The main pyramidal summit, Monte Sarmiento, reaches 2207 m a.s.l. Several glaciers descend from all sides of the MSM, together covering ∼70 km2 (Barcaza et al., 2017). To the south of the MSM, another glacierized area of 39 km2 is centered around Pico Marumby (1253 m a.s.l.). Both ice bodies together represent the MSM study area in this study (Fig. 1). The larger ice field includes both land- and lake-terminating glaciers, whereas the smaller one consists entirely of land-terminating glaciers. Schiaparelli Glacier is the largest glacier of the MSM, with an area of 24.3 km2 in 2016 (Meier et al., 2018). It descends Monte Sarmiento to the northwest almost to sea level and calves into a moraine-dammed proglacial lake, which was formed after strong recession in the 1940s (Meier et al., 2019). Meier et al. (2019) found a continuous average glacier retreat of approximately 5 m yr-1 from 1973 to 2018. Analysis of the surface energy and mass balance of Schiaparelli Glacier with a physically based energy balance model revealed a glacier-wide mean annual climatic mass balance of -1.8±0.36 m w.e. yr-1 in 2000–2017 (Weidemann et al., 2020). The mass balance is dominated by surface melt and precipitation (Weidemann et al., 2020).

The largest glaciers in the study site after Schiaparelli are Pagels, Lovisato, Conway and Emma glaciers. Emma Glacier was the target for studying Holocene glaciation in the MSM, which indicated that the Holocene glacier behavior in Tierra del Fuego and southern Patagonia responds synchronously to the same regional climate change (Strelin et al., 2008). The other glaciers in the MSM are largely unsurveyed, except by remote sensing (Melkonian et al., 2013; Braun et al., 2019; Dussaillant et al., 2019). From the geodetic MB data from previous studies, different patterns are observed. Despite the rather small study site and proximity of the glaciers, the characteristics of the geodetic MB in 2000–2013 (see Sect. 2.4) are very heterogenous (Fig. 2). Lovisato Glacier shows by far the highest mass loss. Satellite images (see Fig. 1) reveal large numbers of icebergs in the proglacial lake, indicating significant calving losses for this glacier. A clear contrast between lake- and land-terminating glaciers is not visible. There are several lake-terminating glaciers in the northern part of the study site (Schiaparelli, Conway, 138, Lovisato); however, land-terminating glaciers in this area show similar MBs. Geodetic MBs are also heterogenous in the southern part of the study site, despite all the glaciers being land-terminating.

We use observational data of two automatic weather stations (AWSs) at Schiaparelli Glacier (Fig. 1). AWS Rock (92 m a.s.l.) is located on rock close to the glacier front. Since the installation in September 2015, it has been measuring air temperature T, relative humidity RH, global radiation G, wind speed U, wind direction DIR and precipitation RRR at hourly resolution. Bucket-based precipitation measurements often show undercatch due to wind and snow (Rasmussen et al., 2012; Buisán et al., 2017), specifically if the bucket is not heated as in this case. Due to the high wind velocities, precipitation measurements are known to be specifically error-prone in Patagonia (Schneider et al., 2003; Weidemann et al., 2018b; Temme et al., 2020). We therefore assume that the measurement instrument only records a fraction of the total precipitation and, thus, the annual amount needs to be increased by 20 %. AWS Glacier (140 m a.s.l. in September 2016) is located on ice in the ablation area of Schiaparelli Glacier. It has been measuring T, RH, G, U, DIR and air pressure PRES at hourly resolution since August 2013 to the present, with some interruptions. Since this AWS is subject to tilting due to melting of the ice surface, we do not use measurements that require a horizontal sensor orientation from this station. In addition, we identified a step change and a multiannual drift in the RH measurements. These measurements were therefore discarded. RH values at AWS Glacier were inferred from AWS Rock assuming identical specific humidity. Values of T, corrected RH and PRES from AWS Glacier are used to inform the statistical downscaling. G and RRR from AWS Rock serve to evaluate modeled radiation and precipitation.

Several ablation stakes, concentrated in the lowest part of the ablation area, deliver information about surface melt. Stakes have been installed at varying locations and for irregular time spans ranging from a few months to almost 1 year. The largest number of stakes was installed in the period November 2018–April 2019 with six stakes at the same time (see Fig. 1). In the other periods the number of measured stakes ranges from one to four (see Fig. S4 in the Supplement). The stake located next to AWS Glacier has been in use for the longest period between August 2013 and April 2019. Additionally, an automatic ablation sensor measured every 150 mm of melt (recording the time point when 150 mm melted) from September 2016 to November 2017, giving temporally higher-resolved information about surface melt.

In April 2016 the ice thickness was measured with a ground-penetrating radar in the ablation area approximately parallel to the glacier front of Schiaparelli Glacier (Fig. 1). Measurements reveal a maximum ice thickness of 324 m with an estimated uncertainty of around 10 % (Gacitúa et al., 2021).

The ERA5 reanalysis data set is the latest global climate reanalysis product of the European Centre for Medium-Range Weather Forecasts (ECMWF). Being a global data set, ERA5 shows high temporal and spatial resolution with an hourly time step and an approximately 31 km horizontal grid over 137 vertical levels (Hersbach et al., 2020). ERA5 and its previous versions have been successfully applied in modeling studies in Patagonia (e.g., Lenaerts et al., 2014; Bravo et al., 2019b; Sauter, 2020; Temme et al., 2020; Weidemann et al., 2020).

ERA5 data are required to extend the time period for our modeling framework beyond the AWS records. Therefore, we infer the local surface conditions near AWS Glacier from the spatially coarse ERA5 data, averaging the four closest grid points to the AWSs. U and cloud cover fraction N are directly taken from ERA5. For T, RH and PRES, quantile mapping (Gudmundsson et al., 2012) was used to relate ERA5 to AWS data (see Sect. 3.1). For downscaling of precipitation, we use a model of orographic precipitation (see Sect. 3.1). It requires the large-scale precipitation and upwind information about geopotential height, air temperature, wind vectors and relative humidity between 850 and 500 hPa which was extracted in a rectangular domain upstream of the study site (54.0–55.0∘ S, 72.0–71.25∘ W).

Glacier outlines from the Monte Sarmiento Massif and their surrounding glaciers are extracted from the two national Chilean inventories generated by the water directorate of Chile (DGA). The first comprehensive glacier inventory of Chile was created from Landsat TM (Thematic Mapper) and ETM+ (Enhanced Thematic Mapper) images acquired in 2004 (Barcaza et al., 2017) and later updated using Sentinel-2 images acquired in 2019 (DGA, 2022). The latter inventory presents an improvement in the glacier catchment areas; however, we observed small inconsistencies in both inventories (ice-covered areas not included in the inventory). Hence, we homogenized both inventories and corrected them. To do so, we used the original and close-date satellite images of both inventories to manually correct these inconsistencies. Moreover, we also generated glacier outlines of 2013 using a composite band of Landsat OLI (Operational Land Imager) images.

We calculate the geodetic MB using digital elevation models (DEMs) from 2000 and 2013. The DEMs from the 2013 TerraSAR-X add-on for the Digital Elevation Measurement mission (TanDEM-X) correspond to a part of the data set generated in the study of Braun et al. (2019), which presents a complete coverage of the study area. Braun et al. (2019) calculated the elevation changes for the entire Tierra del Fuego region from synthetic aperture radar (SAR) DEMs between 2000 and 2011/2015. For this study, the elevation changes were derived from the Shuttle Radar Topography Mission (SRTM) in 2000 and the 2013 TanDEM-X DEMs. In general, the TanDEM-X DEMs were derived using differential SAR interferometry techniques. Details regarding the SAR approach can be found in Braun et al. (2019).

To obtain precise elevation change fields, the TanDEM-X DEMs are horizontally and vertically coregistered to the SRTM (reference) DEM using stable areas (Braun et al., 2019; Sommer et al., 2020). Subsequently, the elevation change differencing is estimated. Data gaps are filled by applying an elevation change versus altitude function by calculating the mean elevation change within 100 m-high bins across the glacier area. Finally, we remove steep slopes (<50∘) to avoid artificial biases introduced by outliers and filter each elevation band by applying a quantile filter (1 %–99 %) (Seehaus et al., 2019; Sommer et al., 2020).

To estimate the geodetic MB between 2000 and 2013, we use the two corresponding abovementioned glacier inventories to take into account the glacier area loss (Sommer et al., 2020). To convert volume to mass changes, a density factor of 900 kg m-3 is applied.

Errors and uncertainties from the geodetic MB (MΔt) were calculated using a standard error propagation Eq. (1) from Braun et al. (2019), which considers the following factors.

Accuracy of the elevation change rates (considering spatial autocorrelation and hypsometric gap filling) (δΔh/Δt)

Multi-mission SAR remote-sensing data were employed to obtain information about glacier speeds. The database covers the period 2001–2021 (ERS-1/2 IM SAR, July–August 2001; ENVISAT ASAR, March–July 2007; ALOS PALSAR, August 2007–September 2010; TerraSAR-X/TanDEM-X, May 2011–February 2021). More detailed information about the sensor specifications can be found in Seehaus et al. (2015).

Atmospheric forcing for the SMB models requires T, RH, U, PRES, G, RRR and N. T, RH and PRES are statistically downscaled to the AWS location and subsequently extrapolated over the study site, while U and N are directly taken from ERA5. G and RRR are produced using additional models for radiation and orographic precipitation, respectively. The statistical performance of all input variables compared to AWS measurements is summarized in Table S1 in the Supplement.

Statistical downscaling of T, RH and PRES is performed via quantile mapping. Quantile mapping is a technique for statistical bias correction of climate model outputs by transferring the cumulative distribution function of the model to the cumulative distribution function of the observation (Gudmundsson et al., 2012; Cannon et al., 2015). This technique has been successfully applied in Patagonia before (e.g., Weidemann et al., 2018a, 2020). Statistically downscaled T and PRES are spatially extrapolated from AWS Glacier over the topography using a linear temperature lapse rate (TLR) and the barometric equation, respectively.

G over the glacier surface is modeled based on the radiation scheme of Mölg et al. (2009a). It calculates both the direct and diffuse parts of the incoming solar radiation from N, T, RH and PRES. Radiation is corrected for the slope and aspect of the respective grid cell. Furthermore, both self-shading and topographic shading are considered, and thus shaded grid cells only receive the diffuse component of the incoming solar radiation (Mölg et al., 2009a, b).

As precipitation events can be short-lived and highly variable in space, it is challenging to infer reliable distributions over complex terrain from coarse global data sets. Furthermore, a direct extrapolation of the sparse AWS measurement network in the CDI using altitudinal lapse rates is critical because measurements in this region are error-prone (Schneider et al., 2003, 2007; Temme et al., 2020). Therefore, we follow a physically motivated approach using an orographic precipitation model, which has been successfully used in glaciological studies before (e.g., Schuler et al., 2008; Weidemann et al., 2018a, 2020). The model is based on the linear steady-state theory of orographic precipitation and includes airflow dynamics, cloud timescales and advection and downslope evaporation (Smith and Barstad, 2004; Barstad and Smith, 2005). In this way, the precipitation resulting from forced orographic uplift over a mountain is calculated (Weidemann et al., 2018a). For a more detailed description, see Smith and Barstad (2004), Barstad and Smith (2005) and Sauter (2020).

The orographic precipitation model assumes stable and saturated conditions, and thus time intervals that do not fulfill these constraints need to be excluded (Smith and Barstad, 2004; Weidemann et al., 2018a). We use relative humidity, Brunt–Väisälä frequency and Froud number as model constraints in order to ensure saturated, stable airflow without flow blocking. A positive zonal wind component guarantees that airflow crosses the mountains from west to east. The total precipitation is calculated by adding the large-scale precipitation (after removing the orographic component from the ERA5 precipitation) to the orographic precipitation calculated in the model. Based on the annual precipitation amounts from AWS Rock, we are able to constrain the relative humidity threshold (90 %) above which orographic precipitation can occur and the large-scale precipitation from ERA5. In this way, we guarantee that the annual total precipitation at the AWS location agrees with the observed amounts. Conversion and fallout timescales of hydrometeors (τ) are varied within the range of previous studies (Table 1) in the latter model calibration (Jiang and Smith, 2003; Barstad and Smith, 2005; Smith and Evans, 2007; Weidemann et al., 2013, 2018a, 2020; Sauter, 2020).

We use four types of SMB models with different complexity. In this way, we can understand which type of model is well-suited and which processes are essential for SMB simulations in the MSM. Calibration parameters for each model are summarized in Table 1. The calibration approach is described in detail in Sect. 3.5.1. For calibration, simulations were limited to the period in which observations are available (April 1999–March 2019). The final SMB simulations have been extended to the period April 1999–March 2022 at the end to produce the most comprehensive and updated results possible. A complete overview of the model setup and fixed parameter values is given in the Supplement (Table S2). In the following we explain the different models.

Redistribution of snow caused by wind plays a key role in the spatial heterogeneity in accumulation. In Tierra del Fuego and Patagonia, where strong winds prevail throughout the year, we hypothesize that snowdrift has a crucial impact on accumulation and the SMB. Thus, a simple parameterization to capture wind-driven snow redistribution based on Warscher et al. (2013) was slightly modified and added to the SMB model types. The scheme determines locations that are sheltered from or exposed to wind by an analysis of the topography and corrects the solid precipitation accordingly: 6Psnow,SD=Psnow×(1+Cwind). The correction factor Cwind for each grid cell is calculated by 7Cwind=U×E×Dmax1-dSVF-1+K. Dmax gives the maximum deposition in millimeters, dSVF is the directed sky-view factor and K is a calibration constant, which was set to 0.1 by Warscher et al. (2013). We vary it in a range from -0.2 to +0.2 in the snowdrift calibration together with the Dmax (Table 1). E is a factor for weighting with elevation (linearly) ranging from 0 to 1, assuming that lower wind speeds prevail at lower elevations, which reduces the snow redistribution. In this study, we additionally include a weighting with prevailing wind speed U to further improve the performance because we observe different wind directions with different velocities and we suppose that more (less) snow is also redistributed during periods of higher (lower) wind velocities. For a more detailed description of the snowdrift scheme, please refer to Warscher et al. (2013).

Ice surface velocity fields are derived from the SAR imagery database by applying intensity offset tracking to co-registered image pairs (Strozzi et al., 2002). Tracking parameters were adjusted depending on sensor specification and acquisition intervals. The tracking is done using multiple tracking patch sizes in order to account for different glacier flow speeds, and a subsequent stacking of the results was applied. More details on the processing, including filtering and error estimation, can be found in Seehaus et al. (2018).

In order to estimate the ice flux, a flux gate was defined along a cross-profile following the thickness surveys in 2016 (Gacitúa et al., 2021). By combining the obtained surface velocity information with the ice thickness measurements along this flux gate, the ice flux was computed following the approach of Seehaus et al. (2015) and Rott et al. (2011). In order to account for ice thickness changes at the flux gate throughout the observation period, the measured ice thickness values were corrected by a surface-lowering rate of -2.8 m yr-1 derived from the annual average elevation change rate between 2000 and 2019 (see Sect. 2.4) near the flux gate. The resulting ice flux through the flux gate is summarized in Table S3.

The combination of the SMB integrated over the glacier area above the flux gate and the mass lost through the flux gate allows us to determine the total mass budget of Schiaparelli Glacier. This value is comparable to the geodetic MB from elevation changes in the area above the flux gate and will be used as one calibration constraint in this study.

Generally, all four models give very similar results of SMB (Fig. 5). The spatial distribution and seasonal/interannual patterns are captured by all the models in a similar way. We will summarize the main characteristics of SMB in the MSM in the following and highlight differences between the models. For this analysis, we include all glaciers at the study site (no area limit) to produce the most comprehensive results possible.

The massif-wide average annual SMB lies just below equilibrium, with the PDD and COSIPY producing more negative values (-0.28 and -0.20 m w.e. yr-1, respectively) than the SEB_Gpot and SEB_G (-0.07 and -0.13 m w.e. yr-1, respectively). For all the models, the SMB is mainly influenced by snowfall (average of +1.66 to +1.79 m w.e. yr-1) and melt (average of -1.87 to -2.55 m w.e. yr-1). Snowfall is almost zero in the lowest parts of the glaciers, indicating melt all year round (Fig. S3). The distribution of snow reflects the topography, increasing strongly towards the summits and showing the largest snow deposition southeast of the mountain peaks and ridges. The highest amounts are found on the wind-sheltered slopes of the Monte Sarmiento summit. For all four SMB models, we see high mass gain due to snowfall in the elevated areas of the massif (up to around 10 m w.e. yr-1) and extreme mass loss at the glacier tongues (up to around -10 m w.e. yr-1) (Fig. 5). Several glaciers have large ablation areas (Schiaparelli, Lovisato, Pagels). However, Schiaparelli Glacier stands out due to its large size, the range of altitude and its huge glacier tongue causing a much larger area of intense ablation compared to the other glaciers in the region. Depending on the model type, the massif-wide equilibrium line altitude is on average between 770 and 794 m a.s.l. during the study period. Equilibrium line altitudes tend to be lower in the east of the massif compared to the west, which can be confirmed by snowline altitudes from satellite observations in the region (Table S4).

Due to the location at the higher mid latitudes, the seasonal variations are huge. In summer, the average SMB is negative up to around 900–1000 m a.s.l., which leaves (almost) no area of mass gain for several glaciers in the region (see Fig. S3). This applies in particular to the southern, lower-elevation massif. In winter, the majority of the MSM area is characterized by a positive SMB. The cooler temperatures cause higher snowfall amounts, and we also observe snowfall over lower altitudes (see Fig. S3). More than 65 % of the total snow accumulates in winter (June to August) and spring (September to November) but only 13 % in summer (December to February).

Over the course of the 22-year study period, we see a phase of more negative and more positive annual SMB that all four models agree on (Fig. 6). Massif-wide, more positive MB values prevail between 2009 and 2015/16, with more negative ones before and after this phase. More negative MBs coincide with over-average temperatures and decreased snowfall and vice versa. All the models agree that the most negative MBs likely occurred in 2003/04, 2005/06, 2016/17, 2019/20 and 2020/21. However, the amplitude of annual mass balances differs significantly between the models. Overall, the PDD and COSIPY tend to simulate more negative MBs; however, COSIPY also simulates more positive MB in several positive years (Fig. 6).

We can compare modeled and observed MB for the individual glacier catchments to assess the performance of the individual models (Fig. 7). The area-weighted RMSE (Table 2) is similar for the PDD, SEB_Gpot and SEB_G (0.17, 0.19 and 0.16 m w.e. yr-1) and largest for COSIPY (0.31 m w.e. yr-1) comparing land-terminating glaciers. The range of uncertainty of the RMSEs is very similar for all four models (see Table S6), with RMSEs lying between 0.16 and 0.34 m w.e. yr-1. Only the PDD stands out, with a maximum RMSE of 0.75 m w.e. yr-1. The BMSMnc range of the 10 best-ranked runs are very similar (range below 0.24 m w.e. yr-1) for the SEB_Gpot, SEB_G and COSIPY. For the PDD, this range is distinctly larger with 0.51 m w.e. yr-1 taking the five (due to the smaller sample size) best-ranked runs.

In order to investigate the importance of including accurate information about incoming radiation, we can directly compare the performance of the SEB_Gpot with the SEB_G. The former relies on the potential radiation, whereas the latter accurately calculates the direct and diffuse parts of incoming shortwave radiation by taking into account cloud cover and shading. Generally, both models tend to overestimate the SMB in the MSM (Fig. 7), which is also reflected in the overestimation of the BMSMnc (-0.33 and -0.32 m w.e. yr-1 for the SEB_Gpot and SEB_G, respectively) (Table 2). The differences between both models are overall minor, although the RMSE for the SEB_G is smaller (Table 2). Thus, for this study site, the improvement by using more accurate instead of potential radiation appears insignificant. This finding further agrees with the fact that the PDD also produces satisfying results.

The second observation available for model evaluation is the stake measurements. However, the agreement between measured and modeled ablation at the stakes is poor for all considered SMB models (see Fig. S4 and S5). Mean RMSEs are in the range between 3.92 and 4.78 m w.e. yr-1 (Table 3), which is about 33 % of the observed melt. The model bias ranges between -0.77 and 3.51 m w.e. yr-1. The best results are achieved with COSIPY, followed by the PDD regarding both RMSE and bias. At the individual ablation stakes (Fig. S4), COSIPY behaves distinctly differently from the other models, for which melt rates are more similar most of the time. Subsequently, COSIPY meets the observations better for the first half of the time when the other models underestimate the melting. However, after 2018, COSIPY clearly overestimates the melt rates degrading the overall statistics. In general, we consider the ablation measurements to be error-prone when looking at individual observations. Thus, we consider the large RMSE and bias values to be caused only partly by poor model performance.

The single-glacier model calibration at Schiaparelli Glacier (Strategy A) results in a TLR of -0.70 ∘C 100 m-1, which is slightly stronger compared to previously reported annual values that vary from -0.60 to -0.67 ∘C 100 m-1 in the southern Patagonian region (Strelin and Iturraspe, 2007; Buttstädt et al., 2009; Koppes et al., 2009; Schaefer et al., 2015; Weidemann et al., 2018a, 2020). Furthermore, calibration suggests a τ of 850 s, which differs significantly from the value used at Schiaparelli Glacier in a recent SMB study (Weidemann et al., 2020). However, it agrees well with values reported in various other applications, including southern Patagonia (Smith and Barstad, 2004; Barstad and Smith, 2005; Smith and Evans, 2007; Schuler et al., 2008; Jarosch et al., 2012; Sauter, 2020). The degree-day factors DDFice and DDFsnow of 6.0 and 3.0 mm d-1 ∘C-1, respectively, lie within the range of previously reported values (Stuefer et al., 2007; Gabbi et al., 2014; Réveillet et al., 2017). At Gran Campo Nevado, Schneider et al. (2007) found a value of 7.6 mm d-1 ∘C-1 for ice in summertime. Calculating the average DDFice directly from measured ablation and a positive degree-day sum at the stake locations (Groos et al., 2017) delivers values very close to the calibrated one, with 5.0 mm d-1 ∘C-1 at the automatic ablation sensor and 6.0 mm d-1 ∘C-1 at the individual stakes.

Going from a single-glacier calibration (Strategy A) to a regional calibration (Strategy B), the TLR and τ need changing, and the DDFice decreases slightly to 5.0 mm d-1 ∘C-1. A TLR of -0.60 ∘C 100 m-1 is required, which is distinctly lower than the result of Strategy A. However, this value is close to values used in the Cordillera Darwin before ranging from -0.60 to -0.63 ∘C 100 m-1 (Strelin and Iturraspe, 2007; Koppes et al., 2009). The τ of 1200 s produces a precipitation field with less orographic contribution compared to Strategy A. The value is still distinctly smaller than in Weidemann et al. (2020). Both changes significantly reduce the snowfall amounts and result in a better match with observed geodetic MBs.

The results suggest that the exclusive use of ablation stakes (Weidemann et al., 2020), which have been installed in the lowest part of Schiaparelli Glacier, for model calibration shows limited utility because no information about accumulation is included. Thus, adding the total mass budget of Schiaparelli Glacier by a flux gate approach brings significant benefit to constrain the drainage basin-wide mass input. Still, the transfer of a SMB model, which has been calibrated to a single glacier, to a regional study site (Strategy A) can imply severe biases in the overall mass budget. This demonstrates that model parameters are not transferable from one single glacier to the surroundings. This shortcoming has been reported similarly in previous studies with various melt models and at many locations (e.g., MacDougall and Flowers, 2011; Gurgiser et al., 2013; Zolles et al., 2019). In general, the SMB in the MSM is excessively overestimated, which indicates that the SMB model either produces too little melt or receives excessive snowfall. The latter seems more likely, since melt is well constrained by the stake measurements at least at Schiaparelli Glacier, whereas the precipitation amounts are generally more uncertain. By the use of a regional calibration strategy (Strategy B), the agreement between the observed geodetic and modeled surface MB can be significantly improved. This highlights the importance of including regional observations for realistic simulations of regional surface mass balance in the Cordillera Darwin.

Considering the regional distribution of the difference of SMB from the geodetic observations (Fig. 4), the model tends to overestimate the MB on the land-terminating glaciers in the northwest (e.g., 149, 152) and underestimate it in the southeast (e.g., Emma, Pagels) of the massifs. This pattern indicates that snowfall amounts are overestimated on the northwestern slopes and underestimated on the southeastern slopes, which may be associated with the neglect of climatic gradients, e.g., in temperature or precipitation. Mass transfer by snowdrift due to the consistent westerlies has been neglected so far. With the addition of a basic snowdrift scheme (Strategy C), the agreement between modeled and observed mass balance can be improved further. Thus, the results show that snowdrift plays an important role for the SMB in the MSM.

The calibration of the SEB models and COSIPY reveals realistic parameter values within the range of previous applications as well (see Table 1). For COSIPY, the calibrated parameters zice and αfirn lie on the margin of the range, implying that a larger range may be beneficial or that a parameter not considered in calibration is not chosen optimally. However, extending the limits of these parameters would result in physically unrealistic values. We have not been able to find a parameter that was neglected in the calibration and that would solve the issue. Apart from the model-inherent parameters, the difficulties with the calibration of COSIPY might alternatively lie in the input data set. Variables that are only considered in COSIPY and not in the other models are, e.g., wind speed and relative humidity, which both affect turbulent heat fluxes and thereby impact the choice of ice roughness length.

A high discrepancy between modeled and observed mass balance is obtained for two lake-terminating glaciers south of Monte Sarmiento (138 and Lovisato) (Fig. 7). Due to the lake termination, it is expected that the modeled SMB will be higher than the geodetic MB. However, the difference is extremely large, especially when considering snow redistribution due to snowdrift. In Sect. 5.4, we will discuss possible explanations for this discrepancy.

The mean annual SMB of -0.79 to -1.20 m w.e. yr-1 (2000–2013) at Schiaparelli Glacier is distinctly less negative than the previous estimate for the period 2000–2017 (-1.8±0.36 m w.e. yr-1) by Weidemann et al. (2020) but in much better agreement with the satellite observations (-0.79±0.19 m w.e. yr-1). The massif-wide average SMB over the full study period (2000–2022) is estimated between -0.28 and -0.07 m w.e. yr-1 depending on the model choice. In the eastern part of the CDI, an average SMB of -0.53 m w.e. yr-1 was simulated between 2000 and 2006 using a PDD model (Buttstädt et al., 2009). Similarly large accumulation amounts over the highest parts of the glaciers and the extreme ablation over the glacier tongues, which we see at our study site, have been reported for the Southern Patagonian Icefield (Schaefer et al., 2015). We can confirm that the SMB of the MSM is controlled by winter accumulation and summer temperature, as has been observed in the Cordillera Darwin before (Weidemann et al., 2020; Mutz and Aschauer, 2022). The orientation of the individual glaciers does not seem to dictate a particular pattern. Glaciers that receive more direct solar radiation (e.g., Schiaparelli, Conway, Pagels) do not show more negative MBs than glaciers with stronger shading (e.g., Lovisato, 138).

We simulate an average equilibrium-line altitude (ELA) at 770–795 m for the MSM. This is close to the mean ELA at 730±50 m simulated at Schiaparelli Glacier in 2000–2017 (Weidemann et al., 2020) but higher than the ELA suggested by Bown et al. (2014) for Ventisquero Glacier at the southwestern edge of the CDI at around 650 m in 2004. At the CDI's northern edge at Marinelli Glacier and the eastern edge at Martial Este Glacier, average ELAs have been reported at around 1100 m (Buttstädt et al., 2009; Bown et al., 2014). The altitude difference can be explained by the more continental conditions due to leeside effects that reduce the precipitation in the east of the CDI (Strelin and Iturraspe, 2007), while the MSM is located at the western edge of the CDI, directly exposed to the moist westerly winds causing abundant precipitation and, thus, higher accumulation amounts (Bown et al., 2014), which results in lower equilibrium lines.

Ice losses due to dynamical adjustment and calving are assumed to play an important role only for a few glaciers in the CDI (Koppes et al., 2009; Bown et al., 2014; Weidemann et al., 2020), like Marinelli Glacier (Porter and Santana, 2003). Weidemann et al. (2020) conclude that mass loss due to SMB processes is the main reason for the recent areal changes of Schiaparelli Glacier. Based on our results, we can confirm that the SMB contributes the largest amount to the ice loss at Schiaparelli Glacier. However, calving is not negligible. Using calibration Strategy A, where the PDD model is tuned to the Schiaparelli Glacier conditions directly, we assess a resulting calving flux of 0.17 m w.e. yr-1, which equals a mass loss of 4.26 Mt yr-1 at Schiaparelli Glacier. The average geodetic MB estimated from elevation changes for the whole study site is with -0.55 m w.e. yr-1 (2000–2013) distinctly more negative than the SMB (-0.28 m w.e. yr-1 by the PDD, in the same period), indicating that calving losses are not insignificant in the region. However, in order to determine the calving flux more accurately, detailed information about the ice thickness and velocities at the glacier fronts is required.

Overall, we achieve a very good agreement between the modeled surface and the observed geodetic MB. For most glaciers, the RMSEs are in a similar range to the uncertainties in geodetic MB (Table S5). We want to highlight the remarkable performance of all four models used under these challenging conditions, with very sparse observations leading to overparameterization issues.

Previous studies (e.g., Six et al., 2009; Gabbi et al., 2014; Réveillet et al., 2017) of melt model comparison have come to the conclusion that more complex, physically based models can achieve more realistic SMB results in case they are based on high-quality and well-distributed in situ observations. If observations are limited or inferred from distant weather stations, the performance decreases rapidly, and less complex, empirical models produce better results (Gabbi et al., 2014; Réveillet et al., 2017). Since we focus on a study area where in situ measurements are extremely limited and, thus, need to infer model input from reanalysis data via downscaling, and furthermore glacier SMB is known to be highly correlated with precipitation and air temperature (Weidemann et al., 2020), we strongly challenge the question of which SMB model can produce the most realistic SMB.

Results are validated against the individual geodetic MBs and the stake measurements. The results of this study show that less complex model types overall outperform COSIPY, although the simulated melt at the ablation stakes is best represented by COSIPY. Both SEB model variants tend to overestimate the SMB in the MSM on average, but the SEB_G achieves the smallest RMSE compared to geodetic observations of the land-terminating glaciers. The MB of Schiaparelli Glacier, the largest glacier of the massif, is also simulated well by both SEB variants. Comparison of the measured against modeled melt at the stakes delivers similar results for all the models with large RMSEs (Table 3). Although COSIPY achieves the overall smallest RMSE and bias, the difference from the other models is small compared to the difference from the measurements.

Gabbi et al. (2014) concluded that models considering the temperature- and radiation-induced melt separately are more suitable for long-term simulation periods because they are less sensitive to temperature. However, shorter time periods might not be able to bring issues like parameter instability to light (Gabbi et al., 2014), which might apply to our study period. The importance of correct radiation information cannot be confirmed even by comparing the two SEB model variants we used. Although the agreement with observations can be increased (see Table S1), including accurate radiation calculation (SEB_G) instead of potential radiation (SEB_Gpot) only produces a minor improvement in the glacier-wide SMBs. Interestingly, the SEB_G always produces slightly larger melt rates at the individual stake locations (Fig. S4), whereas at the automatic ablation sensor we do not see this consistent pattern (Fig. S5).

Overall, due to the small sample size of glaciers, it is not possible to point out the one best-suited SMB model for the MSM. The strong correlation with air temperature and precipitation makes the PDD a good predictor of the SMB. Both SEB model variants show a convincing performance as well, although they tend to produce a less negative BMSMnc. Still, the highest agreement with geodetic MB is achieved using the SEB_G. COSIPY delivers more accurate and confident results (smaller uncertainty) (Table S6) and can best reproduce the melt at the stakes. As in this study, in Schneider et al. (2007) the applied PDD and energy balance model at the Gran Campo Nevado showed very similar results. In order to better understand the interaction between the atmosphere and the glacier surface, a physically based energy and mass balance model like COSIPY is advantageous.

A large discrepancy between the surface and geodetic MB is modeled for glaciers 138 and Lovisato. Both glaciers are calving: thus, a positive anomaly in SMB is to be expected, but the difference seems very high. Including the snowdrift parameterization (Strategy C), the discrepancy gets even larger due to the mainly prevailing northwesterlies during snowfall events. The results from the four different SMB models imply a mass loss through calving of 2.5 to 2.9 m w.e. yr-1 and 1.9 to 2.2 m w.e. yr-1 for glaciers 138 and Lovisato, which equals ice masses of 9.73 to 11.28 Mt yr-1 and 23.88 to 27.65 Mt yr-1, respectively. We will discuss in the following whether these values are realistic.

Assessing satellite images of the last few years, it can be confirmed that Lovisato Glacier has significant calving losses, seen through large numbers of icebergs in the proglacial lake (see Fig. 1). Lovisato Glacier has a frontal width of around 500 m. Satellite observations suggest surface velocities of around 400 m yr-1 in recent years. In order to obtain the suggested ice mass loss of 23.88 to 27.65 Mt yr-1, an ice thickness of around 130–150  m would be necessary. The 2019 consensus estimate gives an ice thickness of up to ∼190 m in this area of Lovisato Glacier (Farinotti et al., 2019). Other ice thickness reconstructions estimate a thickness between 144 and 200 m (Carrivick et al., 2016; Millan et al., 2022). Subsequently, the high calving rates suggested by our results are realistic for Lovisato Glacier.

For glacier 138, however, we do not see any major icebergs that would indicate a significant calving flux. Surface velocities are below 20 m yr-1 and maximum ice thickness between 50 and 70 m (Farinotti et al., 2019; Millan et al., 2022). This would result in calving flux magnitudes smaller than implied by our results. Therefore, we reject the calving explanation for glacier 138. It is one of the smallest glaciers that we included in the comparison with satellite observations. Due to the small size, the uncertainty in the observed elevation change rate is large. Furthermore, the DEMs used for the calculations have large gaps over this glacier, specifically in the accumulation area. Thus, we assume that in reality the uncertainty for glacier 138 is even larger, which could cause the large difference between model and observation in this case.

Other factors that could explain the large discrepancy between geodetic and surface MB are limitations in the snowdrift parameterization. The snowdrift scheme does not track the snow on its way from one location to another but identifies locations sheltered from or exposed to wind and, subsequently, corrects the snowfall amounts based on that. Looking at the study site, the question can be asked where the snow deposited at glacier 138 should come from. The main snowdrift direction is towards the southeast. There is no area directly northwest of glacier 138, where we would expect much snowfall that could be blown to and deposited at glacier 138. This highlights one limitation of the snowdrift parameterization. However, even without snowdrift (Strategy B), our results require a calving flux of more than 1.42 m w.e. yr-1 (Table 2). Thus, limitations are given by the SMB model itself and the climatic forcing as well.

Using one TLR throughout the whole study site and throughout the year is a major simplification. SMB models are highly sensitive to the air temperature field. It is known that the TLR over mountainous terrain varies not only temporally, but also locally (Gardner and Sharp, 2009; Gardner et al., 2009; Petersen et al., 2013; Ayala et al., 2015; Heynen et al., 2016; Shaw et al., 2016; Shen et al., 2016; Hanna et al., 2017). Bravo et al. (2019a) found that the observed lapse rates at the SPI are steeper in the east compared to the west and that differences exist between the lower and upper sections of glaciers. Thus, it is possible that a northwest-to-southeast gradient in temperature (lapse rate) prevails in the MSM, affecting the SMB. However, since we do not have any measurements of TLR at the study site that would allow a more realistic estimate, a constant and linear lapse rate is applied.