Glacial isostatic adjustment (GIA) is a major source of uncertainty for ice and ocean mass balance estimates derived from satellite gravimetry. In Antarctica the gravimetric effect of cryospheric mass change and GIA are of the same order of magnitude. Inverse estimates from geodetic observations hold some promise for mass signal separation. Here, we investigate the combination of satellite gravimetry and altimetry and demonstrate that the choice of input data sets and processing methods will influence the resultant GIA inverse estimate. This includes the combination that spans the full GRACE record (April 2002–August 2016). Additionally, we show the variations that arise from combining the actual time series of the differing data sets. Using the inferred trends, we assess the spread of GIA solutions owing to (1) the choice of different degree-1 and C

The quantification of recent and current sea level changes plays a crucial role for local, regional, and global projections. Mass changes of the Greenland and Antarctic ice sheets are responsible for approximately 20 % of the global mean sea level rise between 1991 and 2010

Large uncertainty in the ice mass change estimates derived from space gravimetry is related to viscoelastic deformation of the solid Earth by glacial isostatic adjustment (GIA). This is the deformation of the solid Earth due to loading variations through sequences of past glacial advance and retreat over many millennia. The manifestation of ice sheet and GIA mass change signals is superimposed and is of the same order of magnitude over Antarctica

One approach to determine the GIA signal is forward modelling

In an alternative approach, satellite gravimetry and altimetry are combined to separate the GIA and ice-related mass signals

Recent studies separate the individual processes of the ice sheet and the underlying bedrock with statistical modelling

We analyse the sensitivity of inverse GIA estimation on the choice of data input and methodology, thereby identifying both the possible causes of discrepancies and the uncertainty. Our inverse GIA estimation is based on the approach of

Another focus of this research is on the use of ice altimetry data. Different altimeter missions such as Envisat, ICESat, or CryoSat-2 use different observation techniques and differ in their spatial and temporal coverage. The multi-mission (MM) altimetry data set delivered by

Section 2 derives and describes in detail the combination approach, bias corrections using the low-precipitation zone (LPZ) of East Antarctica, estimation of the mass balance, and filtering. Afterwards, we explain how the errors for the firn process models are characterized and how the sensitivity analysis is performed. Furthermore, the approach is adapted to extract a more nuanced and self-consistent combination of input-data time series. Section 3 describes the products employed, processing steps, and additional assumptions. Section 4 presents results of derived uncertainties of the firn process models, the sensitivity analysis, and the time-series-based combination. Finally, the results are discussed and the most important findings are summarized in the conclusions.

Analogously, the overall linear SEC derived from altimetry

The process-related elevation and area density changes are linked with effective density assumptions (

In

The following steps are performed in sequence.

In Step 2, the

In Step 3, the

In Step 4, the debiased ice mass trend is calculated as

For the necessary noise suppression we use GRACE data with a de-striping filter applied (

In Eqs. (

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

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.

By assumption the GIA signal in the resulting time series

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.

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

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

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

This section specifies the data sets and processing steps used in the sensitivity experiments which are summarized in Table

The SECs from

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

Part of the altimetry-derived SEC is caused by the elastic BEC of the solid Earth due to present-day ice mass change (

GRACE-derived monthly mass variations are calculated from the ITSG-Grace2016 monthly gravity field solutions up to a degree and order of 90

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

Furthermore, the influence of the flattening term C

A critical point is filtering because the monthly solutions are noisy and have a correlated error pattern

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.

Information on variations in the firn layer is required in the combination approach (Eq.

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 (

The used firn model IMAU FDM

The LPZ (Fig.

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

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

The density mask for

Overview of all performed experiments of the sensitivity analysis (Sects.

There are considerable differences between the time series of cSMBA from the RACMO2 and MAR SMB products for each cell. Figure

Cumulated surface mass balance anomalies (cSMBAs) of the regional climate models RACMO2.3p2 (blue;

Three uncertainty assessments for the area density change (ADC) trend induced by cumulated surface mass balance anomalies (cSMBA).

Results from the sensitivity experiments. This table is structured like Table 2 in

To extract the dominant error patterns, a spectral decomposition of the 32 7-year trend differences (see Sect.

Inverse GIA estimates are calculated using different choices of (1) degree-1 solutions, (2) C_{RE} (Eq.

Biased total-mass changes for different C

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

Applying the approach to different time intervals April 2002–August 2016 and July 2010–August 2016 leads to debiased total-mass changes of

The addition of the EOFs (Sect.

Mass change results for the entire AIS over the interval March 2003–October 2009 from experiments with different data products and methodological choices. The LPZ-based bias correction was applied. Debiased total-mass change (solid black lines) is separated into debiased GIA-related mass (red) and ice mass change (blue). Dotted lines show the total-mass changes that arise when no bias corrections are applied. The case of no bias correction is further illustrated in Fig. S7.

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.

We used the values of

The GIA-related mass time series of the AIS (with 400 km buffer zone) resulting from the combination of the monthly gridded time series (July 2010–August 2016) with (blue) and without (red) LPZ-based bias correction of the determined GIA signal.

For the July 2010–August 2016 time period.

Since the aim of this study is to examine the sensitivity of the inverse approach to several data input and methodological choices, differences from the reference experiment are discussed on the basis of the selected processing parameters.

To test our data processing we performed a run with similar input data as used in

In general our GIA estimate (Fig.

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

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 comparison of integrated mass changes calculated in this study and those published in

The use of several degree-1 and C

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

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

A crucial point in the combination approach is the case distinction for

The

The propagation of the empirically determined error patterns (EOFs 1–3) of the firn process models (Sect.

Uncertainties assumed in

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.

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 investigated a combination method to isolate the GIA signal from satellite gravimetry and altimetry data. Our analysis is an extension of ideas presented by

The comparison between the data sets used in this study shows impressive similarities in terms of the spatial pattern of determined trends (Fig.

Following

The definition of

A critical feature of the combination approach is the observational constraints that are imposed on the inversions by the limitations of the actual geodetic satellite sensors. On the one hand, altimetry enables the derivation of SEC with a high resolution. However, observations are missing in some areas, especially in areas of high topographic relief, such as valleys and mountainous coastal regions. In many of these regions lateral ice mobility may have a more complex relationship to ice heights that are extracted from altimetry as SEC. On the other hand, GRACE records all mass changes, albeit with lower resolution and signal-to-noise ratio. Because of the availability of the MM altimetry from

For the integrated mass changes over the AIS area, results of our sensitivity analysis are as follows. (1) The use of different degree-1 and C

Our results do not fully address the uncertainty introduced by input parameters. Especially important may be the assumption of an equilibrium state assumed in the firn model. In future work, improvement is needed for the correction of apparent biases and for the separation of processes in the firn and the ice layer. This might improve the self-consistency of GIA inverse estimates from satellite observations and generate a more appropriate time-series-based estimate.

GRACE monthly solutions:

The supplement related to this article is available online at:

MOW and MH conceptualized the study. MOW performed the investigation, computation, and visualization tasks and wrote the manuscript. LS provided the ice altimetry time series. AG supported the GRACE data processing and the development of the data combination approach. SRML, PKM, and MRvdB provided the SMB output from RACMO2.3p2, the FDM, and assistance in their uncertainty characterization. All co-authors discussed and improved the manuscript.

Michiel R. van den Broeke is a member of the editorial board of the journal. The authors declare that they have no conflict of interest.

We thank Cécile Agosta (Université de Liège, Belgium) for providing the SMB output and density fields of the MAR model, Erik Ivins (JPL, Caltech, Pasadena, USA) for improving the manuscript, and Olga Engels (Universität Bonn, Germany) for discussion on details of data combination strategies. We thank the editor Pippa Whitehouse and the two anonymous referees for their constructive reviews which helped to improve the manuscript. We acknowledge the German Space Operations Center (GSOC) of the German Aerospace Center (DLR) for providing continuously and nearly 100 % of the raw telemetry data of the twin GRACE satellites. Further, we thank the developers of Matplotlib

This work was supported in part through grant no. HO 4232/4-1 “Reconciling ocean mass change and GIA from satellite gravity and altimetry (OMCG)” from the Deutsche Forschungsgemeinschaft (DFG) as part of the Special Priority Program (SPP)-1889 “Regional Sea Level Change and Society” (SeaLevel).

This paper was edited by Pippa Whitehouse and reviewed by two anonymous referees.