were the first to suggest the combination of satellite geodetic methods – gravimetry and altimetry – to estimate GIA. We use the analytical approach from to explain gravity changes by mass changes projected into a spherical layer (with radius a) – termed area density changes (ADCs) or surface density changes. Note that a change of mass is with respect to a reference mass distribution. Based on GRACE solutions given in the spherical harmonic domain, the conversion of changes in Stokes coefficients with degree n and order m (Δcnm) into spherical harmonic coefficients of ADC (Δκnm) is 1Δκnm=2n+11+kn′ME4πa2Δcnm, where ME is the total mass of the Earth, a the equatorial radius of the reference ellipsoid, and kn′ the second-load Love number to account for the deformation potential of the solid Earth induced by the mass redistribution. The linear ADC κ˙nm is synthesized into spatial domain m˙grav, which is the superposition of the ADC through GIA, and processes in the ice (ID) and firn layer: 2m˙grav=m˙GIA+m˙ID+m˙firn. While GIA is not a process of ADC, m˙GIA is defined as the apparent ADC that would induce a gravity field effect equal to the GIA-induced gravity field effect. We refer to m˙GIA (as well as spatial integrals of m˙GIA) as GIA-related mass change. ID summarizes all processes which are weighted with ice density, e.g. ice-dynamic flow or basal melt. We summarize the ice-induced, or cryospheric, area density trend as m˙ice=m˙ID+m˙firn.

Analogously, the overall linear SEC derived from altimetry h̃˙alt is the sum of the linear SEC through ID, firn, GIA, and elastic BEC: 3h̃˙alt=h˙GIA+h˙elastic+h˙ID+h˙firn. Note that GIA refers to the viscoelastic deformation of the solid Earth. The elastic BEC (h˙elastic) triggered by present-day ice mass changes needs to be subtracted from the overall SEC observed by altimetry h̃˙alt prior to the combination. We define h˙alt=h̃˙alt-h˙elastic. Doing this, the SEC signals in h˙alt are consistent with ADC signals in m˙grav.

The process-related elevation and area density changes are linked with effective density assumptions (ρGIA, ρID): 4m˙GIA=ρGIA⋅h˙GIA,5m˙ID=ρID⋅h˙ID. Rearranging Eq. () to 6h˙ID=h˙alt-h˙firn-h˙GIA and substituting it together with Eqs. () and () into Eq. () leads to 7m˙grav=ρGIAh˙GIA+ρID(h˙alt-h˙firn-h˙GIA)+m˙firn, which can be solved for 8h˙GIA=m˙grav-ρID(h˙alt-h˙firn)-m˙firnρGIA-ρID.

In , Eq. () is modified with a criterion to include assumptions about the difference h˙alt-h˙firn using a priori uncertainties. ρID is replaced by ρα to permit the following case distinction: 9h˙GIA=m˙grav-ρα(h˙alt-h˙firn)-m˙firnρGIA-ρα, where

10ρα=ρID,(I)ifh˙alt-h˙firn<0and|h˙alt-h˙firn|>2σhρfirn,(II)ifh˙alt-h˙firn>0and|h˙alt-h˙firn|>2σh0,(III)otherwise with 11σh=σh˙alt2+σh˙firn2. The case distinction accounts for uncertainties in altimetry and in the firn densification model (FDM) as well as a priori knowledge on ice sheet processes. The GIA-induced BEC is in the millimetre per year range, whereas h˙firn and h˙ID can be in the centimetre to metre per year range. If altimetry and FDM are perfect, h˙alt-h˙firn would leave essentially h˙ID (apart from a very small h˙GIA). The following case distinctions are made.

Case I. If differences between h˙alt and h˙firn are significantly negative, an ice-dynamic-induced SEC is assumed (glacial thinning). argue that only one region in Antarctica is known to show glacial thickening: the area of the Kamb Ice Stream . This region is therefore treated separately by a mask which sets ρα to ρID. The mask is generated from positive SEC from altimetry in this area.

The following steps are performed in sequence.

Step 1. Estimation of biased h˙GIA using the data combination approach (Eq. ).

For the necessary noise suppression we use GRACE data with a de-striping filter applied (FDS(m˙grav)) in addition to the filtering implied by the spherical harmonic truncation. Ideally, the data and models involved in the combination should have consistent spatial resolution; that is, they should be filtered consistently. This is not strictly possible for the quotient (m˙grav)/(ρGIA-ρα) in Eq. () because no unfiltered m˙grav is available that could be divided by (ρGIA-ρα) before filtering. Pragmatically, components with a similar spatial resolution are combined before they are filtered with a Gaussian filter F. Hence, we obtain a filtered GIA-induced BEC: 16F̃(h˙GIA)=F(FDS(m˙grav))F(ρGIA-ρα)-Fρα(h˙alt-h˙firn)-m˙firnρGIA-ρα. For integrating mass trends in space, the signal redistribution (leakage) is taken into account by a buffer zone equal to the half-response width of the Gaussian filter appended to the grounding line of the ice sheet (Sect. ). We do not correct for leakage through ocean mass signal separately as it amounts to only 4.5 Gt a-1 . This ocean mass leakage is the same in every experiment, because we do not test the sensitivity to filters.

In Eqs. () and (), assumptions on uncertainties of the FDM and altimetry are crucial. take σh˙alt from the formal uncertainty of the least-squares estimation. σh˙firn can be derived in the same way from the estimated trend of FDM SEC for the observation period. Note that both uncertainties are derived from stochastic information of the least-squares estimation rather than from an uncertainty characterization of the measurements and the model. have also performed an uncertainty analysis of the combination result. For this purpose, they define the SMB-related uncertainty as 10 % of the estimated trend value, referring to . Note that the uncertainty assessment by , which amounts to 10 %–30 % of the signal, is applied to a different physical quantity than h˙firn: namely to the snow accumulation in a drainage basin.

Because there is no comprehensive regional climate model ensemble, we quantify the error of firn process models by statistics on differences between two models. We use differences of trends of cumulated surface mass balance anomalies (cSMBAs) and of firn thickness trends. We assume those differences are due to modelling errors. This characterization comprises only a part of the full uncertainty, because it is based on two alternative climate model products.

Previous studies combining gravimetry and altimetry are based on linear seasonal deterministic models over certain periods . However, signals in the firn and ice layer over the Antarctic Ice Sheet (AIS) show inter-annual changes . In theory, combining observations on a time series level will lead to a linear GIA signal. For T months the vector 17mgrav={mgrav(t=1),…,mgrav(t=T)} contains the differences in mass at month t=1,…,T with respect to a reference mass distribution. The combination of all time series is 18hGIA=mgrav-ρID(halt-hfirn)-mfirnρGIA-ρID. This requires that all data are available as monthly gridded products. To simplify, we assume that effective densities do not change over time. To be consistent with the combination of trends, ρID is replaced with ρα from the trend-based approach.

The data and models of every month are filtered in the same way as for the trend-based approach to make the resolution consistent (Sect. ). Afterwards they are combined according to Eq. (), which results in a GIA time series for each grid cell.

By assumption the GIA signal in the resulting time series hGIA is linear over decades of satellite observations e.g.. A fitted trend to hGIA is h˙GIA. We are aware that for regions with a low-viscosity asthenosphere, e.g. Pine Island Bay, the viscoelastic deformation associated with GIA may be non-linear even for decadal periods . In this case, the assumption of a linear GIA-induced BEC introduces an error.

The sensitivity of inverse GIA estimates to different data, models, and assumptions is quantified. Starting from a reference experiment, certain parameters are changed. Every experiment is performed with and without the two LPZ-based bias corrections to demonstrate their effect. It is examined how different altimetry data (Sect. ), degree-1 and C20 products (Sect. ), and the empirically determined errors of the firn process models (Sect. ) affect the GIA solution. Analogous to and , a Gaussian filter (half-response width = 400 km) is applied to all data sets. For the integration of mass trends over the AIS, the West Antarctic Ice Sheet (WAIS), and the East Antarctic Ice Sheet (EAIS), we use a buffer zone of 400 km grounding-line distance to mitigate leakage. The Antarctic Peninsula (AP) is not considered separately here.

For each inverse GIA solution, the integrated mass change is calculated. In addition, a root-mean-square (rms) difference with respect to the reference experiment is determined, hereinafter referred to as the rms difference from reference experiment (RMSRE), 19RMSRE=1N∑i=1Nh˙GIA,comp,i-h˙GIA,ref,i2. Here, N is the number of grid cells of a Cartesian grid in the polar stereographic projection of the AIS area (EPSG: 3031) including the buffer zone. h˙GIA,comp refers to the GIA solution which is compared to the reference experiment (h˙GIA,ref). The RMSRE values are sensitive to regional differences, which may be hidden in the comparison of integrated mass trends.

The sensitivity to the choice of firn process models is investigated as follows: based on the comparison of two firn process models, empirical samples of error patterns are generated. They are added to h˙firn and m˙firn and propagated to the empirical GIA estimates. Additionally, all identified trend differences of cSMBAs are added to h˙firn and m˙firn.

Furthermore, the dependency on differing time periods is investigated. Under the assumption that GIA is linear in time, the used time interval should have negligible influence. While the time interval for the reference experiment is March 2003–October 2009 (according to ), alternative periods are the main GRACE observation period (April 2002–August 2016) and the overlap period between GRACE and CryoSat-2 (July 2010–August 2016).

The SECs from are based on a repeat-altimetry analysis in a multi-mission altimetry (MM altimetry) framework. Data from the Seasat, Geosat, ERS-1, ERS-2, Envisat, ICESat, and CryoSat-2 missions are combined, resulting in a monthly sampled time series on a 10 km grid. The reader is referred to for details on processing and background information. In order to combine the altimetry time series with GRACE, we use the monthly results from April 2002 at the earliest to August 2016 at the latest. This period involves observations of ERS-2, Envisat, ICESat, and CryoSat-2 missions (Fig. a). The altimetry missions have a different spatial and temporal sampling, e.g. ICESat's campaign-style temporal sampling. Further, the data quality varies over mission lifetime. For this reason every month of the combined time series differs in spatial coverage. We obtain a linear rate over the respective intervals by adjusting an offset and a linear trend to the MM time series for each cell of the 10 km grid. For the reference experiment no annual periodic signal is co-estimated in order to be consistent with . We apply weights according to the uncertainty estimates of each epoch of the MM time series. We took the criterion that the trend would only be estimated for a grid cell if more than 5 months with observations are available and at least 80 % of the selected total time span is covered. This criterion should avoid outlier trends through insufficient sampling. The uncertainty σh˙alt used in Eq. () is the a posteriori standard deviation derived from the least-squares adjustment of the MM time series.

To investigate how the choice of altimetry products affects the GIA estimation, single-mission time series are calculated for Envisat and ICESat. They consistently use the same processing steps as the MM altimetry from , with the exception that the final step of weighted spatio-temporal smoothing is applied to single-mission data rather than multi-mission data. In total three different altimetry time series are used for testing the gravimetry–altimetry combination approach. To assess the sensitivity of results to the co-estimation of seasonal signals, an additional version of the MM altimetry trends is calculated by co-estimating the annual sinusoidal signal (MM seasonal in Table ). This is consistent with the treatment of GRACE and the firn process models.

Part of the altimetry-derived SEC is caused by the elastic BEC of the solid Earth due to present-day ice mass change (h˙elastic), which needs to be subtracted from the altimetry observations (h̃˙alt) prior to the combination (Eq. ). We estimate h˙elastic to be -1.5 % of h̃˙alt . Hence, the elastic-corrected altimetry-derived SEC is 20h˙alt=h̃˙alt-h˙elastic≈1.015⋅h̃˙alt. The approximative nature of this elastic correction leaves an error, but its influence on the GIA estimate is negligible .

GRACE-derived monthly mass variations are calculated from the ITSG-Grace2016 monthly gravity field solutions up to a degree and order of 90 using Eq. (). Monthly solutions from other processing centres are not considered because ITSG-Grace2016 is identified through internal comparison as the gravity field solution series with a high signal-to-noise ratio. This is supported by , who found that the precursor ITSG-Grace2014 show a lower noise level compared to solutions from other processing centres. The influence of the different GRACE monthly solutions on the inverse GIA result was shown and discussed in . We do not use solutions after August 2016. Those solutions show a much higher noise level due to accelerometer issues.

GRACE monthly solutions need to be complemented by the degree-1 term of the spherical harmonic coefficients, as this is not observed by GRACE. Three different products to replace the degree-1 coefficients are evaluated. (1) A product is determined following using ITSG-Grace2016 monthly solutions (d1_ITSG). (2) A satellite laser ranging (SLR) product by (d1_SLR) and (3) degree-1 coefficients by are used (d1_ITG).

Furthermore, the influence of the flattening term C20 is investigated. Because C20 is poorly determined by GRACE , external products are compared. (1) SLR-based time series are used from the Center for Space Research at the University of Texas, USA (c20_SLR_CSR; ). (2) SLR-based time series from the German Research Centre for Geosciences, Potsdam, Germany are used (c20_SLR_GFZ; ). (3) A time series from the Delft University of Technology, Delft, Netherlands (c20_TU_Delft), which is derived from GRACE observations themselves and an ocean model is used .

A critical point is filtering because the monthly solutions are noisy and have a correlated error pattern . A de-striping filter is applied in the spherical harmonic domain .

A linear seasonal model is adjusted to fit the filtered Stokes coefficients (offset, linear, annual periodic, and 161 d periodic). The trend is synthesized from the spherical harmonic into the spatial domain on the altimetry grid with 50 km resolution. In this way for each grid cell a linear area density trend in kilogrammes per square metre per year is determined (Fig. b).

Information on variations in the firn layer is required in the combination approach (Eq. ). SMB is the sum of precipitation, snow drift, sublimation, and meltwater runoff. The SMB components are numerically simulated with the RACMO2.3p2 model, which contains a multilayer snow model developed by the Royal Netherlands Meteorological Institute (KNMI) and the Institute for Marine and Atmospheric Research, Utrecht, Netherlands (IMAU) . These results are compared to the MAR model of the Laboratory of Climatology, Liège, Belgium . The regional climate models are forced at their lateral boundaries with the ERA-40 and ERA-Interim reanalyses. Mass fluxes (snowfall, snow drift, sublimation, erosion–deposition, and surface melt) as well as surface temperature are then used to force an offline firn densification model that includes firn compaction, vertical meltwater transport and refreezing, and thermodynamics of the firn layer.

The RACMO2 and MAR SMB products are appropriate for comparison as both are similar in terms of temporal (monthly) and spatial resolution (RACMO2: 27 km; MAR: 35 km). Moreover, both variants considered here use the same forcing. There is no independent knowledge (in a spatial resolution similar to that of SMB models) about the ice flow contribution to ice mass balance and hence about the degree of balance or imbalance between SMB and ice flow. Therefore, the modelled SMB is only used to derive SMB-induced mass variations with respect to any background signal of mass change. The unknown background signal of mass change is the possible imbalance between the mean SMB over a multi-year reference period and the mean effect of ice flow over the same reference period. The considered SMB-induced mass variations hence arise from the temporal cumulation of SMB anomalies with respect to the mean SMB over the reference period. Here, we define the reference period to be the entire model period for RACMO2.3p2 and MAR (January 1979–December 2016). For the satellite observation periods (e.g. April 2002–August 2016) the surface mass trend (m˙firn), or literally the trend of cumulated surface mass balance anomalies (cSMBAs), is estimated (co-estimated with bias and annual periodic signal).

The used firn model IMAU FDM is forced at the upper boundary by SMB components from RACMO2. The firn layer is initialized by forcing the FDM repeatedly with the 1979–2016 surface mass fluxes and temperature, until an equilibrium firn layer is established. This implies that present-day conditions represent a state of equilibrium and that there is no net firn thickness change over the model period January 1979–December 2016. One result of the actual model run is the firn-elevation-change time series. A linear seasonal model (bias, trend, annual periodic signal) of firn-process-induced SEC is adjusted to fit the FDM time series for the observation periods under investigation (Fig. c).

The LPZ (Fig. c) is defined based on the ECMWF ERA-Interim reanalysis precipitation product. We use 20 mm a-1 annual precipitation as a threshold for low precipitation , rather than 21.9 mm a-1 used by .

The trend differences between RACMO2.3p2 and MAR SMB products are used for uncertainty characterization of firn process models. In order to gain statistical information on possible trend differences over a 7-year interval, we calculate trend differences over 32 intervals of 7 years (January 1979–December 1965; January 1980–December 1966;...; January 2010–December 2016) covered by RACMO2.3p2 and MAR. The 7-year length is the approximate length of the observation period of our reference experiment (March 2003–October 2009) defined by the ICESat observation period. A FDM forced with MAR SMB does not exist. However, the RACMO2.3p2 SMB and the derived FDM are directly linked to each other. For this reason we assume that derived conclusions on errors of SMB are transferable to the FDM as a lower bound. Pseudo FDM trend differences are estimated out of the cSMBA trends by 21Δh˙firn,j=Δm˙firn,jρMAR. Δm˙firn,j is the jth trend difference between cSMBA from RACMO2 and cSMBA from MAR. ρMAR is calculated from MAR density fields by taking their average over the near-surface layers (0–1 m) and over the whole model period. This does not consider the evolution of the firn layer, as an independent FDM driven by MAR outputs would consider it. Furthermore, uncertainties associated with equilibrium assumptions are not considered.

Prior to the combination, cSMBA and FDM trends are linearly interpolated to the polar stereographic grid. The high-resolution products (altimetry and firn process models) are modified as follows. NaN-Grid cells on the grounded part of the ice sheet (missing data) are treated as case III in Eq. ().

The ratio between volume changes and area density changes is defined by the effective densities ρGIA, ρfirn, and ρID for GIA-related, firn-related, and ice-related processes, respectively. We use a ρID of 917 kg m-3. The firn density is variable in space and time. The location-dependent estimation for ρfirn is calculated using the empirical Eq. (2) in .

The density mask for ρGIA is generated as follows: The ratio between the GIA-induced BEC and the GIA-induced ADC is about 3700 kg m-3 . We use 4000 kg m-3 over the Antarctic continent and 3400 kg m-3 under the ice shelves and the ocean with a smooth transition (according to ). These numbers account for the redistribution of ocean mass through GIA and are derived from forward-model results. This density is not a density in a material-science sense. It is an effective value which sets GIA-induced BEC and the ADC in relation. The term rock used in the literature might be misleading.

There are considerable differences between the time series of cSMBA from the RACMO2 and MAR SMB products for each cell. Figure  shows the integrated values for the AIS. Note that a ∼400 Gt cSMBA difference in 1987 (8 years after model start) represents a 50 Gt a-1 difference in SMB, which is ∼2 % of the total grounded ice sheet SMB. The integrated SMB from RACMO2.3p2 is 2229 Gt a-1 with an interannual variability of 109 Gt a-1 . We use the 32 trend differences from the moving 7-year intervals to quantify discrepancies of derived cSMBA trends between both models. Figure  shows (1) the rms of all trend differences and compares it with (2) the formal uncertainty we derive from the least-squares estimation and with (3) the 10 % uncertainty assumption (Sect. ). The last two are derived from the estimated cSMBA trends of the RACMO2.3p2 SMB product over the ICESat observation period (March 2003–October 2009). The formal uncertainty and the 10 % assumption are similar in spatial pattern and magnitude. The rms of trend differences is similar in spatial pattern, too, but approximately 3 times larger in magnitude.

To extract the dominant error patterns, a spectral decomposition of the 32 7-year trend differences (see Sect. ) is carried out using principal-component analysis (using singular value decomposition). Hence, the dominant empirical orthogonal functions (EOFs) and accompanying principal components are computed. From this analysis we obtain the dominant error patterns that are uncorrelated to each other and capture characteristic features of uncertainty. The first three EOFs of the trend differences explain ∼68 % of the total variance (Fig. a–c). The normalized EOF is scaled with the square root of the particular eigenvalue. Figure d shows the principle components indicating the scaling of the corresponding EOF. For instance, EOF-1 is dominated by variations in the WAIS. EOF-2 shows more variations on smaller scales. Without an attempt to further interpret the patterns of trend differences between the two models, the explored trend differences are used here to investigate the sensitivity of the inverse GIA estimates to these differences characterizing firn process uncertainty. For this purpose, (1) we add the EOFs to the firn process trends (m˙firn, h˙firn), which we use as input for the data combination. Because a FDM forced with MAR products does not exist, we transfer the cSMBA-derived EOFs to FDM EOFs by calculating pseudo EOFs using MAR density fields (see Sect. , Eq. ). The pseudo EOFs account for a lower bound of uncertainties of the firn thickness trends. True firn thickness trend differences are presumably higher as they would contain the potential mis-modelling of firn densification. From the added EOFs we get three GIA estimates to be compared with our reference solution. (2) Moreover, we add each trend difference separately to the cSMBA trend and each pseudo trend difference separately to the firn thickness trend. The pseudo firn thickness trend differences are likewise calculated using MAR density. This results in another 32 GIA estimates.

Inverse GIA estimates are calculated using different choices of (1) degree-1 solutions, (2) C20 substitutions, (3) altimetry products, (4) empirical orthogonal functions (EOFs) of firn process errors, and (5) time intervals (Table ). The reference experiment refers to the time period March 2003–October 2009 and uses the MM-altimetry-derived SEC, ITSG-Grace2016 monthly solution (degree-1: d1_ITSG, C20: SLR_CSR) and the firn process trends from RACMO2.3p2 over this period. The rms of the reference GIA-induced BEC estimate is 2.2 mm a-1. The estimated ρα (Eq. ) is shown in Fig. a. Apart from the gridded GIA-induced BEC (Figs. b, S5 in the Supplement), we compare the integrated trends m̃˙grav, m̃˙GIA, and m̃˙ice corresponding to total-mass change (from GRACE), GIA-related mass change, and ice mass change, respectively. The results are summarized in Table . Furthermore, the RMS_RE (Eq. ) quantifies the discrepancy to the reference experiment GIA estimate. Figure  shows the mass balance estimates for March 2003–October 2009.

Biased total-mass changes for different C20 and degree-1 products vary between -43 Gt a-1 (c20_TU_Delft) and +25 Gt a-1 (d1_SLR), a range of 68 Gt a-1. Debiased total-mass change (Eq. ) only differ by 6 Gt a-1 for the same time period (Table ). Figure  illustrates biased and debiased total-mass changes of the entire AIS. Note that the biased total-mass change of 0 Gt a-1 in Table  arises coincidentally.

The biased GIA-related mass change of the AIS with MM altimetry (reference experiment) is very close to the Envisat-only estimate (174 vs. 172 Gt a-1). The biased ICESat-only result differs from the reference experiment by about 30 Gt a-1 (142 vs. 172 Gt a-1). Debiased estimates that use Envisat-only or ICESat-only results differ from the estimate of the reference experiment by 10 and 15 Gt a-1, respectively. The differences due to the co-estimation of seasonal components are marginal (∼2 Gt a-1).

Applying the approach to different time intervals April 2002–August 2016 and July 2010–August 2016 leads to debiased total-mass changes of -121 and -181 Gt a-1, respectively (biased estimates: -48 and -70 Gt a-1).

The addition of the EOFs (Sect. ) propagates to differences in the GIA solution of up to 7 Gt a-1 for the debiased GIA-related mass change and up to 18 Gt a-1 for the biased GIA-related mass change. Additionally, Fig. S6 shows the standard deviation of the 32 GIA estimates resulting from propagating the 32 trend differences between RACMO2 and MAR.

Our time-series-based combination takes advantage of the fact that gravimetry, altimetry, SMB, and FDM are available as monthly gridded products with sufficient spatial coverage from July 2010 to August 2016, owing to the availability of GRACE, CryoSat-2, and RACMO2.3p2. and only use ICESat altimetry data, which does not allow a monthly sampling, as it has only 2–3 months of observation per year.

We used the values of ρα estimated from the trend-based combination during the same time interval (Fig. S4I) to be consistent for comparison. Figure  shows the GIA-related mass change time series for the AIS (with 400 km buffer-zone). For applying the LPZ-based GIA bias correction, the linear GIA trend in the LPZ is estimated (offset and trend only). Figure A shows the debiased GIA-induced BEC based on the time series combination. Figure c shows its formal uncertainty from least-squares estimation, which should be considered as a lower bound. For comparison, Fig. B shows the GIA-induced BEC following the trend-based combination approach. The GIA-related mass changes from the time-series-based and trend-based combinations are 39 and 67 Gt a-1 for the AIS, 17 and 37 Gt a-1 for the WAIS, and 23 and 30 Gt a-1 for the EAIS, respectively (Table ). The ice mass changes are -220 and -248 Gt a-1 for the AIS, -206 and -227 Gt a-1 for the WAIS, and -14 and -21 Gt a-1 for the EAIS, respectively. The integrated formal uncertainty of the GIA-related mass change for the AIS with a 400 km buffer zone is 25 Gt a-1 (Fig. c).

To test our data processing we performed a run with similar input data as used in . We used GFZ RL05 GRACE solutions, ICESat Altimetry, the RACMO2.1 SMB product, and the corresponding IMAU FDM. Table  shows the comparison of both results. AIS total-mass, GIA-related mass, and ice mass change estimates reproduce results by to within 6, 5, and 1 Gt a-1, respectively. Those differences might be attributed to a slightly different LPZ, altimetry processing, and the missing ocean mass leakage correction. indicate that the uncertainty for the GIA-related mass change and ice mass change from various GRACE solutions and filtering variants is 40 and 44 Gt a-1, respectively.

In general our GIA estimate (Fig. b) shows a similar spatial pattern compared to estimates by . Nonetheless, notable differences appear in the AP, Marie Byrd Land (MBL), Victoria Land (VL), and Queen Mary Land (QML).

In the AP, altimetry-derived SECs are available for a part of the area only (Fig. S1). As a result of missing altimetry data, GRACE-derived area density changes are attributed mainly to GIA-related mass change. The result is a negative GIA-induced BEC. Although negative GIA-induced BECs are predicted by forward models for other regions e.g., we consider it unphysical for this particular region because we cannot find any further indications to substantiate it. Furthermore, the missing altimetry leads to unconsidered elastic deformation. The negative signal in MBL is of a similar order of magnitude as in and . A negative GIA signal in QML can be found in . The uncertainty of the GIA signal is sometimes so large that even its sign cannot be determined.

For example, propagating trend differences between RACMO2.3p2 and MAR cSMBA products to GIA estimates (Fig. S6) leads to a high standard deviation of the GIA signal in MBL and Victoria Land (VL). This issue cannot be resolved by considering the results of forward models because they also show large variations and sign differences in the predicted spatial pattern of the GIA-induced BEC .

The use of several degree-1 and C20 products for the GRACE processing leads to a differing total-mass trend for the AIS . The Supplement showed the influence of two different degree-1 products. Here we show how the bias corrections eliminate those differences in total-mass and GIA-related mass change (Sect. , Table ). The RMSRE of all debiased GIA estimates amounts to only 0.1 mm a-1 (Table ). As discussed in Sect. , any GIA signal over the LPZ would be removed erroneously in the method of , but the uncertainty in low-degree harmonics is assumed to be much higher than a potential GIA signal within the LPZ. The bias correction regionalizes the GIA estimate; i.e. derived mass changes are always given relative to the mean LPZ mass change. The bias correction defines how the total-mass change is decomposed into mass signals and is made to ensure that the combination approach produces robust mass estimates. The large uncertainty introduced by degree-1 and C20 is suppressed at the cost of global consistency.

Several objections can be made to the assumption that over the LPZ the mean GIA-induced BEC, the mean total-mass change, and hence the mean ice mass change are zero. (1) The precipitation of the last 40 years is not directly linked to GIA. (2) Areas are included which show quite relevant GIA-induced BEC in forward models, e.g. close to the Ross Ice Shelf . (3) The threshold for low precipitation is arbitrary and cannot be based on physical reasons in relation to GIA. For a given threshold, the definition of the LPZ still depends on the precipitation product used. (4) The LPZ is a large area in which even a low GIA effect can cause several gigatonnes per year of mass changes. (5) The LPZ bias correction does not allow for a simple transfer of the approach to Greenland or to a global framework. Nevertheless, the calibration over the LPZ is at least one possibility to consider the presumably existing biases.

Fig. 3 show large differences in the EAIS mass change estimates derived from satellite gravimetry and altimetry. In principle, the question of quantifying GIA in the EAIS arises. For this discussion, the reader is referred to  and , for example.

The choice of the altimetry product has a major effect on the GIA estimate. Using ICESat-only and Envisat-only products leads to a RMS_RE of 1.1 and 0.8 mm a-1, respectively (Table ). Both missions use different observation methods and have different spatial coverage. The radar altimetry time series of Envisat is sampled monthly but only to a latitude of 81.5∘ south. ICESat uses laser altimetry and its polar gap is smaller (south of 86∘). These differences affect the results across Kamb Ice Stream where a dominant ice-dynamic signal is expected . ICESat's campaign-style temporal sampling may affect the trend estimation significantly. For the time period March 2003–October 2009 the MM altimetry product uses mainly observations from ICESat and Envisat. The trend derived from the MM altimetry product shows a spatial discontinuity at the 81.5∘ latitude limit of Envisat coverage (Figs. S1A, a). We attribute this to the sparse time sampling of the ICESat mission. The spread of debiased GIA-related mass change estimates of the AIS using various altimetry products is 15 Gt a-1 (Table ). Furthermore, differences in the spatial GIA pattern are remarkable in MBL and VL (Fig. S5f, g). The co-estimation of an annual seasonal signal in altimetry only leads to small changes in the overall result (Sect. , RMSRE: 0.1 mm a-1) but is more consistent with processing of other data and models.

A crucial point in the combination approach is the case distinction for ρα (Eq. ). As mentioned in Sect. , it accounts for the uncertainty of altimetry and the FDM but does not account for the uncertainty of GRACE and the cSMBA trends. The resulting map of ρα (Figs. a, S4) does not agree with predefined, physically reasonable density maps. For example, ρα is set to ice density in large areas of the EAIS where dynamically induced ice mass losses are not plausible. The values of ρα largely depend on used data sets (Fig. S4b, c). An alternative to the ρα approach could be the formal approach shown in Eq. (). Technically this would be correct. However, it results in an ice density weight for the whole AIS. We are aware that this is not correct either because presumable processes in the firn layer are not completely considered by input data and models. Another strategy may use a predefined density mask similar to , but with a predefined significance criterion for all input data sets. This would need further investigation.

The ρα approach (Eq. ) to assign height changes to either ice dynamics or firn processes may be a source of bias. For example, if a negative SEC is firn-related, but erroneously attributed to the density of ice by Eq. (), this will lead to a higher ice mass decrease assigned to altimetry. GRACE would sense the true smaller ice mass decrease. Through combination of both, this discrepancy in ice mass change would be assigned to a positive GIA signal. We suppose this is qualitatively visible for ice-density-weighted regions in the EAIS (Fig. a, b), e.g. the sector between a longitude of 30 and 100∘ (Dome F). We presume this erroneously introduced positive GIA signal explains a part of the GIA bias.

The propagation of the empirically determined error patterns (EOFs 1–3) of the firn process models (Sect. ) shows small effects on the spatial pattern of inverse GIA estimates (Fig. S5i–k). The RMSRE for the EOF 1, EOF 2, and EOF 3 experiments is 0.5, 0.3, and 0.3 mm a-1, respectively (Table ). Note that this deviation arises solely from differences in similar climate models that use the same forcing data.

Uncertainties assumed in for σh˙firn are very small compared to our results (Sect. , Fig. ). In addition, any long-term trend in firn mass and firn thickness is ignored by the equilibrium assumption made by the firn modelling. SEC from Altimetry and the IMAU FDM show major differences even with a different sign for some areas, such as the AP and QML (Fig. a, c). These differences may indicate that the equilibrium assumption of the FDM (Sect. ) is not fulfilled for those areas of the AIS, i.e. that net firn thickness changes occur over the modelling period.

We also investigate a GIA solution derived from data sets over almost the entire GRACE period (April 2002–August 2016) and the approximately 6-year period of CryoSat-2 overlapping with GRACE (July 2010–August 2016). The variability of these estimates cannot be attributed to a single processing choice. On the one hand, different data sets are used (depending on assembled altimetry missions). On the other hand, cSMBA trends and FDM-derived SEC differ largely depending on the selected time interval (Sect. , Fig. S3). Ice mass change estimates are very high for the time interval July 2010–August 2016 (Table ). The quality of input data varies over time, e.g. due to the changing availability of data. Therefore the GIA estimates show large discrepancies among different time intervals, which is incompatible with the assumption of a constant linear rate of GIA-induced BEC. However, regions (e.g. Pine Island Bay) are known where a non-linear deformation through GIA is plausible during decadal periods .

The combination of time series leads to similar results compared to the trend-based approach for the same July 2010–August 2016 interval (Sect. ). We combined time series only for this time period, where CryoSat-2 and GRACE data are available with monthly sampling and sufficient spatial coverage. A closer examination of the time series approach is the aim of ongoing research. It needs to account for monthly uncertainties in all input data sets. Similar to the trend-based combination, challenges include (1) the consideration of uncertainties of all data sets, (2) differences in spatio-temporal sampling of both sensors, and (3) dealing with the resolution discrepancies including the consideration of signal leakage in GRACE observations. For further discussion of the challenges associated with combining geodetic time series, the reader is referred to , for example. It should be noted that state-space approaches in geodetic Earth system research show promising results dealing with time-variable geophysical signals in observational time series .