We use along-track altimetry measurements from ICESat 633 Level 2, providing high-resolution elevation change observations for the period February 2003 to October 2009. Two corrections are applied to this data set: the range determination from transmit-pulse reference-point selection (centroid vs. Gaussian; Borsa et al., 2014) available from the National Snow and Ice Data Center (NSIDC) and the inter-campaign correction (Hofton et al., 2013). The centroid–Gaussian correction is a well-established correction and has been incorporated into the latest ICESat release (634). Concerning the ICESat inter-campaign bias (ICB) correction, uncertainties are available at Hofton et al. (2013). Furthermore, several studies have determined this correction from different methodologies. For a summary of published ICESat ICB corrections, see Scambos and Shumman (2016). Note that these corrections are part of a widely accepted procedure, and their effect on the elevation rates and uncertainties caused by varying the processing choices have not been evaluated. Because the ICESat tracks do not usually overlap, a regression approach is used in which topographic slope (both across-track and along-track) and the rate of surface-elevation change yICESath are simultaneously estimated using the “plane” method (Howat et al., 2008) over areas that span 700 m and are a few hundred meters wide. A regression is only performed if a plane has at least 10 points from four different tracks that span at least 1 year. Regression was carried out twice. First, individual elevation measurements with corresponding residuals outside the range of two standard deviations were detected; then, the regression was repeated omitting these outliers. The standard deviation of the regression coefficient, here taken as the uncertainty in the elevation rate, σh (here, ICESat) is calculated by the propagation of the residuals of the input data and the estimated topographic heights, s^ICESat=∑ei2(n-2)∑xi-x‾2, to the trend parameter (see Eq. 1 in Hurkmans et al., 2012), where e is the vector of residuals, n is the sample size (i=1,2,…,n) and x is the vector of input elevations with mean x‾. This standard deviation (σICESat) takes into account the sample size and the variance in both input data and residuals of the regression (Hurkmans et al., 2012). The residuals of the regression are used as they quantify the approximation of fitting the data with a plane. The exact ICESat observation periods are shown in the Appendix (A1, Table A1). Then, the elevation rate and its uncertainty are interpolated (bilinearly) to a common 10 km ×10 km grid in polar-stereographic projection (central latitude 71∘ S; central longitude 0∘ W; origin at the South Pole; WGS-84 reference ellipsoid).

We use a time series of elevation changes derived from along-track Envisat radar altimetry data for the interval January 2003 to October 2009 (coeval to ICESat time span). Elevation rates yEnvisath are obtained at points every 1 km along track by binning all the echoes within a 500 m radius. Then, a 10-parameter least squares model is fitted in order to correct for the across-track topography and changes in snowpack properties. The least square model is defined in Flament and Remy (2012). The estimated parameters include parameters determined for the backscatter, leading-edge width and tailing-edge slope, the mean altitude, quadratic surface slope parameters to define surface curvature, and a linear time trend. A digital elevation model was not used for the correction of the topographic slope. For processing reasons, the temporal resolution is resampled from 35 days to monthly periods for each grid cell before estimating the elevation rates. This has a minor effect on the elevation rate estimate (smaller than ±1 cm yr-1) and reduces the standard deviation by about 14 %. As with ICESat, the elevation rate is interpolated to a common 10 km ×10 km polar-stereographic grid (and 20 km ×20 km for download in the archive), and the standard deviations of the rates within each grid cell are taken as an estimate of the measurement uncertainty, σEnvisat, according to Eq. (1).

We produce a combined rate of surface-elevation change product from the ICESat and Envisat data sets for the Antarctic ice sheet, yh. The aim is to take advantage of the high spatial resolution of ICESat data and the high temporal resolution and high-track density of the Envisat data.

We combine the two altimetry data sets based on their common 10 km ×10 km polar-stereographic grid. At each location, the elevation rate with the smallest standard deviation is chosen from either Envisat or ICESat data sets. We prefer this masking procedure instead of a weighted average, in order to avoid introducing possible biases associated with gridded elevation rates of very high uncertainty.

Figure 1 shows the resulting mask underlying the combination. It is evident that some grid points are only represented by either ICESat or Envisat. Most prominent is the narrowing of the polar gap with ICESat data, resulting from the 81.5∘ S latitude limit for Envisat compared to 86∘ S for ICESat due to satellite orbit inclination. On the Antarctic Peninsula, Envisat picks up some points that are not present due to a sparser track coverage in the ICESat data set. As expected, ICESat outperforms Envisat in terms of the uncertainty in the elevation rate over steep topographic slopes and along the ice sheet margins. This is due to the smaller footprint of the laser altimeter, its higher accuracy and lower slope-dependent uncertainty (e.g., Brenner et al., 2007). On some flat areas and over some faulty ground tracks, where ICESat data measurements are scarce, however, Envisat provides better temporal and spatial coverage, leading to better accuracy of the resulting elevation rates. The resulting combined data set of surface-elevation rates and its uncertainties are shown in Fig. 2.

The elevation rates derived from ICESat and Envisat are corrected for changes in the firn layer thickness using the firn compaction model of Ligtenberg et al. (2011), which is driven by the regional atmosphere and climate model RACMO2/ANT (Lenaerts et al., 2012). We determine the firn compaction for January 2003 to October 2009, with respect to the mean of the years 1979 to 2002, and estimate a temporal linear trend, hcomp. The model output is re-gridded onto the 10 km ×10 km common grid using nearest-neighbor interpolation. The standard deviation of the re-gridding is less than 1 cm yr-1, causing a maximum change of 2 % of the firn compaction rate. Note that the firn compaction model has a spatial resolution of 27 km, potentially neglecting finer-scale processes relevant for the altimetry data. Clearly, the re-gridding uncertainty stated above is merely a minimum estimate, neglecting, for example, uncertainties in the calibration or the atmospheric forcing of the firn compaction model.

The data were resampled from every 2 days to monthly mean time periods for every grid cell before estimating elevation rates. As with the Envisat and ICESat data, no seasonal terms were co-estimated and removed (i.e., annual and semiannual). We do not apply an a priori correction for surface mass balance (SMB) trends, in accordance with GRACE processing (Sect. 5), which requires defining a climatological reference period. Note that applying the commonly used reference period (1979 to present) leads to spurious accumulation anomalies in the altimetry data (see Appendix A2, Fig. A1). The derivation of an adequate climatological reference epoch in the RACMO2/ANT simulations is in itself challenging and beyond the scope of this paper.

The total uncertainty in the rate of elevation change from satellite altimetry is calculated by σh=σEnvisat/ICESat2+σFirn2, where the standard deviation of the firn correction, σFirn, is the formal regression uncertainty (neglecting model uncertainties, as these are not available), and we assume the error sources to be uncorrelated. It is recognized that neglecting uncertainties in the firn model leads to underestimated values of σh. However, the magnitude of the firn correction itself is small (see Appendix A2) compared to the observational uncertainties, and the associated underestimation of σh is likely to be small.

Annual elevation trends from a combination of Envisat and ICESat data are provided for the time period between February 2003 and October 2009. Trends have been corrected for firn densification processes using RACMO2/ANT. Elevation trends are provided in a 20 km polar-stereographic grid (central meridian 0∘, standard parallel 71∘ S) with respect to the WGS84 geoid. X and Y are given in kilometers, and the elevation rate and its standard deviation are given in meters per year.

The altimetry data and related ancillary data are directly accessible in the PANGAEA repository: http://hs.pangaea.de/model/Sasgen-etal_2017/Ice_sheet_topographic_change.zip.

Next, we briefly compare the uplift rates at individual sites (data span: decimal year 1995.0–2013.7) derived from the GPS processing described above with those available from three previous studies: Thomas et al. (2011) (data span 1995.0–2011.0), Argus et al. (2014) (data span: decimal year 1994–2012) and the more geographically limited set of Wolstencroft et al. (2015) (data span 2006–late 2013, focused on Palmer Land). It should be noted that the REGINA and Wolstencroft et al. (2015) rates are in ITRF2008 (International Terrestrial Reference System 2008), the Thomas et al. (2011) rates are in ITRF2005 (which has negligible scale or translation differences to ITRF2008) and the Argus et al. (2014) rates are in a reference frame specific to their paper, which they note yields 0.5 mm yr-1 more uplift than ITRF2008 at high southern latitudes. All rates from Argus et al. (2014) in Tables 2, 3 and 4 are shown as given in the original paper.

Due to the large number of Antarctic sites (in total 118), we focus the comparison on the uplift rates and uncertainties derived by the methods “cats, cats” (Table 2) and “prop, prop” (Table 3). Uplift rates resulting from our study are provided in Appendix A4 for all sites (Table A2). Table A3 shows comparisons with the values of Thomas et al. (2011) and Argus et al. (2014) for “prop, eman” sites not shown in the main text. All uplift rates, yu , are in mm yr-1, with uncertainties reflecting 1σ standard deviations, σu. Sites with particularly complex nonlinear time series, such as those at O'Higgins (ohi2, ohig) and Palmer (palm) in the Antarctic Peninsula, are omitted here, as comparison with different studies is potentially misleading due to the effects of different measurement time periods. Table 2 shows data for selected sites with long time series, where uplift rate and uncertainty were derived using the CATS software (Williams, 2008). Uplift rates at the majority of the GPS sites agree within uncertainty, except syog (Syowa), where the REGINA value is between that from the other two studies. The uncertainty limits for the REGINA value and the Argus et al. (2014) just meet at 0.9 mm yr-1, even when allowing for the ∼ 0.5 mm difference in reference frames, but the Thomas et al. (2011) value does not. This may be due to the fact that Thomas et al. (2011) estimate two offsets in the series. Table 3 shows uplift rate comparisons for sites where the “prop, prop” method was used; the noise characteristics are derived from median values from CATS software results for longer site records and then propagated in the parameter estimation in which annual and semiannual parameters were also estimated along with the trend. Again, the rates agree within uncertainty, except for site belg where there is disagreement with Thomas et al. (2011). This may be due to their shorter data span. Table 4 shows comparisons for sites where the REGINA rates and uncertainties have been manually evaluated based on the spread of rates obtained by subsampling the time series (“rman” method). There is a large difference (over 10 mm yr-1) in the values at capf (Cape Framnes) between the REGINA value (4.0 ± 1.4 mm yr-1) and the Argus et al. (2014) value (15.0 ± 4.2 mm yr-1). Interestingly, the Wolstencroft et al. (2015) rate values for bean, gmez, lntk, mkib and trve are all systematically higher than the REGINA values, by an average of just over 3 mm yr-1, and the uncertainties we assigned are also several times larger. The systematic differences between Wolstencroft et al. (2015) and the REGINA values for Palmer Land are currently unexplained and a matter of ongoing investigation.

The GPS data and related code are directly accessible in the PANGAEA repository: http://hs.pangaea.de/model/Sasgen-etal_2017/In_situ_GPS_uplift_rates.zip.

Due to effects like the propagation of measurement noise and temporal aliasing, a large proportion of the variations contained in the monthly solutions is related to noise. The noise of the monthly solutions is lowest close to the pole and exhibits a characteristic north–south-oriented stripe pattern. This is visible in the gravity field rate and the propagated root-mean-square (rms) uncertainties shown in Fig. 3. In order to remove the stripe pattern, we apply the de-correlation filter of Swenson and Wahr (2006) (hereinafter, “Swenson filter”) specifically tuned to optimize the recovery of the gravity field rate over the region of Antarctica, which is detailed in Sect. 4.1. Figure 3 shows that the de-striping procedure reduces the rms uncertainty in the rate by approximately 1 order of magnitude.

For isolating gravity field rates, the second step in the processing is the reduction in de-trended variations in the surface mass balance, caused by accumulation events. The data set used for this purposes is the RACMO2/ANT (Lenaerts et al., 2012) converted into monthly sets of spherical harmonic coefficients. The reduction in these accumulation variations does not change the temporal linear trend, but it reduces rms uncertainties especially in coastal regions (Fig. 3). Details are provided in Sect. 4.2.

The performance of the GRACE satellite system was weaker in the early mission phase due to issues with the star cameras of the satellites (Christoph Dahle, GFZ German Research Centre for Geosciences, personal communication, 2015; Fig. 5). A rate estimate with uniform weighting of all months does not account for these variations. Therefore, in the last step, month-dependent uncertainties are estimated and applied as weights during the linear regression of the temporal linear trend. This slightly changes both the resulting rate estimate, as well as its rms uncertainties. Details are provided in Sect. 4.3.

Finally, after post-processing and evaluation of the gravity field rate (Sect. 4.4), we select the GRACE release and cutoff degree providing the lowest uncertainty level (Sect. 4.5) as a reference input for our joint inversion for present-day ice-mass change detailed in the second REGINA paper (Sasgen et al., 2017a).

The Swenson filter has been proven to effectively reduce the typical north–south-correlated error structures of GRACE monthly solutions. The filter is based on the observation that these structures correspond to correlated patterns in the spherical harmonic domain, namely correlations within the coefficients of the same order and even or odd degree (Swenson and Wahr, 2006). The standard way of fitting and removing these patterns is by adjusting polynomials to the respective sequences of spherical harmonic coefficients, independently for individual months. Parameters to choose are the degree of the polynomial npol and the minimum order mstart, starting from which this procedure is applied. In principle, a higher-degree polynomial reduces the variability in coefficients of even/odd degree and results in stronger filtering also at a lower minimum order – however, the behavior of the filter may differ for regional applications, as discussed below. Note that the tuning of other parameters has been presented previously, e.g., the window width (Duan et al., 2009) or the degree range to which the filter is applied. Chambers and Bonin (2012) have assessed these parameter options with regard to the new GRACE RL05 solutions and global oceanic signals. Here, we perform a detailed analysis of the choice of the Swenson filter parameters in order to optimize the signal-to-noise characteristics of the rate of the gravity field change over Antarctica.

We assess signal corruption by applying the filter to a synthetic test signal, which is based on high-resolution elevation rates from satellite altimetry and reflects the prevailing signatures of present-day ice change with sufficient realism. For each choice of filter parameters, the signal corruption is assessed as the rms difference between the original and the filtered synthetic signal (rmssignal). The rms is evaluated in terms of water-equivalent height per year for the signal components within the region south of 60∘ S latitude.

For assessing the noise and noise reduction in the filtered fields, we face the task of separating the noise from the geophysical signals in the gravity field rates derived from GRACE. Here we attempt such a separation by reducing a priori information on the rate of ice-mass change from the GRACE fields and considering the residual as an upper bound representation of noise. The a priori information is, again, based on elevation rates. For the noise assessment we then take the rms of the residual rates in terms of water-equivalent height per year, rmsnoise, again for the region south of 60∘ S latitude. Since the residual gravity field rates may still contain some geophysical signal, we consider this noise estimate as an upper bound for the true GRACE uncertainties. It should be stated that, after the Swenson filtering, an additional Gaussian filtering is applied to the signal and noise models with a 200 km filter width, which was determined to be the optimal smoothing half width for the signal-to-noise ratio in the GRACE spectra by Wiener optimal filtering (Sasgen et al., 2006) as reflected in the degree-amplitude spectrum.

Figure 4 shows the assessed signal corruption and noise reduction as a function of the two Swenson filter parameter choices, the polynomial degree npol and the minimum order mstart. The results are shown for the gravity field expanded to degree and order jmax=90 of the GFZ RL05a coefficients, even though using jmax=50 and CSR RL05 yields similar results. As expected, the signal corruption (rmssignal) increases with increasing strength of the Swenson filter, that is with increasing npol and the decreasing minimum order mstart. In terms of noise reduction, we see as expected that stronger filtering (increasing npol; decreasing mstart) decreases the rmsnoise (Fig. 4). However, we find that for filter parameters mstart<10, this pattern is reversed, and rmsnoise increases again. A closer analysis indicates that the consideration of the low orders in the Swenson filtering transfers energy (both from signal and noise) from low latitudes to midlatitudes to the polar regions, which leads to a considerable signal corruption over the region of interest. We avoid this degradation by limiting the range of filter parameters in the subsequent optimization of the gravity trends to mstart≥10.

To define the optimal filter parameters a quadratic sum of the signal corruption and noise reduction is computed, allowing us to balance both effects. The optimal values are mstart=12 and npol=7, as indicated in Fig. 4c. These filter parameters are subsequently used. For comparison it is stated that Chambers and Bonin (2012) found mstart=15 and npol=4 to be optimal for oceanic applications. Note that the signal corruption is assessed only to optimize the de-striping filter. Possible signal degradation due to de-striping is not included in the uncertainty estimate of the optimally filtered GRACE trends. However, signal loss due to the additional smoothing with a 200 km Gaussian filter is accounted for by applying the same filter to the viscoelastic response functions, as well as the altimetry-based input fields (Appendix A5).

The temporal variations in the Antarctic gravity field show a strong year-to-year fluctuation, apart from the linear trend (Wouters et al., 2014). A large portion of the nonlinear signal in geodetic mass and volume time series is well explained by modeled SMB fluctuations (Sasgen et al., 2010; Horwath et al., 2012). Towards the ultimate goal of isolating the linear GIA signal from time series of mass change, we removed nonlinear effects of modeled SMB variations from the GRACE time series; for this we calculate the monthly cumulative SMB anomalies with respect to the time period 1979 to 2012 obtained from RACMO2/ANT (Lenaerts et al., 2012).

We then transfer the monthly cumulative SMB anomalies in terms of their water-equivalent height change into the spherical harmonic domain and subtract them from the monthly GRACE coefficients. In principle, the reduction in the SMB variations from the GRACE time interval has two effects: first, it may change the overall gravity field rate derived from GRACE, depending on the assumption of the SMB reference period. Ideally, the reference period reflects a state of the ice sheet in which input by SMB equals the outflow by ice discharge, and SMB anomalies estimated for today reflect the SMB component of the mass imbalance. However, any bias in the SMB in the reference period leads to an artificial trend in the ice sheet mass balance attributed to SMB. This is an undesired effect, and to avoid it, we de-trend the cumulative SMB time series for the time interval coeval to the GRACE analysis (February 2003 to October 2009) before subtracting it from the GRACE gravity fields, yielding zero difference in the gravity field rates before and after processing Step 2 (Fig. 3). The second effect is the reduction in the post-fit rms residual for this known temporal signal variation. After reducing the SMB variations, the propagated rms uncertainty in the derived gravity field rate becomes closer to the uncertainty level of the GRACE monthly solutions (Fig. 3).

The quality of GRACE monthly solutions changes with time, for example due to changing orbital sampling patterns (Swenson and Wahr, 2006). Figure 5 shows the temporal evolution of rms uncertainties in the monthly GRACE gravity fields in the Antarctic region; shown are residual mass anomalies, integrated over Antarctica, after the grid-based removal of the temporal linear trend and annual oscillation components and after the application of the filtering described in Step 1 and the removal of the SMB fluctuations in Step 2. Note than an annual oscillation component is included to remove possible seasonal fluctuations in SMB not captured by the regional climate model. However, omitting the annual oscillation component yields similar results. The residual monthly mass anomalies are attributed to noise and are used as weights in our least-squares linear regression, applied as Step 3 of the GRACE processing. Figure 5 shows that these uncertainties are higher during early 2003. Applying the monthly dependent weighting has the effect of reducing the influence of the first months of the year 2003 on the estimated gravity field rate, which is similar to shortening the time series, given the relatively large uncertainties. As expected, the post-fit rms uncertainty associated with the rate reduces if the early months of the year 2003 are excluded. Altogether, the month-dependent weighting reduces the magnitude of stripe patterns characteristic of the uncertainty in GRACE monthly solutions and yields a more realistic estimate of the uncertainty associated with the gravity field rates (Fig. 3).

Figure 6 shows the estimated rms uncertainty in the gravity field rate over Antarctica, after post-processing. It is evident that the largest uncertainties are located in a ring south of -80∘ S latitude. This is explained by the design of the Swenson filter; little or no noise reduction is achieved close to the poles, as the gravity field is represented by near-zonal coefficients, which pass the filter mostly unchanged (mstart=12). Extending the kernel of the Swenson filter to these near-zonal coefficients (mstart≤10) creates high signal corruption and is not suitable for the optimal rate estimate over Antarctica (see Sect. 4.1). Larger uncertainties are also estimated for the Ronne and Ross ice shelf areas, which are most likely a consequence of incomplete removal of the ocean tide signal during the GRACE de-aliasing procedure (Dobslaw et al., 2013). It should also be stated that the rms uncertainty estimate does not include possible systematic errors in the GRACE solutions, e.g., due to long-term drift behavior of the observing system.

Our evaluation of the monthly GRACE uncertainties (Fig. 5), as well as the propagated rms uncertainty in the temporal linear trend (Fig. 6) indicates that the lowest noise level for the Antarctic gravity field rate (February 2003 to October 2009) is currently achieved with GRACE coefficients of CSR RL05, expanded to jmax=50. We therefore refrain from including coefficients with jmax>50 in order not to compromise the rate estimates by unnecessarily increasing the noise level (see Appendix A5, Fig. A3). We adopt CSR RL05 with jmax=50 as our preferred solutions for the representation of the gravity field rates over Antarctica, even though GFZ RL05 with jmax= 50 yields very similar rates (Fig. 6). This choice is supported by the joint inversion, as CSR RL05 with jmax=50 provides the highest level of consistency (lowest residual misfit) with the altimetry and GPS data sets (see the second REGINA paper, Sasgen et al., 2017a, Supplement Sect. S3), which we interpret to indicate a minimum of spurious signals in the trends. To account for the uncertainty related to our choice of the solution, we consider not only rms uncertainties in the GRACE rates but also solution differences, in the uncertainty in the final GIA estimate (Fig. 6). The solution differences represent the absolute deviation between trends from GFZ RL05 and CSR RL05 (February 2003 to October 2009; cutoff degree jmax= 50). These are then summed up and squared with the propagated rms uncertainties. It is acknowledged that the solution differences contain systematic noise arising from the GRACE processing; the pattern and magnitude may change over time. However, they provide a measure of how much the results will change if a GRACE release alternative to CSR RL05 is considered. The difference between GRACE rates filtered with Gaussian smoothing of 200 km and the optimized Swenson filter together with Gaussian smoothing of 200 km is shown in the Appendix A5, Fig. A4.

The gravity data and related code are directly accessible in the PANGAEA repository: http://hs.pangaea.de/model/Sasgen-etal_2017/Geoid-height_change_from_GRACE_satellite.zip.

The load function σt,ϑ is disc shaped with a constant radius of ca. 63 km. The radius of 63 km matches the mean radius of the discs south of 60∘ S of the geodesic grid (here, the Icosahedral Nonhydrostatic (ICON) Model 1.2 grid, status 2007, e.g., Wan et al., 2013), which underlies the joint inversion of the altimetry, gravimetry and GPS observations (see the second REGINA paper, Sasgen et al., 2017a). The resolution of the geodesic grid is chosen to allow for an adequate representation of the load and viscoelastic response with regard to the input data sets while minimizing the computational cost. The disc load experiment consists of a linear increase in the ice thickness at a rate of 0.5 m yr-1 continuing until a new dynamic equilibrium state between load and response is reached. After the application of the constant loading rate, two extra time steps are done with no loading change to give the purely viscoelastic response. For West Antarctica, the loading rate is held constant for 2000 years, for East Antarctica it is 15 000 years, which are longer times than needed to reach dynamic equilibrium (see Appendix A8). With reference to the assumed ice density of 910 kg m-3, this thickness increase corresponds to a mass gain of ca. 5.6 Gt yr-1. Then, to obtain the signal component of the viscous Earth response only, the elastic response and the direct gravitational attraction of the load are subtracted.

The experiment is designed as an increasing load, for example representative of the ceasing motion of the Kamb Ice Stream (Ice Stream C; Retzlaff and Bentley, 1993), West Antarctica. Due to the linearity of the viscoelastic field equations, it is not necessary to calculate separately the equivalent unloading experiment, -σ(tϑ), for example corresponding to the past and present glacier retreat of the Amundsen Sea sector, West Antarctica (Bentley et al., 2014 and Rignot et al., 2014, respectively). Among others, the combined inversion of the altimetry, gravity and GPS data (the second REGINA paper, Sasgen et al., 2017a) solves for the magnitude and the sign of the load, allowing for ice advance as well as ice retreat.

We set up an ensemble of 58 simulations representing different parameterizations of the viscosity structure (Table 5), split into West Antarctica (56 simulations) and East Antarctica (2 simulations). The ensemble approximately covers the range of values of the viscosity and lithosphere thickness inferred from Priestley and McKenzie (2013). For West Antarctica, varied parameters are the lithosphere thickness, hL (30 to 90 km in steps of 10 km), the asthenosphere viscosity (1×1018 to 3×1019 Pa s in four steps) and the presence of a DL, with 1018 Pa s. For East Antarctica, we employ parameter combinations appropriate for its cratonic origin with hL of 150 and 200 km and an asthenosphere viscosity equivalent to the upper-mantle viscosity of 5×1020 Pa s. These values lie in the range of previously applied viscosity values in Antarctica (Nield et al., 2012; Whitehouse et al., 2012; van der Wal et al., 2015). For the radial layering of the elastic properties, we adopt the Preliminary Reference Earth Model (PREM; Dziewonski and Anderson, 1981).

Later, in the joint inversion, the distribution of viscoelastic response functions is based on the Earth structure model of Priestley and McKenzie (2013). Priestley and McKenzie (2013) provide a global distribution of viscosity values up to a depth of 400 km, which is sampled at the location of the geodesic grid. We then define a threshold value for the viscosity (here, 1022 Pa s) above which the Earth response is considered purely elastic and infer the associated thickness of the elastic lithosphere. The impact on the final joint inversion estimate of changing the threshold value of 1022 Pa s is presented in the second REGINA paper (Fig. S4 in Sasgen et al., 2017a). Note that the Earth response in the equilibrium state only depends on the lithosphere thickness (independent of viscosity), which is therefore considered as the main Earth model parameter in the joint inversion. Further details are presented in the second REGINA paper, Sasgen et al. (2017a).

The calculated response functions for surface deformation (radial displacement) and gravity (geoid-height change) are discretized along 1507 latitudinal points within the range 0≤ϑ≤90. Simulations are typically run over 2 kyr with a temporal resolution of Δt=10 years (plus two time steps with constant load thickness). For East Antarctic parameterizations, the simulation period was extended to 20 kyr due to the higher upper-mantle viscosities and associated slower relaxation. However, note that the ratio of geoid-height change versus radial displacement falls off to a factor of 1/e2 relative to the initial value after ca. 2 kyr of simulation (Appendix A6, Fig. A5). The forcing expected in central East Antarctica is an increase in accumulation towards present-day conditions after ca. 7 ka BP (van Ommen et al., 2004), also justifying the use of equilibrium kernels for East Antarctica. The time derivatives of the radial displacement yu and of the geoid-height change yg are calculated with a central difference scheme.

Examples of response functions to the loading detailed in Sect. 5.1 for the rate of radial displacement, yu, and rate of geoid-height change, yg, are shown in Figs. 7 and 8, respectively. Instantaneously, the increasing load, σ˙(t)=const., induces an elastic response that is characterized by subsidence and an increase in the direct gravitational potential (dashed lines in Figs. 7 and 8, respectively). This is the elastic response function adopted in the joint inversion. Note that the elastic response function will not differ between East and West Antarctica, as it is entirely based on the distribution of densities and elastic parameters provided by the PREM. As the load buildup continues, the instantaneous response is followed by the viscoelastic response, which depends in timing and magnitude on the underlying lithosphere and viscosity structure, further increasing the displacement rates, yu (blue to red lines in Fig. 7). The compensation by solid Earth deformation is reflected in the decreasing geoid rate, yg (Fig. 8). After a certain time, which depends on the value of the asthenosphere viscosity, a new dynamic equilibrium state is reached at which yu and yg do not change in time any more. In the last two time steps, the load is kept constant (yg(t)=) and the responses in yu and yg are only caused by the relaxation of the Earth's viscoelastic deformation (solid black line in Figs. 7 and 8), which is the viscoelastic response function adopted in the joint inversion.

Figure 9 shows the response of yu for four end-member sets of Earth model parameters with a thick lithosphere, weak asthenosphere (TkWk: hL=90 km; ηAS=1×1018 Pa s); a thick lithosphere, strong asthenosphere (TkSg: hL=90 km; ηAS=3×1019 Pa s); a thin lithosphere, weak asthenosphere (TnWk: hL=30 km; ηAS=1×1018 Pa s); and a thick lithosphere, strong asthenosphere (TnSg: hL=30 km; ηAS=3×1019 Pa s), without a ductile layer, DL. In this context, thick/thin and strong/weak refer to values in comparison to the “average” value of the ensemble for West Antarctica; an elastic lithosphere of thickness 90 km (here, “Tk”) is in the range of the global average continental lithosphere usually applied in GIA studies (e.g., Peltier, 2004) or that of East Antarctica (150 to 200 km). Figure 10 shows the response in yu for the same end-member set of Earth model parameters with a DL included. It should be stated that the Earth structure with hL=30 km and a DL is considered very extreme because in this case the ductile layer extends down to the asthenosphere and an elastic mantle lithosphere is missing.

Figures 9 and 10 show that for the weak asthenosphere (ηAS=1×1018 Pa s), viscoelastic deformation is visible already after 1 decade of loading (or unloading), leading to considerably larger subsidence rates compared to the purely elastic case even on very short timescales. For these Earth model parameters, a new dynamic equilibrium state is achieved within a few centuries. The rates of subsidence in this equilibrium then primarily depend on the support provided by the flexure of the elastic lithosphere.

For the extreme TnWk case, equilibrium rates of -45 mm yr-1 are achieved at the load center, and considerable subsidence of -20 mm yr-1 already occurs after 10 years of loading (Fig. 9). Increase in asthenosphere viscosity (TnSg case) reduces the viscous material transport and leads to a slower adjustment towards the dynamic equilibrium state, which takes more than 1 kyr. It should be stated that in our definition of the ensemble parameters, reducing the lithosphere thickness in turn increases the thickness of the asthenosphere (bottom depth of asthenosphere is fixed), which facilitates lateral material transport inside the asthenosphere.

The consideration of the DL in the Earth structure causes a thinning of the effective elastic lithosphere. As a consequence, greater and more localized subsidence rates are produced for all sets of parameters (Fig. 10). Interestingly, in the case of a thick elastic lithosphere (90 km), the radial displacement exhibits a local minimum at around 120 and 160 km distance from the load center (Fig. 10), which is a consequence of the viscous material transport inside the ductile layer. The maximum equilibrium rate of -76 mm yr-1 is achieved for the ThWk case with DL, where the viscous deformation leads to rates of -25 mm yr-1 already after 10 years of loading.

Although the approach of modeling response functions to axisymmetric disc loads and subsequently superposing them is very efficient in terms of the computational cost, this simplification introduces some limitations. First, the superposition of response functions representing different Earth structures neglects the transmission of stresses between these regions – a problem that can only be resolved with fully three-dimensional solid Earth modeling (e.g., van der Wal et al., 2015). The largest impact for the displacement rates is expected in regions with lateral contrasts in lithosphere thickness and mantle viscosity such as the Transantarctic Mountains. Second, the constant disc radius of about 63 km implies that finer-scale deformation cannot be resolved. Although this resolution is adequate for interpreting GRACE data (spatial half wavelength of ca. 200 km) smaller-scale loading excitement may be necessary for interpreting local GPS measurements near the loading, particularly for the elastic response to present-day glacial changes. Furthermore, the viscoelastic response functions describe the Earth response in an equilibrium state for a constant rate of load change; if the load exhibits more complex temporal variations, this assumption is violated. Finally, it is assumed that the lithosphere thickness, upper- and lower-mantle viscosities are approximately known.

The viscoelastic response functions and related ancillary data are directly accessible in the PANGAEA repository: http://hs.pangaea.de/model/Sasgen-etal_2017/Viscoelastic_response_functions.zip.