. However, following bias correction against station data, both interpolation methods resulted in similar cumulative mass balance simulation (Fig. 8a).

Despite the low correlation between the NARR and AWS wind speed, the simulated point mass balance in 2015 was not sensitive to using either the downscaled NARR or AWS wind speed forcings (Fig. 8b, c). Replacing the downscaled NARR global radiation and relative humidity with their AWS counterparts had also a small effect on the model validation. Despite its strong correlation with AWS observations and low residual error after bias correction, the simulated point mass balance in 2015 was most sensitive to downscaled NARR air temperature (Fig. 8b, c). This shows that air temperature has a large influence on the energy balance calculations and that the simple scaling correction could probably be improved to better represent the effect of the glacier wind on air temperature on the glacier (Shea and Moore, 2010).

The approach used in this study could be extended to other reanalysis products, especially the new global ERA5 reanalyses (Hersbach and Dee, 2016). While its spatial resolution (0.25∘, ∼28 km) is only slightly finer than the NARR (0.30∘, ∼32 km), ERA5 has an hourly resolution compared to the 3-hourly NARR resolution.

Despite the physical nature of the model, some assumptions remain simplistic, such as a constant precipitation lapse rate and spatially invariant ice albedo, aerodynamic roughness and wind speeds. Despite these limitations, the interannual variability in mass balance was relatively well simulated by the model, with NSE values of 0.83 to 0.91 for direct point observations (Fig. 6). Point mass balance measurements with the glaciological method are affected by several uncertainties related to errors in ablation stake height measurements, stake self-drilling into the ice or firn, and snow–firn density measurements (Zemp et al., 2013). Errors are 0.14 m w.e. a-1 for ablation measurements on ice, 0.27 m w.e. a-1 for ablation measurements on firn and 0.21 m w.e. a-1 for snow measurements in the accumulation area (Thibert et al., 2008). The root mean squared error (RMSE) on the simulated bw was 0.24 m w.e. a-1 (median relative error of 15 %) – on the same order as the typical measurement error for snow and firn. RMSE values, however, were higher than typical measurement errors for bs (0.87 m w.e. a-1, relative error = 22 %) and ba (0.77 m w.e. a-1, relative error = 24 %), due in part to the restricted number of available observations for validation (Fig. 6). The reconstructed mass balance also compared favourably against the independent geodetic estimates (see Sect. 4.5 and Fig. 8). The simulated cumulative mass loss (-26.79 m w.e.) was close to the geodetic estimate (-25.59 ± 8.44 m w.e.), despite the large uncertainties in the geodetic balance introduced from 2000 onward due to vertical uncertainties in the SRTM DEM. The long-term consistency between geodetic and modelled mass balance gives further confidence that the bias-corrected NARR forcings do not suffer from systematic biases.

The model sensitivity to uncertain model parameters showed that the simulated mass balance was most sensitive to uncertainties in the precipitation lapse rate, followed by the ice aerodynamic roughness, while the sensitivity to the snow aerodynamic roughness and ice albedo were lower. This demonstrates that the precipitation lapse rate must be carefully evaluated using ancillary meteorological data, which can be difficult in regions with no permanent weather station network nearby, a conclusion also reached for the Himalayas by Immerzeel et al. (2014). As the model only accepted a constant lapse rate, we used a value (15.6 ± 4 %) representative of the period during which most of the snow accumulation occurs, i.e. when the glacier toe is above the 0 ∘C isotherm (Fig. S1 in the Supplement). Including shoulder months (April and September) which have mixed precipitations would slightly lower this gradient. The extrapolation of the gradient beyond the highest weather station (2000 m) is also a common but hazardous practice, and validation against snow courses (Avanzi et al., 2021) or winter mass balance surveys (Carturan et al., 2012) offers a way to check the validity of the gradient. The gradient used in this study resulted in accurate simulations of winter mass balance (Fig. 6a), which strengthens our confidence in extrapolating the gradient to the glacier. However, there were no observations beyond 2900 m to constrain the gradient further. The area of the glacier above 2900 m represents only 8.8 % of the total area, so extrapolation errors in this unsampled area would have a small impact on the glacier-wide mass balance. Further development of the model should also consider including a time-varying precipitation lapse rates, as it was shown for example that the lapse rate was smaller during the dry year 2016 (Fig. 6a).

A high sensitivity to the ice aerodynamic roughness has been reported in several studies (e.g. Brock et al., 2000; Hock and Holmgren, 1996; MacDonell et al., 2013; Munro, 1989). It remains one of the most challenging parameters to constrain in glacier mass balance models, and the assumption of a spatially and temporally constant z0 value is a simplistic representation of reality (Fitzpatrick et al., 2019). This parameter is indeed often calibrated in the absence of direct observations (Hock, 2005). In this study, observations from the nearby Peyto Glacier allowed using a representative value which yielded good results; however the uncertainty range in the values reported by Munro (1989) (± 1 mm) was sufficient to induce a ± 17% error in the simulated cumulative balance (Fig. 8d). Advances in deriving aerodynamic roughness from remote sensing could help in the future to improve the calculation of turbulent fluxes in distributed glacier models (Chambers et al., 2020; Fitzpatrick et al., 2019; Smith et al., 2020). The use of remotely sensed albedo maps also contributed to constrain a representative value for ice albedo (see Sect. 3.3.2), but the simulated mass balance was not very sensitive to the uncertainty around this estimate (Fig. 8d). Nevertheless, only an average value was used, when in fact significant heterogeneity was found within the ablation zone (in the Supplement). Decreasing ice albedo can occur over the course of the melt season due to impurities of geogenic origin concentrating at the surface (Cuffey and Paterson, 2010), cryoconite development (Takeuchi et al., 2001) and more discrete events not taken into account in the model, such as algal mat development (Lutz et al., 2014) or wildfires that bring black carbon and ash onto the glacier and decrease the albedo (Marshall and Miller, 2020). Long-term darkening has also been observed on glaciers of the European Alps, which questions the use of fixed albedo values in long historical and future mass balance simulations (Oerlemans et al., 2009). Further efforts could look to assimilate such remotely sensed albedo maps within distributed models.

The multitemporal DEMs used in the study allowed for quantifying the impact of glacier elevation changes on long-term mass balance (Fig. 7). The conventional mass balance simulation with the multitemporal DEMs showed a maximum difference of ∼1.5 m w.e., or 5.6 % of the 1979 reference cumulative balance. This is a small difference overall, which shows that glacier recession has had a minor impact on the mass balance of Saskatchewan Glacier. This is expected for this setting in particular, since the glacier margin is at the bottom of the occupying valley and glacier retreat has occurred over a restricted elevation range – thereby limiting negative feedback effects between glacier retreat and mass balance. This study has focused on the static climate sensitivity of mass balance, which ignores future dynamical feedbacks. Static or reference mass balances calculated over a constant glacier hypsometry have been proposed to be better suited for climatic interpretation (Elsberg et al., 2001; Harrison et al., 2009). But from a hydrological perspective, future glacier retreat towards higher elevations would mitigate an increasing portion of the simulated mass loss, gradually increasing the difference between the reference (2010) and conventional mass balance and progressively decreasing the volume of meltwater released annually (Huss and Hock, 2018; Huss et al., 2012). An increase in dynamical adjustment effects on mass balance was already visible on Saskatchewan Glacier from 2000 onward (Fig. 7a).

The simulated glacier-wide energy balance regime of Saskatchewan Glacier showed that energy inputs are dominated by the sensible-heat flux, flowed by net radiation and latent-heat fluxes. This is different than commonly reported for mid-latitude glaciers in continental climates, where net radiation dominates over turbulent fluxes (e.g. see compilation by Smith et al., 2020). However, most studies reporting energy flux partitioning relied on summer observations in the ablation zones of glaciers. Hence, the often-reported high contribution of net radiation to melting energy is biased by the season (values are commonly reported for July–August – when net radiation is high) and to the ablation zone of glaciers, where most micrometeorological studies have been done and where again net radiation is higher due to the lower albedo. Year-round, glacier-wide values are rarely published and are only available from distributed energy balance models. On Saskatchewan Glacier, the glacier-wide contribution to melting energy was 26.1 % for net radiation, 57 % for sensible heat (QS) and 16.9 % for the latent-heat flux (QL) during the ablation period (July–August). The energy partitioning was quite different when looking at the ablation zone only, with net radiation contributing 57 %, QS contributing 32 % and QL contributing 11 % of the melting energy at the AWS, midway up the ablation zone. The lower glacier-wide contribution of net radiation reflects the fact that much of Saskatchewan Glacier is covered in snow and later firn in summer. This is also accordance with Klok and Oerlemans (2002), who showed that net radiation dominates over QS in the lower part of the glacier, while QS dominates in the higher part of Morteratsch Glacier, Switzerland. Studies that reported glacier-wide energy partitioning include Storglaciären, Sweden (summer net radiation: 38 %–57 %, QS: 42 %, QL: up to 17 %) (Hock and Holmgren, 2005); Brewster Glacier, New Zealand (annual net radiation: 45 %, QS+QL: 52 %, turbulent fluxes dominating in summer) (Anderson et al., 2010); Arolla Glacier, Switzerland (summer net radiation: 82 %, QS: 25 %) (Arnold et al., 1996); Donjek Range glaciers in the Yukon, Canada (summer net radiation: 60 %–83 %, QS: 20 %–45 %, QL: -4 % to -9 %) (MacDougall and Flowers, 2011); and Haig Glacier in Alberta, Canada (summer net radiation: 70 %, QS: 30 %) (Marshall, 2014). Point measurements in late June–early July on nearby Peyto Glacier showed that net radiation contributed 63 % and 42 % of the melt energy over ice and snow surfaces, respectively, while sensible heat contributed 34 % (ice) and 50 % (snow) (Munro, 2006). Hence, the contribution of sensible- and latent-heat flux to summer melting on Saskatchewan Glacier is higher than common values for mid-latitude temperate glaciers with a continental climate and closer to that encountered for glaciers in more humid climates (Anderson et al., 2010; Smith et al., 2020). The contribution of turbulent fluxes to melting energy was however not so different from the earlier measurements reported by Munro (2006) at Peyto: 32 %–57 % at Saskatchewan vs. 34 %–50 % at Peyto for QS and 11 %–17 % vs. 3 %–9 % for QL. The higher contribution of turbulent fluxes to melting on Saskatchewan Glacier, together with a simulated balance ratio (BR = 3.10; see Sect. 4.4.1) that is closer to values from more humid climates (Rea, 2009), may thus reflect the locally wetter and cloudier climate and high accumulation rates resulting from the efficient interception of moist polar maritime air masses from the west by the high and extensive plateau of the Columbia Icefield (Demuth and Horne, 2018; Tennant and Menounos, 2013). This “icefield weather” frequently wraps the Columbia Icefield in clouds, while surrounding valleys are cloud-free.

The simulated mass balance sensitivity to a +1 ∘C warming was -0.65 m w.e. a-1 ∘C-1. This value is comparable to other mid-latitude glaciers: -0.60 m w.e. a-1 ∘C-1 for the Illecillewaet Glacier in the Selkirk Mountains of British Columbia (Hirose and Marshall, 2013), -0.66 m w.e. a-1 ∘C-1 for the Haig Glacier in the Canadian Rocky Mountains (Ebrahimi and Marshall, 2016), -0.65 ± 0.05 m w.e. a-1 ∘C-1 for small (<0.5 km2) glaciers in Switzerland (Huss and Fischer, 2016), -0.60 m w.e. a-1 ∘C-1 for the larger Morteratsch Glacier in Switzerland (Klok and Oerlemans, 2004), and -0.61 m w.e. a-1 ∘C-1 for Storglaciären in Sweden (Hock et al., 2007). Higher sensitivities are found in more humid climates – e.g. -0.86 m w.e. a-1 ∘C-1 for the South Cascade Glacier, Washington (Anslow et al., 2008), and up to -2.0 m w.e. a-1 ∘C-1 on Brewster Glacier, New Zealand (Anderson et al., 2010) – and lower sensitivities in drier climate – e.g. -0.44 m w.e. a-1 ∘C-1 on Urumqi Glacier No. 1 in the Chinese Tien Shan (Che et al., 2019). Earlier work by Braithwaite (2006), Oerlemans and Fortuin (1992), and Oerlemans (2001) showed that the mass balance sensitivity to temperature scales with mean annual precipitation, due to larger albedo and precipitation phase feedbacks and longer melt seasons on glaciers in wetter climates.

We found that the albedo feedback is the main contributor (mean = 44 %) to the temperature sensitivity of mass balance on Saskatchewan Glacier (Fig. 13). Increases in net shortwave radiation caused by a reducing snow cover and ensuing reduced glacier albedo account for 39 %–47 % of the increase in melt energy across the various warming scenarios (Table 1). A similar finding was reported on Haig Glacier by Ebrahimi and Marshall (2016), who found that introducing albedo feedbacks doubles the net energy sensitivity to warming. This value (44 %) is significantly high but less than the 80 % reported recently by Johnson and Rupper (2020) for the summer-accumulation type Chhota Shigri Glacier in High Mountain Asia. As shown by Fujita (2008), higher sensitivities are found for glaciers located in a summer-precipitation climate, where albedo feedbacks on ablation are stronger, than for glaciers located within a winter-precipitation climate. Atmospheric warming itself contributed only 24.3 % to the mass balance sensitivity to temperature across all warming scenarios, through sensible heat and longwave radiation transfer to the glacier (Table 1). A significant air humidity feedback was also found, with latent-heat fluxes representing an average of 22 % of the temperature sensitivity across all warming scenarios. Keeping the relative humidity constant under warming scenarios may be plausible for the high-elevation Columbia Icefield. The icefield receives moist air masses from the British Columbia interior and the Pacific Ocean uplifted onto the icefield, as the region is subject to upslope conditions derived from convergent upper air masses as low-pressure systems spin by the south of the region. Under a stable atmospheric moisture regime, increasing atmospheric warming would lead to an increasing humidity feedback on ablation (Table 1). Other glaciers subjected to subsiding air masses could experience drier weather in the future, which would decrease their melt sensitivity to warming (Ebrahimi and Marshall, 2016). The large contribution of latent-heat fluxes to melting under warming scenarios points to the necessity of considering changes in specific air humidity when simulating glacier melt under future climates. This conclusion is in line with the recent findings by Harpold and Brooks (2018), who showed that atmospheric humidity plays a critical role in local energy balance and snowpack ablation under warmer climates, with latent and longwave radiant fluxes cooling the snowpack under dry conditions and warming it under humid conditions. The precipitation phase feedback, on the other hand, contributed the remaining 10 % of the mass balance temperature sensitivity (Table 1).

The mass balance sensitivity of Saskatchewan Glacier to a ± 10 % change in precipitation under the current temperature regime was 1.01 (unitless: m w.e., of mass change per m w.e., of precipitation change). A value of 1 would occur if all precipitation were snowfall and there were no albedo feedbacks on Ba. With the snowfall fraction being 0.81 under the present climate (Fig. 13c), the albedo feedback on ablation contributes 19 % to the mass balance sensitivity to precipitation. With the mean annual precipitation on the glacier being 1880 mm for the reference period 1981–2010, the maximum +20 % precipitation increase projected from ensemble climate scenarios for the end of the century would add a maximum of 0.4 m w.e. a-1 if all new precipitation falls as snow, which is small compared to the mean temperature sensitivity of -0.65 m w.e. ∘C-1. As such, precipitation increases can only buffer up to +0.5 ∘C-1 of warming on Saskatchewan Glacier. As warming causes snowfall to shift to rainfall at a rate of ∼5 % ∘C-1 (Fig. 13c), the buffering effect of a +20 % increase precipitation would decrease accordingly, i.e. from 0.29 m w.e. a-1 for a +1 ∘C warming to 0.17 m w.e. a-1 for a +7 ∘C warming.