The poorly known correction for the ongoing deformation of the
solid Earth caused by glacial isostatic adjustment (GIA) is a major
uncertainty in determining the mass balance of the Antarctic ice sheet from
measurements of satellite gravimetry and to a lesser extent satellite
altimetry. In the past decade, much progress has been made in consistently
modeling ice sheet and solid Earth interactions; however, forward-modeling
solutions of GIA in Antarctica remain uncertain due to the sparsity of
constraints on the ice sheet evolution, as well as the Earth's rheological
properties. An alternative approach towards estimating GIA is the joint
inversion of multiple satellite data – namely, satellite gravimetry,
satellite altimetry and GPS, which reflect, with different sensitivities,
trends in recent glacial changes and GIA. Crucial to the success of this
approach is the accuracy of the space-geodetic data sets. Here, we present
reprocessed rates of surface-ice elevation change (Envisat/Ice, Cloud,and
land Elevation Satellite, ICESat; 2003–2009), gravity field change (Gravity
Recovery and Climate Experiment, GRACE; 2003–2009) and bedrock uplift (GPS;
1995–2013). The data
analysis is complemented by the forward modeling of viscoelastic response
functions to disc load forcing, allowing us to relate GIA-induced surface
displacements with gravity changes for different rheological parameters of
the solid Earth. The data and modeling results presented here are available
in the PANGAEA database (

Glacial isostatic adjustment (GIA), the viscoelastic deformation of the solid Earth in response to climate-driven ice and water mass redistribution on its surface, is poorly constrained in Antarctica. The primary reason is the sparseness of geological evidence for the past ice sheet geometry and local relative sea-level change. These are important constraints on the glacial forcing exerted and on the viscoelastic structure of the lithosphere and of the mantle, which together determine the signature of GIA (e.g., Peltier, 2004; Ivins and James, 2005; Whitehouse et al., 2012; van der Wal et al., 2015). The predictions of GIA in Antarctica remain ambiguous (Shepherd et al., 2012) and cause a large uncertainty in gravimetric mass balance estimates of the ice sheet of the order of the estimate itself (Martín-Español et al., 2016b). Measurements of bedrock uplift by GPS are inconsistent with the predictions of existing GIA. In many regions, uplift rates and thus mass increase due to GIA is overpredicted (Bevis et al., 2009), biasing estimates of present-day Antarctic ice-mass loss from Gravity Recovery and Climate Experiment (GRACE) to more negative values. However, for regions with a weak Earth structure, large uplift signals are recorded by GPS (e.g., Groh et al., 2012), which are likely caused by load changes within the past few thousand year, and are often not accurately represented in GIA predictions (Wolstencroft et al., 2015).

Much progress has been made in reconstructing the ice sheet evolution from geomorphological evidence (Bentley et al., 2014) and inferring the underlying Earth structure from seismic observations (An et al., 2015; Heeszel et al., 2016). However, an independent approach to constraining GIA is to make use of the different sensitivities of the various types of satellite data to recent glacial changes and GIA, respectively. Separating both signals in a joint inversion approach has been pursued by, e.g., Wahr et al. (2000), Riva et al. (2009), Wu et al. (2010), Gunter et al. (2014) and Martín-Español et al. (2016a). Another approach used regional patterns of GIA from forward modeling and adjusted them to GIA uplift rates in Antarctica (Sasgen et al., 2013).

In this paper, we present methods and data inputs in preparation to solve the joint inversion problem for GIA in Antarctica. As the GIA process is gradual, causing an approximately constant rate of change within a decade, we first process the satellite data to recover optimal temporal linear trends. We focus on the trends derived for the time period 2003–2009 in which GRACE and Ice, Cloud,and land Elevation Satellite (ICESat) operated simultaneously. Note that the stationarity of the trend is a key assumption underlying our approach when including GPS rates covering a longer time span (1995–2013). However, limiting the GPS data to the time span 2003–2009 leads to a significant reduction in the number of stations for which reliable trends can be estimated and, hence, to a loss of spatial coverage. For comparison, the reader is referred to the data archive, in which GPS uplift rates for the time periods 2003–2009 and 2003–2013 are made available.

In this paper, we present refined procedures for estimating trends in the data sets on surface-ice elevation changes, surface displacement and gravity field changes. The rates of surface-ice elevation change from Envisat and ICESat satellite altimetry are improved by combining both data sets based on their respective uncertainties, increasing the spatial coverage and accuracy of the elevation rates (Sect. 2). Bedrock displacement from in situ networks of GPS stations in Antarctica are improved in coverage by allowing for campaign-based data and carefully assessing the uncertainty in the trend with a noise model (Sect. 3). Compared to the rates in Thomas et al. (2011) more stations and longer time series are also included The gravity field changes from GRACE are refined compared to previous work by optimizing the de-striping filtering for the region of Antarctica (Sect. 4). The processing aims at fulfilling the requirement of the joint inversion to combine input data based on the same time period (not possible for GPS without having to ignore a large number of stations) and covering all of Antarctica, accompanied by a realistic description of the uncertainties.

We also present forward-modeling results of viscoelastic response functions to disc load forcing for the range of Earth structures likely to prevail in Antarctica (Sect. 5). The viscoelastic response functions allow us to combine the surface displacement and gravity changes based on the physical description of the Earth's viscoelastic response for a specified Earth structure. In addition, the response functions enable us to combine data sets of different spatial resolutions, as is the case for GPS, GRACE and altimetry.

The determination of viscoelastic response functions is a classic topic in solid Earth modeling (e.g., Peltier and Andrews, 1976), though uncommon in the application to joint inversion studies of satellite data. Although this paper focusses on Antarctica, the response functions and data processing techniques presented here are applicable to other regions. The response kernels represent a wide range of Earth structures and can be used for the separation of superimposed present-day (elastic) and past (viscoelastic) signatures of mass change in other regions with a similar Earth structure, for example hydrological storage changes and GIA in North America and Alaska. The response functions give insight into the temporal and spatial scales of deformation expected for Antarctica and are crucial when combining the input data streams.

The data sets and modeling results presented in this paper are accessible
in the PANGAEA archive (

The actual method of the joint inversion is described in a second contribution of the Regional glacial isostatic adjustment and CryoSat elevation rate corrections in Antarctica (REGINA) project team (Sasgen et al., 2017a). In this second paper, the resulting GIA estimate is also compared to previous studies.

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

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

We produce a combined rate of surface-elevation change product from the
ICESat and Envisat data sets for the Antarctic ice sheet,

We combine the two altimetry data sets based on their common 10 km

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

Mask for the combination of Envisat and ICESat. ICESat but not Envisat available –
yellow;

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,

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

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

The altimetry data and related ancillary data are directly accessible in the
PANGAEA repository:

The data set is provided in a 10 km grid in polar-stereographic projection
(central meridian 0

The data set is provided in a 10 km grid in polar-stereographic projection
(central meridian 0

Elevation changes have been corrected for firn densification processes using
a firn densification model. The data set is provided in a 10 km grid in
polar-stereographic projection (central meridian 0

An acceleration term in areas with dynamic thinning was added to the linear
trend to obtain annual rates. Elevation trends are provided at 10 km
resolution in a polar-stereographic grid (central meridian 0

Annual firn densification rates over 2003–2013 rates obtained from RACMO2.3.
Data are provided in a 27 km polar-stereographic grid (central meridian
0

To perform the conversion of volume change to mass change, a density map is
provided at 20 km resolution in a polar-stereographic grid (central meridian
0

The mask used for combining ICESat and Envisat is provided with a 10 km resolution in polar-stereographic coordinates..

4: only Envisat was available

3: only ICESat was available

2: ICESat lower errors

1: Envisat lower errors.

The aim of the GPS time series analysis is to derive uplift rates,

The median values of the white-noise scale (1.6) factor and the power-law
noise amplitude (13.4 mm), derived from these long time series, were then
used to propagate rates and uncertainties for the shorter time series, for
which CATS cannot produce reliable estimates of the error model. For the
propagation, the time series with fewer than 2000 epochs are additionally
subdivided into two categories; continuous sites (

Number of sites for each GPS uplift rate and uncertainty estimation method.

Finally, for each site, the uplift rate

Table 1 summarizes the rate estimation methods and the number of sites for each. For further details and full information on individual rates and time series, see Petrie et al. (2018a) for a full description of the processing and ensemble evaluation and Petrie et al. (2018b) for details of time series analysis and rate and uncertainty estimation. Further work was undertaken to combine or “cluster” the rates regionally for inclusion in the estimation process – see Section 3.2 of the second REGINA paper (Sasgen et al. 2017a) for more details..

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

Uplift rates

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,

Uplift rates

The GPS data and related code are directly accessible in the PANGAEA
repository:

Bedrock uplift rates derived for the REGINA project are available in the text
file “REGINA_rates_full.txt”, as presented in Tables A2 and A3 of the
Appendix A4. The files “REGINA_rates_03-13.txt” and
“REGINA_rates_03-09.txt” contain subsets of the data, with the temporal
coverage limited to 2003–2013 and 2003–2009, respectively. The files are
organized as follows: longitude (

These *.txt files are the input to the clustering script described below. No elastic correction has been applied.

Uplift rates

In addition to the uplift rates for individual GPS sites, we provide a bash
script “cluster.sh” for clustering the heterogeneous data according to
their geographic locations, for a predefined threshold value. The idea is to
reduce stochastic and geophysical noise of neighboring stations in order to
obtain uplift rates that are better regional representations of the length
scale recovered with GRACE (ca. 200 km). In an iterative procedure, the
script selects neighboring sites within a threshold ranging from 10 to
220 km and calculates the weighted average of the uplift rates and a simple
average of the station locations. Input to the script is the REGINA rate
files, specified in the previous Sect. 3.2.1. Further details and the
application to the GPS data set can be found in the second REGINA paper (Sasgen et
al., 2017a). Note that the script relies on the open-source program suite
Generic Mapping Tools (^{®} or its open-source alternative GNU
Octave.

The GPS time series were created as part of the RATES project (UK NERC grant NE/I027401/1), not solely the REGINA study. The data can be obtained upon request from coauthor Elizabeth Petrie.

We investigate the Release 5 (RL05) GRACE coefficients of the Centre for
Space Research (CSR; Bettadpur, 2012) and the German Research Centre for
Geosciences (GFZ; Dahle, 2013), provided up to spherical harmonic degree and
order

The determination of the rate of the gravity field change over Antarctica follows the scheme sketched in Fig. 3. The rate of the gravity field change, expressed as equivalent water height variations, is estimated in the spatial domain by adjusting a six-parameter function consisting of a constant, a temporal linear trend, and annual and semiannual harmonic amplitudes. A quadratic term was not co-estimated due to the project's focus on the rates (i.e., temporal linear trends). It should be stated that including a quadratic term would slightly reduce the residual uncertainties, particularly in the Amundsen Sea sector, where an ice-dynamic acceleration of mass balance rates occurs that is not accounted for by nonlinear SMB variations in the ice sheet (see Sect. 4.2).

Post-processing steps applied to the GRACE gravity fields; shown is
the impact on the gravity field rate

The post-processing of the GRACE coefficients follows three main steps below.

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).

Effect of Swenson filter parameters

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

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 (rms

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, rms

Residual mass anomaly of monthly GRACE gravity fields for 2003–2011, averaged over the Antarctic region south of

Figure 4 shows the assessed signal corruption and noise reduction as a
function of the two Swenson filter parameter choices, the polynomial degree

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

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

Linear trend in the GRACE gravity fields for the years 2003–2009;

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

The gravity data and related code are directly accessible in the PANGAEA
repository:

The monthly GRACE gravity field solutions from the data system centers GFZ
and CSR are available at

Degree

The Matlab^{®} function “KFF_filt” performs
decorrelation filtering for sets of spherical harmonic coefficients,
typically from GRACE gravity field solutions, following the idea of Swenson
and Wahr (2006). An open-source alternative to
Matlab^{®} is GNU Octave:

The Earth structure of Antarctica is characterized by a strong dichotomy
between east and west, separated along the Transantarctic Mountains (e.g.,
Morelli and Danesi, 2004). Recent seismic studies have produced refined maps
of crustal thicknesses also showing slower upper-mantle seismic velocities in
West Antarctica, indicating a thin elastic lithosphere and reduced mantle
viscosity (An et al., 2015; Heeszel et al., 2016). Moreover, yield strength
envelopes of the Earth's crust and mantle suggest the possibility of a viscously deforming, ductile layer (DL) in the lower part of the crustal lithosphere
(Ranalli and Murphy, 1987), a few tens of kilometers thick and with
viscosities as low as

The choice of the viscoelastic modeling approach used to determine load-induced surface displacements and gravitational perturbations is governed by three main requirements: (i) to accommodate lateral variations in Earth viscosity, (ii) to allow for Earth structures with a thin elastic lithosphere and low viscosity layers, in particular including a DL, and (iii) to provide viscoelastic response functions for the joint inversion of the satellite data described in the second REGINA paper (Sasgen et al., 2017a). With regard to point (iii), it should be mentioned that the viscoelastic response functions provide a geophysically meaningful way to relate surface displacement and gravity field changes, considering also dynamic density changes within the Earth's interior . Moreover, it allows us to consider the changes in the ratio of surface-displacement and gravity field changes caused by the Earth structure, in particular, the lithosphere thickness. Another advantage is that different filtering can be applied to the viscoelastic response functions in order to match the filtering of the input data set, avoiding the introduction related biases (Appendix A5).

To meet these requirements, we adopt the time-domain approach (Martinec, 2000) for calculating viscoelastic response functions of a Maxwell continuum to the forcing exerted by normalized disc loads of constant radius. Then, the magnitudes and spatial distribution of the surface loads are adjusted according to the satellite data to obtain the full GIA signal for Antarctica. The forward modeling of viscoelastic response functions is a classic topic in solid Earth modeling (e.g., Peltier and Andrews, 1976); however, their application to inverting multiple-satellite observations for present and past ice sheet mass changes is new and applicable to other regions, such as Greenland or Alaska.

The viscoelastic response function approach allows for high spatial
resolution at a low computational cost in the numerical discretization of the
Earth structure as well as in the representation of the load and the
response. In addition, we can accommodate a high temporal resolution, which
is required when considering low viscosities and associated relaxation times
of only a few decades. The spherical harmonic cutoff degree for the
simulations shown in the following is

Displacement rates over the simulation period of 2 kyr, for an exemplary set of Earth model parameters

The load function

Same as Fig. 7 but for the rate of geoid-height change and Earth
model parameters

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,

Earth model parameters associated with the disc load ensemble simulations.
The viscoelastic parameterization of the Earth model is discretized in six radial layers;
upper and lower crust, mantle lithosphere, asthenosphere, upper and lower mantle. The lower
mantle extends down to the core mantle boundary (CMB; at the depth of 2763 km). Elastic
layers are represented by a quasi-infinite viscosity of 10

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,

Same as Fig. 7 but for four end-member sets of Earth model
parameters, without a DL and lithosphere thickness/asthenosphere viscosity of

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, 10

Same as Fig. 9:

The calculated response functions for surface deformation (radial
displacement) and gravity (geoid-height change) are discretized along 1507
latitudinal points within the range

Examples of response functions to the loading detailed in Sect. 5.1 for the
rate of radial displacement,

Figure 9 shows the response of

Figures 9 and 10 show that for the weak asthenosphere
(

For the extreme TnWk case, equilibrium rates of

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

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:

Output files contain 1507 latitudinal points (0

The results are stored independently for the rheology of East and West
Antarctica, the latter with and without a ductile layer in the elastic part
of the lithosphere. The data are stored in a Matlab^{®} file
format, which is also readable with GNU Octave

“Viscoel_response_WA_with_DL.mat”: response functions for West Antarctica with ductile layer

“Viscoel_response_WA_no_DL.mat”: response functions for West Antarctica without ductile layer

“Viscoel_response_EA_no_DL.mat”: response functions for East Antarctica without ductile layer

“Time_EA_rheo.mat/Time_WA_rheo.mat”: time (kyr) file related to response file for East and West Antarctica

“Coord_Co-Latitude.mat”: colatitude (

HL: lithosphere thickness (30 km, 40 km, …, 90 km; seven entries)

AV: asthenosphere viscosity (

LAT: latitude grid node, corresponding to file “Coord_Co-Latitude.mat” (1537 entries)

TIME: time, corresponding to file “Time_WA_rheo.mat” (202 entries)

VAR: variable type (1: rate of radial displacement in mm yr

Note that the asthenosphere and upper mantle viscosity is constant at

The spectral resolution underlying these fields is spherical harmonic cutoff degree 2048. The user should apply an adequate smoothing filter when using for inverting GRACE gravity fields. Filtered kernels are available upon request by the author.

The computation of the geodesic grid is not an original contribution of the
authors but based on the grid generator of the ICON GCM project
(

The files format is as follows:

vert-7.mask.cont_and_shelf.re.dat: longitude
(

vert-7.mask.cont_and_shelf.re.proj.dat:

The thickness of the elastic lithosphere at the locations of the geodesic
grid for different values of the viscosity threshold is applied to the data
set of Priestley and McKenzie (2013).

lith_thresh_21.disc.txt (threshold 10

lith_thresh_22.disc.txt (threshold 10

lith_thresh_23.disc.txt (threshold 10

The open-source software package SELEN allows the computation of the Maxwell
viscoelastic Earth response to user-defined ice sheet evolutions, in
particular also a simplified disc load forcing as presented in this paper.
The program is downloadable at:

The altimetry, gravimetry, GPS and viscoelastic modeling data used in this
project are available at

In this paper, we have presented refined temporal linear trends in surface
elevation, gravity field change and bedrock displacement based on
Envisat–ICESat (2003–2009), GRACE (2003–2009) and GPS (1995–2013),
respectively. In addition, we have performed forward modeling of the
viscoelastic response of the solid Earth to a disc load forcing. These
response functions are particularly suited to represent the distinct
geological regimes of East and West Antarctica in the joint inversion of
multiple satellite data. Similarly, the functions can be applied to the other
geographical regions as well. The data and code necessary to reproduce our
results, or apply our approach to a different problem, are provided at

We have refined surface-elevation rates for the Antarctic ice sheet for the time interval 2003–2009 by combining Envisat and ICESat altimetry data. The straightforward approach performs a grid-based comparison of the noise in the elevation rates obtained from Envisat and ICESat. For large parts of the ice sheet, the elevation rate is based on ICESat data, particularly for the rough terrain along the coast as well as close to the South Pole (polar gap of Envisat). Envisat contributes in some low-relief areas in East Antarctica and along the Antarctic Peninsula, as well as along single spurious ICESat tracks. Thus, the composite elevation rates are maximized in terms of spatial coverage and minimized in terms of the uncertainties.

The GPS processing carried out as part of the RATES and REGINA projects has produced a comprehensive data set of 118 Antarctic GPS records, which, for continuous sites, span a longer time interval (1995–2013) than those of previous studies (Thomas et al., 2011: 1995–2011; Argus et al., 2014: 1994–2012; Martín-Español et al., 2016b: 2009–2014). The ensemble processing done for the REGINA project has allowed us to assess the contribution of systematic error sources. In addition, for sites where there is potential doubt about the quality of the metadata or the behavior of the site, we have adopted a “conservative but realistic” approach to assigning new confidence limits. The screening of GPS data for outliers involved careful manual assessment, encompassing the review of measurement logs and notes on problems in the field. The data quality is reflected in the uncertainty estimates for the GPS rates, which therefore represents more reliable input data than GPS rates based on processing without manual intervention. Note, however, that SMB variations might also contribute to the GPS uplift rates given that the time spans of these data vary.

We have optimized the post-processing sequence for estimating the temporal linear trend and its uncertainty in the GRACE gravity field solutions for the region of Antarctica. In particular, we have derived optimal parameters for de-striping the monthly gravity fields over Antarctica according to Swenson and Wahr (2006). In addition, we have removed de-trended SMB fluctuations from the GRACE time series, to obtain a more representative uncertainty estimate based on the post-fit rms residual. We have included month-dependent weighting in the least-squares estimate of the gravity field rates to account for the varying quality of the monthly GRACE solutions. The optimization of the de-correlation filter of Swenson and Wahr (2006) to the signals expected in Antarctica reduced the residual uncertainty and improved the reliability of inferred mass anomalies.

With the aim of joining the multiple satellite data using the knowledge of the geophysical processes involved, we have calculated elastic and viscoelastic response functions of the solid Earth. The viscoelastic response functions represent the gravity field change and surface displacement to a disc load forcing for a variety of Earth model parameters; particularly, values of mantle viscosity and lithosphere thickness strongly varying between the distinct geological regimes of West and East Antarctica.

In particular, we have investigated the effect of a ductile layer in the
crustal lithosphere on the viscoelastic rebound signature. We show that for
moderate load changes of 0.45 m yr

The advantage of the viscoelastic response kernels is that a meaningful ratio of the rate of the gravity disturbance versus the rate of the surface displacement is calculated for each choice of the Earth model parameters, avoiding the approximation with an average rock density (e.g., Riva et al., 2009; Gunter et al., 2014). Using the response functions allows us to reconcile GIA signatures with measurements of large bedrock uplift and small gravity field increase in the Amundsen Sea Embayment, associated with weak Earth structures. Clearly, the response functions adopted here represent only the viscoelastic equilibrium state and, thus, are considered only an intermediate step to full dynamic modeling of the GIA response. Nevertheless, this approximation represents a significant improvement of other joint inversion methods, as it bases the joint inversion on physically meaningful response kernels. With extra data on the past ice evolution, such as paleo-thickness rates, our approach can be expanded to address the temporal evolution as well.

In the second REGINA paper (Sasgen et al., 2017a), we perform the joint inversion for present-day ice-mass changes and GIA in Antarctica, based on the input data sets and viscoelastic response functions presented here. We validate our results using forward-modeling results and other empirical models and show the impact on CryoSat-2 volume and GRACE mass balances, respectively. Note, however, that the post-processing methods and viscoelastic functions presented here are applicable also to other geographical regions with superimposed present-day mass change and GIA signatures.

ICESat 633 Level 2 data for the time span February 2003 until October 2009 used in this study.

We apply rates of firn compaction,

Rate of elevation change

Flowchart showing the estimation process for the temporal linear trends in the bedrock for Antarctic GPS site time series.

GPS uplift rates for this study. The columns
are as follows: site name, estimated uplift rate

Continued.

Continued.

Comparison of “prop, eman” GPS uplift rates for this study with rates from other studies.

In this study, we identify GRACE coefficients of CSR RL05 up to degree and order 50 appropriate to yield the most robust gravity field rates over Antarctica. Figure A3 provides another indication based on the degree-power spectrum of the geoid rates. It is visible that GFZ RL05 and CSR RL05 are very similar up to degree and order 50, where the power spectra show minima. For higher degrees, however, the power of the gravity field recovered with GRACE increases due to increasing noise; for the unfiltered coefficients, this increase is faster for GFZ RL05 than for CSR RL05.

Degree-amplitude spectrum of the rate of geoid-height change (mm yr

Spatial rate of geoid-height change

The filtering of the GRACE gravity fields was optimized for reducing noise over Antarctica. The effect on the rms uncertainties is shown in Fig. 3. Additionally, Fig. A4 presents the difference between the GRACE rates filtered only with a Gaussian smoothing filter of 200 km and with the optimized Swenson filter. It is visible that the differences in the rate of geoid-height change and the associated rate of equivalent water-height change show a stripe-like noise pattern. This suggests that the de-striping is superior to conventional Gaussian smoothing, even at high latitudes, where GRACE ground-track spacing is very dense. It is also important to note that the filter does not introduce any magnitude bias or change the spectral content of the gravity field rates, which is important when applying only a Gaussian smoothing of 200 km (without Swenson filtering) to the altimetry data set and response kernels.

The viscoelastic response kernels employed (Sect. 5) describe the
viscoelastic equilibrium state for the forcing with a disc load of constant
radius and constant rate of mass increase (likewise mass loss). We neglect
transitional changes in the solid Earth for load changes that have not
reached the equilibrium state in terms of geoid-height change and surface
displacement. Although the deformation and gravity signature in equilibrium
eventually only depends on the lithosphere thickness, the time to reach the
equilibrium is controlled by the viscosity parameters chosen. Figure A5 shows
the evolution of the standardized ratio of the geoid-height change vs.
surface displacement over time, calculated as

Standardized ratio of the rate of geoid-height change versus the rate of radial displacement for different values of the asthenosphere viscosity. Note that the ratio is calculated at the load center.

We assess the impact of SMB fluctuations on the uplift rate at the GPS
station locations using the modeled SMB of RACMO2 for the years 1979–2010.
We compute the elastic deformation related to cumulative monthly SMB,
de-trended for the entire simulation period 1979–2010. We then estimate the
temporal linear trends at the GPS station locations for a moving window of
varying width from 3 to 16 years. Then, for each window width, we estimate
the standard deviation of the apparent trend induced by SMB for selected
stations (Fig. A7). Typically, the uncertainty in uplift rate due to SMB
variability is below 0.4 mm yr

Standard deviation (

The load increases, with a fixed radius, at a constant rate of ca.
5.6 Gt yr

Load function applied to obtain the viscoelastic response functions.

IS conceived, managed and summarized this study with support from MRD. AME, BW and JLB performed the altimetry analysis. AH, MH and RP undertook the gravity field analysis, with contributions from IS. EJP and PJC analyzed and clustered the GPS data with critical input from TW. VK and HK performed the viscoelastic modeling, with contributions from IS. All authors were involved in writing and reviewing this manuscript.

The authors declare that they have no conflict of interest.

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