GPR is a widely used geophysical method for glacier thickness and internal structure investigations (Bogorodsky et al., 2012; Karušs et al., 2019, 2022; Lamsters et al., 2020a, b, 2024; Navarro and Eisen, 2009; Pellikka and Rees, 2009; Sevestre et al., 2015). The main factors influencing GPR performance in glacial environments include the specifications of a GPR system (e.g. frequency, power) and the physical properties of ice, such as density and water content, which affect dielectric permittivity and signal attenuation (Ogier et al., 2023; Schroeder et al., 2020; Watts and England, 1976). Temperate Alpine glaciers commonly contain liquid water, resulting in strong radar signal attenuation and limited bedrock detectability. As shown by Ogier et al. (2023), liquid water contents as low as ∼0.2 % (typical for Alpine glaciers) can significantly impede bed detection for ice thicknesses exceeding ∼100 m.

In this study, we conducted a GPR survey using a terrestrial Zond-xLF GPR system manufactured by Radar Systems, Inc. Given the temperate nature of Alpine glaciers, maximizing penetration depth was a priority, therefore, we used an antenna with the lowest possible frequency (38 MHz) compatible with our radar system, as higher-frequency signals are more susceptible to absorption in temperate ice conditions (Murray et al., 2007; Ogier et al., 2023). We selected a time window of 2560 ns with a range per sample of 1250 ps for shallower parts and 2500 ps for deeper sections, and 2048 samples per trace, which allowed for a theoretical maximum penetration depth of ∼210 m, assuming a dielectric permittivity of 3.2 (cold-ice value, Church et al., 2020; Robin, 1975; Rutishauser et al., 2016).

We oriented the GPR profiles mainly transverse to the ice flow direction with 200 m spacing in between to optimize data collection within the available timeframe and to gather enough data points for subglacial valley mapping. Transverse profiles were preferred over along-flow profiles for two reasons: they better characterise the cross-sectional shape of the subglacial valley, and they maximise the chance of crossing englacial conduits oriented parallel to flow, producing characteristic hyperbolic reflections that allow these features to be identified (Jol, 2008). Additionally, we recorded one profile parallel to the ice flow direction (Fig. 1). Prior to the GPR survey, we created a waypoint plan with 50 m step size to ensure we could precisely and easily follow our predetermined profile layout on the glacier. During the survey, we used a Garmin Montana 610 GNSS receiver to navigate along the waypoints, while Emlid Reach RS2 GNSS receivers provided precise positioning of the GPR profiles.

We processed the gathered GPR data using Prism 2.7 software. Our processing workflow included the application of manually adjusted time-dependent signal gain function and Ormsby band-pass filter with a low frequency cut-off at 20 MHz and a high frequency cut-off at 70 MHz. We applied topographical corrections using the elevation data from the DSM. During data interpretation, we extracted the two-way travel time (TWTT) for basal reflections from each GPR-profile along its entire length or where visible, picking individual travel-time measurements approximately every 10 m along the profile. Two GPR specialists independently interpreted the bed reflection across the full profile set. As the two interpretations were generally consistent for profiles with clear basal reflections but diverged in sections with weaker or ambiguous reflections, we retained the complete set of picks from the more conservative interpreter throughout, to ensure picking uncertainty remained consistent and comparable across all profiles. Due to intense internal signal scattering, we could not apply a hyperbola matching function for GPR signal velocity determination, and an equipment malfunction prevented us from conducting a common-midpoint survey. Consequently, we set the GPR propagation speed to 0.168 mns-1, which corresponds to a dielectric permittivity of 3.2 – the typical value for cold ice conditions (Robin, 1975). Our assumed value of 0.168 mns-1 is consistent with other GPR studies on temperate Alpine glaciers, e.g. Church et al. (2020) and Rutishauser et al. (2016), who used 0.1689 and 0.168 mns-1, respectively, for Swiss Alpine glaciers. The implications of this assumption are discussed in Sect. 3.3.

We interpolated glacier thickness point data derived from the GPR survey using cubic interpolation algorithms. We chose cubic interpolation as a compromise between more simplistic methods (e.g. inverse-distance weighting) and more geostatistical approaches (e.g. kriging), providing a relatively smooth thickness field and honouring the measured values. To prevent unrealistic values along the glacier margins, we assigned a uniform boundary thickness of 0.1 m to points along the glacier outline. In addition to the GPR-derived ice thickness measurements, we evaluated three widely-used, open-source ice thickness models: GlabTop2 (Frey et al., 2014), OGGM v1.6.2. (Maussion et al., 2019), and Millan et al. (2022) global model. These models differ in implementation algorithms and input data, although they ultimately derive ice thickness from glacier surface geometry and simplified ice flow assumptions.

The GlabTop2 model calculates ice thickness from surface slope using a shear-stress based approach, while the OGGM model inverts ice thickness along the central flowline based on mass conservation laws, calibrated against WGMS direct glaciological mass balance observations for HEF. The Millan et al. (2022) model inverts ice thickness from satellite-derived glacier surface velocities acquired in years 2017–2018, utilizing regionally calibrated ice rheology parameters and geometry-based regularization.

For the GlabTop2 and OGGM models, we used open-source Python implementations with glacier outline and digital elevation model (DEM) as input, applying default parameters throughout for both models – including the OGGM ice dynamics parameters (inversion_glen_a=2.4×10-24 s-1Pa-3, inversion_fs=0). For the Millan et al. (2022) ice thickness was extracted directly from the published dataset. However, since Millan et al. (2022) dataset represents glacier conditions for the 2017–2018 period, for the accuracy assessment, the thickness values were adjusted to account for the temporal offset to our GPR data acquisition date of August 2023. We derived the correction from elevation-band cumulative mass balances reported by the WGMS for the 2018–2023 period, converted to surface elevation change assuming an ice density of 900 kgm-3. We applied this correction only for the statistical comparison at GPR measurement points, while all visual materials (maps, difference maps, and scatterplot) retain the original 2017–2018 geometry.

All scripts used for model setup and execution are provided in the Zenodo repository (Švinka et al., 2026) to ensure full reproducibility. All generated maps are in WGS84/UTM zone 32N coordinate system and ellipsoidal height.

We did all uncertainty and accuracy calculations based on the measured GPR point data (n=1313), while we used the interpolated thickness and bedrock maps solely for visualization purposes to avoid introducing interpolation artefacts.

In our survey GPR velocity could not be measured in-situ, therefore a standard cold-ice (0 % water content) velocity value of 0.168 mns-1 was applied, introducing a systematic bias in our GPR-derived thickness values. Given that Hintereisferner is classified as a temperate glacier and some level of internal scattering is visible in all the GPR-profiles, possibly indicating high water saturation, the true EM velocity likely is lower, as it decreases with increasing liquid-water content (Bradford and Harper, 2005; Delf et al., 2022; Murray et al., 2000, 2007; Navarro and Eisen, 2009). This would result in shallower true ice thickness than calculated using the cold-ice value of 0.168 mns-1.

To bound the potential effect of water content, we compare our assumed cold-ice velocity against the temperate Falljökull glacier, Iceland Murray et al. (2000), where a mean EM velocity of 0.156 mns-1 was measured. This value represents an exceptionally water-saturated case that provides a conservative lower bound on velocity. The intense englacial scattering present in our GPR profiles suggests HEF might also contain significant water content, justifying inclusion of this lower bound on EM velocity. Using the Falljökull value, our assumed value would correspond to +7 % systematic overestimation in ice thickness. In addition, uncertainty in the resolution of bed-reflection picking introduces a locally variable random error of ±λ/2, which for our 38 MHz antenna and EM velocity of 0.168 mns-1 equals ±2.2 m (∼2 %–3 % of the mean ice thickness). Considering both sources of uncertainty – the dominant positive velocity bias and the smaller, spatially variable picking error – the worst-case overestimation of ice thickness is likely up to ∼10 %, whereas the typical uncertainty represents a positive bias of only a few percent.

To evaluate the accuracy of modelled ice thickness data, we compared the modelled and measured (GPR-derived) values using mean error or bias (ME), relative error (RE), and root-mean-square error (RMSE), defined as: 1ME=1n∑i=1n(hmod,i-hGPR,i),2RE=hmod-hGPRhGPR⋅100%,3RMSE=1n∑i=1n(hmod,i-hGPR,i)2, where hmod and hGPR are the modelled and measured ice thickness, respectively. These metrics were chosen to summarize the direction and magnitude of deviation from the GPR measurements in a simple and interpretable way.

The interpolated ice thickness map (Fig. 3c) reveals a distinct subglacial trough aligned with the glacier flow. The maximum detected ice thickness of ∼160 m occurs along the central flowline at an elevation of ∼2870 m, while ice gradually thins toward both the margins and the terminus. The mean ice thickness of the surveyed area is 81.1 ± 37.4 m (±1 standard deviation (σ)). The derived bedrock topography (Fig. 3d) reveals a U-shaped valley with bedrock elevations ranging from ∼2540 to 2940 m. The subglacial valley is slightly asymmetrical with the NW slope being steeper, while at the upper reaches of the study area it is the opposite. The longitudinal glacier cross-section (Fig. 3d, inset), extracted along the central flowline from the interpolated bedrock elevation map, indicates a down-glacier gradient with an evident rise in bedrock topography at around 1.8 km from the terminus. This coincides with an apparent local thinning in the interpolated thickness map (Fig. 3c) and a steeper slope visible in the DSM (Fig. 3a). We also found the slightly undulating bed up-glacier and more notable overdeepening down-glacier. However, the glacier bed is not fully identifiable in the GPR profiles starting from the bedrock rise, visible as discontinuities in the GPR thickness point-measurements (Fig. 3b) and corresponding to the boundary between well-defined and obscure basal reflections (Figs. 1 and 3c). Therefore, the apparent ice thinning and morphology of the rise should be interpreted with caution. We do not further expand the probable connection and causes between the bedrock rise and bedrock signal weakening up-glacier due to the lack of other evidence and to avoid overinterpretation. But it is worth noting that the bedrock rise and overdeepening has been reported in other studies (Förtsch and Vidal, 1956; Span et al., 2005). Overall, the spatial pattern of interpolated thickness and derived bedrock topography is best represented in the lower half of the survey area, where bed visibility was highest, and the uncertainty increases toward the upper half of the survey area, where interpolation relied on fewer measurement points.

The modelled ice thickness distributions from GlabTop2, OGGM and Millan et al. (2022) represent the entire glaciated area but were clipped to match our GPR-surveyed region. All three models generally reproduce similar geometries, with maximum thickness in the lower half of the glacier, coinciding with the GPR-surveyed area (Fig. 4a, c, and e). The maximum modelled ice thickness is 167 m for GlabTop2, 246 m for OGGM, and 247 m for Millan et al. (2022).

The thickness difference maps between modelled and GPR-interpolated thickness (Fig. 4b, d, and f) reveal a consistent pattern of overestimation, particularly along the glacier margins. Maximum positive deviations range from ∼146 m for GlabTop2, 173 m for OGGM, and 207 m for Millan et al. (2022). Negative deviations are comparatively small, reaching -27 m for GlabTop2, -10 m for OGGM and -68 m for Millan et al. (2022). Ice thickness tends to be slightly underestimated along the centre flowline for the GlabTop2 and Millan models, although these values are several times smaller than the overestimations. The potential influence of the DSM temporal offset on these patterns is discussed in Sect. 5.2.

The mean modelled ice thickness and corresponding error statistics for all three models are summarised in Table 1. These results reveal a consistent positive bias across all three models, reflecting a general pattern of ice thickness overestimation, also visible in measured vs. modelled ice thickness scatterplots (Fig. 5). Here, GlabTop2 shows a relatively narrow range of modelled thickness, mostly between 100 and 150 m, with a clear horizontal banding pattern aligned with elevation. This indicates that GlabTop2 tends to produce spatially uniform thickness distribution, and fails to represent the local extremes, as revealed by GPR data. In contrast, OGGM and Millan et al. (2022) show a wider spread of modelled values, although both, like GlabTop2, still substantially overestimate ice thickness across the survey area and present some degree of elevation banding as well. The largest spatial variability in model bias occurs for Millan et al. (2022), whose bias standard deviation (±57.0 m) exceeds its mean bias (Table 1).

As an additional metric, total ice volume for the surveyed area is 0.103 km3 for GPR, 0.161 km3 for GlabTop2 (RE 56.7 %), 0.194 km3 for OGGM (RE 89.0 %), and 0.175 km3 for Millan et al. (2022), without adjusting for temporal offset (RE 70.9 %). These volume estimates should be interpreted with caution, as they were calculated from each model's native raster resolution, which differs substantially between models (particularly for Millan et al., 2022), introducing additional uncertainty unrelated to model performance itself.

The comparison between GPR-derived ice thickness and the three modelling approaches used in this study reveals a systematic ice thickness overestimation across the surveyed area of HEF, with broadly comparable mean error magnitudes across all three models (Table 1). The models differ primarily in the spatial distribution of this bias. The OGGM model best reproduces the overall geometry of the GPR-derived thickness field, with thickness maximised along the central flowline and thinning towards the margins (Figs. 4c and 3c). This likely reflects the valley-confined geometry of HEF, for which the flowline-based algorithm and enforced flux continuity of this model are particularly well suited (Maussion et al., 2019). However, this correct geometric representation does not translate into accurate thickness magnitudes: OGGM shows the largest mean bias of the three models (Table 1), indicating that reproducing the correct spatial pattern alone is insufficient to constrain absolute ice thickness. GlabTop2's spatially uniform bias field, by contrast, reflects a documented sensitivity to slope and DEM quality, alongside the relative robustness of OGGM to these same factors (Chen et al., 2022; Pelto et al., 2020). Millan et al. (2022) shows a qualitatively similar spatial pattern to GlabTop2 – underestimating thickness in the central trough while overestimating near the margins – but at a substantially larger magnitude: the bias standard deviation for Millan et al. (2022) (±57.0 m) is nearly three times that of GlabTop2 (±19.9 m; Table 1), despite comparable mean bias (46.0 m vs. 40.6 m). Margin-related overestimation of the kind seen here has similarly been reported for GlabTop2 and OGGM elsewhere, attributed to interpolation procedures and simplified glacier geometry near glacier boundaries (Vergnano et al., 2025), where it was suggested that this effect would be less pronounced for elongated, valley-confined glaciers compared to wider, more complex geometries. However, despite the valley-confined geometry of HEF, margin overestimation is still present (Fig. 4), suggesting that valley confinement alone does not necessarily constrain this bias.

Although the three models differ in complexity and input data, they remain fundamentally constrained by glacier surface geometry and slope-dependent stress balance assumptions. In the absence of glacier-specific constraints on basal sliding and ice rheology, variations in ice viscosity must be balanced by the inferred thickness. Deviations from commonly assumed rheological parameters, such as a Glens' stress exponent n and rate factor A expected for temperate, water-rich ice, would reduce the thickness required to reproduce the observed velocities. Recent studies show that temperate ice may behave substantially differently from cold ice, with the stress exponents approaching n≈1 (Schohn et al., 2025) and strong weakening associated with meltwater content and strain localization (Haseloff et al., 2019). If these effects are not accounted for, deformation-based inversions tend to misattribute ice motion to increased thickness, resulting in systematic overestimation. Although local underestimation occurs in places, the net effect across the surveyed area is a positive bias. We note that these rheological arguments represent possible contributing factors rather than demonstrated mechanisms. A post-hoc sensitivity analysis, i.e. tuning parameters such as the rate factor A after the true thickness is known, would not meaningfully address this, as the whole point of these models is to estimate thickness without prior knowledge of it. Tuning a model to match known observations on one glacier tells us little about how well it would perform elsewhere, where no thickness data exist. For models capable of forward projections, such as OGGM, the magnitude of thickness overestimation documented here suggests that projections initialised from the default model state may inherit a substantially biased ice volume, which is worth considering when interpreting projected glacier evolution – including for glaciers in the same region as HEF (Hartl et al., 2025) and at broader scales (Rounce et al., 2023; Shafeeque et al., 2026). Systematic evaluation of rheological assumptions and their effect on projection reliability across diverse glacier settings therefore remains a valuable direction for future work.

A different source of bias applies to Millan et al. (2022), whose inversion instead relies on observed surface velocities. Millan et al. report an accuracy of ∼10 myr-1 for their satellite-derived velocity mosaics, which are provided on a 50 m grid. For HEF, however, surface velocities during the past decade are generally below ∼10 myr-1 (Stocker-Waldhuber et al., 2024), with only localized areas possibly reaching higher values. As a result, the velocity signal is close to the uncertainty level of the velocity mosaic and provides only weak constraint on ice thickness magnitude. Under these conditions, ice thickness becomes the primary variable used to satisfy the flux continuity, leading to spatially inconsistent thickness estimates, consistent with the large bias variability observed for this model. This sensitivity to input data quality is consistent with broader model intercomparison results, where models relying on additional datasets such as surface velocity fields showed particularly high sensitivity to input data quality (Farinotti et al., 2017).The contrasting behaviour of the Millan et al. (2022) model across different glaciological settings further highlights this sensitivity. While the model overestimates thickness for the temperate, valley-confined HEF, Lamsters et al. (2024) reported substantial underestimation of up to ∼80–90 m along the central flowline for small, narrow, cold outlet glaciers of the Qaanaaq Ice Cap in NW Greenland.

In addition to these model-specific bias sources, we considered several potential sources of systematic error when interpreting the reported model biases. Regarding GPR-derived thickness uncertainty: as mentioned in Sect. 3.3, our cold-ice velocity assumption results in a worst-case overestimation of GPR-derived thickness of up to 7 %, meaning the true model bias is likely larger than reported here. Our bias estimates are therefore conservative. Regarding the DSM temporal offset, it is important to note that the DSM has no effect on GPR-derived ice thickness, as thickness is calculated directly from the two-way travel time between the surface and bed reflections and does not depend on any elevation reference. The DSM temporal offset is therefore solely relevant as a potential source of uncertainty in the model inputs for GlabTop2 and OGGM. The DSM–GNSS elevation differences at GPR profile locations average 2.8 m and reach a maximum of 4.9 m (Fig. A2), while maximum positive model deviations reach 146, 173, and 207 m for GlabTop2, OGGM, and Millan et al. (2022), respectively. The DSM offset therefore represents less than 3 % of the maximum overestimation and cannot account for the observed spatial pattern of thickness bias along the glacier margins. To further quantify this, we reran both GlabTop2 and OGGM with the DEM uniformly lowered by 5 m – exceeding the maximum observed DSM–GNSS difference of 4.9 m – and found that mean modelled thickness changed by less than 1 % for both GlabTop2 and OGGM models, confirming that the DEM temporal offset has a negligible effect on modelled thickness relative to the reported biases.

The spatial extent of GPR survey in this study must be considered when interpreting model performance. The comparison is restricted to the middle-to-lower sections of HEF, where surface slopes are relatively gentle. The steeper upper-glacier area and potential local thickness maxima were not fully covered, and bed visibility decreased towards higher elevations. Inclusion of these areas could potentially reduce the overall magnitude of overestimation, but no definitive conclusions can be drawn.

Even ground-based GPR systems, generally considered to offer the highest data quality and bedrock detection capability on temperate Alpine glaciers (Forte et al., 2019), can suffer from substantial bedrock signal loss. Across a large dataset of Swiss Alpine glaciers, Rutishauser et al. (2016) found that the bedrock interface could be identified in only 12 %–69 % of profiles, depending on the glacier. Similar limitations in bedrock visibility have been reported elsewhere, particularly where ice thickness exceeds ∼100 m (Binder et al., 2009; Ogier et al., 2023). Although winter surveys are sometimes used to reduce attenuation from liquid water and improve bedrock detectability (Church et al., 2020), such conditions pose substantial logistical and safety challenges for ground-based GPR on Alpine glaciers due to deep snow cover and limited accessibility. In such cases, combined interpretation of GPR measurements and modelled thickness has been recommended to reduce uncertainty (Vergnano et al., 2025); however, the pronounced model biases identified in this study indicate that such approaches must be applied with caution and cannot be universally recommended.