Global glacier mass changes and their contributions to sea-level rise from 1961 to 2016

Abstract

Glaciers distinct from the Greenland and Antarctic ice sheets cover an area of approximately 706,000 square kilometres globally¹, with an estimated total volume of 170,000 cubic kilometres, or 0.4 metres of potential sea-level-rise equivalent². Retreating and thinning glaciers are icons of climate change³ and affect regional runoff⁴ as well as global sea level^5,6. In past reports from the Intergovernmental Panel on Climate Change, estimates of changes in glacier mass were based on the multiplication of averaged or interpolated results from available observations of a few hundred glaciers by defined regional glacier areas^7,8,9,10. For data-scarce regions, these results had to be complemented with estimates based on satellite altimetry and gravimetry¹¹. These past approaches were challenged by the small number and heterogeneous spatiotemporal distribution of in situ measurement series and their often unknown ability to represent their respective mountain ranges, as well as by the spatial limitations of satellite altimetry (for which only point data are available) and gravimetry (with its coarse resolution). Here we use an extrapolation of glaciological and geodetic observations to show that glaciers contributed 27 ± 22 millimetres to global mean sea-level rise from 1961 to 2016. Regional specific-mass-change rates for 2006–2016 range from −0.1 metres to −1.2 metres of water equivalent per year, resulting in a global sea-level contribution of 335 ± 144 gigatonnes, or 0.92 ± 0.39 millimetres, per year. Although statistical uncertainty ranges overlap, our conclusions suggest that glacier mass loss may be larger than previously reported¹¹. The present glacier mass loss is equivalent to the sea-level contribution of the Greenland Ice Sheet¹², clearly exceeds the loss from the Antarctic Ice Sheet¹³, and accounts for 25 to 30 per cent of the total observed sea-level rise¹⁴. Present mass-loss rates indicate that glaciers could almost disappear in some mountain ranges in this century, while heavily glacierized regions will continue to contribute to sea-level rise beyond 2100.

Main

Changes in glacier volume and mass are observed by geodetic and glaciological methods¹⁵. The glaciological method provides glacier-wide mass changes by using point measurements from seasonal or annual in situ campaigns, extrapolated to unmeasured regions of the glacier. The geodetic method determines glacier-wide volume changes by repeated mapping and differencing of glacier surface elevations from in situ, airborne and spaceborne surveys, usually over multiyear to decadal periods.

In this study, we used glaciological and geodetic data from the World Glacier Monitoring Service (WGMS)¹⁶, complemented by new and as-yet-unpublished geodetic assessments for glaciers in Africa, Alaska, the Caucasus, Central Asia, the Greenland periphery, Iceland, New Zealand, Scandinavia, Svalbard and the Russian Arctic. At present, this data set includes observations from 450 and 19,130 glaciers for the glaciological and the geodetic samples, respectively, which correspond to sample sizes of, respectively, less than 1% and 9% of the total number of glaciers¹. We estimated regional mass changes for the 19 first-order regions of the Randolph Glacier Inventory (RGI)¹ (Fig. 1). The observational coverage ranges from less than 1% to 54% of the total glacier area per region for the glaciological sample, and from less than 1% to 79% for the geodetic sample (Extended Data Fig. 1). In each region, we combined the temporal variability from the glaciological sample—obtained using a spatiotemporal variance decomposition—with the glacier-specific values of the geodetic sample (see Methods). We then extrapolated the calibrated annual time series from the observational to the full glacier sample to assess regional mass changes, taking into account regional rates of area change (see Methods). Uncertainties originate from four independent error sources. These relate to the temporal changes assessed from the glaciological sample, to the long-term geodetic values, to the extrapolation to unmeasured glaciers, and to estimates of regional glacier area. To estimate regional mass changes, we spatially interpolated the specific mass changes from the observational sample to all glaciers in the region. We estimated the related error from the deviations of this approach to regional (specific) mass changes, calculated as arithmetic averages or as area-weighted averages of the observational sample (see Methods).

Fig. 1: Regional glacier contributions to sea-level rise from 1961 to 2016.

The cumulative regional and global mass changes (in Gt, represented by the volume of the bubbles) are shown for the 19 first-order regions¹ (outlined with bold black lines). Specific mass-change rates (m w.e. yr⁻¹) are indicated by the colours of the bubbles. In the background, the locations of glaciological and geodetic data samples are plotted over the glacier polygons from RGI 6.0. The grey plus signs mark latitudes and longitudes. As an example, glaciers in Alaska (ALA) show the largest contribution to sea-level rise, with a total mass change of approximately −3,000 Gt or 8 mm sea-level equivalent (s.l.e.) from 1961 to 2016, because of a strongly negative specific mass-change rate (−0.6 m w.e. yr⁻¹) combined with a large regional glacier area. Note that South Asia West (ASW, blue bubble) is the only region in which glaciers slightly gained mass. ACN, Arctic Canada North; ACS, Arctic Canada South; ANT, Antarctic and Subantarctic; ASC, Central Asia; ASE, South Asia East; ASN, North Asia; CAU, Caucasus and Middle East; CEU, Central Europe; GRL, Greenland; ISL, Iceland; NZL, New Zealand; RUA, Russian Arctic; SAN, Southern Andes; SCA, Scandinavia; SJM, Svalbard and Jan Mayen; TRP, Low Latitudes; WNA, Western Canada and USA (see Table 1).

Over the full observation period from 1961 to 2016, global glacier mass changes cumulated to −9,625 ± 7,975 Gt (1 Gt = 10¹² kg). This corresponds to a contribution of 27 ± 22 mm to global sea level, or a contribution of 0.5 ± 0.4 mm yr⁻¹ when a linear rate is assumed. The total mass change excluding peripheral glaciers in Greenland and Antarctica sums to −8,305 ± 5,115 Gt, corresponding to a contribution to sea level of 0.4 ± 0.3 mm yr⁻¹. Cumulative mass changes and corresponding contributions to global sea level were largest from the heavily glacierized regions, with approximately one third originating from Alaska (Fig. 1). Additionally, large contributions originate from regions with less glacierization but strongly negative specific mass changes, such as Western Canada and the USA (Extended Data Fig. 2). South Asia West was the only region that exhibited mass gain over the full observation period. Cumulative specific mass changes over this period, from 1961 to 2016, were most negative in the Southern Andes, followed by Alaska, the Low Latitudes, Western Canada and the USA, New Zealand, the Russian Arctic and Central Europe (Extended Data Fig. 2a). When annual rates are averaged over pentads (that is, periods of five years; Fig. 2), sea-level contributions ranged between 0.2 ± 0.5 mm yr⁻¹ and 0.3 ± 0.4 mm yr⁻¹ until the 1980s, and then increased continuously to reach 1.0 ± 0.4 mm in the latest pentad (2011–2016). Over corresponding periods, our estimates show that global glacier mass loss is approximately equivalent to various mass-loss estimates from the Greenland Ice Sheet (between 2003 and 2012)¹², and it exceeds present contributions to sea-level rise from the Antarctic Ice Sheet (2012–2017: 219 ± 43 Gt yr⁻¹, including the Antarctic Peninsula)¹³ by 62%. Hence, glaciers contributed between 25% and 30% of the observed global mean sea-level rise, which ranged between 2.6 mm yr⁻¹ and 2.9 ± 0.4 mm yr⁻¹ over the satellite altimetry era (1993 to mid-2014)¹⁴.

Fig. 2: Global glacier contributions to sea-level rise from 1961 to 2016.

Annual and pentadal mass-change rates (left vertical axis) and equivalents of mean global sea-level rise (right vertical axis) are shown with related error bars (indicated by shading) corresponding to 95% confidence intervals. Annual errors originate from independent sources: glaciological sample, geodetic sample, spatial interpolation and glacier area. Over the five-year periods, the individual error terms are cumulated separately and then the multiyear terms are combined according to the law of random error propagations, and divided by the number of years (see Methods).

Glacier mass changes were negative in all regions over the latest observational decade, from 2006 to 2016 (Table 1)—that is, covering the hydrological years¹⁵ from 2006/07 to 2015/16. Glaciers in South America had the most negative specific mass changes, with rates exceeding −1.0 m water equivalent (w.e.) per year, followed by glaciers in the Caucasus, Central Europe, Alaska, and Western Canada and the USA, with rates of less than −0.8 m w.e. yr⁻¹ (Fig. 3a; 1 m w.e. = 1,000 kg m⁻²). The least negative specific mass changes were found for glaciers in the Antarctic periphery (−0.1 m w.e. yr⁻¹) and in South Asia West, with glaciers close to balanced-budget conditions^17,18. Again, regions with large ice cover and negative specific mass changes showed the largest total losses (Fig. 3b). Record mass losses are thus found in Alaska, with rates of −73 Gt yr⁻¹, followed by other heavily glacierized regions (that is, with glacier areas of more than 29,000 km²) such as Arctic Canada North (−60 Gt yr⁻¹), the Greenland periphery (−51 Gt yr⁻¹), and the Southern Andes (−34 Gt yr⁻¹; Table 1). Exceptions are Central Asia and South Asia West, with limited mass losses (−7 Gt yr⁻¹ and −1 Gt yr⁻¹) despite their large glacier areas. Of the regions with smaller glacierization, Western Canada and the USA and Iceland lost the most mass, at rates of −12 Gt yr⁻¹ and −8 Gt yr⁻¹, respectively.

Table 1 Annual rates of glacier change by region from 2006 to 2016

Fig. 3: Regional estimates of glacier mass change for the period 2006–2016.

a, b, Annual mass-change rates in metres of water equivalent per year (a) and in gigatonnes per year (b) as estimated from spatial interpolation (blue circles), area-weighting (black triangles) and arithmetic averaging (orange squares). The spatial-interpolation approach is the reference, with results provided in Table 1. Error bars correspond to 95% confidence intervals, and consider uncertainties related to the temporal variability of the glaciological sample, the geodetic value, the regional interpolation, the regional glacier area, and a second-order crossed term. We estimate the error related to the regional interpolation from the differences between the three interpolation approaches (see Methods section ‘Uncertainty estimates’).

We calculated the relative annual ice loss (Extended Data Fig. 3) by comparing present mass-change rates (2006–2016) with total estimated ice volumes for each region². Nine out of nineteen regions lost between 0.5% and 3% of their total ice volume per year. The other regions featured smaller loss rates. Under present ice-loss rates, most of today’s glacier volume would thus vanish in the Caucasus, Central Europe, the Low Latitudes, Western Canada and the USA, and New Zealand in the second half of this century. However, the heavily glacierized regions would continue contributing to sea-level rise beyond this century, as glaciers in these regions would persist but continue to lose mass. It is worthwhile noting that a substantial part of the future ice loss is already committed owing to the imbalance of most glaciers with the present climate^19,20, and that numerical models are required to fully assess future glacier changes in view of climate-change scenarios^20,21.

The total error bars related to regional mass changes (Fig. 3b) reflect a composite of different error sources. In most regions, the geodetic error accounts for the largest contribution, followed by the error related to temporal changes assessed from the glaciological sample (Extended Data Fig. 4). The extrapolation to unmeasured glaciers contributes substantially to the overall error only in regions with large differences between interpolation methods. The reasons for these differences are region specific and depend on various factors, such as the observational sample, the glacier size distribution, or a bias towards large tidewater or surge-type glaciers. Uncertainties related to glacier areas and their changes contribute only minimally to the overall error. However, considering area changes is important despite their small contribution to random errors, as a constant glacier area over time would result in a systematic error that increases with the length of the time series and the rate of the area change^22,23.

Our new approach, in combination with major advances in observational evidence, allows for a sound assessment of global glacier mass changes independently of satellite altimetry and gravimetry. This is a basic requirement for the comparison of regional results and the detection of potential biases over the satellite era. By comparison with the Intergovernmental Panel on Climate Change’s Fifth Assessment Report (IPCC AR5)^11,24, the greatest improvement herein is in the geodetic sample: it has been boosted from a few hundred glaciers⁷ to more than 19,000 globally, with an observational coverage exceeding 45% of the glacier area in 11 out of 19 regions (Extended Data Fig. 1). Our approach, combining the temporal variability from the glaciological sample with large-scale observations from the geodetic sample, facilitates the inference of mass changes at annual resolution for all regions, back to the hydrological year 1961/62. This represents a major development compared with IPCC AR5^11,24, which had to focus on the satellite altimetry and gravimetry era (2003–2009) and relied on estimates modelled using climate data or on interpolated values from scarce and mostly uncalibrated observational samples for earlier time periods (see Methods).

Our central estimate for the global rate of glacier mass loss is 47 Gt yr⁻¹ (or 18%) larger than that reported in IPCC AR5 (section 4.3.3.3, table 4.4)^11,24 for the period 2003 to 2009 (Extended Data Fig. 5). A direct comparison of our results is possible for the seven regions (all with less than 15,000 km² of ice cover) with estimates based on glaciological and geodetic samples in IPCC AR5^11,24. In these regions, our mass-change estimates are systematically less negative (Extended Data Fig. 5a). This suggests that our new approach of calibrating regional glaciological mass-change time series with geodetic observations has overcome an earlier reported negative bias in the glaciological sample¹¹. Regions with estimates based on satellite altimetry and gravimetry in IPCC AR5^11,24 featured absolute differences of the same order of magnitude but with varying signs. The more negative global mass changes result mainly from heavily glacierized regions where we estimate larger mass losses (for example, Alaska, peripheral Greenland and Antarctic, the Russian Arctic and Arctic Canada North), and are partly offset by smaller mass-loss estimates for a few other regions with abundant ice cover (for example, Arctic Canada South, Iceland, South Asia West, and Central Asia; Extended Data Fig. 5b) and by the above-mentioned bias in regions with less glacierization. Our error bars are considerably larger than and overlap with those reported in IPCC AR5 (section 4.3.3.3, table 4.4)^11,24. However, a direct comparison is challenging, because the uncertainties in the earlier study¹¹ were based on a combination of regionally different methods and data sources. A detailed comparison will require a regional assessment of glacier changes and related uncertainties, including scaling issues from glacier-wide observations (this study) and results from satellite altimetry (regional averages of repeat-path measurements) and gravimetry (coarse resolution of sensor and hydrological models). However, our error estimates are methodologically consistent and consider all known relevant sources of potential errors. We consider the relative differences of our error bars between the regions to be plausible and their absolute values to be upper bounds.

Improvements in global glacier mass-change assessments are still possible and necessary. First, the observational database needs to be extended in both space and time. We currently see the most urgent need for closing observational gaps being in regions where glaciers dominate runoff during warm/dry seasons, such as in the tropical Andes and in Central Asia⁴, and in regions that dominate the glacier contribution to future sea-level rise, that is, Alaska, Arctic Canada, the Russian Arctic, and peripheral glaciers in Greenland and Antarctica. Second, a systematic assessment of regional area-change rates²⁵ will improve the estimate of corresponding impacts on regional mass changes. Finally, more research is required to better constrain the observational uncertainties at individual glaciers²⁶ and for regional mass-change assessments. Despite these remaining challenges, our assessment of global glacier mass changes provides a new observational baseline for a sound comparison with estimates based on other methods²⁷, as well as for future modelling studies of glacier contributions to regional runoff and global sea-level rise.

Methods

Glaciological and geodetic mass changes

The glaciological method usually provides glacier-wide surface mass balance (B_sfc) over an annual period related to the hydrological year. In line with ref. ¹⁵, we use the unit m w.e. for the specific mass change (1 m w.e. = 1,000 kg m⁻²) and the unit Gt for the mass change (1 Gt = 10¹² kg), with mass balance and mass change as synonymous terms. Results are reported as cumulative values over a period of record or as annual change rates (yr⁻¹). The geodetic balance is the result of surface (sfc), internal (int) and basal (bas) mass changes and—in the case of marine-terminating or lacustrine glaciers—of calving (D) in the unit m w.e.:

$${B}_{{\rm{geod}}}=\Delta M={B}_{{\rm{sfc}}}+{B}_{{\rm{int}}}+{B}_{{\rm{bas}}}+D$$

In practice, the geodetic (specific) mass change is calculated as the volume change, ΔV, over a survey period between t₀ and t₁, from differencing of DEMs, over the glacier area multiplied by a volume-to-mass conversion factor:

$${B}_{{\rm{geod}}}=\frac{\Delta V}{\bar{S}}\times \frac{1}{\left({t}_{1}-{t}_{0}\right)}\times \frac{\bar{\rho }}{{\rho }_{{\rm{water}}}}$$

where $\bar{S}$ is the average glacier area of the two survey times (t₀, t₁) assuming a linear change through time²⁶, and $\bar{\rho }$ is the average density of ΔV with a commonly applied value²⁸ of 850 ± 60 kg m⁻³. The glaciological method is able to satisfactorily capture the temporal variability of the glacier mass change even with only a small observational sample^29,30. However, its cumulative amount over a given time span is sensitive to systematic errors, which accumulate with the number of annual measurements^31,32. The geodetic method provides mass changes covering the entire glacier area and large glacier samples. However, the method requires a density conversion and surveys are typically carried out at multi-annual to decadal intervals only. For both measurements, we use the latest version (wgms-fog-2018-06) of the Fluctuations of Glaciers (FoG) database from the WGMS¹⁶. The glaciological sample was recently updated with observations from latest years, consolidated by adding results from approximately 100 additional glaciers⁷, and the entire mass-balance series was replaced after reanalysis^{33,34,35,36,37}. The geodetic sample was increased recently by the inclusion of large-scale assessments from several mountain regions^{17,38,39,40,41,42}.

For the present study, we complemented the dataset from the WGMS with an additional 70,873 geodetic volume change observations computed for 6,551 glaciers in Africa, Alaska, the Caucasus, Central Asia, Greenland’s periphery, Iceland, New Zealand, the Russian Arctic, Scandinavia and Svalbard (Extended Data Table 1). This was achieved by calculating geodetic mass changes from ASTER DEMs processed using MMASTER⁴³ which were co-registered using off-glacier elevations from the Ice, Cloud, and land Elevation Satellite (ICESat) as common frame following ref. ⁴⁴. Where available, we used ArcticDEM 2-m strips, SPOT5-based DEMs from IPY-SPIRIT or the High Mountain Asia 8-m DEMs⁴⁵ to increase spatial and temporal coverage, after resampling to 30-m resolution to match the resolution of the ASTER DEMs. Pairs of DEMs (for example, ASTER/ASTER or ASTER/ArcticDEM) were automatically chosen on the basis of at least 40% overlap and a time separation of at least eight years. This time separation, together with the selection of DEMs towards the end of the ablation period (Extended Data Table 1), aimed to reduce the effect of seasonal variations in the surface elevation and minimizes differences to glaciological survey dates. On the basis of the selected DEM pairs, glacier elevation changes were computed for various time periods between 2000 and 2018. We used the local hypsometric method⁴⁶ to fill voids in the DEMs. For each glacier outline with an area of at least 0.6 km² from RGI 6.0, we calculated the glacier hypsometry using 100-m elevation bins. For each DEM pair, we calculated the mean elevation difference per elevation bin, and multiplied this by the glacier hypsometry to obtain a volume change. The longest available differences with at least 70% data coverage were then used for each glacier to obtain the geodetic mass change. For the peripheral glaciers in Western Greenland, geodetic mass changes were calculated using the Aero DEM⁴⁷ from 1985 and a prerelease of the 2010–14 TanDEM-X Global DEM. Before differencing, all DEMs were co-registered to each other⁴⁴. We estimated the glacier volume change with the local hypsometric method⁴⁶, using elevations derived from TanDEM-X and the RGI 6.0 outlines. Again, only glaciers with at least 70% data coverage were used. We estimated uncertainties in geodetic mass changes on the basis of off-glacier differences between the two DEMs after co-registration, following the approach of ref. ¹⁷.

Glacier inventory

We derived the global distribution of glaciers from the RGI^1,48, which is a snapshot glacier inventory derived from the Global Land Ice Measurements from Space (GLIMS) database⁴⁹ and a large compilation of national and regional sources compiled by the RGI consortium¹. We used glacier area and its distribution with elevation (that is, glacier hypsometry) for the 215,547 glaciers in RGI 6.0, covering a total area of 705,739 km², mainly for survey years between 2000 and 2010. Improvements with respect to earlier RGI versions as used in IPCC AR5^11,24 (168,331 glaciers, 726,258 km²) include the separation of glacier complexes (for example, ice fields or ice caps) into individual glaciers, replacement of nominal glaciers (that is, size-equivalent circles) by real glacier outlines, assignment of glacier-specific survey dates, and the introduction of glacier-specific hypsometries (Extended Data Fig. 1). The latter come as a list of elevation-band areas (at a resolution of 50 m in height) in the form of integer thousands of the glacier’s total area¹. Note that at present the RGI includes peripheral glaciers surrounding the Greenland Ice Sheet⁵⁰ but not the peripheral glaciers on the Antarctic Peninsula⁵¹ and in the McMurdo Dry Valleys⁵². For future versions of the RGI, the inclusion of these peripheral glaciers in Antarctica should be considered in order to reach global completeness and consistency with the classification of peripheral glaciers in Greenland⁵⁰.

Changes in glacier area

For hydrological and sea-level applications, it is the conventional mass balance that is relevant—that is, the mass change calculated over a constantly changing area and hypsometry of a glacier¹⁵. While the changes in hypsometry are implicitly captured, the changes in glacier area need to be explicitly accounted for by both the glaciological and the geodetic methods²⁶. In contrast with earlier approaches, we considered the impact of changes in glacier area over time on regional mass-change estimates. Therefore, we used a collection of relative area changes from IPCC AR5 (chapter 4, figure 4.10 and table 4.SM.1 of ref. ²⁴), extended with additional literature^53,54,55 to obtain area change rates for all first-order glacier regions.

Glacier volume estimates

Regional estimates for glacier volumes are based on ref. ², updated to the glacier outlines of RGI 6.0.

Spatial regionalization

For regional analysis, it is convenient to group glaciers by proximity. We achieved this by using the latest version of glacier regions as available from the Global Terrestrial Network for Glaciers⁵⁶. These 19 first-order and more than 90 second-order regions derive ultimately from glacier regions proposed by the GLIMS project around the year 2000 and from studies dealing with global glacier distribution^57,58, and are implemented in both the RGI and the FoG databases. For mass-balance studies, the 19 first-order regions seem to be appropriate because of their manageable number and their geographical extent, which is close to the spatial correlation distance of glacier mass-balance variability (that is, several hundred kilometres)^59,60. We further divided these regions for areas that are known to feature large diversities in mass-balance gradients and where sufficient data coverage allowed (Extended Data Table 2).

Extraction of temporal variability from the glaciological sample

In a first step, we subdivided the sample of glaciological series into spatial clusters. We started from the smallest possible units (second-order glacier regions) and then extended them until the number and completeness of the time series was acceptable to ensure a proper variance decomposition based on visual and quantitative criteria, such as a common mass-balance temporal variability (that is, a high correlation between annual mass-balance series) and spatial consistency (that is, a cluster cannot be geographically too wide). The resulting 20 regional clusters correspond to first- and second-order glacier regions or a combination thereof (Extended Data Table 2). For half of these clusters, the available mass-balance series cover the full survey period with only minor data gaps of a few years. For the other half of the clusters, we complemented the glaciological sample with a few long-term series from neighbouring regions that feature a similar mass-balance variability (Extended Data Table 2). For the few clusters without glaciological data before the mid-1970s, we used the mean value of the geodetic sample (that is, neglecting interannual variability) for these years and set the related uncertainty to twice the average value of the first decade with glaciological observations.

In the second step, we extracted the temporal mass-balance variability for each cluster using a variance decomposition model⁶¹, which is a further development of the approach of ref. ³⁰, based on Bayesian techniques^62,63 and applied to a regional sample of glacier-wide mass balances instead of to a series of point measurements. For this model, we defined the specific mass change for a given glacier i and year t as:

$${B}_{{\rm{glac}},i,t}={\alpha }_{0}+{\alpha }_{i}+g\left(t\right)+z\left(t\right)+{\varepsilon }_{i,t}$$

(1)

where α₀ is the cluster’s annual average and α_i is the glacier-specific site deviation of the (specific) mass change from the cluster’s average. The variables g(t) and z(t) are the long-term trend and annual fluctuations, respectively, of the time deviation from the average, and ε_i,t are residuals. The variable g(t) was taken as a smooth nonparametric trend and z(t) as a white-noise term. Their sum is the annual deviation of the glaciological sample from the average α₀, which is further used in the analysis:

$${B}_{{\rm{glac}},{\rm{cluster}}}=g\left(t\right)+z\left(t\right)$$

(2)

Model inference was performed using Bayesian simulation techniques, giving access, for any parameter or combination of parameters, to a point estimate and to a credibility interval quantifying the related uncertainty. This especially applies to cumulated temporal deviations $\mathop{\sum }\limits_{t={t}_{1}}^{{t}_{2}}(g(t)+z(t))$ over any time interval [t₁,t₂], such as the full period from 1961 to 2016 (Extended Data Fig. 6)

Calibration to mass-change values from the geodetic sample

For each cluster (Extended Data Table 2), we calibrated the temporal mass-balance variability as derived from the glaciological sample B_glac,cluster to the values from the geodetic methods²⁶. Owing to the differences in length of the geodetic survey periods, we carried out the calibration individually for all glaciers with available geodetic balances. If more than one geodetic survey was available per glacier, we combined those with the longest survey periods by arithmetic averaging of annual change rates. For each glacier i, we calculated the mean annual deviation $\bar{{\beta }_{t}}$ between the glaciological balance of the cluster B_glac,cluster and the glacier-specific geodetic balance B_geod,i over a common time period of N years between t₀ and t₁:

$$\bar{{\beta }_{t}}=\frac{{B}_{{\rm{geod}},i}-{\sum }_{{t}_{o}}^{{t}_{1}}{B}_{{\rm{glac}},{\rm{cluster}}}}{N}$$

(3)

The annual calibrated specific mass change for every glacier i and year t was then calculated as:

$$\Delta {M}_{{\rm{cal}},i,t}={B}_{{\rm{glac}},{\rm{cluster}},t}+\bar{{\beta }_{t}}$$

(4)

As a result, for each glacier with available geodetic data we obtained a calibrated specific mass-change series that features the temporal variability of the glaciological cluster but is adjusted to the glacier-specific geodetic value (Extended Data Fig. 7).

Regional mass changes and contributions to sea level

To estimate the total mass change, we need to scale the results from the sample with available (geodetic) data to all glaciers of a region (from RGI 6.0). We followed three different approaches to calculate the regional specific mass change ΔM_region (in units of m w.e. yr⁻¹): arithmetic averaging ΔM_region,AVG, area-weighting ΔM_region,AW, and spatial interpolation ΔM_region,INT. For the approaches ΔM_region,AVG and ΔM_region,AW, we assigned the arithmetic and glacier area-weighted average, respectively, of the annual specific mass change of the observational sample to all unobserved glaciers in the region. For our reference approach ΔM_region,INT, we spatially interpolated the individual specific mass-changes to all glacier locations in the region using an inverse distance weighting function. For all approaches, we calculated the regional mass change ΔM_region (in units of Gt yr⁻¹) as the product of the specific mass change multiplied by the regional glacier area from RGI 6.0, applying the relative area change rates of the corresponding region. Global mass changes, ΔM_global, were calculated as the sum of all regional mass changes. For conversion to sea-level equivalent, we assumed a total area of the ocean of 362.5 × 10⁶ km² (ref. ⁶⁴).

Uncertainty estimates

The random error of the regional mass change, σ_regional, is composed of the errors related to: first, the temporal changes in the regional glaciological sample σ_glac; second, the geodetic values of the individual glaciers σ_geod; third, the extrapolation from the observational to the full sample σ_{extrapolation}; fourth, the glacierized area σ_area of the region; and fifth, a second-order crossed term related to the calculation of the regional mass change (as the product of specific mass change multiplied by the glacierized area):

$${\sigma }_{{\rm{regional}}}=\sqrt{{\sigma }_{{\rm{glac}}}^{2}+{\sigma }_{{\rm{geod}}}^{2}+{\sigma }_{{\rm{extrapolation}}}^{2}+{\sigma }_{{\rm{area}}}^{2}+{\sigma }_{{\rm{crossed}}}^{2}}$$

(5)

The variable σ_glac can be rigorously estimated from the variance decomposition (Extended Data Fig. 6). However, we used a less computationally intensive approach to estimate it for any subperiod (pentad, decade) from the full study period. Specifically, the annual standard deviations of the temporal deviation (g(t) + z(t)) as obtained from the variance decomposition model were summed up according to the law of random error propagation. Hence, the standard deviation of any subperiod was evaluated as if annual deviations would be independent. The variable σ_glac implicitly accounts for errors related to differences in the glaciological survey period, because the sample contains results from various time systems (for example, fixed-date, floating-date and stratigraphic)¹⁵.

The variable σ_geod is the uncertainty from the geodetic method. We calculated the annual values as rates—that is, dividing the reported (multiyear) uncertainties by the number of years between the two surveys. It includes the observation uncertainty σ_{geod.observation} as reported with the geodetic results. In addition, we considered the uncertainty introduced by calibrating annual mass-balance variability with geodetic values, σ_calibration (Equation (4)), which was inferred for each glacier individually on the basis of randomly superimposing σ_glac and extracting the standard deviation of average balances over the reference period. The uncertainties related to density conversion factor σ_density were set to ± 60 kg m⁻³ according to ref. ²⁸. The overall geodetic uncertainty was calculated from these terms, assuming them to be uncorrelated, and was divided by the square root of the number of independent items n of information in the sample:

$${\sigma }_{{\rm{geod}}}=\sqrt{\frac{{\sigma }_{{\rm{geod}}.{\rm{observation}}}^{2}+{\sigma }_{{\rm{calibration}}}^{2}+{\sigma }_{{\rm{density}}}^{2}}{n}}$$

In the ideal case, n would be equal to the number of geodetic series in the regional sample. For spaceborne surveys, however, the geodetic uncertainty is usually derived from the stable terrain in between a group of glaciers. We thus assumed geodetic uncertainties uncorrelated for samples larger than 50 glaciers, and estimated n by dividing the regional geodetic sample size by 50. Note that, for the geodetic sample, we do not explicitly formulate uncertainties related to differences in the survey date. For individual glaciers, a corresponding rigorous estimate would be possible using seasonal mass-balance information, meteorological data, and numerical modelling^7,26,33. These studies show that the corresponding uncertainties can be relevant for individual years but tend towards zero for longer periods of records and larger samples.

To estimate σ_{extrapolation}, we used the regional mass change from spatial interpolation (ΔM_region,INT) as a best guess and calculated the extrapolation uncertainty as 1.96 standard deviations of the results from the three approaches (ΔM_region,INT, ΔM_region,AVG and ΔM_region,AW). As for σ_glac, we evaluated σ_{extrapolation} over any subperiod by the square root of the number of survey years, assuming that annual values are uncorrelated.

For the regional glacier area, we assumed a general uncertainty of ± 5% for the total area derived from RGI 6.0, given earlier single-glacier and basin-scale uncertainty estimates⁴⁸ and in line with the latest GCOS product requirements (table 25 of ref. ⁶⁵; terrestrial essential climate variable (ECV) product requirements). This uncertainty was combined with an error related to the regional area changes σ_area.change, which was estimated as 1.96 standard deviations of the different approaches used to calculate regional change rates. For a given region, the first approach, used as reference, weights multiple published change rates by the total ice cover of the corresponding glacier samples. The second approach weights multiple results by the length of the survey periods. The uncertainties related to the total area and to area changes were assumed to be uncorrelated and, hence, cumulated according to the law of random error propagation.

Over multiyear periods, in contrast with σ_glac, σ_{extrapolation} and σ_crossed, the errors related to the geodetic values and glacier areas (σ_geod and σ_area) cumulate linearly. Consequently, the individual terms need to be cumulated separately, followed by a combination of the multiyear terms according to the law of random error propagation (see equation (5)). For global sums, the overall error was calculated by cumulating the regional errors according to the law of random error propagation.