Antarctic ice-mass balance 2003 to 2012: regional reanalysis of GRACE satellite gravimetry measurements with improved estimate of glacial-isostatic adjustment based on GPS uplift rates

We present regional-scale mass balances for 25 drainage basins of the Antarctic Ice Sheet (AIS) from satellite observations of the Gravity and Climate Experiment (GRACE) for time period January 2003 to September 2012. Satellite gravimetry estimates of the AIS mass balance are strongly influenced by mass movement in the Earth interior caused by ice advance and retreat during the last glacial cycle. Here, we develop an improved glacial-isostatic adjustment (GIA) estimate for Antarctica using newly available GPS uplift rates, allowing us to more accurately separate GIA-induced trends in the GRACE gravity fields from those caused by current imbalances of the AIS. Our revised GIA estimate is considerably lower than previous predictions, yielding an estimate of apparent mass change of 53± 18 Gt yr−1. Therefore, our AIS mass balance of −114± 23 Gt yr−1 is less negative than previous GRACE estimates. The northern Antarctic Peninsula and the Amundsen Sea sector exhibit the largest mass loss ( −26± 3 Gt yr−1 and −127± 7 Gt yr−1, respectively). In contrast, East Antarctica exhibits a slightly positive mass balance (26 ± 13 Gt yr−1), which is, however, mostly the consequence of compensating mass anomalies in Dronning Maud and Enderby Land (positive) and Wilkes and George V Land (negative) due to interannual accumulation variations. In total, 6 % of the area constitutes about half the AIS imbalance, contributing 151± 7 Gt yr−1 (ca. 0.4 mm yr −1) to global mean sea-level change. Most of this imbalance is caused by ice-dynamic speed-up expected to prevail in the near future.


Introduction
The current mass balance of the Antarctic Ice Sheet (AIS), and its response to a changing global climate, is challenging to assess due to the spatio-temporal gaps in the meteorological and glaciological instrumental records.Although satellite measurements have considerably improved our knowledge on the state of the AIS, estimating an accurate mass balance and associated contribution to global sea-level change is difficult due to incomplete spatial coverage of the data sets, and/or the diverse processes influencing the satellite measurements.For example, surface-elevation trends of the AIS acquired with laser or radar altimeters need to be corrected for the spatially and temporally heterogenous firn compaction (e.g.Helsen et al., 2008) to infer mass trends.The input-output method (e.g.Rignot et al., 2008Rignot et al., , 2011;;Joughin et al., 2010) also relies on estimates of the surface velocity and ice thickness close to the grounding line of variable quality.There also may be a bias in the extrapolation to areas of relatively poor data (Rignot, 2008), and there is some uncertainty in converting surface velocity to depthaveraged velocity.
While determining mass trends comparably directly from satellite gravimetry data of the Gravity and Climate Experiment (GRACE) has substantial advantages over other measurements, the accuracy of AIS mass balances from GRACE has been limited by a poorly constrained glacial-isostatic adjustment (GIA).The change in volume and extent of the AIS during the last glacial cycle(s) imposed a varying load on the Earth's surface, inducing mass movement and surface deformation.Since the mantle material acts as a highly viscous fluid on these millennial timescales, the GIA of the Earth is delayed with respect to the forcing, where the induced response is governed by the viscosity of the Earth's mantle and the temporal evolution of the ice sheet.Despite that the major ice retreat associated with the last glacial cycle has ceased in Antarctica, GIA continues, causing an inflow of mantle material and an upward bending of the lithosphere in large areas of the former glacial loads.In the periphery of the ice sheet or in areas with comparably recent accumulation increase, also subsidence may occur due to the collapse of the peripheral forebulge and ongoing adjustment to additionally imposed ice loads, e.g. in East Antarctica (Ivins and James, 2005;Whitehouse et al., 2012b;Ivins et al., 2013); a rather complex GIA pattern is expected that very much depends on the poorly known lithosphere and mantle structure beneath the AIS.Nevertheless, GIA-induced trends in the Earth's gravity field and in the surface deformation are more and more clearly revealed in Antarctica by space-and geodetic observing systems, such as GRACE and GPS, respectively.
Several glacial reconstructions have been proposed for predicting GIA using viscoelastic Earth models.These are based on geomorphologic constraints on the past ice height and extent (e.g.Ivins and James, 2005), thermomechanical ice sheet modelling (e.g.Huybrechts, 2002;Ritz et al., 2001), and -considering GIA-induced surface deformation and gravity field changes of the Earth -on indicators of the past relative sea level (e.g.Lambeck and Chappell, 2001;Peltier, 2004), as well as a combination of these approaches (e.g.Bassett et al., 2007;Whitehouse et al., 2012a, b).However, due to the sparsity of constraints on the ice sheet evolution during the last glacial cycle, both in space and time, the ambiguity introduced by the poorly known mantle viscosity beneath Antarctica, and the complexity of the ice-dynamic processes involved, the reconstructions and associated GIA predictions substantially differ in their magnitude and spatial pattern, causing a large uncertainty in the mass balance estimates from GRACE (e.g.Barletta et al., 2008;Chen et al., 2009;Thomas et al., 2011).
In this context, GPS uplift rates in Antarctica are an important constraint on GIA.Records of surface deformation dating back to the late 1990s are available from stations of the International GNSS Service (IGS) located near research stations along the coast of Antarctica.Inland stations began to be deployed only after austral spring of 1995 (e.g.Raymond et al., 2004).The analysis of GPS data now collected are beginning to provide a robust complement to the longer IGS time series (Thomas et al., 2011), as they bound -although with larger uncertainty due to shorter records -GIA in regions where the signal is expected to be large.Currently, however, the longest, and hence most precise, GPS records remain along the coastal perimeter.
In addition to GPS, also GRACE may represent a constraint on GIA in certain areas of Antarctica.During the last glacial cycle, the dominant amount of ice mass retreated from the major ice-shelf areas, inducing a peak GIA signal in the gravity field.At the same time, contemporary ice-mass variations of and on floating ice shelves can be considered "transparent" in the GRACE data, as the floating ice freely seeks a freeboard height oceanward of the grounding line.Nevertheless, the reliability of the GRACE estimate on Antarctic GIA remains limited due to superposition with the signal from continental ice-mass changes or trends in the ocean beneath the ice shelves.
The aim of the following investigation is to provide more accurate regional mass balances of the AIS based on an improved correction for GIA.We develop this improved GIA estimate by rigorous analysis of available space-geodetic measurements that measure the unique signal standout of the process itself.Although our approach resembles the global inversion of GRACE and GPS data presented by Wu et al. (2010b), it includes more accurate and spatially dense data regionally.Furthermore, here we base the inversion on a richer ensemble of GIA forward models.It also differs from the approach followed by Ivins and James (2005), Whitehouse et al. (2012b) and Ivins et al. (2013), which is based on selecting from a suite of GIA scenarios those that fit geologic and relative sea-level constraints and -in the case of the W12a modification (Whitehouse et al., 2012b) in the southern Antarctic Peninsula -GPS uplift rates, without attempting to formally minimize the misfits to both space gravimetry and terrestrial GPS data.In contrast to the approach of Riva et al. (2009), altimetry data are not used in our inversion due to the persisting problem of relating surfaceelevation trends to mass trends.Unless stated otherwise, all GRACE mass balance and acceleration values provided represent error-weighted means with 2-sigma uncertainties for the results based on the GRACE coefficients CSR RL05 and GFZ RL05 for the time period January 2003 to September 2012.

GRACE filtering and inversion
Here, we use 113 monthly mean solutions of the Earth's gravity field derived from data of the GRACE satellites spanning  Flechtner, 2007), and the Centre for Space Research at University of Texas, Austin, USA (CSR RL05; Bettadpur, 2007), which are publicly available as Stokes potential coefficients complete to degree and order 90 and 60, respectively, at http://isdc.gfz-potsdam.de/.Following the recommendation of Bettadpur (2007), the poorly determined GRACE coefficient of degree 2 and order 0 is replaced in CSR RL05 by an estimate from satellite laser ranging (SLR; Cheng and Tapley, 2004), whereas the degree 1 coefficients are completed with estimates from SLR tracking (Cheng et al., 2010), accessible via http://grace.jpl.nasa.gov/data/degree1/.It should be stated that global GPS data are involved in the SLR-based determination of the degree 1 coefficients, due to the sparse and inhomogeneous coverage of SLR tracking stations.
In this paper, we apply the band-pass-filtering function presented in Sasgen et al. (2012a), as well as the coefficients of the forward model, to regionalize the representation of the gravity field and reduce noise in the uncertain low-and high degree and order coefficients (see Supplement).Barletta et al. (2012) have shown a considerable influence of the current mass loss trends (and accelerations) in Greenland and Antarctica on the degree 1 coefficients.The dominant trend, however, is caused by GIA in North America, causing a geocentre motion rate between 0.1 and 1 mm yr 1 , depending on the mantle viscosity and the glacial history (Klemann and Martinec, 2011).Considering that observational estimates for the degree 1 coefficients are uncertain and show large deviations between difference methods (e.g.Barletta et al., 2012), we confine the adjustment to coefficients of degree and order 2 to 60.The geocentre motion velocity of the adjusted forward model, however, is shown to agree with the SLR estimate by Cheng et al. (2010) (see Supplement).
The temporal variations in the gravity field are inverted for mass changes of the AIS using the forward-modelling approach detailed in Appendix A of Sasgen et al. (2010).A priori, this involves the calculation of the gravity field changes induced by a prescribed mass distribution within 25 drainage basins (Fig. 1); here, surface-ice velocity fields used for the input-output method (IOM; Rignot et al., 2008) are considered as an indication of where mass changes should be expected, assuming that recent imbalances primarily occur in regions of fast glacier flow.The main effect is that mass changes are concentrated along the margin of the ice sheet, which is a more realistic approximation for ice-dynamic as well as accumulation-driven mass imbalances than assuming a uniform mass distribution within each basin.The forward model is then regionally adjusted by the least-squares method to fit the GRACE observations.The inversion method is similar to the one used by Schrama and Wouters (2011) in the sense that a modelled signal is fitted to the spatial GRACE monthly solutions.The inversion results are weakly depen-  (after Rignot et al., 2008;Zwally and Giovinetto, 2011) dent on the definition of a priori mass distribution and accurate to < 10 % (Sasgen et al., 2012b).

GPS data
The GPS uplift rates used in our study are those presented and provided by Thomas et al. (2011).The rates are obtained from time series of vertical motion, with the time span varying from station to station, the longest being from the year 1995 to 2010.We use the two sets of elastic corrections provided in Thomas et al. (2011), which are based on mass balance estimates from the IOM and ice-mass trends derived from ICESat satellite laser altimetry.Although Shepherd et al. (2012) showed that mass balance estimates from both methods agree within their uncertainty for largescale averages over the AIS, results are divergent for regional to local scales; the elastic correction differs up to about ±1.5 mm yr 1 , particularly over the Filchner-Ronne Ice Shelf region and East Antarctica.Another problem arises because the elastic correction rates from IOM and ICESat are not based on the same time span as the GPS uplift rates, giving concerns about an inconsistently reflecting interannual accumulation-driven elastic deformation.Nevertheless, we consider the IOM method, which contrasts the average accumulation between 1980 and 2004 with the glacial discharge in 2006 (Rignot et al., 2008), to be most appropriate for correcting the long-term GPS records for the elastic deformation.The ICESat-based elastic deformation provided is applied as an alternative correction to capture some of the uncertainty related to contemporary mass variations.
The GPS stations of the northern Antarctic Peninsula (OHI2, ROTB and PALM) tend to exhibit a kink in the time series of the vertical component after the Larsen Ice Shelf breakup in 2001 (Thomas et al., 2011).Here, we include estimates of the vertical motions for these stations prior to the www.the-cryosphere.net/7/1499/2013/The Cryosphere, 7, 1499-1512, 2013 breakup event of 2002, though the crustal motion is likely to be a mixture of viscous and elastic responses that have memory of the losses prior to 2002 (Rignot et al., 2005).The complexity of the response is exacerbated by the quite low asthenospheric viscosity that occurs in mantle adjacent to the Bransfield Strait and a young mantle slab window (Ivins et al., 2011;Simms et al., 2012;Nield et al., 2012).Also, for SMRT, only GPS uplift rates prior to 2002 are included, despite the fact that the station record does not exhibit a significant change of the trend from 2002 until ceasing measurement in early 2005 (Thomas et al., 2011).We thus include 46 GPS estimates of uplift rates for 35 mostly near-coastal locations along with their uncertainties as a new constraint on GIA.We assume uncorrelated errors, also for co-located GPS sites, despite that the GPS processing may rely on the same clock and orbit estimates causing correlated station estimates.The GPS uplift rates are corrected for surface deformation arising from the Northern Hemisphere GIA (and present-day ice-mass balance in Alaska, Greenland and Ellesmere Island) that are related to two effects: (i) a shift of the centre of figure with respect to the centre of mass of the Earth, in which the GPS data are supplied, as well as changes in the Earth's rotation; and (ii) surface deformation caused by the uplift of all continents by the ocean loading since the Last Glacial Maximum.Using the first-order global inversion estimate from GRACE, we estimate this correction to amount to 0.03 ± 0.08 mm yr 1 at the location of the GPS stations.

Improved estimate of Antarctic glacial-isostatic adjustment
In the following, we will distinguish between a GIA prediction, obtained by applying a glacial reconstruction to a viscoelastic Earth model assuming a set of Earth model parameters, and a GIA estimate, obtained by inversion of (space-)geodetic measurements.In this sense, the load histories of Ivins and James (2005) and Huybrechts (2002) and Peltier (2004) et al., 2012b).These results can be considered a GIA formal prediction.It should be emphasized that we do not attempt to evaluate the glacial histories our GIA predictions are based upon.But we aim at providing a new empirical estimate of Antarctic GIA along with its uncertainties hereinafter called the Antarctic glacial-isostatic adjustment estimate version 1 (AGE1).Due to a broader sampling of the parameter space compared to Wu et al. (2010a), AGE1 is more independent from assumptions on the viscosity distri-bution or glacial reconstruction taken there.However, it still relies on three roughly similar glacial reconstructions (not including all geomorphological data available today) and a limited range of mantle viscosity distributions; including regional advance and retreat scenarios, which are not captured by the glacial histories, or a more complex rheological structure underneath Antarctica such as a ductile crustal layer (e.g.Schotman and Vermeersen, 2005), may influence the resulting AGE1 GIA estimate and its uncertainty range.Nevertheless, AGE1 represents a GIA estimate, alternative to the predictions of Ivins and James (2005) or Whitehouse et al. (2012a), for correcting GPS, GRACE and altimetry trends in Antarctica.

Modelling of the GIA in Antarctica
We predict GIA with the viscoelastic Earth model of Martinec (2000), which solves the governing equations of a Maxwell-viscoelastic continuum with the spectral-finite element approach and an explicit time scheme.Rotational deformation is implemented, as well as the sea-level equation, allowing for the migration of coastlines (Hagedoorn et al., 2007).Here, the Earth model is run with spatial resolutions of spherical-harmonic degree and order 170 (equivalent to 118 km).We consider as free parameters of the model the viscosity of the upper and lower mantle, ⌘ UM and ⌘ LM , respectively, as well as the thickness of the elastic lithosphere h L .We force our viscoelastic Earth model with three load histories, derived from three published glacial reconstructions of the AIS, LH1 (after Huybrechts, 2002, version digitized from publication), LH2 (after Peltier, 2004, publicly available) and LH3 (after Ivins and James, 2005, personal communication).For LH2, the maximum ice height of the discshaped loading centred at the pole was reduced from 765 to 444 m in order to obtain a smooth transition to neighbouring regions.To obtain regional retreat histories, we subdivide the AIS into five sectors (see Fig. 1 in Supplement): Antarctic Peninsula (AP), Filchner-Ronne Ice Shelf (FRIS), Ross Ice Shelf (RIS) and Amery Ice Shelf (AMIS), and the remaining parts into East Antarctica (EAIS).The criteria for the division are to capture areas with substantial ice retreat in all load histories LH1, LH2 and LH3, and to encompass the main clusters of GPS stations recording the regional GIA signals.That is 6 stations in AP, 14 in FRIS, 13 in RIS, 4 in AMIS, and 9 in EAIS .We then predict the global GIA-induced rate of radial displacement, u r (in the centre of mass), and rate of geoid-height change, e r (in the centre of figure), subject to the forcing of each per-sector subdivision (r = 1 through 5, corresponding to AP, FRIS, RIS, AMIS and EAIS) of each load history LH1, LH2 and LH3.The calculation is repeated for each per-sector load history adopting four different radial-symmetric viscosity distributions VD1 through VD4 (Table 1).The thickness of the elastic lithosphere is held constant at 100 km, except for EAIS (150 km) and AP (60 km), where seismic tomography suggests considerably greater and lesser lithosphere thicknesses, respectively (Danesi and Morelli, 2001; Kobayashi and Zhao, 2004), even though there is evidence for a thinner lithosphere in AP (Yegorova et al., 2011).

First-order global inversion of GRACE trends
In this paper, we perform a two-step procedure towards improving Antarctic GIA estimates from GRACE and GPS data (Fig. 2).First, we estimate the temporal linear trends in the GRACE gravity fields, e GRACE , for the time interval January 2003 to September 2012.We then perform a first-order global inversion by fitting a forward model of the rate of geoid-height change, e pred., to the peak signal in the GRACE trends (see Supplement Fig. 5).The model superimposes s = 1 through 35 components describing the major trends due to (i) present-day ice-mass changes in Greenland (eight basins), Ellesmere Island, Alaska and Antarctica (23 basins) and (ii) GIA over North America and entire Antarctica (s = 35), where stands for the spherical colatitude # and longitude ', and hence = (#, '), and q refers to all possible combinations of LH and VD (Table 1) for Antarctica (here, q = 1 through 12).We adopt a global solution domain, 0  #  180 , 180  '  180 .The scalar parameter S q s is obtained by minimizing the difference between the e GRACE and e q Pred. in a least-squares sense over the 35 adjustment areas encompassing the peak anomalies of e q s () (Sasgen et al., 2010).The Antarctic GIA signal is estimated from latitude-and longitude-limited adjustment area centred over the Filchner-Ronne Ice Shelf; the associated scaling factor is henceforth referred to as S q FRIS .The forward models of i) involve a priori information of the distribution of mass within each region based on ICESat surface-elevation changes (Greenland, Sørensen et al., 2011), airborne laser measurement (Alaska, Arendt et al., 2002) and surface-ice velocities measured by radar for Antarctica (Rignot et al., 2008).The GIA predictions for the Northern Hemisphere are obtained by using the four viscosity profiles (Table 1) together with the glacial reconstruction NAWI (Zweck and Huybrechts, 2005).Although the quality of the glacial reconstruction NAWI has not been assessed with, for example, palaeo-sea-level indicators in the nearfield of the ice sheet, it has the advantage of being mostly independent of assumptions on the viscosity distribution.Both the total sea-level variation during the last glacial cycle and the GIA signal over North America are constrained at a sufficiently accurate level (Sasgen et al., 2012b) for isolating and removing this influence on time-varying geoid heights and crustal displacements in Antarctica.Due to the approximate linearity of the GIA response with respect to the forcing, the scaling factors can be interpreted as adjustment factor on the the ice heights of the glacial reconstructions.
From the scaling factors, the mean Northern Hemisphere contribution to surface displacement in Antarctica is estimated according to ûNH () = 1 12 12 P q=1 u q NH (), where u q NH () is the modelled rate of radial displacement associated with the rate of geoid-height change e q NH () for the Northern Hemisphere components only.In step 2, which is described in the following, the mean field ûNH obtained from step 1 is used to correct the Antarctic GPS uplift rates for surface displacement arising from mass changes in the Northern Hemisphere, while S q FRIS is employed as constraint on parameter estimates, which are from GPS uplift rates (Fig. 2).

Refinement of Antarctic GIA estimates with GPS uplift rates
In the second step, we fit GPS uplift rates, u GPS , by predictions of GIA-induced surface displacement in Antarctica, u pred., obtained by the linear combination of the GIA predictions of per-sector loading histories (LH1, LH2 and LH3) and viscosity distributions (VD1 through VD4), Note that q 0 now represents all combinations of LH1 through LH3 and VD1 through VD4 for all sectors AP, FRIS, RIS, AMIS and EAIS, generating an ensemble with 3 5 ⇥ 4 5 members.As opposed to step 1, we now fit r = 1 though 5 scaling parameters for per-sector Antarctic GIA predictions, relaxing on the condition that the relative proportion between the per-sector loads is unchanged or that the viscosity profile is the same for the entire Antarctic continent.It should be stated that although different viscosity distributions are applied to different sectors, the predictions rely on a radialsymmetric distribution of Earth model parameters, neglecting possible effects caused by lateral heterogeneities in the Earth's structure.The scalar parameters S q 0 r in Eq. ( 2) are obtained by minimizing in the least-squares sense the misfit of u q 0 Pred.() to the GPS uplift rates, u ⇤ GPS (), which are beforehand corrected for the Northern Hemisphere contribution, u ⇤ GPS () = u GPS () ûNH ().For each ensemble member q 0 , five scaling parameters S q 0 r , r = 1 through 5, are determined according to (e.g.Tarantola, 2005) where the symbols are as follows: r ( i ) (design matrix), dependent on ensemble realization q 0 C GPS covariance matrix of GPS observations u GPS = (u ⇤ GPS ( 1 ), ..., u ⇤ GPS ( 46 )) T .The design matrix F contains the GIA-induced uplift rates at the i = 1...46 GPS station locations predicted by each of the five per-sector load histories and four viscosity profiles for a specified ensemble member q 0 .It should be noted that although the forcing from each load history for AP, FRIS, RIS, AMIS and EAIS is confined by distinct boundaries, the GIA response in surface deformation extends beyond each sector, on the one hand because the elastic lithosphere acts as a low-pass filter, and on the other hand because the Earth response produces a peripheral forebulge along the margin of the load change.This implies that the fit of each parameter S q 0 r depends on all GPS uplift rates, u GPS , as well as on the specific ensemble member q 0 underlying in F.
The GIA-estimate satisfying both GRACE and GPS observations according to their respective errors is obtained by the constrained least-squares approach (e.g.Tarantola, 2005).This approach provides a parameter estimate under the condition that it is close to an a priori value -the deviation being governed by the balance of the uncertainties of the data and the a priori parameter (constraint).Here, the a priori value is the scaling factor, S q FRIS , derived in step 1 from the GRACE signal over the FRIS area.The constrained solution is ob-tained by where the symbols additional to Eq. ( 3) are It should be noted that F and S q FRIS in Eq. ( 3) are dependent on the ensemble members q 0 and q, respectively; for the constraint estimate, the scaling factor S q FRIS of members with matching LH and VD are selected; for example, if AP is predicted with (LH1, VD3), the scaling factor with (LH1, VD3) is adopted from step 1.

Statistical approach to mean GIA estimate
With Eq. ( 2), we calculate our best unconstrained (i.e.GPS only) estimate of Antarctica GIA, AGE1a, for the rate of geoid-height change, e AGE1a , and rate of radial displacement, u AGE1a , from the arithmetic mean of the ensemble according to u AGE1a () e AGE1a () r (). (5) For the constrained estimate (i.e.GRACE and GPS), AGE1b, this becomes u AGE1b () e AGE1b () In Eqs. ( 5) and ( 6), n stands for the total number of members in our ensemble, which relies on 1. load history (LH1, LH2, LH3) and viscosity distribution (VD1 through VD4) for each sector (3 5 ⇥ 4 5 possibilities), 2. elastic corrections for GPS uplift rates (two possibilities, based on input-output method and ICESat) (Thomas et al., 2011), 3. GRACE release (two possibilities: CSR RL05 and GFZ RL05), resulting in an ensemble of n = 995 328, where (1) influences the design matrix F and the GRACE constraint S q FRIS , (2) the GPS observation vector u GPS and (3) again the GRACE constraint.The estimates from GPS, S q 0 , are affected only little by the GRACE release permutation -merely due to subtracting a different estimate of the Northern Hemisphere contribution to the observed GPS uplift rates.It is worth noting that Fig. 3. Observed minus predicted rate of surface deformation at GPS sites.Shown are the residuals in GPS-measured (mean of InSAR and ICESat-based elastic and Northern Hemisphere GIA correction applied) minus GIA estimated uplift rates, based on GPS (left) and GRACE and GPS observations (right).Residuals < 0 (> 0) indicate overestimated (underestimated) GIA with respect to the GPS uplift rates.The residuals are separated for each sector: Antarctic Peninsula (AP, red), Filchner-Ronne Ice Shelf (FRIS, dark blue), Ronne Ice Shelf (RIS, light blue) and Amery Ice Shelf (AMIS, yellow), and the remaining parts as East Antarctica (EAIS, green).Also indicated are the mean bias (bias, not weighted), as well as the standard deviation (std, not weighted).
the method effectively results in non-physical ice sheet representation at the boundaries of the sectors; that is, jumps in the ice thickness, which are, however, of minor importance because of the elastic lithosphere acting as an effective low-pass filter.Finally, the apparent rate of ice-mass change associated with Antarctic GIA estimates is calculated for 25 basins and the entire AIS from the ensemble mean of the rate of geoidheight change e AGE1a and e AGE1b .
Since the combination of GRACE and GPS observations in the scaling parameter T q 0 is sensitive to the parameter and data uncertainties, some care has to be taken in estimating meaningful (co-)variance matrices C GRACE and C GPS .For the scaling factor inferred from GRACE, we estimate errors due to (i) leakage of present-day signal by estimating the scaling factor with and without adjusting for contemporary ice-mass changes in basins 4 to 25; a leakage error is estimated to 29 %, (ii) sensitivity with respect to the choice of the adjustment area (choice of the adjustment area in the FRIS variability introduced by subdividing the adjustment area in four sectors: 9 %), (iii) remaining aliasing periods of oceanic tides underneath the FRIS (with and without estimating S 2 with 161.5 day and K 2 with 1395.7 day periods in temporal decomposition: < 5%), (iv) difference between two data sets of GRACE coefficients (GFZ RL05 vs. CSR RL05: 9 %), and (v) formal GRACE coefficient uncertainties (< 2 %), adding up to a total uncertainty of 32% for S q FRIS .Uncertainties for the GPS trends are taken from Thomas et al. (2011).The sensitivity of our results to the choice of the GPS and GRACE uncertainties is discussed below.

Apparent ice-mass change of GIA correction
The GRACE signal over the FRIS area requires a downward adjustment of the initial GIA predictions mainly for LH1 and LH2, for most combinations of load histories and viscosity distributions, whereas the signal of LH3 already reconciles with GRACE over the FRIS area.In principle, a scaling factor could also be obtained for the RIS area; however, here, we determine only a single factor based on the FRIS, which is intended to compensate for the trade-off between the viscosity distribution and magnitude of the load.This factor is then applied (for a specified viscosity distribution) to all other areas, meaning that the spatial pattern of the GIA signal is entirely governed by the model.Although the adjustment reduces spread for different viscosity distributions for each load history to < 30 Gt yr 1 , the differences between load models remains large due to their distinct spatial patterns (90 Gt yr 1 between minimum and maximum estimate).By the sectorwise adjustment to the GPS uplift rates, the load histories are homogenized, reducing the deviation to 38 Gt yr 1 .
Figure 3 shows the residuals of the uplift rates at the GPS stations after subtracting the GIA estimate.For each sector, the distribution of residuals is centred around zero (standard deviation of 2.7 mm yr 1 ), even though for FRIS there is an indication that the subtracted GIA is slightly underestimated.The apparent mass change associated with this GIA correction is 50 ± 26 Gt yr 1 .For the GIA estimate constrained by GRACE and GPS, the GIA estimate increases in magnitude to 53 ± 18 Gt yr 1 .The mean bias slightly increases ( 0.1 mm yr 1 ), but GPS uplift residuals for the stations in the FRIS and AMIS centre slightly better around zero.This is an indication that the GRACE-constrained GIA estimate reproduces data better, which have short records and uncertain trends and are given a low weight in the GPS-only adjustment (Fig. 4).In general, the fit to the GPS uplift rates is dominated by the long term, and hence most accurate station records.Due to the comparably large error of the GRACEbased scaling factor (32 %), the contribution to the combined www.the-cryosphere.net/7/1499/2013/The Cryosphere, 7, 1499-1512, 2013 estimate is small, and the mean of the GIA estimates based on GPS as well as GRACE and GPS are very similar (Fig. 4).
It should be noted that varying the lithosphere thickness influences the pattern of the regional GIA signals, particularly in the peripheral region of the former ice sheet, and therefore may also affect the fit to individual GPS stations.It is expected, however, that after scaling, this will mainly influence the spread of the GPS uplift residuals and apparent mass change values, and not so much their mean.
Figure 5 shows the distributions of the GIA-induced apparent mass change for each of the 25 drainage basins of the AIS and the total AIS for GIA estimate AGE1b (GRACE and GPS).The largest GIA-induced mass change is obtained for the basins in the vicinity of the large ice shelves: 4 to 6 Gt yr 1 for basins 17, 18 and 19 (RIS) as well as basins 1 and 3, and 4 Gt yr 1 for basin 2 and for the southern part of AP (basin 24).For many basins, the scatter of the values are similar to a Gaussian distribution.But since sub-sector GIA signal is mostly governed by the shape of the ice histories LH1, LH2 and LH3, systematic clusters appear for some basins (e.g.basin 25 of the AP, basin 16 in East Antarctica) -differences between the load histories, which are small on average for each sector, again become important.It becomes clear that although LH1, LH2 and LH3 include some of the variety obtained of different reconstructions, further regionally refined glaciation histories will alter the GIA pattern, and therefore the influence basin-scale apparent mass change.
The reader is encouraged to apply the GIA correction directly to the GRACE coefficients.We therefore provide the GIA estimate AGE1a (GPS only) and AGE1b (GRACE and GPS) of the rate of geoid-height change and rate of radial displacement as fully normalized spherical-harmonic coefficients (Heiskanen and Moritz, 1967) in the Supplement of this paper.

Regional-scale trends and accelerations from GRACE
Table 2 presents rates and accelerations of mass changes for the 25 basins of the AIS from GRACE for the time period January 2003 to September 2012.The mass balance of the AIS is characterized by strong losses along the Antarctic Peninsula and Amundsen Sea sector ( 140 ± 16 Gt yr 1 : basins 1, and 18 to 25) and moderate gain of mass for East Antarctica (26 ± 13 Gt yr 1 : basins 2 to 17), adding up to total of 114 ± 23 Gt yr 1 .Major mass loss in West Antarctica occurs in basin 21 (Thwaites glacier system: 57 ± 3 Gt yr 1 ) and basin 22 (Pine Island glacier: 28 ± 3 Gt yr 1 ).Mass loss along the Antarctic Peninsula is concentrated in the north, basin 25 ( 26 ± 3 Gt yr 1 ).This compares well to GRACE estimates (January 2003 to March 2009) that are slightly higher at 32 ± 6 Gt yr 1 by Ivins et al. (2011) and this difference is possibly attributable to a different approach to incorporating the GPS data into the GIA estimation.East Antarctica exhibits a bimodal pattern of mass increase in Dronning Maud and Enderby Land (basins 3 to 8: 60 ± 7 Gt yr 1 ) and mass decrease in Wilkes Land (basins 12 to 15: 31 ± 4 Gt yr 1 ).
The situation is more diverse for the acceleration estimates from GRACE presented also in Table 2, here with respect to the midpoint of the time interval January 2003 to September 2012.Acceleration of mass loss (negative in sign) is observed for the Antarctic Peninsula -here, Palmer Land (basin 24: 6 ± 2 Gt yr 2 ) as well as for the Amundsen Sea sector, in particular the Pine Island, Thwaites and Getz/Hull/Land glacier systems (basins 22, 21 and 20, respectively: 17 ± 6 Gt yr 2 ).For the northern Antarctic Peninsula, the acceleration term is not statistically significant.For East Antarctica, mass loss acceleration is observed for Wilkes Land (basin 12: 2 ± 1 Gt yr 2 ), while deceleration (positive in sign: decrease of mass loss) is observed in Dronning Maud Land and Enderby Land (basins 4, 5, 6 and 7: 14 ± 4 Gt yr 2 ).For the entire AIS, mass loss acceleration arising in West Antarctica ( 21 ± 10 Gt yr 2 ) is counterbalanced by about half by mass loss deceleration in East Antarctica (12 ± 6 Gt yr 2 ), adding up to a total of 16 ± 12 Gt yr 2 .Figure 6 presents the basin-scale mass balance estimates of the AIS from GRACE (GIA correction AGE1, GRACE&GPS), ordered according to the expected signalto-noise ratio of present-day ice-mass balance value and the sum of propagated GRACE coefficient errors, filtering and inversion uncertainties, and uncertainties of the GIA correction from Table 2. Additionally, the cumulative sum of the basin-scale mass balances are shown.The most dominant imbalances originate from the northern Antarctic Peninsula (basin 25) and the Amundsen and Bellinghausen Sea sector (basins 20, 21 and 22).Due to the rather weak influence of our GIA correction in these basins -which is, however, in contrast to the finding of Groh et al. (2012), who attribute 34 ± 12 Gt yr 1 to GIA in the Amundsen Sea sector -and the strong imprint in the GRACE gravity fields, the sum of imbalances amounting to 153 Gt yr 1 is resolved with an accuracy of ±7 Gt yr 1 (5 %).Representing only 6 % of the area of the ice sheet, more than half of the mass imbalances (53 %), positive or negative, occurs in these wellresolved basins.But even if all increase in mass observed with GRACE is attributed to snow accumulation, and not GIA, the total AIS mass balance remains significantly negative ( 61 ± 15).However, mass trends in East Antarctica are strongly influenced by interannual accumulation variability along the coast, limiting the significance of extrapolating the total AIS mass balance into the future.
The acceleration terms inferred for each of the 25 basins for January 2003 to September 2012 are shown in Fig. 7,  GPS and GRACE (GRACE and GPS comb.).The apparent ice-mass change is calculated by applying the gravimetric inversion method for the present-day ice-mass changes to each estimate of the GIA-induced gravity field.
which are ordered identically to the trend estimates depicted in Fig. 6 (not according to their signal-to-noise ratio).In West Antarctica, substantive accelerations of mass loss (negative in sign) occurs mainly in the Thwaites ( 8 ± 1 Gt yr 2 : basin 21) and the Getz/Hull/Land glacier systems ( 6 ± 6 Gt yr 2 : basin 20), and to a lesser extent in the Pine Island glacier (basin 22: 3 ± 1 Gt yr 2 ) in the Amundsen Sea sector.Evidence of glacier retreat and acceleration of ice flow in these regions (Rignot et al., 2011) suggests that the GRACE trends and accelerations reflect longterm responses of the ice sheet, caused by melting of ice shelves by wind-driven penetration of warm ocean water, decreasing buttressing of tributary ice streams (Pritchard et al., 2012).In contrast, for northern Graham Land (basin 25), no statistically significant acceleration is found, despite a strong imbalance in this region.East Antarctica apparently compensates 12 ± 6 Gt yr 2 of the mass loss acceleration.Here, however, a preliminary comparison with output from the regional atmospheric climate model (RACMO2/ANT; Helsen et al., 2008;Lenaerts et al., 2012) suggests that the changes in Dronning Maud Land and Enderby Land (basins 4 to 7: 14 ± 4 Gt yr 2 ), Wilkes Land (basins 12 and 13: 4 ± 1 Gt yr 2 ), and also those in Palmer Land, Antarctic Peninsula (basin 24: 6 ± 1 Gt yr 2 ), are nearly completely explained by accumulation variations within the comparably short observation period.

Discussion
Our mass balance for the AIS of 114 ± 23 Gt yr 1 for the time period January 2003 to September 2012 and our new GIA estimate AGE1b (GRACE and GPS) is considerably less negative than early GRACE estimates of Velicogna (2009) ( 143 ± 73 Gt yr 1 : 2002-2009), who applies a mean GIA correction of 176 ± 76 Gt yr 1 based on the reconstructions of Ivins and James (2005) and Peltier (2004) as well as a suite of viscosity distributions.This is mainly a result of correcting GIA with only 53 ± 18 Gt yr 1 .Our study confirms the estimate of 109 ± 48 Gt yr 1 (Horwath and Dietrich, 2009), based on the shorter time interval August 2002 to January 2008.It also supports the previous joint inversion estimate for the total AIS based on GRACE and GPS data (Wu et al., 2010b) of 87 ± 43 Gt yr 1 (2002-2008), even though with a very different separation between East and West Antarctica -that is, 116 ± 15 Gt yr 1 and 26 ± 13 Gt yr 1 (this study) versus 64 ± 32 Gt yr 1 and 23 ± 29 Gt yr 1 (Wu et al., 2010b), respectivelymost likely owing to regional differences between the GIA estimates.And our estimate lies within the range of 87 ± 43 Gt yr 1 (2000-2011) provided the multi-satellite ice sheet mass balance inter-comparison exercise (IMBIE, Shepherd et al., 2012), using the average of the most recent GIA corrections of Whitehouse et al. (2012b) and Ivins et al. (2013).Compared to the recent estimate of King et al. (2012) with 69 ± 18 Gt yr 1 , based on the new GIA prediction W12a (Whitehouse et al., 2012b), our results are with 114 ± 23 Gt yr 1 significantly more negative, even though excellent agreement is obtained for single glacier systems in the Amundsen Sea -for example, Thwaites: 57 ± 3 Gt yr 1 (this study) and 54 ± 5 Gt yr 1 (King et al., 2012); and Pine Island glacier: 28 ± 3 Gt yr 1 (this study) and 24 ± 7 Gt yr 1 (King et al., 2012).Differences mainly reside in East Antarctica, for which King et al. (2012) propose a mass gain of 60 ± 13 with a GIA correction close to zero (3 Gt yr 1 : W12a model), however, with upper and lower bounds of 56 Gt yr 1 and 26 Gt yr 1 , respectively, which also encompass our GIA estimate of 30 ± 11 Gt yr 1 for East Antarctica (Table 2).Without GIA correction, our apparent GRACE mass balance for East Antarctica is 56 ± 7 Gt yr 1 , in agreement with the 63 Gt yr 1 provided by King et al. (2012).Possibly, the uncertainty range of W12a in East Antarctica of 82 Gt yr 1 could be reduced by including GPS uplift rates.
With the GIA estimate AGE1b (GRACE and GPS), GRACE indicates a modest mass increase for East Antarctica (26 ± 13 Gt yr 1 ), supporting estimates from radar altimetry 22 ± 39 Gt yr 1 rather than from the mass budget method 30 ± 76 (Shepherd et al., 2012, October 2002 to December 2008).However, comparing different time periods is of limited validity due to the strong influence accumulation variations in EA, as discussed above.For the northern Antarctic Peninsula (basin 25), our results of 26 ± 3 Gt yr 1 show excellent agreement with the most recent GRACE-based estimates of ( 33 ± 3 Gt yr 1 : August 2002 to December 2012, King et al., 2012), and a previous estimate of 32 ± 6 Gt yr 1 for the time period January 2003 to March 2009 (Ivins et al., 2011).
Compared to other recent GRACE estimates of the AIS mass balance, we obtain stronger losses, even if a similar GIA correction is applied; for example, Ivins et al. (2013) correct for a GIA-induced apparent mass change of 55 ± 13 Gt yr 1 based on the revised version of glacial history from Ivins and James (2005), resulting in a mass loss of the AIS of 57 ± 34 Gt yr 1 .Both methods use very different approaches towards regionalizing, as well as towards removing leakage from and to the region of Antarctica.In particular, our treatment of the degree 1 terms is different from Ivins et al. (2013) and the procedure agreed upon in IMBIE (Shepherd et al., 2012); due to the uncertainty of the degree 1 coefficients estimate from SLR and the large influence of far-field signal (e.g.GIA from the Northern Hemisphere), we exclude these coefficients from the adjustment of our forward model, which is, however, complete for spherical-harmonic degree and order 0 to 512 (see Supplement).If the predetermined approach used in IMBIE is applied, this may weaken the estimate by about 30 Gt yr 1 (Ivins et al., 2013).
As shown in Fig. 3, AGE1b (GRACE and GPS) fitted the GPS uplift rates with a mean bias of 0.1 mm yr 1 and a standard deviation of 2.2 mm yr 1 .This is a significant improvement with respect to the bias of 1.2 mm yr 1 associated with the GIA prediction of (Whitehouse et al., 2012a, b).Due to our statistical approach, AGE1a and AGE1b are rather insensitive to the viscosity distribution and to the glacial history -at least when integrating over a sector -as deviations are mostly scaled out by the loading adjustment.However, the uncertainty of the GIA correction (Fig. 4, Supplement) www.the-cryosphere.net/7/1499/2013/The Cryosphere, 7, 1499-1512, 2013 depends to a large extent on the availability and accuracy of GPS uplift rates.For example, both AGE1a and AGE1b suggest the largest GIA anomaly in the RIS sector due to very sparse GPS data (Fig. 3), which is in contrast to more recent geomorphological evidence on the ice sheet retreat in the RIS sector (Ivins et al., 2013).The uncertainties of AGE1b (Fig. 4, Supplement) should be kept in mind when applying it as a GIA correction to the GRACE data.
Limitations of AGE1 also apply to the representation of the sub-sector (i.e.basin-scale) GIA -arising from unknown regional retreat history, which are not included in the uncertainty estimate for AGE1b.For example, Groh et al. (2012) presented evidence for a GIA-induced uplift in the Amundsen Sea sector (part of the FRIS sector in our study) ranging for different locations between 14.1 ± 6.7 and 22.9 ± 6.7 mm yr 1 , causing a mass increase of 34 ± 12 Gt yr 1 .These uplift rates are exceptionally large compared to the trends measured by Thomas et al. (2011), and, if included in our adjustment, cannot be fitted by our GIA sectorial patterns; we obtain a GPS residual of 13 to 22 mm yr 1 for the additional stations, compared to a maximum deviation of 8 mm yr 1 for the stations of Thomas et al. (2011).Another example is the subsidence due to a substantial ice-thickness increase in the late Holocene predicted by Whitehouse et al. (2012a) in Coats Land (basin 3) of our East Antarctic sector.Clearly, further detailed research on the regional Antarctic GIA signal is needed.

Conclusions
We have provided a revised GIA estimate for Antarctica, AGE1, based on numerical simulations and newly available GPS uplift rates, as well as GRACE trends beneath the Filchner-Ronne Ice Shelf.The residual misfit of surface deformation associated with AGE1b (GRACE and GPS) and measured GPS uplift rates in Antarctica is 0.1 mm yr 1 , which represents an improvement with respect to the GIA prediction, for example, of Whitehouse et al. (2012b) ( 1.5 mm yr 1 mean bias at 46 GPS stations of W12a model, optimum Earth model).The apparent ice-mass change of 53 ± 18 Gt yr 1 associated with AGE1b is considerably lower than previous estimates, in particular compared to the earlier correction 176 ± 76 Gt yr 1 applied by Velicogna and Wahr (2006) based on a combination of ICE5G (Peltier, 2004) and IJ05 (Ivins and James, 2005), but in line with more recent, independently derived GIA corrections of Whitehouse et al. (2012b) and Ivins et al. (2013).The implication is significantly weaker negative AIS mass balance of 114 ± 23 Gt yr 1 estimated from GRACE for the time period January 2003 to September 2012.
Our regional GIA and GRACE mass balance estimates clearly show that more than half of current Antarctic sealevel contribution (positive or negative) arises from 6 % of the area of the ice sheet; mass loss along the northern Antarc-tic Peninsula and the in Amundsen Sea sector amount to 151 ± 7 Gt yr 1 .East Antarctica, in contrast, has a slightly positive mass balance (26 ± 12 Gt yr 1 ), exhibiting a bipolar signature of accelerating mass increase in Dronning Maud Land and Enderby Land (basins 5, 6 and 7: 12 ± 4 Gt yr 2 ) and accelerating mass loss in Wilkes Land and George V Land (basin 13 and 14: 4 ± 2 Gt yr 2 ).The preliminary comparison with output from RACMO2/ANT suggests that the temporal signatures in East Antarctica (and Palmer Land, Antarctic Peninsula) are mainly due to interannual accumulation variability; enhanced precipitation in the years 2005 and 2007 as part of variability in the large-scale atmospheric circulation has induced these mass anomalies, not changes in ice-dynamic flow.The strong imbalance and acceleration observed for the northern Antarctic Peninsula and the Amundsen Sea sector ( 151 Gt yr 1 and 22 Gt yr 2 , respectively), however, clearly reflect more vigorous ice flow (Scambos et al., 2004;Rignot et al., 2008) and are more likely to be a sustained sea-level contribution of AIS.

Fig. 2 .
Fig. 2. Scheme of the two-step procedure to derive GIA estimates based on GPS only (AGE1a) as well as GRACE and GPS combined (AGE1b) based on an ensemble of forward models.

Fig. 4 .
Fig. 4. Rate of radial displacement (mm yr 1 ) and rate of geoidheight change (mm yr 1 ), respectively, (a) and (c) for AGE1a (GPS only) and (b) and (d) for AGE1b (GRACE and GPS).Sphericalharmonic cut-off degrees are 0 to 170 for (a) and (b) and 2 to 60 for (c) and (d).Also indicated are the GPS uplift rates (after the correction for the Northern Hemisphere contribution) according to Thomas et al. (2011)

Fig. 5 .
Fig. 5. Distribution of the rate of apparent ice-mass change (Gt yr 1 ) induced by the GIA for the total AIS and the 25 basins, obtained by constraining the ensemble of per-sector combinations (995 328 samples) with GPS and GRACE(GRACE and GPS comb.).The apparent ice-mass change is calculated by applying the gravimetric inversion method for the present-day ice-mass changes to each estimate of the GIA-induced gravity field.

Fig. 6 .Fig. 7 .
Fig. 6.Rate of basin-scale ice-mass change from GRACE (Gt yr 1 ) for the drainage basins of the Antarctic Peninsula (red), West Antarctica (blue) and East Antarctica (green).Numbers in the bottom part of the plot refer to the drainage basins in Fig. 1 and Table 2. Grey bars reflect 1-sigma uncertainties.The drainage basins are sorted according to the estimated signal-to-noise ratio of the linear trend component.GIA correction AGE1b (GRACE and GPS) applied.Statistically insignificant temporal components are indicated with dashed lines.The cumulative sum over the basins is provided in the top part of the figure, depicting that nearly all mass loss originates from a very small portion of the AIS.cale ice-mass change from GRACE (Gt/yr) for the drainage basins of the Antarctic Peninsula (red), West Antarctica tica (green).Numbers in the bottom part of the plot refer to the drainage basins in Figure 1 and Table 2. Grey bars inties.The drainage basins are sorted according to the estimated signal-to-noise ratio of the linear trend component.(GRACE&GPS) applied.Statistically insignificant temporal components are indicated with a dashed lines.The he basins is provided in the top part of the Figure, depicting that nearly all mass loss originates from a very small the time interval January 2003 to September 2012.We adopt the GRACE gravity field solutions of release version 5 (RL05) of the processing centres German Research Centre for Geosciences GFZ, Potsdam, Germany (GFZ RL05; The Cryosphere, 7, 1499-1512, 2013 www.the-cryosphere.net/7/1499/2013/

Table 1 .
Upper and lower mantle viscosity values (Pa s) for the four applied viscosity distributions.

Table 2 .
Rate and acceleration of basins-scale ice-mass change from GRACE and revised GIA estimate AGE1b (GRACE and GPS).The GRACE estimates represent error-weighted values of GFZ RL05 and CSR RL05 estimates.⇤ denotes statistical significant acceleration terms in both GFZ RL05 and CSR RL05, while ⌃ denotes linear trends that are not statistically significant in both releases (95 % confidence interval: before correcting for GIA).Time period is January 2003 to September 2012.