We use time series of GMSL anomalies derived from satellite altimetry observations. For the period January 1993–December 2015, the GMSL record is version 2.0 of the ESA (European Space Agency) Sea Level CCI project (https://climate.esa.int/en/projects/sea-level/, last access: 13 January 2022). The CCI sea-level record combines data from the TOPEX/Poseidon, Jason-1 and Jason-2, GFO (Geosat Follow-On), ERS-1 and ERS-2 (European Remote Sensing), Envisat, CryoSat-2, and SARAL (Satellite with ARgos and ALtiKa) ALtiKa (Ka-band altimeter) missions and is based on a new processing system (Ablain et al., 2015, 2017a; Quartly et al., 2017; Legeais et al., 2018). It is available as a global gridded 0.25∘ × 0.25∘ dataset over the 82∘ N–82∘ S latitude range. It has been validated using different approaches including a comparison with tide gauge records as well as ocean reanalyses and climate model outputs. While our study focusses on utilising CCI products, the CCI sea-level product did not cover the year 2016. We therefore extended the GMSL record with the Copernicus Marine Environment Monitoring Service dataset (CMEMS, https://marine.copernicus.eu/, last access: 13 January 2022) from January 2016 to December 2016.

The TOPEX-A instrumental drift due to ageing of the TOPEX-A altimeter placed in the TOPEX/Poseidon mission from January 1993 to early 1999 was corrected for in the GMSL time series following the approach of Ablain et al. (2017b). It was derived by comparing TOPEX-A sea-level data with tide gauge data. The TOPEX-A drift value based on this approach amounts to -1.0 ± 1.0 mm yr-1 over January 1993 to July 1995 and to 3.0 ± 1.0 mm yr-1 over August 1995 to February 1999 (see also WCRP Global Sea Level Budget Group, 2018).

For the SLBC_cci project, the gridded sea-level anomalies were averaged over the 65∘ N–65∘ S latitude range. The GMSL time series was corrected for GIA, applying a value of -0.3 mm yr-1 (Peltier, 2004). Annual and semi-annual signals were removed through a least-squares fit of 12- and 6-month-period sinusoids.

Figure 1a shows the record of GMSL anomalies. The well-known, sustained GMSL rise has a linear trend of 3.05 ± 0.24 mm yr-1 over P1. An overall increase of the rate of sea-level rise over the 24 years is visible (cf. Nerem et al., 2018). The overall GMSL rise is superimposed by interannual variations like the temporary GMSL drop between 2010 and 2011 by about 6 mm (cf. Boening et al., 2012) with a subsequent return to the rising path.

Over recent years, several articles (Ablain et al., 2015, 2017b; Dieng et al., 2017; Quartly et al., 2017; Legeais et al., 2018) have discussed sources of errors in GMSL trend estimation. Ablain et al. (2019) extended these previous studies by considering new altimeter missions (Jason-2 and Jason-3) and recent findings on altimetry error estimates. We use the uncertainty assessment by Ablain et al. (2019), which can be summarised as follows.

Three major types of errors are considered in the GMSL uncertainty: (a) biases between successive altimetry missions characterised by bias uncertainties at any given time; (b) drifts in GMSL due to onboard instrumental drifts or long-term drifts such as the error in the GIA correction, orbit, etc. characterised by a linear-trend uncertainty; and (c) other measurement errors such as geophysical correction errors (wet tropospheric, orbit, etc.) which exhibit temporal correlations and are characterised by their standard deviation. These error sources are assumed to be independent of each other.

For each error source, the variance–covariance matrix over all months is calculated from a large number of random trials (> 1000) of simulated errors with a standard normal distribution. The total error variance–covariance matrix is the sum of the individual variance–covariance matrices of each error source. The GMSL uncertainties per epoch are estimated from the square root of the diagonal terms of the total matrix. The covariances are rigorously propagated to assess the uncertainties of multi-year linear trends. In the present study we use standard uncertainties, while Ablain et al. (2019) quote 1.65σ uncertainties to characterise the 90 % confidence margins. Ablain et al. (2019) refer to GMSL anomalies with respect to the mean over a 1993–2017 reference period, while our study uses the 2006–2015 reference period. We neglect the effect of this difference on the uncertainties.

Figure 1b shows the GMSL anomaly uncertainties per epoch. They are larger during the TOPEX/Poseidon period (3 to 6 mm) than during the Jason period (close to 2.5 mm). This is mainly due to uncertainties of the TOPEX-A drift correction. Long-term drift errors common to all missions also increase the uncertainties towards the interval boundaries.

Since the Argo-based steric product developed within SLBC_cci (see Sect. 3.2.2 below) does not cover the full P1 period, for P1 we resort to the ensemble mean steric product by Dieng et al. (2017), updated to include the year 2016. It comprises the following three datasets for the period 1993–2004: the updated versions of Ishii and Kimoto (2009), the NOAA dataset (Levitus et al., 2012), and the EN4 (version 4 of the Met Office Hadley Centre “EN” series) dataset (Good et al., 2013). Over recent years, these datasets have integrated Argo data from IPRC (International Pacific Research Center, http://apdrc.soest.hawaii.edu/projects/Argo/data/gridded/On_standard_levels/, last access: 13 January 2022), JAMSTEC (Japan Agency for Marine-Earth Science and Technology, http://www.jamstec.go.jp/ARGO/argo_web/argo/, last access: 13 January 2022), and Scripps (Scripps Institution of Oceanography (http://sio-argo.ucsd.edu/RG_Climatology.html, last access: 13 January 2022). Annual and semi-annual signals were removed. The uncertainty was characterised from the spread between the ensemble members and, where available, from uncertainties given for the individual ensemble members.

Figure 2 shows the ensemble mean steric time series. It exhibits an overall rise, modulated by interannual fluctuations which are within uncertainties prior to 2005 but exceed assessed uncertainties later, e.g. in 2010/2011 (due to the smaller size of the latter).

Within SLBC_cci, the calculation scheme for the steric-sea-level change based on Argo data was updated from that described by von Schuckmann and Le Traon (2011). Formal propagation of uncertainty was included following JCGM (2008, their Eq. 13) in which an overall uncertainty estimate is obtained by propagating and combining the evaluations of uncertainty associated with each source.

The steric-thickness anomaly for a layer l of water with density ρl is hl′=hl-hl,c, where hl,c is the steric thickness of a layer with climatological temperature and salinity and hl=1ρl-1ρ0ρ0Δzl0 is the “steric thickness” of the layer relative to a layer of reference density ρ0 and reference height Δz0. The thickness hl′ can therefore be written in terms of layer density ρl and climatological density for the layer ρl,c as 11hl′=1ρl-1ρl,cρ0Δzl0. The monthly mean steric-thickness anomaly for layer l is found as the optimum combination of the steric-thickness anomaly calculations from all the valid profiles in the grid cell for the month. Let the individual anomaly calculations be collected in a vector xl. The optimum estimate is then given by the following collection of equations: 12hl′=wlTxl,13wl=1iTSxl-1iSxl-1i, where i is a column vector of ones, wl is the vector of weights appropriate to a minimum error variance average, and Sxl is the error covariance matrix of the steric-thickness anomaly estimates.

The error covariance matrix Sxl is needed for the optimal calculation of the monthly average in Eq. (12), as well as for the evaluation of uncertainty discussed below. To estimate this matrix, we need to be clear about what “error” means here: it is the difference between the steric-thickness anomaly for the layer from a single profile (Argo or climatological) and the (unknown) true cell-month mean. This difference therefore has two components: the measurement error in the profile, characterised by an error covariance Sxm, and a representativeness error arising from variability within the cell month Sxr. The measurement error covariance is the smaller term and was modelled to be independent between profiles within the cell (neglecting the fact that on occasion a single Argo float will contribute more than one profile within a given cell in a month). The representativeness error covariance was modelled assuming that this error has an exponential correlation form with a length scale of 2.5∘ and timescale of 10 d.

It is relatively common to have layers with no observations, sometimes in the upper ocean and often at depth. Conditional climatological profiles were used as an additional “observation” to fill in information for missing-data layers. The climatology of profiles was conditioned by the observed SST from Merchant et al. (2019), which essentially has negligible sampling uncertainty at monthly cell-average scales. The SST information constrains the upper-ocean profile to a degree determined by the vertical correlation of variability, which is variable in time and place according to the mixed-layer depth. The uncertainty in the conditional climatological profile is the variability. Examples of an unconditional and conditional climatological profile are shown in Fig. 3. For this particular month (August), year (2003), and location (30.5∘ N, 9.5∘ E), the SST is about 2 ∘C below the climatological value. The conditioning is strong for the upper ∼ 50 m of the ocean, and within this modest depth range the conditioned profile is realistic given the SST (approximately isothermal over a mixed layer). The uncertainty is reduced at the surface, where the cell-month SST is well known from the satellite data. Below about 150 m, the effect of conditioning decays towards zero (conditioned and unconditioned profiles converge).

Including SST information slightly reduces uncertainty and affects steric height in the mixed layer often enough to influence the global mean. Over the period 2005–2018 the trend in steric height is larger using SST-conditioned climatological profiles than when using a static climatology as the prior. The use of static climatology to fill gaps in Argo profiles has been shown to cause systematic underestimates of trends in the literature (e.g. Ishii and Kimoto, 2009). Inclusion of SST conditioning to the climatology mitigates the over-stabilising effect.

The steric-sea-level height anomaly (SSLHA) was calculated for every month from January 2002 to December 2017 in a global grid of 5∘ × 5∘ resolution. For a given cell month, the SSLHA is the sum of the layer-by-layer estimates of steric-thickness anomaly, i.e. h′=∑lhl′. By concatenating vectors wlT and xl for all the layers into a vector of weights w and a vector of thicknesses x, we can write the following equivalent equation: 14h′=wTx. This form makes it more clear how to estimate the uncertainty in the SSLHA, which is 15uh′=wTSxw0.5. To evaluate the uncertainty we need to formulate Sx. The diagonal blocks corresponding to each layer in Sx are the matrices Sxl that have already been calculated on a layer-by-layer basis. Assumptions about the error correlations between layers are then required in order to complete the off-block-diagonal elements of Sx. Conservative assumptions were made:

Measurement errors are perfectly correlated vertically in a given profile; this is equivalent to saying that the sensor calibration bias dominates all other sources of measurement uncertainty in each profile.

Figure 2a shows the ΔSLSteric time series from the SLBC_cci product. While from 2005 onwards, the trends of the SLBC_cci product and the ensemble mean product (cf. Table A1) as well as their interannual behaviour are similar, the SLBC_cci product shows little change prior to 2005. This difference and its reflection by the uncertainty characterisation are discussed further below.

The uncertainties of the available cell-month mean SSLHA estimates were propagated to ΔSLSteric. In any given month, there are missing SSLHA cells, through lack of sufficient Argo profiles. Using ΔSLSteric estimated from the available SSLHA cells as an estimate for the global steric-sea-level anomaly introduces a global representativity uncertainty. Moreover, the global sampling errors are correlated month to month because the sampling distribution evolves over the course of several years towards near-global representation. To evaluate the global representativity uncertainty and serial correlation, the sampling pattern of sparse years was imposed on near-complete fields: the standard deviation of the difference in global mean with the two sampling patterns is a measure of uncertainty. The correlation between the sample-driven difference in consecutive months was found to be 0.85. The time series of the global representativity uncertainty, the uncertainty propagated from the gridded SSLHA uncertainty (which has no serial correlation), and this correlation coefficient combine to define a full error covariance estimate to be obtained for the ΔSLSteric time series.

Figure 2b shows the two components of uncertainty (global representativity and propagated SSLHA uncertainty) together with the total uncertainty. The global representativity uncertainty dominates prior to 2005 and is very large in 2002 and 2003. This reflects how sparse and unrepresentative the sampling by the Argo network was at that early stage.

Given the large representativity uncertainty prior to 2005, the absence of an increase in the SLBC_cci steric record during that time is thus understood to arise from global sampling error and is consistent with the global sampling uncertainty. The SLBC_cci steric time series and the ensemble mean steric time series are consistent given the evaluated uncertainties throughout the record. In addition, the evaluated uncertainties for the two time series with their different ways of uncertainty assessment are remarkably similar for the period of the established Argo network starting in 2005, giving confidence in the validity of two very distinct approaches to uncertainty characterisation.

The use of a formal-uncertainty framework allows separation of distinct uncertainty issues, namely, our ability to parameterise and estimate the various uncertainty terms, our ability to estimate the error covariance, and the model for propagation of error at each successive step.

Two aspects of the uncertainty model are recognised to be potentially optimistic: the modelling of measurement errors as independent between profiles rather than platforms and the use of only 10 years for assessing interannual variability. Two assumptions are potentially conservative: measurement errors in salinity and temperature were combined in their worst-case combination, and representativity errors in profiles are assumed to be fully correlated vertically, whereas in reality they are likely to decorrelate over large vertical separations.

A significant output of the uncertainty modelling of the steric component is the error covariance matrix for the time series. It enables proper quantification of the change during the time series. We employed the time-variable uncertainties to determine the linear trend in a weighted regression according to Eq. (10). Without weighting, the global sampling error prior to around 2005, noted above, would bias any fitted trend result. Use of the error covariance matrix enables proper quantification of the uncertainty in the trend calculation by propagating the error covariance matrix through the trend function. Without this, the serial correlation in the global sampling error would be neglected, and the calculated trend uncertainty would be an underestimate.

For the deep ocean below 2000 m depth, the steric contribution was assessed as a linear trend of 0.1 ± 0.1 mm yr-1 following the estimate by Purkey and Johnson (2010) corroborated by Desbruyères et al. (2016). Note that this estimate is based on sparse in situ sampling. Corresponding evolutions of the ocean observing system are under way (Roemmich et al., 2019). This deep-ocean contribution is included in the ensemble mean steric product described in Sect. 3.2.2. This deep-ocean component is added to the Argo-based SLBC_cci steric product described in Sect. 3.2.1 (which is for depths < 2000 m) in order to address the full-ocean steric contribution.

Time series of ocean-mass change (OMC), in terms of anomalies with respect to the 2006–2015 reference period, were generated from monthly gravity field solutions of the GRACE mission (Tapley et al., 2019). Similar to previous analyses (Johnson and Chambers, 2013; Uebbing et al., 2019) we used spherical harmonic (SH) GRACE solutions in order to have full control over the methodology and uncertainty assessment. Greater detail is provided by Horwath et al. (2019).

The following GRACE monthly gravity field solutions series were considered:

ITSG-Grace2018 (Kvas et al., 2019; Mayer-Gürr et al., 2018) from Institut für Geodäsie, Technische Universität Graz, Austria, with maximum SH degree 60 (data source: http://ftp.tugraz.at/outgoing/ITSG/GRACE/ITSG-Grace2018/monthly/monthly_n60, last access: 13 January 2022)

Gravity field changes were converted to equivalent water height (EWH) surface mass changes according to Wahr et al. (1998). The total mass anomaly over an area like the global ocean was derived by spatial integration of the EWH changes. We used the unfiltered GRACE solutions in order to avoid damping effects from filtering. A 300 km wide buffer zone along the ocean margins was excluded from the spatial integration. Around islands, the buffer was applied if their surface area exceeds a threshold, which was set to 20 000 km2 in general and 2000 km2 for near-polar latitudes beyond 50∘ N or 50∘ S. The integral was subsequently scaled by the ratio between the total area of the ocean domain and the buffered integration area. This scaling is based on the assumption that the mean EWH change in the buffer equals the mean EWH change in the buffered ocean area. Effects of violations to this assumption are included in the uncertainty assessment (see further below).

Modelled short-term atmospheric and oceanic mass variations are accounted for within the gravity field estimation procedure (Flechtner et al., 2014; Dobslaw et al., 2013) and are not included in the monthly solutions. To retain the full mass variation effect, the monthly averages of the modelled atmospheric and oceanic dealiasing fields were added back to the monthly solutions by using the so-called GAD products (Flechtner et al., 2014). We subsequently removed the spatial mean of atmospheric surface pressure over the full-ocean domain. Our investigations confirmed findings by Uebbing et al. (2019) on the methodological sensitivity of this procedure. If the GAD averages were calculated over only the buffered area, OMC trends would be about 0.3 mm yr-1 higher than for our preferred approach.

GIA implies redistributions of solid-Earth masses and (to a small extent) of ocean masses. We corrected the gravity field effect of GIA-related mass redistributions by using three different GIA modelling results: the model by A et al. (2013), based on ICE-5Gv2 glaciation history from Peltier (2004); the model ICE-6G_D (VM5A) by Peltier et al. (2015, 2018); and the mean solution by Caron et al. (2018). The correction was applied on the level of the SH representation. Our preferred GIA correction is the one by Caron et al. (2018). It is based on the ICE-6G deglaciation history (Peltier et al., 2015), while the model by A et al. (2013) is based on its predecessor model, ICE-5G. Furthermore, while the models by A et al. (2013) and Peltier et al. (2015) are single GIA models, the solution by Caron et al. (2018) arises as a weighted mean from a large ensemble of models, where the glaciation history and the solid-Earth rheology have been varied and tested against independent geodetic data to provide probabilistic information. Table 1 demonstrates the sensitivity of GRACE OMC solutions to the GIA correction.

In order to include the degree-one components of global mass redistribution (not determined by GRACE), we implemented the approach by Sun et al. (2016), which combines the GRACE solutions for degree n ≥ 2 with assumptions on the ocean-mass redistribution. The results depend on the input GRACE solution series and, more importantly, on the adopted GIA model. While GRACE Technical Note TN13 provides degree-one time based on the Sun et al. (2016) method, their input is fixed to the CSR, GFZ, or JPL solutions and the ICE-6G_D GIA model. By our own implementation we generate degree-one series that are consistent with our choice of GRACE solution series (such as ITSG-Grace2018) and GIA models (such as by Caron et al., 2018). We also replaced GRACE-based C20 (the Earth flattening component of SH degree 2 and order 2) components with results from satellite laser ranging (SLR) (Loomis et al., 2019a, https://podaac-tools.jpl.nasa.gov/drive/files/allData/grace/docs/TN-14_C30_C20_GSFC_SLR.txt, last access: 13 January 2022).

Figure 4a shows our preferred time series of the mass contribution to sea level (see Fig. 10 for a time series where the seasonal signal is subtracted). The overall trend at 2.62 ± 0.26 mm yr-1 over P2 is superimposed by a seasonal signal with 10.3 mm amplitude of annual sinusoid and by interannual variations like a drop by about 6 mm sea-level equivalent between 2010 and 2011.

Overall, integrated OMC time series were generated from four series of SH GRACE solutions, using three GIA corrections (and the option of no GIA correction), for the global ocean and the ocean domain between 65∘ N and 65∘ S. For comparison, we also considered OMC time series from two external sources: global OMC time series from CSR, GFZ, and JPL SH solutions by Johnson and Chambers (2013), updated by Don​​​​​​​ Chambers on 6 November 2017 and made available by the author on 26 January 2018, and Goddard Space Flight Center (GSFC) mascon solutions RL06v01 (release 6, version 1) (Loomis et al., 2019b), dedicated for ocean-mass research (data source: https://earth.gsfc.nasa.gov/geo/data/grace-mascons, last access: 13 January 2022). Time series of total OMC from GSFC mascons were derived by the weighted integral over all oceanic points, using the ocean–land point-set mask contained in the solutions (but excluding the Caspian Sea). Integrated OMC was then divided by the total area over the corresponding oceanic mascons.

The following sources of uncertainty are relevant (cf. Groh and Horwath, 2021):

GRACE errors: errors in the GRACE observations as well as in the modelling assumptions applied during GRACE processing propagate into the GRACE products.

The noise was assessed from the GRACE OMC time series themselves as detailed by Groh et al. (2019a). The detrended and deseasonalised time series were high-pass-filtered in the temporal domain. The variance of the filtered time series was assumed to be dominated by noise. This variance was scaled by a factor that accounts for the dampening of uncorrelated noise variance imposed by the high-pass-filtering process. The assessed noise component comprises uncorrelated errors from all uncertainty sources except for GIA, which is considered purely linear in time.

The systematic errors of the linear trends are assumed to originate from errors in degree-one components, C20, the GIA correction, and leakage. The related uncertainties were assessed for each source individually and summed in quadrature.

Trend uncertainties associated with GIA, degree-one, and C20 were assessed individually based on the spread of a small ensemble of different options to incorporate these effects. The ensemble standard deviation was taken as the associated standard uncertainty. The GIA uncertainty assessment used the ensemble of the three GIA correction options mentioned above and in Table 1. For the degree-one uncertainty assessment we made a choice of 10 different series in the attempt to represent a balanced sample of different methods and input datasets. This choice includes three series from our implementation of the method by Sun et al. (2016) using the ITSG-Grace2018 GRACE solutions and the GIA model by A et al. (2013), Peltier et al. (2018), and Caron et al. (2018), respectively; the three TN13 series using CSR, GFZ, and JPL GRACE solutions, respectively; the series originally provided by Sun et al. (2016); the series derived from SLR by Cheng et al. (2013); the series from the global combination approach by Rietbroek et al. (2016); and the series derived earlier for CSR RL05 solutions according to the method by Swenson et al. (2008). Likewise, for the C20 uncertainty assessment we made a choice of the following seven series: five series based on SLR by, respectively, Loomis et al. (2019a), Cheng and Ries (2017), Bloßfeld et al. (2015), König et al. (2019), and Cheng et al. (2013); one series from a combined analysis of SLR and GRACE by Bruinsma et al. (2010); and one series based on the GRACE model combination by Sun et al. (2016).

To estimate the uncertainty that arises from leakage, in conjunction with buffering and rescaling, we performed a simulation study based on synthetic mass change data from the ESA Earth System Model (ESM; Dobslaw et al., 2015). The ESM data were processed according to the settings of the SLBC_cci OMC analysis, and the results (simulated observations) were compared with the OMC that arises from the full-resolution ESM data (simulated truth). In order to derive statistics for multi-year trends, we calculated linear trends of the simulated observations and of the simulated truth and of their misfit for every interval of a length between 9 and 12 years contained in the ESA ESM period. The weighted RMS (root mean square) of misfits over all intervals was taken as the estimate of the leakage error uncertainty.

Results of the uncertainty assessment for the ITSG-Grace2018-based OMC solutions are summarised in Table 2. Figure 4b shows the time-dependent uncertainties associated with the ocean-mass contribution time series. They reflect their construction by Eq. (16), where away from 2011.0, the uncertainty of the linear trend contributes an increasing share.

The GRACE-based OMC products described and used here adopt recent standards concerning the degree-one and C20 series used as well as the inclusion of ICE-6G_D (instead of ICE-6G_C) in our comparison of GIA corrections. The products are an update to previous estimates by Horwath et al. (2019). Our estimated global OMC trends (for identical periods) are larger than the previous estimates. The difference (updated trend minus previous trend) is mainly due to the updated degree-one treatment and depends on the GIA model involved in the degree-one estimation. Incidentally, this difference is largest for our preferred choice of the GIA model by Caron et al. (2018), where the difference amounts to 0.43 mm yr-1. This difference is outside the assessed 1σ uncertainty but inside the 1.65σ range that would correspond to a 90 % confidence range. The sensitivity of GRACE-based OMC trends observed here and in previous studies (Blazquez et al., 2018; Uebbing et al., 2019; Dobslaw et al., 2020) corroborates the uncertainty on the order of a few tenths of a millimetre per year that remains associated with GRACE-based OMC trends.

The glacier mass change estimate was derived by updating the global glacier model (GGM) of Marzeion et al. (2012). Annually reported direct mass balance observations (using the glaciological method) are available for only a few hundred of the roughly 215 000 existing glaciers (Zemp et al., 2019). Global-scale geodetic, altimetric, and gravimetric observations are limited to the most recent decades (e.g. Bamber et al., 2018). Estimates of geodetic glacier mass balance back into the 1960s are available only at regional scale and are more disperse (e.g. Maurer et al., 2019; Zhou et al., 2018). The overall objective of the model approach is to use observations of glacier mass change for calibration and validation of the glacier model, which then translates information about atmospheric conditions into glacier mass change, taking into account various feedbacks between glacier mass balance and glacier geometry. This enables a reconstruction of glacier change that is complete in time and space and that has higher temporal resolution than the observations (here, we use monthly output). In our analysis, we included all glaciers outside of Greenland and Antarctica and separately reconstructed the glacier change for Greenland peripheral glaciers.

As initial conditions, we used glacier outlines obtained from the Randolph Glacier Inventory (RGI) version 6.0 (updated from Pfeffer et al., 2014). The timestamp of these outlines differs between glaciers but typically is around the year 2000. To obtain results before this time, the model uses an iterative process to find that glacier geometry in the year of initialisation (e.g. 1901) that results in the observed glacier geometry in the year of the outline's timestamp (e.g. 2000) after the model was run forward.

The model relies on monthly temperature and precipitation anomalies to calculate the specific mass balance of each glacier. It uses the gridded climatology of New et al. (2002) as a baseline. Here, we used seven different sources of atmospheric conditions (as well as their mean) as boundary conditions (Harris et al., 2014; Saha et al., 2010; Compo et al., 2011; Dee et al., 2011; Kobayashi et al., 2015; Poli et al., 2016; Gelaro et al., 2017). Temperature is used to estimate the ablation of glaciers following a temperature-index melt model and to estimate the solid fraction of total precipitation, which is used to estimate accumulation. Glacier area and length change are estimated following mass change based on volume–area–time scaling, allowing for a delayed response of glacier geometry to glacier mass change. A detailed description of the model is provided by Marzeion et al. (2012).

There are four global model parameters that need to be optimised: (i) the air temperature above which melt of the ice surface is assumed to occur; (ii) the temperature threshold below which precipitation is assumed to be solid; (iii) a vertical precipitation gradient used to capture local precipitation patterns not resolved in the forcing datasets; and (iv) a precipitation multiplication factor to account for effects from (among other processes) wind-blown snow and avalanching, which are not resolved in the forcing dataset. For each of the eight forcing datasets cited above, we performed a multi-objective optimisation for these four parameters, using a leave-one-glacier-out cross validation to measure the model's performance on glaciers for which no mass balance observations exist. We used annual in situ observations from about 300 glaciers, covering a total of almost 6000 mass balance years (WGMS, 2018). In the optimisation, the temporal correlation of observed and modelled mass balances is maximised; the temporal variance of modelled mass balances is brought close to that of observed mass balances (aiming for a realistic sensitivity of the model to climate variability and change); and the model bias is minimised (to avoid an artificial trend in modelled glacier mass). Using the mean of the seven atmospheric datasets described above results in the overall best model performance. Compared to the results in Marzeion et al. (2012), the correlation of annual glacier mass change was increased from 0.60 to 0.64; the bias was changed from 5 to -4 kg m-2 (both statistically indistinguishable from zero); and the ratio of the temporal variance of modelled and observed mass balances was improved from 0.83 to 1.00.

The model output for each glacier is aggregated on a regular 0.5∘ × 0.5∘ grid, where the mass change of each glacier is assigned to the grid cell that contains the glacier's centre point, even if the glacier might cover several grid cells (the GGM does not calculate the spatial distribution of mass changes of a glacier so that a more accurate spatial assignment to the grid is not possible). Regional or global values of glacier mass change were obtained by summing over the region of interest.

Figure 5a shows the global glacier contribution to GMSL anomalies (see Figs. 10 and 12 for time series after subtraction of the seasonal signal). The glacier contribution has a linear trend of 0.64 ± 0.03 mm yr-1 over P1, where the positive rate increases from the first half to the second half of the period. Interannual variations are less pronounced than for other budget elements.

The root mean square error obtained during the cross validation was propagated through the model. Since the evaluation of the model results does not indicate any temporal or spatial correlation of the model errors, the uncertainty of temporal and spatial mass change aggregations was calculated assuming independence of the model errors, i.e. by taking the root of the summed squares of each glacier's (and year's) uncertainty.

Uncertainties of mass anomalies with respect to the mean over the 2006–2015 interval were approximated by uncertainties of anomalies with respect to the centre of the interval, 2011.0. Uncertainties of yearly mass change rates were aggregated (as the root sum square) forward or backward from 2011.0 to the specific epochs. Figure 5b shows the uncertainties per epoch, reflecting this aggregation from 2011.0. Uncertainties of multi-year linear trends were calculated as follows: the uncertainties of yearly rates of mass change were aggregated in time over the interval of interest, leading to an uncertainty of cumulated mass change over the interval of interest, which was subsequently divided by the length of the interval. That is, the trend uncertainty was calculated as the root sum square of yearly rate uncertainties, divided by the interval length.

The GRACE-based product developed at DTU Space (Technical University of Denmark) within the Greenland Ice Sheet CCI project is used to provide mass change estimates for the GrIS from GRACE monthly gravity field solutions. The quasi-monthly GRACE-based mass anomaly estimates (grids and basin time series) are available at https://climate.esa.int/en/projects/ice-sheets-greenland/ (last access: 13 January 2022). Comprehensive descriptions and references are given by Barletta et al. (2013), Horwath et al. (2019), and Mottram et al. (2019).

An inversion technique was used to obtain monthly mass anomalies for all of Greenland from each of the available GRACE monthly solutions, with the approach described in Barletta et al. (2013). An icosahedral grid of point masses, each representing an area of ∼ 20 km radius, was inverted in order to fit the gravity observations at the satellite altitude. The limited ∼ 300 km resolution of the GRACE monthly solution requires inversion for ice-mass changes over the whole GrIS including peripheral glaciers, whose contribution cannot be isolated independently. For this work the CSR RL06 monthly solutions were used, with a maximum degree and order of 96, and prior to the inversion the prescribed C20 and degree-one corrections were applied, together with an anisotropic filtering (DDK3 from Kusche et al., 2009). Mass changes of glaciers outside Greenland were co-estimated to minimise their leakage into the Greenland ice-mass change estimates.

Our inversion did not include a GIA correction. We separately calculated the effect of GIA on our inversion. Based on the Caron et al. (2018) GIA solution (chosen for consistency with the OMC estimate; cf. Sect. 3.3) we obtained a GIA effect of 7.5 Gt yr-1. This linear trend was subtracted from the time series.

GRACE-based products are provided with error estimates based on the approach developed by Barletta et al. (2013). The uncertainties were propagated from the errors in GRACE monthly solutions, leakage errors due to GRACE-limited spatial resolutions, and errors in the models used to account for degree-one contributions and the GIA correction. In detail, the uncertainty related to GRACE solutions was obtained in a Monte Carlo-like approach, with 200 simulations for Stokes coefficients selected from a zero-mean normal distribution and the standard deviation from the GRACE CSR RL06 Level 2 solution.

The surface elevation changes estimates are based on satellite radar altimeter observations for the period 1992–2017 and include data from the missions ERS-1, ERS-2, Envisat, and CryoSat-2. The temporal evolution of surface elevation is estimated by a combination of crossover, repeat-track, and least-squares methods covering the entire GrIS and for the entire time span covered by the different missions, on a 5 km common uniform grid (Sørensen et al., 2018). The different characteristics of the missions (the conventional radar altimetry of the ERS-1, ERS-2, and Envisat missions and the novel interferometric SAR – synthetic-aperture radar – altimetry of CryoSat-2) and the different orbital characteristics call for special care in the combinations of the different datasets and in the determination of the uncertainties.

Before an ice-sheet-wide estimate of volume change can be converted into ice sheet mass balance, contributions which are not related to ice-mass change must be corrected for. These contributions include factors such as changes in firn compaction rates, GIA, and elastic uplift. Such a correction method was first applied for satellite ICESat (Ice, Cloud and land Elevation Satellite) lidar observation (Sørensen et al., 2011). As Ku-band radar altimetry is subject to weather-induced changes in subsurface penetration depth of snow-covered areas (Nilsson et al., 2015), we here chose to apply a different calibration procedure instead of the direct correction fields. This approach follows that of Simonsen et al. (2021). The calibration period is the era of ICESat (2003–2009), where the ICESat laser altimeter provides precise estimates of surface elevation change without surface penetration and ENVISAT provides similar estimates subject to surface penetration. The spatial differences between the ICESat and ENVISAT mass estimates provides the input for a calibration field (initial radar-volume mass balance) which can be applied to the full time series of elevation changes based on satellite radar altimeter observations for the period 1992–2017.

Following the calibration procedure described above, we computed monthly grids of mass change rates at 100 × 100 km2 resolution for the entire main GrIS. The peripheral glaciers (connectivity level 0 and 1 according to Rastner et al., 2012) were excluded from the grid. For each epoch, the mass rates of the grid cells were added to derive monthly mass change rates of the entire ice sheet. Time series of ice sheet mass anomalies with respect to the reference interval of 2006–2015 were then generated by cumulating the mass change rates in time and subtracting the mean over 2006–2015 from the cumulated time series.

The monthly grids were derived by applying a temporal window to aggregate the radar observations. For the ERS-1, ERS-2, and ENVISAT mission, this window is 5 years long. For CryoSat-2 the window is 3 years long. The monthly grids referred to the centre of the time window. This result in a smoothing of the time series to resolve climatic changes and not seasonal weather.

The error of the traditional altimetry-based mass-change estimates originates from different sources: uncertainty in the interpolation from point changes to ice-sheet-wide changes, uncertainty in the bedrock movement and in the firn compaction model, uncertainties due to the neglect of basal melt contributions, and of the possible ice accumulation above the equilibrium line altitude due to ice dynamics. For observations from radar altimetry, an additional source of uncertainty is the changing radar penetration in the firn column. The latter was reduced by the calibration approach applied here.

The overall uncertainty in the altimetry-derived mass change time series is provided as a conservative estimate based on converting the radar altimetry volume error into mass by ascribing ice densities to all grid cells. This estimate is assumed to be slightly overestimating the combined error of the five error sources.

Uncertainties of cumulated mass changes (in space as well as in time) were derived as follows: for the cumulation in space, standard uncertainties from all grid cells were added linearly. For the cumulation in time, uncertainties of mass change rates were aggregated (as the root sum square) forward or backward from 2011.0 to the specific epochs. Uncertainties of multi-year trends were calculated as the aggregated uncertainties of mass change rates over the interval of interest, divided by the interval length.

Unlike the GRACE-based assessment for Greenland, the altimetry-based assessment does not include Greenland peripheral glaciers. We therefore take the sum of the altimetry-based estimates for the ice sheet and GGM-based estimates for the peripheral glaciers to represent the total ice-mass changes in Greenland. The GGM methods and products and the related uncertainty assessments described in Sect. 3.4 were applied. The uncertainties of the sum of the two products were calculated as the root sum square of the uncertainties of the two summands.

The synthesis of assessed Greenland GMSL contributions in Fig. 6a shows that both the proper ice sheet and peripheral glaciers contribute significantly (0.43 ± 0.04 and 0.17 ± 0.02 mm yr-1, respectively, over P1). The rates of change vary interannually, peaking in 2011 and 2012. This is consistently reflected in the GRACE-based estimate and in the altimetry–GGM combination. The altimetry–GGM combination shows a somewhat larger trend over P2 than the GRACE-based estimate (0.89 ± 0.07 mm yr-1 versus 0.78 ± 0.02 mm yr-1) and does not resolve the annual cycle in the same way as GRACE, as the annual cycle is not resolved in the altimetry-based time series. The time-variable uncertainties of the altimetry-based and GGM-based time series (Fig. 6b) reflect the cumulation of uncertainties of rates of change backward and forward from the reference interval centre. The uncertainties of the GRACE-based time series reflect the superposition of a linear-trend uncertainty and an individual uncertainty for each monthly GRACE solution.

The GRACE-based product developed at Technische Universität Dresden within the Antarctic Ice Sheet CCI project is used to provide mass change estimates for the AIS from GRACE monthly gravity field solutions (Groh and Horwath, 2021). Quasi-monthly GRACE-based mass anomaly estimates (grids and basin time series) are available at https://climate.esa.int/en/projects/ice-sheets-antarctic/ (last access: 13 January 2022) or https://data1.geo.tu-dresden.de/ais_gmb (last access: 13 January 2022).

The AIS GRACE-based products were derived from the SH monthly solution series by ITSG-Grace2016 by Technische Universität Graz (Klinger et al., 2016; Mayer-Gürr et al., 2016) following a regional integration approach with tailored integration kernels that account for both the GRACE error structure and the information on different signal variance levels on the ice sheet and on the ocean (Groh and Horwath, 2021). The GIA correction adopted by these products was based on the regional model by Ivins et al. (2013). In Sect. 7.2 we address the trade-off between using global or regional GIA models for Antarctica.

The uncertainty assessment (Groh and Horwath, 2021) is analogous to that described for the GRACE OMC assessment in Sect. 3.3. For the AIS, the dominant source of uncertainty is the GIA correction. Uncertainties in the degree-one components and the C20 component of the gravity field are also important.

We computed Antarctic mass change from 1992 to 2017 using observations from four different satellite radar altimetry missions – ERS-1, ERS-2, ENVISAT, and CryoSat-2 – following the methodology described by Shepherd et al. (2019). For each mission, we computed elevation change from repeated elevation measurements during fixed epochs of 140 d on a polar stereographic grid using a plane fit method (McMillan et al., 2016). We applied a backscatter correction to remove the short-term fluctuations in elevation change correlated with changes in backscatter, and we combined the time series from different missions together by applying a cross-calibration technique. To convert our elevation change time series into a mass change time series, we first identified areas of ice dynamical imbalance in order to discriminate between changes occurring at the density of snow and ice. We defined these regions as areas with persistent elevation change that is significantly different from firn thickness change estimates derived from a semi-empirical firn densification model (Ligtenberg et al., 2011). Areas with an accelerated rate of ice thickness change were allowed to evolve through time. Based on this empirical classification, we converted our elevation change time series to a mass change time series by using a density of 917 kg m-3 in areas classified as ice and using spatially varying snow densities from the firn densification model in areas classified as snow. The mass anomalies for the West Antarctic Ice Sheet (WAIS), the East Antarctic Ice Sheet (EAIS), and the Antarctic Peninsula (APIS) at a 140 d resolution from 1992 to 2016 are available at http://www.cpom.ucl.ac.uk/csopr/ (last access: 13 January 2022).

Figure 7a shows the AIS GMSL contributions from the altimetry-based assessment as well as from the GRACE-based assessment. Over P1 (assessed from altimetry), the AIS contribution to GMSL is 0.19 ± 0.04 mm yr-1. Rates of change are much smaller from 1995 to 2006 and larger from 2006 onwards. Mass losses are dominated by mass losses in West Antarctica due to changing ice flow dynamics (cf. Shepherd et al., 2018). Over P2, the evolution of the AIS GMSL contribution from altimetry and GRACE is similar, with linear trends at 0.34 ± 0.04 and 0.27 ± 0.11 mm yr-1, respectively, overlaid by noise as well as a common interannual signal.

We assessed the uncertainties of our elevation change time series and convert them to a mass change uncertainty using the same time-evolving mask of areas of ice dynamical imbalance described in the previous section. At each epoch, we estimated the overall error of our elevation change as the sum in quadrature of systematic errors, time-varying errors, errors associated with the calibration between the different satellite missions, and errors associated with snowfall variability. The systematic errors refer to errors that affect the long-term elevation change trend. These may arise from short-term changes in the snowpack properties or from short-lived accumulation events that may not be accounted for in our plane fit model. We quantified the systematic errors as the standard error of the long-term rate of elevation change. The time-varying error refers to errors in the satellite measurements that might hinder our ability to measure elevation change at one particular epoch due to the measurement's precision or non-uniform sampling. We calculated these errors as the average standard error of elevation measurements. The inter-satellite bias uncertainties were computed as the standard deviations between modelled elevations during a 2-year period centred on each mission overlap. Finally, we quantified the snowfall variability uncertainty based on estimates from a regional climate model.

Cumulated mass changes and their uncertainties were originally generated with respect to the reference epoch of 1993.0, separately for the EAIS, the WAIS, and the APIS. To refer the product to the reference interval of 2006–2015, we subtracted the respective mean from the mass anomaly time series. We calculated uncumulated uncertainties by taking the differences between the uncertainties of consecutive epochs. We re-cumulated these uncertainties with respect to the centre of the reference interval, 2011.0, by linearly cumulating the uncumulated uncertainties, forward or backward, from 2011.0. Uncertainties of linear trends were calculated by linearly cumulating the uncumulated uncertainties over the interval of interest and division by the interval length. Uncertainties for the mass changes of the entire AIS were calculated as the root sum square of uncertainties for the EAIS, WAIS, and APIS.

Figure 7b shows the time-dependent uncertainties resulting from the cumulation with respect to the reference interval centre. The uncertainties of the GRACE-based estimates also shows reflect the model analogous to Eq. (16).

The LWS contribution is assessed with the global hydrological model (GHM) WaterGAP (Döll et al., 2003; Müller Schmied et al., 2014) in its latest version, WaterGAP2.2d (Müller Schmied et al., 2021). The model simulates daily water flows and water storage anomalies including the effects of human water use on a 0.5∘ × 0.5∘ grid (55 km × 55 km at the Equator and ∼ 3000 km2 grid cell) covering the whole land area except for Antarctica (we excluded model outputs over Greenland to avoid double-counting). Note that the Caspian Sea is not part of the model grid (based on the WATCH–CRU – Climatic Research Unit – land–sea mask) and thus not included in the assessment of the LWS component. Water flows are routed through a series of individual water storage compartments (Fig. 2 in Müller Schmied et al., 2021). Following the stream network defined by the global drainage direction map DDM30 (Döll and Lehner, 2002), streamflow is laterally routed until reaching the ocean or an inland sink. The model is calibrated against observed mean annual streamflow at 1319 gauging stations (Müller Schmied et al., 2021). The LWS anomaly (LWSA) is the aggregation of the anomalies in all individual water storage compartments: 17LWSA=SnWSA+CnWSA+SMWSA+GWSA+LaWSA+ReWSA+WeWSA+RiWSA, where WSA is the water storage anomaly in snow (Sn), canopy (Cn), soil moisture (SM), groundwater (G), lake (La), reservoir (Re), wetland (We), and river (Ri) storages. The model does not account for anomalies related to glacier mass variations. Land areas that in reality are covered by glaciers are represented as non-glacier-covered land areas where hydrological processes (evapotranspiration, runoff generation, groundwater recharge, etc.) are simulated. In terms of OMB assessment, adding the glacier contribution (Sect. 3.4) and the LWS contribution has the implication of “double-counting” the land areas covered by glaciers, which are then included in both contributions. In a recent study (Cáceres et al., 2020), time series of glacier mass variations computed by the GGM of Marzeion et al. (2012) were integrated as an input to WaterGAP; this resulted in a non-standard version of the model that explicitly accounts for glaciers. The aggregated water storage anomalies computed by this model version were compared to the result of adding LWSA computed by the standard WaterGAP and anomalies related to glacier mass variations computed by the GGM. The comparison of these two approaches showed that the impact of double-counting glacier-covered areas is insignificant at a global scale.

Human water use is accounted for through the representation of the impact of water impoundment in man-made reservoirs and of net water abstractions (i.e. total abstractions minus return flows) on water flows and storages. The reservoir operation algorithm implemented in WaterGAP is a slightly modified version of the generic algorithm of Hanasaki et al. (2006) (Döll et al., 2009). Based on a preliminary version of the Global Reservoir and Dam (GRanD) database (Lehner et al., 2011), the model accounts for the largest 1082 reservoirs. The reservoir filling phase is simulated based on the first operational year and the storage capacity. Net water abstractions are simulated for five water use sectors (irrigation, livestock farming, domestic use, manufacturing industries, and cooling of thermal power plants) and subsequently subtracted from the surface water and groundwater storage compartments (Müller Schmied et al., 2021; Döll et al., 2014).

In the framework of this study, we used monthly globally averaged (over 64 432 0.5∘ × 0.5∘ grid cells) LWSA time series extending from January 1992 to December 2016. Anomalies are relative to the mean over the period January 2006 to December 2015. The model was forced with daily WATCH Forcing Data (WATer and global CHange) methodology applied to ERA-Interim data (WFDEI, Weedon et al., 2014). Two different variants of this climate forcing were used. In one of them, precipitation was bias-corrected using monthly precipitation sums from the Global Precipitation Climatology Centre (GPCC, Schneider et al., 2015), and, in the other one, it was bias-corrected using monthly precipitation sums from the Climate Research Unit (CRU, Harris et al., 2014); hereafter, we refer to these climate forcings as WFDEI–GPCC and WFDEI–CRU, respectively. In addition, we considered two different assumptions in relation to consumptive irrigation water use in groundwater depletion regions. Typically, consumptive irrigation water use is calculated by assuming that crops receive enough water for actual evapotranspiration to be equivalent to the potential evapotranspiration value (Döll et al., 2016). We assumed consumptive irrigation water use to be either optimal (i.e. 100 % of water requirement) or 70 % of optimal in groundwater depletion areas (for more details, see Döll et al., 2014). Consequently, an ensemble of four LWSA time series corresponding to two climate forcings and two irrigation water use variants was considered. The unweighted mean of the four ensemble members was used in the SLB assessment.

A comparison of the monthly time series of total water and ice storage anomaly over the continents (except Greenland and Antarctica) as derived from GRACE and from the non-standard WaterGAP version with glacier integration showed a very good fit, with a modelling efficiency of 0.87 (Cáceres et al., 2020). The GRACE trend during 2003–2016, however, was 26 % weaker than the trend from the non-standard WaterGAP version. More recently this difference was significantly reduced after the GRACE analysis for continental total water storage was made more consistent with the GRACE OMC analysis (Gutknecht et al., 2020).

Figure 8a shows the monthly time series of the LWSA contribution to GMSL. It is characterised by the highest seasonal amplitude of all ocean-mass contributions due to seasonal climate variations. (See Fig. 12 for a time series where the seasonal signal is subtracted.) The overall positive trend (0.40 ± 0.10 mm yr-1 over P1) is caused mainly by groundwater and surface water depletion that more than balances increased land water storage due to the filling of new reservoirs.

Uncertainties are characterised by the spread between the four model runs. For each month, the standard deviation of the values from the four time series was taken as the standard uncertainty. Figure 8b shows these time-variable uncertainties of the LWSA. They reflect month-to-month differences in the spread between the ensemble members. Since the LWS anomalies referred to the 2006–2015 mean value and the four ensemble members show different trend, the uncertainty is lowest around 2011.0 and tends to increase towards the beginning (1993) and the end (2016). The standard deviation of the linear trends calculated for each ensemble member was taken as the standard uncertainty of the linear trend of the ensemble mean.