Concerning the elevations of Alpine glaciers, past and present surface heights are derived from aerial photography, digitized topographic maps and spaceborne synthetic aperture radar (SAR) DEMs. For the Austrian Alps, aerial photographs of all glacierized areas were acquired with a mean picture scale of 1:30000 between September and October 1969 (Patzelt, 1980). The original images were later digitized and co-registered to derive the photogrammetric DHM69 elevation model (Lambrecht and Kuhn, 2007) that is used in this study. Historic glacier elevations of the Swiss Alps are available via the DHM25 dataset provided by the Federal Office of Topography swisstopo (Anonymous, 2005). The DHM25 elevation model was created from the Swiss national topographic map (scale 1:25000) by digitizing contour lines and spot heights which were then interpolated to a 25 m resolution grid. We use the original DHM25lvl1 product, because glacier areas were updated with surface heights from winter 2000/2001 in the DHM25lvl2. Most map tiles covering the central Swiss Alps are from the 1980s (Anonymous, 2005), but the dates of glacier heights differ from the DHM25 specifications. Therefore, we refer to a detailed manual reconstruction (Fischer et al., 2015a) of the specific reference years (1961–1991) of the DHM25 to derive the map date of each glacier. For the early 21st century, glacier surface topography is extracted from the 1 arcsec void-filled C-band SAR DEM of the Shuttle Radar Topography Mission (SRTM; Farr et al., 2007; Podest and Crow, 2013), which was acquired during February 2000. Recent glacier surface heights are provided by the bistatic TanDEM-X (TerraSAR-X add-on for Digital Elevation Measurement) satellite mission, operated by the German Aerospace Center (DLR) (Krieger et al., 2007; Zink et al., 2016). We use SAR DEMs from winter 2013/2014 (Sommer et al., 2020) and 2018/2019 to derive respective elevation mosaics. The 2018/2019 TanDEM-X DEMs were created from ∼160 co-registered single-look slant-range complex (CoSSC) acquisitions using differential interferometry and were vertically and horizontally co-registered according to the workflow described by Sommer et al. (2020). In this study, all elevation datasets are resampled to 30 m grids.

The 1969–2019 elevation changes in Austrian glaciers are inferred from the DHM69 and TanDEM-X. The TanDEM-X DEM and the DHM69 are vertically and horizontally co-registered, using non-glacierized and flat terrain (slope < 15∘) outside glacierized areas and waterbodies (Braun et al., 2019). Eventually, the DEMs are differenced and elevation change rates (δHδt) are calculated from the TanDEM-X and DHM69 acquisition dates. As the DHM69 was acquired between September and October 1969, we use the average date (1 October 1969) as a reference date. For the Swiss Alps, glacier elevation changes are obtained by differencing the TanDEM-X 2018/2019 DEM mosaic and the DHM25. As for the DHM69, the TanDEM-X DEM and the DHM25 are vertically and horizontally co-registered to minimize elevation offsets. Thereafter, we use the individual glacier reference years of the DHM25 (Fischer et al., 2015a) (Sect. 2.3.1) and the TanDEM-X acquisition dates to convert the height difference into elevation change rates. The median observation period of all Swiss glaciers is 1975–2019, with glacier-specific periods varying between 1961–2019 and 1991–2019. For the later reconstruction period (2000–2014), we use glacier elevation change rates derived from differencing SRTM C-band (February 2000) and TanDEM-X DEMs of winter 2013/2014 (Sommer et al., 2020). In most cases, data voids in the elevation change maps are caused by SAR layover and shadows. Therefore, we apply a bilinear interpolation, as recommended by a recent study (Seehaus et al., 2020). Finally, all elevation change fields are bilinearly resampled to a spatial resolution of 30 m.

The mean vertical precision of the glacier elevation change rate is derived as slope-dependent standard deviations on non-glacierized areas. All elevation change values outside glacier areas are aggregated within 5∘ slope bins. Thereafter, standard deviations of each slope bin are calculated and weighted by the respective total glacier area to derive the region-wide mean uncertainty. Further details on the elevation change error calculation are described in Sommer et al. (2020). For the DEM differences of the DHM25, DHM69 and the 2019 TanDEM-X acquisitions, the mean regional elevation change uncertainty is ±0.26 and ±0.18 m a-1, respectively, which represents an uncertainty of ∼39 % relative to the measured absolute elevation change over the entire observation period. The slope-derived δHδt error of the 2000–2014 period is ±0.39 m a-1 (Sommer et al., 2020), with a relative uncertainty of ∼53 %. Therefore, we use an average uncertainty in the elevation change fields of ±0.3 m a-1. It should be noted that we did not attempt a correction for height offsets between the optical/topographic and SAR DEMs due to signal penetration into the glacier surface. Particularly, for the DEM difference between the 2018/2019 TanDEM-X DEMs of winter 2018/2019 and DHM69 of autumn 1969, glacier surface elevations were acquired during different seasons, and the presence of signal penetration is likely for the TanDEM-X DEMs. However, we assume that the bias in elevation change due to radar penetration is small, as the snow cover of winter 2018/2019 is probably partially invisible for the X-band SAR, and the observation period is very long (∼40 years). In the case of the DHM25, no correction is applied because the exact mapping dates of the glacier areas are unknown.

The 20th century outlines of Alpine glaciers are extracted from the 1969 Austrian glacier inventory (GI1) and the 1973 Swiss glacier inventory (SGI1973). The 1969 Austrian GI1 was originally compiled from the same aerial photographs as the DHM69 (Patzelt, 1980) and later digitized (Lambrecht and Kuhn, 2007). Outlines of the SGI1973 are based on aerial photographs of September 1973 (Müller et al., 1976; Maisch et al., 2000) that were digitized and georeferenced by Paul et al. (2004). Due to the varying timestamps of the DHM25, many glacier outlines of the SGI1973 are older than the respective surface heights of the DHM25. However, the glacier area change between 1973 and 1985 was very small (∼-1 %) (Paul et al., 2004). Several glacier inventories covering the entire Alps have been created from satellite images and semiautomatic classification. The Randolph Glacier Inventory (RGI) (Pfeffer et al., 2014) of the European Alps was mostly created from band ratios of optical 2003 Landsat imagery with a resolution of 30 m (Paul et al., 2011). The 2013–2015 glacier extents were mapped from Landsat 8 images (Sommer et al., 2020). The most recent Alps-wide inventory is based on 10 m resolution Sentinel-2 acquisitions (Paul et al., 2020) from the years 2015–2017. Additionally, recent regional inventories, based on the manual delineation of glacier outlines from high-resolution optical images and elevation models, are available for the Austrian Alps for 2015 (Buckel et al., 2018) and for the Swiss Alps for 2013–2018 (SGI2016) (Linsbauer et al., 2021). For each glacier inventory, adjacent glacier boundaries are removed and the respective glaciers are merged into continuous geometries, thereby avoiding inconsistencies in ice thickness across divides and ridges (Fürst et al., 2017).

Reference ice thickness observations of Alpine glaciers are available via the Glacier Thickness Database (GlaThiDa v3) (GlaThiDa Consortium, 2020), which is a standardized collection of remote sensing and in situ measurements, and a recent publication on helicopter-borne ground-penetrating radar (GPR) measurements of almost all larger Swiss glaciers (Grab et al., 2021). While the GlaThiDa database of Alpine glaciers includes a large number of observations from different time periods and measurement techniques, the dataset by Grab et al. (2021) provides mainly GPR tracks between 2016 and 2020 but also older, so far not publicly available, measurements. Most of the in situ thickness observations are very densely spaced. Therefore, we removed observations that were less than 30 m apart, resulting in a total number of ∼53000 in situ measurements. The mean measurement uncertainty of all in situ observations is 8.2 m or ∼10 % relative to the mean glacier thickness of all in situ observations. For ∼30 % of the observations, the error is unknown because there is no information on the measurement uncertainty within the GlaThiDa database. For those measurements, the uncertainty is approximated as 20 % of the respective ice thickness value.

Thickness observations of the glacier margins are derived from glacier areas that became ice-free, as delineated by the multi-temporal outline and elevation information. Therefore, absolute elevation change values (δH) are extracted at glacier retreat areas that were inferred from differencing the respective glacier inventories. Additionally, an inner and outer buffer of 30 m is applied to the glacier retreat areas, and a slope threshold of 25∘ is enforced to exclude values close to the glacier outlines or on steeper slopes, as these values tend to be less reliable. In summary, we derive approximately 140 000 thickness observations from glacier areas that became ice-free between 1969 (Austria – AT) and ∼1970 (Switzerland – CH) and 2019. For the 2000–2014 period, we obtain 70 000 observations. The uncertainty in the δH measurements can be estimated from the errors in the δHδt fields (Sect. 2.4.2) and the respective observations periods of the DEM differences. Hence, the mean vertical δH uncertainties for the ∼1970–2019 period are ±10.3 and ±8.9 m for Swiss and Austrian glaciers, whereas they are ±5.5 m for the 2000–2014 period. Compared with the respective mean δH, the relative uncertainty in the retreat thickness measurements is ∼34 % (AT) and 38 % (CH) for ∼1970–2019 and ∼25 % (Alps) for 2000–2014.

SMB estimates for all glaciers of the European Alps (referring to the RGI v6.0) are available from the Global Glacier Evolution Model (GloGEM) (Huss and Hock, 2015; Zekollari et al., 2019). The model describes the main processes of mass balance – accumulation, melt and refreezing – and provides the annual SMB for 10 m elevation bins between 1951 and 2019. The model was driven by the E-OBS dataset (Cornes et al., 2018) and has been calibrated to match glacier-specific mass changes for 2000–2019 (Hugonnet et al., 2021). For the ice thickness reconstruction in this study, we averaged the annual SMB of each glacier over the 1969–2019 and 2000–2014 periods, respectively. The elevation-binned SMB information is transferred to 30 m grids using linear interpolation. To account for variations in glacier area and elevation over time, we use the SRTM DEM and RGI outlines for the 2000–2014 period and the Swiss and Austrian glacier inventories (GI1 and SGI1973) as well as surface elevations of the DHM69 and DHM25 for ∼1970–2019. For the historic glacier inventories (GI1 and SGI1973), we spatially matched the glacier areas and the IDs of the RGI by comparing the respective outlines. In cases where the historic outline overlapped with more than one RGI ID, due to differences in the delineation of ice divides or the disintegration of a once continuous glacierized area into several smaller glaciers, we used the SMB values of the RGI geometry that had the largest spatial overlap. In addition, we had to apply hypsometric extrapolation in some cases to vertically extend the SMB information, as the glacier outline of the historic inventories covered a larger elevation range than the respective RGI geometry.

For the early period from 1969 to 1973, glacier heights are extracted from the Swiss DHM25 and Austrian DHM69, and glacier areas are taken from the SGI1973 and GI1. The acquisition dates of the respective glacier outlines mostly refer to the years 1973 (CH) and 1969 (AT). Regarding the glacier surface topography, the Austrian DHM69 represents surface heights for the same year as the glacier inventory, whereas the Swiss DHM25 elevations refer to regionally different acquisition years (see Sect. 2.3.1). Thus, in the following sections, the reconstruction of the historic Swiss and Austrian ice thickness is denoted as HSIA1970, according to the approximate acquisition dates of the glacier outlines and DEMs (1970). As input fields for the HSIA1970 reconstruction, elevation change rates are derived from the difference between the DHM25/DHM69 and TanDEM-X mosaic for winter 2018/2019. The HSIA1970 is applied for two experimental set-ups using different samples of thickness observations and viscosity scaling factors.

For the initial reference run (HSIA1970in situ), the available in situ thickness observations (GlaThiDa Consortium, 2020; Grab et al., 2021) of glaciers in the Swiss and Austrian Alps are divided into two equal subsamples. All glaciers with in situ measurements (304 glaciers total) are grouped into four classes of equal size using the glacier area quantiles. Thereafter, half of the glaciers in each class are selected randomly to create a set of calibration and validation glaciers (152 glaciers each). Based on those randomly selected glaciers, the in situ thickness observations are divided into a set of calibration (ncalin situ=24677 measurements) and validation (nvalin situ=25753 measurements) points. The thickness measurements of these observational datasets are rather equally distributed across lower and upper sections of the glaciers and provide a good representation of both the thick and central parts as well as the glacier margins. A detailed overview of the random selection of calibration and validation glaciers is provided in Tables S1 and S2 and Figs. S1 and S2 in the Supplement. All in situ thickness observations of the 152 calibration glaciers are used for HSIA1970in situ. No scaling of the ice viscosity is applied (Figs. 1, 2a).

The second calibration set-up (HSIA1970retreat) is exclusively based on thickness observations from glacierized areas that became ice-free between the 1970s and the present (n1970retreat). Unlike the in situ measurements, the majority of these thickness observations are derived at lower elevations and close to the margin or terminus, as those glacier parts typically show higher retreat rates than the accumulation areas. For the HSIA1970retreat set-up, we perform three iterations of the thickness reconstruction (Figs. 1, 2b–d): (1) without ice viscosity scaling (HSIA1970retreat01), (2) glacier surface slope-based viscosity scaling (HSIA1970retreat02; see Sect. 3.2), and (3) glacier surface elevation- and slope-based viscosity scaling (HSIA1970retreat03; Sect. 3.3). For each iteration of HSIA1970retreat, the reconstructed viscosity and thickness are compared to the respective HSIA1970in situ values. The initial iteration (HSIA1970retreat01) includes no additional scaling of the ice viscosity, which is similar to HSIA1970in situ. Based on the difference in viscosity between HSIA1970in situ and HSIA1970retreat01 (Sect. 3.2), empirical slope-based scaling factors can be derived that are then applied to the viscosity estimation during the second calibration iteration (HSIA1970retreat02) according to Eq. (5): 5ξηslope=yηslope×α-αηthres for α≤αηthres. Here, ξηslope is the slope-based viscosity scaling factor and depends on the local surface slope (α). Calibration parameters are a slope gradient factor (yηslope; units per degree) and the respective slope threshold (αηthres) beyond which the viscosity ratio equals one.

Additionally, the slope-based scaling of HSIA1970retreat02 (Fig. 2c) is extended with an elevation-based viscosity scaling (HSIA1970retreat03) to avoid unrealistically high thickness values in the upper-glacier parts. As the vertical extents of Alpine glaciers vary significantly, the elevations of each continuous glacier area are normalized between the lowest (hmin=1) and highest glacier elevation (hmax=0). An additional quantile filter is applied to the elevation range that defines the lowest and highest 2 % of elevation values as one and zero, respectively. By this means, we compensate for uncertainties in the glacier area delineation, as the lowest and highest points of the glacier outline are often difficult to identify due to debris coverage or firn. The second scaling factor is applied based on the linear regression between the ice viscosity and the normalized glacier elevation range (Eq. 6): 6ξηelevation=1.0+yηelevation×h̃-h̃ηthres for 0≤h̃≤1. Here, ξηelevation is the empirical elevation-range-based correction, which is derived from the normalized local glacier elevation (h̃), the slope of the linear regression (yηelevation) and the vertical threshold (h̃ηthres) where no correction is applied. By applying Eq. (6), η is corrected and the final flux field and ice thickness (Fig. 2d) is recalculated considering the slope- and elevation-dependent viscosity ratios (HSIA1970retreat03).

For the early 21st century (2003), the glacier volume of all Alpine glaciers (HSIA2003retreat03) is estimated based on RGI glacier areas, the SRTM DEM, and surface elevation changes and mass balance data for the 2000–2014 period. Retreat thickness observations are extracted from glacier areas that have become ice-free since 2000 (n2003retreat). Additionally, HSIA2003in situ&retreat03 is derived from the same input data. However, all available in situ thickness observations (nallin situ) are integrated as well as observations from glacier areas that have become ice-free since 2000 (n2003retreat). For both reconstructions, viscosity correction factors are transferred from the estimates of the 1970s Swiss and Austrian glacier volumes (Fig. 1).

The 1970 reference ice thickness (HSIA1970in situ) of Swiss and Austrian glaciers is estimated from the historic Swiss and Austrian glacier inventories (SGI1973 and GI1), DEMs (DHM25 and DHM69), and respective surface elevation change and mass balance data. In addition, all ncalin situ thickness observations are included to constrain the reconstructed ice thickness distribution. In most cases the survey dates of the in situ measurements differ from the acquisition dates of the DEMs. To derive the respective ice thickness at the acquisition date of the DEM, the in situ observations have to be temporally homogenized. Therefore, we exclusively permit measurements that include both thickness and surface elevation and, thus, provide information on the local basal elevation beneath the glaciers. This is the case for all of the GPR thickness observations (Grab et al., 2021) and almost all of the GlaThiDa entries. For the thickness homogenization and the reconstructions, we assume negligible changes in the basal elevation and subtract it from the reference DEM in 1970 and 2000. The estimated ice volume for the Swiss and Austrian Alps (VSIA1970in situ) is 121.6±24.7 km3, corresponding to a glacierized area of 1792.9 km2. This is equivalent to 96 % of the total glacier area of the 1973 Swiss and 1969 Austrian inventory.

The initial reconstruction of retreat areas (HSIA1970retreat01) is based on the same input data as the reference thickness (HSIA1970in situ); however, instead of the in situ observations, thickness values are extracted from deglacierized areas (Sect. 2.4.4). No temporal homogenization of the observations has to be applied, as the ice thickness is directly derived from the state of the reference DEMs. Compared with the HSIA1970in situ reconstruction, the HSIA1970retreat01 underestimates the total glacier volume (VSIA1970retreat01) by approximately 40 % (Table 1) due to a strong negative bias in the estimated ice thickness (Fig. 3b). The largest differences are found at the troughs of large valley glaciers where the observed ice thickness can be twice as high as the reconstructed thickness (Fig. 1). These observations are very similar to the “low elevation bias” configuration used in ITMIX2. The participating models showed large deviations when the available thickness observations were limited to the low and thin glacier parts, where the ice flux is most likely underestimated due to substantial thinning rates and downwasting of the glacier termini (Farinotti et al., 2021).

To estimate the bias in viscosity between HSIA1970retreat01 and HSIA1970in situ, viscosity values are extracted at locations with in situ observations (ncalin situ) where the ice thickness and viscosity is known. Viscosity values are then aggregated within 2∘ slope bins to derive the ratio of HSIA1970retreat01 and HSIA1970in situ viscosities (Fig. 4). Average HSIA1970retreat01 viscosities are generally lower than HSIA1970in situ, but the difference increases at slopes smaller than 25∘. Using a linear regression, the slope-scaling parameters of Eq. (5) are yηslope=-0.08 and αηthres=43.75∘. The total ice volume (VSIA1970retreat02) is 122.5±24.5 km3 (Table 1).

The total ice volume VSIA1970retreat02 is similar to VSIA1970in situ. However, the modelled ice thickness tends to overestimate the observed glacier thickness at high altitudes, whereas the lower glacier parts continue to remain too thin (Fig. 2c). The overestimation at high altitudes is likely caused by large firn and ice areas with a small surface slope, where the corrected viscosity is overestimated. Compared with HSIA1970retreat01, the strong negative bias between estimated and observed ice thickness (Fig. 3a) is reduced. Nevertheless, there is a remaining negative offset for glacier parts with an observed ice thickness of more than 300 m (Fig. 3c). Figure 5 shows the mean viscosity ratio of HSIA1970retreat02 and HSIA1970in situ versus the normalized glacier elevation range (Sect. 2.5.1). The ratio of HSIA1970retreat02 to HSIA1970in situ is close to 1 at 0.5–0.6 normalized elevation, which is approximately equal to the glacier median elevation. However, a distinct offset is noticeable at the lowest and highest elevations, where the HSIA1970retreat02 ice viscosity is under- and overestimated, respectively. To compensate for this viscosity biases, yηelevation=-2.14 and hηthres=0.61 are applied based on the linear regression (Eq. 6).

As shown in Fig. 3d, the deviation between observed and estimated ice thickness of thick glacier parts (>300 m ice thickness) further decreases after applying Eq. (6). However, the median difference in HSIA1970retreat03 between observed and estimated ice thickness also increases by ∼10 m compared with HSIA1970retreat02, indicating a slight overestimation of the ice thickness by Eq. (6). The reconstructed total glacier volumes of the different reconstruction steps are shown in Table 1. While there is little difference in the modelled glacier volume of HSIA1970retreat03 and HSIA1970in situ (∼2 %), the inclusion of the elevation-based scaling further improves the spatial thickness distribution (Fig. 2d).