Glaciers AX010 (27.725° N, 86.555° E) and AX000 (27.713° N, 86.542° E) are small glaciers located in the Shorong region of Nepal (Table 1 and inset in Fig. 1a). Glacier AX010 has been observed intermittently since 1978 (Fig. S1 in the Supplement), and its volume changes up to 2008 have been previously estimated as a geodetic mass balance .

Glacier AX000 is located in a different catchment from Glacier AX010. However, because it is easily accessible from Glacier AX010 and isolated from other glaciers in the catchment, it was provisionally assigned the number “000”. Changes in the terminus position of this glacier have been observed from 1978 to 1989 , but the ground-based volume change was not estimated so far. We conducted our GNSS observation in 2008, but the results were not published.

In this study, we conducted a drone photogrammetry survey for both glaciers in 2023, and obtained volume changes between 2008–2023.

In this study, we generated 1 m resolution DEMs from our global navigation satellite system (GNSS) survey data acquired in 2008, and derived the geodetic mass balance by extrapolating and interpolating the differences between the 2008-DEMs and the 2023-DEMs obtained from drone photogrammetry surveys within the 2008 glacier area.

To reconstruct the past annual mass balance, we adopted the GLacIer energy Mass Balance model (GLIMB) that calculates the surface energy balance (Qm≥0) as: 1Qm=(1-α)RSd+RLd+RLu+HS+HL-Gg, where α is the surface albedo; RSd is the downward shortwave radiation, RLd and RLu are the downward and upward longwave radiations; HS and HL are the sensible and latent heat; Gg is the conductive heat flux into the glacier ice, respectively. Unit of all variables is Wm-2 except for albedo (no dimension). The surface albedo is calculated using a scheme that applies an exponential, temperature-dependent attenuation with time after a fresh snowfall . Annual glacier mass balance at a given elevation (bz, mw.e.) is calculated as: 2bz=∑dPs-tdQmlm+EV+RF/ρw where Ps is the solid precipitation; td is the length of a day in seconds (86 400 s); lm it the latent heat for ice melt (3.33×105Jkg-1); EV is the daily amount of evaporation (mmw.e.d-1), which is estimated by a bulk method; RF is the daily amount of refreezing water (mmw.e.d-1), which is estimated by calculating heat conduction and water percolation; and ρw is the water density (1000kgm-3) for the unit conversion (mm to m). The daily mass balance is summed over a given period (an observation period or a year). The glacier-wide mass balance (B, mw.e.) is calculated using the hypsometry as: 3B=∑zazbz∑zaz where az is the glacier area for a given 20 m elevation band. Detailed descriptions of the model are available in and . The glacier hypsometry was obtained from in-situ surveys and the UAV photogrammetry of this study (Fig. S3). Elevations of the past hypsometry are calibrated using GNSS-collected benchmarks. The annual hypsometry was prepared by interpolating each elevation band. For the mass-balance reconstruction before 1978, we used the 1978 hypsometry. The effect of the changing glacier geometry is evaluated in the Sect. 3.5.

We extracted daily mean meteorological variables from the ERA5 reanalysis data as model input. Besides the 2 m height temperature in the ERA5 data (T2m, °C), air temperature at a given elevation (Tz, °C) was estimated from the pressure level temperatures at the closest geopotential heights containing the target elevation . Downward longwave radiation (RLdz, Wm-2) at a given elevation (z, ma.s.l.) was calibrated with the effective emissivity (εe, no dimension), which can be defined by the downward longwave radiation (RLdERA5, Wm-2) and 2 m height temperature (T2m) based on the Stefan–Boltzmann equation as: σεe=RLdERA5T2m+273.154,4RLdz=σεeTz+273.154, where σ is the Stefan–Boltzmann constant (5.67×10-8Wm-2K-4).

The variables were compared with those observed at a nearby site (Trakarding AWS, 16 km from the glacier, Fig. S4 and Table S2) for the period 2022–2023, and calibration equations were then obtained. The extracted ERA5 covers both the Trakarding AWS and the glacier in a single cell (0.25°×0.25°).

The ERA5 precipitation is also compared with the AWS data. However, it has the greatest uncertainty among the ERA5 variables, and the spatial differences between the Trakarding site and AX010 are unknown. Therefore, we estimated the precipitation parameter by applying a multiplier (rP, dimensionless) to the ERA5 precipitation . We determined rP to yield the same value as the observed geodetic mass balance (Bgeod) by an iterative calculation (Fig. S5). A period-weighted rP was then obtained from five rP corresponding to the geodetic observations. Because the glacier's elevation range is so small (440 m, even at its maximum extent in 1978 during the observation period, Fig. S3), we did not account for the precipitation gradient with elevation in the simulation.

Figure 1a shows orthomosaics of Glaciers AX010 and AX000 generated from the 2023 drone photogrammetry. Figure 1b shows the same orthomosaic overlaid with the 2008 GNSS tracks. The elevations from the 2023DEM and 2008GNSS outside the glacier had biases of -0.703 m (AX010) and -0.118 m (AX000), respectively. Considering a 15-year duration and ice density, these biases (+0.042 mw.e.a-1 for AX010 and +0.007 mw.e.a-1 for AX000, respectively) were taken into account when calculating the volume changes of the glacier. The relative accuracy between the 2023 DEM and the 2008 DEM was estimated as 1 standard deviation; 0.604 m for Glacier AX010 and 0.387 m for Glacier AX000 by comparison with the 2008 GPS data on moraine ridges (inset in Fig. 1b). These values are comparable to those obtained in other UAV surveys and are more than two orders of magnitude lower than that derived from satellite observations (∼15 m) .

By interpolating/extrapolating the elevation difference, the surface elevation changes in 2023 relative to 2008 were obtained (Fig. 1c). As results, we update the geodetic mass balance (Bgeod, -1.214±0.036 mw.e.a-1 for AX010 and -1.060±0.023 mw.e.a-1 for AX000, respectively) of the glaciers for the 15-year period from 2008 to 2023 (Fig. 1d and Table 1). The associated error (0.036 mw.e.a-1) is substantially smaller than that of previous estimates derived from sparse survey points (0.084 mw.e.a-1) and is also much smaller than those based on satellite observations (∼0.600 mw.e.a-1, ). For comparison, we also show mass balance changes from 2000 to 2020 based on ASTER data for the same glaciers (Fig. 1d) . While no large systematic bias is evident, the satellite-based results appear to slightly underestimate the negative mass balance, particularly during the period 2010–2014. This could be attributed to the fact that the glacier area treated by is much larger (0.400 km2) than those we observed (0.339 km2 in 2008 and 0.185 km2 in 2023), meaning that areas where ice had already been lost were included when calculating surface elevation changes from satellite-based DEMs.

We first compared the daily mean variables in the ERA5 reanalysis data with those observed at the Trakarding AWS site (Fig. S6 and Table S3). The 2 m height temperature (T2m) shows a bias (5.56 °C in Table S3) due to elevation setting in the ERA5 data (blue dots in Fig. S6a), while that derived from the pressure-level data (Tp) shows good consistency with the observational temperature (orange dots in Fig. S6a; bias of -1.07 °C in Table S3). The downward longwave radiation calibrated with Tp (RLdcalib, orange dots in Fig. S6c; bias of 0.2 Wm-2 in Table S3) shows better consistency with the observed one though the coefficient of determination and root mean square error of the linear regression are slightly worse than those of the ERA5 longwave radiation (RLd, blue dots in Fig. S6c; bias of 25.6 Wm-2 in Table S3). The ERA5 shortwave radiation shows the worse coefficient of determination among the variables (R2=0.533), probably due to inaccurate cloud representation in the reanalysis (Fig. S6d). The ERA5 wind speed is significantly underestimated (Fig. S6e), while both relative humidity and precipitation in the ERA5 data are overestimated (Fig. S6b and f). The regressions summarized in Table S3 are used to calibrate the variables for the mass balance simulation, whereas precipitation is estimated using the geodetic mass balance and the model. Similar biases and RMSEs of ERA5-Land have been confirmed through the comparison with the local meteorological variables observed in the Khumbu region, immediately east of the studied site .

The air temperatures estimated using the pressure-level temperatures and geopotential heights applied in this study were compared with temperatures observed near Glacier AX010 in 1978 and in the 1990s, and were found to agree very well (Fig. S7). This consistency further supports the validity of the estimation method based on pressure-level data.

Precipitation parameters (rP) were estimated for each period over which geodetic mass balance was observed (Table S4). The parameters weighted by the length of each period yielded a value of 1.48±0.08. In comparison with the AWS near Trakarding Glacier, the precipitation parameter was 0.264 (Fig. S6f and Table S3), suggesting that ERA5 overestimates precipitation there. In contrast, at Glacier AX010, located 16 km to the southeast of the AWS site, ERA5 underestimates precipitation (3.49 mw.e.a-1 at AX010 against 2.36 mw.e.a-1 in ERA5 for 2023). There, precipitation is more than five times greater than at the Trakarding AWS site (0.62 mw.e.a-1 for 2023). This implies that strong precipitation contrasts exist over short distances within a single ERA5 grid cell (0.25° resolution).

While the estimated temperature from pressure levels showed good agreement with observational data from both Trakarding and AX010 (Figs. S6 and S7), observational data for other variables from Glacier AX010 were unavailable, so the relational formulas from the Trakarding AWS were applied (Fig. S6 and Table S3). Therefore, site-to-site bias may remain in these variables. Although previous studies have shown that the mass balance is insensitive to changes in variables other than temperature , the possibility that the bias correction is affecting the reconstructed precipitation parameters and mass balance cannot be ruled out.

Figure 2a shows the annual mass balance from 1940 to 2023 reconstructed using GLIMB and the calibrated ERA5 data. For the period from 1978 to 2023, hypsometry derived from geodetic observations was interpolated and applied, thereby accounting for glacier-wide shrinkage into account. For the period prior to 1978, the 1978 hypsometry was applied without modification. Because the precipitation parameter was tuned to match the geodetic observations, the modeled results naturally show good agreement with the observations (Fig. S8a). On Glacier AX010, stake-based observations were conducted in 1978 and during 1995–1999 . Comparison of mass-balance profiles corresponding to these observation periods indicates that the model reproduces the observed profiles well (Fig. S9). Figure S8b shows the comparison of the glacier-wide mass balance based on the linear regressions of the stake-based mass balance profile and simulation (Table S5). In particular, the pronounced negative mass balance in 1998, estimated to be the most negative over the past 80 years, is also well captured by the simulation, suggesting that ERA5 temperature data and the adjusted precipitation are appropriate for reconstructing the glacier mass balance.

In contrast, the mass balance calculated without adjusting precipitation became strongly negative (gray dots in Fig. S8a). This is because insufficient precipitation leads to inadequate accumulation, and at the same time, the surface albedo cannot be maintained at high values, which enhances melt. This can be regarded as a characteristic of glaciers influenced by the summer monsoon . For more quantitative evaluation, we compared mass balance, accumulation, and melt averaged over the period from 1978 to 2023 (Table S6). Annual accumulation increased from 1.315 mw.e.a-1 based on the ERA5 precipitation to 1.948 mw.e.a-1, an increase of 0.633 mw.e.a-1 (exactly a factor of 1.48, which is the same value as the precipitation parameter used in the simulation), whereas the mass balance increased from -1.809 to -0.919 mw.e.a-1, an increase of 0.890 mw.e.a-1. The difference between these increases (0.257 mw.e.a-1) can be interpreted as a melt-suppression effect, which is evaluated to be -0.187 mw.e.a-1, mediated through changes in albedo.

Because the observation periods of satellite-based geodetic mass balance (Bsat) do not coincide with those of the ground-based observations in this study (Bgeod), the satellite estimates were compared with the simulation results (Bsim) (Fig. S8c and Table S7). Bsat has increasingly overestimated ice mass loss in recent years. This is likely because, as glacier shrinkage has progressed, surface areas that have already become off-glacier are still included in estimates of surface lowering as addressed in Sect. 3.1. To accurately estimate mass changes of small glaciers, it is therefore essential to carefully track changes in glacier area. The bias in the satellite-based Bgeod due to glacier shrinkage has been pointed out for glaciers in Alaska and the US Rocky Mountains .

Regarding long-term trends (1941–2023), both mass balance and air temperature exhibit clear decreasing and warming trends (-0.153 mw.e.a-1 per decade and 0.089 °C per decade, both p<0.001), respectively (Fig. 2a and b, and Table S8). Breakpoint analysis indicates a change point in 1971 for both variables (-0.245 mw.e.a-1 per decade and 0.172 °C per decade for the period 1971–2023, both p<0.001). In addition, both variables shows another change point in 1999, revealing an acceleration of mass loss in more recent years (-0.473 mw.e.a-1 per decade and 0.351 °C per decade for the period 1999–2023, both p<0.001). The melt amount (M) shows variations synchronized with summer mean temperature (Fig. 2c), including the break point in 1971 (0.126 mw.e.a-1 per decade for the period 1971–2023, p<0.001) though the long-term trend is weak (0.039 mw.e.a-1 per decade, p=0.012). The positive degree day (PDD), which is usually used in temperature-index models , also shows a significant increasing long-term trend (10.81 °Cd per decade, p<0.001), with the rate of increase becoming particularly large since the beginning of the 21st century (50.67 °Cd per decade, p<0.001 for the period 2001–2023) (Table S8). Interestingly, however, there is no significant long-term trend in the number of melt days per year (dmelt, 0.413 d per decade, p=0.299). This suggests that rising temperatures are accelerating glacier shrinkage, not by prolonging the melting period, but by increasing melt intensity. In contrast, precipitation shows little overall change (Fig. 2c). Although a slight decreasing long-term trend is evident over the entire period (-0.049 mw.e.a-1decade-1, p<0.007), precipitation has in fact shown an increasing tendency since 1974 (0.101 mw.e.a-1decade-1, p<0.004 for the period 1974–2023), after a weakly detected change point. However, snowfall and rainfall display statistically significant decreasing and increasing long-term trends (-0.095 mw.e.a-1 per decade, p<0.001 and 0.046 mw.e.a-1 per decade, p=0.001), respectively, with the increase in rainfall after 2007 being particularly pronounced (0.376 mw.e.a-1 per decade, p<0.001 for the period 2007–2023). As a result, the snow accumulation has been consistently less than the melt amount since the late-1970s (Fig. 2c). This suggests that rising air temperatures promote glacier melt not only directly, but also indirectly by changing precipitation phase from snow to rain . Likewise, winter snowfall has been reported to be decreasing in the neighbouring Khumbu region . In contrast, future projections for the Langtang region suggest that increased precipitation may offset the reduction in river discharge that follows glacier shrinkage . Taken together, over the past 45 years at Glacier AX010, there is little doubt that the primary driver of glacier shrinkage has been rising air temperatures. No long-term trend in precipitation is evident, and it is clear that precipitation has neither suppressed nor accelerated glacier shrinkage driven by temperature increases.

Nevertheless, for Trambau Glacier, which lies within the same ERA5 grid cell, the reconstructed mass balance consistently shows negative values but no clear trend despite being forced by the same meteorological data . Moreover, while mass balance at Trambau Glacier shows no correlation with summer mean temperature (r=-0.21) and a significantly positive correlation with annual precipitation (r=0.77, p<0.001), Glacier AX010 exhibits a strong correlation with summer mean temperature (r=-0.81, p<0.001) and weaker correlation with annual precipitation (r=0.33, p<0.003). These correlations indicate that the response of glacier mass balance to variations in temperature and precipitation can differ substantially even among neighboring glaciers. In addition, for Mera Glacier, which lies 33 km east of Glacier AX010, the mass-balance anomaly correlates significantly with the anomalies of annual mean temperature (r=-0.79, p<0.001) and annual precipitation (r=0.87, p<0.001) . Although more detailed analyses are required to identify the causes of these differing responses and trends, our results suggest that it is problematic to naively extrapolate trends and climate-mass balance relationships derived for individual glaciers to the mountain-range scale.

We defined an air temperature at which the glacier mass balance becomes zero (Tequi), i.e. the glacier is in equilibrium, by iterative calculations using GLIMB (purple line in Fig. 2b). The variability and trend of Tequi reflect those of precipitation; however, no statistically significant trend was detected (Table S8). In recent years, air temperature has increased by nearly 1 °C above the level at which the glacier could be maintained in equilibrium.

To reconstruct the annual mass balance, interpolated hypsometries were used. To assess how changes in glacier size affect the reconstructed results, we recalculated the mass balance using hypsometry corresponding to the maximum (1978) and minimum (2023) glacier extents (Fig. S10). The results show that the impact of the hypsometry setting is comparable to the uncertainty in mass balance (±0.149 mw.e.a-1) arising from the precipitation parameter (±0.08). This is because, even though the glacier has shrunk dramatically over the past 45 years, the medians of the two hypsometries differ by only 20 m in elevation (Fig. S3), which is equivalent to the mass-balance uncertainty due to the precipitation assumption.

Figure S2b shows the spatial distribution of glacier ice thickness, which is derived from the cross section estimation (Fig. S11). Comparison with ice thickness measured by ice radar in 1995 indicates differences of +25 to -15 m relative to the 2023 bed elevation (Table S9). Given that the ice radar used in 1995 operated at 5 MHz, corresponding to a wavelength of approximately 60 m, these differences can be considered within the measurement uncertainty. Based on these results, the remaining ice volume in 2023 is estimated at 2.27×106m3, with a mean thickness of 13.7 m and a maximum thickness of 38.2 m. The ice loss from 2008 to 2023 is estimated at 7.17×106m3, indicating that only about one quarter of the 2008 ice volume remains. Assuming an ice density of 890kgm-3, the maximum ice thickness corresponds to 34.0 mw.e. If the geodetic mass balance observed from 2008 to 2023 (-1.214mw.e.a-1) were to continue, the glacier would disappear by 2050. In contrast, if the pronounced acceleration in ice loss observed since 1999 persists (Byr=-0.047yr+93.695; yr denotes year), the glacier is estimated to be lost by 2040.