The Antarctic Ice Sheet is the largest potential source of future sea-level
rise. Mass loss has been increasing over the last 2 decades for the West
Antarctic Ice Sheet (WAIS) but with significant discrepancies between
estimates, especially for the Antarctic Peninsula. Most of these estimates
utilise geophysical models to explicitly correct the observations for
(unobserved) processes. Systematic errors in these models introduce biases in
the results which are difficult to quantify. In this study, we provide a
statistically rigorous error-bounded trend estimate of ice mass loss over the
WAIS from 2003 to 2009 which is almost entirely data driven. Using altimetry,
gravimetry, and GPS data in a hierarchical Bayesian framework, we derive
spatial fields for ice mass change, surface mass balance, and glacial
isostatic adjustment (GIA) without relying explicitly on forward models. The
approach we use separates mass and height change contributions from different
processes, reproducing spatial features found in, for example, regional
climate and GIA forward models, and provides an independent estimate which
can be used to validate and test the models. In addition, spatial error
estimates are derived for each field. The mass loss estimates we obtain are
smaller than some recent results, with a time-averaged mean rate of

Changes in mass balance of the Antarctic Ice Sheet have profound implications
on sea level. While there is a general consensus that West Antarctica has
experienced ice loss over the past 2 decades, the range of mass-balance
estimates still differ significantly (compare, for example, estimates in
Shepherd et al. (2012), Tables S8 and S11, which range from

Different approaches have different sources of error. A key error in the
gravimetry-based estimates is a result of incomplete knowledge on glacial
isostatic adjustment (GIA), which constitutes a significant proportion of the
mass-change signal but leakage and GRACE errors are also important (Horwath
and Dietrich, 2009). For satellite altimetry, uncertainties arise from
incomplete knowledge of the temporal variability in precipitation (Lenaerts
et al.,2012; Frezzotti et al., 2012) and the compaction rates of firn
(Arthern et al., 2010; Ligtenberg et al., 2011), quantities which play a
central rôle in determining the density of the observed volume change.
For the IOM, the main sources of error stem from the surface mass balance
(SMB) estimates used (usually obtained from a regional climate model) and
uncertainties in ice discharge across the grounding line. Recent improvements
in regional climate modelling have reduced the uncertainty in the SMB
component but differences between estimates for the Antarctic Ice Sheet as a
whole still exceed recent estimates of its mass imbalance. For example, a
recent update of the commonly used regional climate model, RACMO, has
resulted in a change in the integrated ice-sheet-wide SMB of about
105 Gt yr

In an attempt to reduce the dependency on forward models, recent studies have combined altimetry and GRACE to obtain a data-driven estimate of GIA and ice loss simultaneously (Riva et al., 2009; Gunter et al., 2014). Here, we extend these earlier approaches in a number of ways. We provide a model-independent estimate not only of GIA but also of the SMB variations, firn compaction rates, and the mass loss/gain due to ice dynamics (henceforward simply referred to as ice dynamics). In doing so, we eliminate the dependency of the solution on solid-Earth and climate models. The trends for ice dynamics, SMB, GIA, and firn compaction are obtained independently through simultaneous inference in a hierarchical statistical framework (Zammit-Mangion et al., 2014). The climate and firn compaction forward models are used solely to provide prior information about the spatial smoothness of the SMB-related processes. Systematic biases in the models have, therefore, minimal impact on the solutions. In addition, we employ GPS bedrock uplift rates to further constrain the GIA signal. In future work the GPS data will also be used to constrain localised ice mass trends that cause an instantaneous elastic response of the lithosphere (Thomas and King, 2011). The statistical framework uses expert knowledge about smoothness properties of the different processes observed (i.e. their spatial and temporal variability) and provides statistically sound regional error estimates that take into account the uncertainties in the different observation techniques (Zammit-Mangion et al., 2014). The study reported here was performed as a proof of concept for a time-evolving version of the framework for the whole Antarctic Ice Sheet, which is currently under development. The time-evolving solution will use updated data sets and, as explained above, will also solve for the elastic signal in the GPS data. In addition, it will provide improved separation of the processes because of the additional information related to temporal smoothness that can be incorporated into the framework (discussed further in Sect. 5).

In this section we describe the data employed, which are divided into two groups. The first group contains observational data which play a direct rôle in constraining the mass trend. These include satellite altimetry, satellite gravimetry, and GPS data (Sects. 2.1–2.3). The second group comprises auxiliary data (both observational and information extracted from geophysical models), which we use to help with the signal separation (differentiating between the different processes we solve for accounting for their spatial smoothness) (Zammit-Mangion et al., 2014). These are discussed in Sect. 2.4.

We make use of two altimetry data sets in this study, obtained from the Ice,
Cloud and land Elevation Satellite (ICESat) and the Environment Satellite
(Envisat). In this study, we used ICESat elevation rates (d

The Gravity Recovery and Climate Experiment (Tapley et al., 2005) has provided temporally continuous gravity field data since 2002. Different methods have been used to provide mass-change anomalies from the Level 1 data. Most are based on the expansion of the Earth's gravity field into spherical harmonics; however, to make the data usable for ice mass-change estimates, it is generally necessary to employ further processing methods. These include the use of averaging kernels (Velicogna et al., 2006), inverse modelling (Wouters et al., 2008; Sasgen et al., 2013), and mass concentration (mascon) approaches (Luthcke et al., 2008). Spherical harmonic solutions usually depend on filtering to remove stripes caused by correlated errors (Kusche et al., 2009; Werth et al., 2009).

Error estimates for the GRACE mascon solutions, derived from a regression of the data (Zammit-Mangion, et al., 2014).

In this paper, we used a mascon approach (Luthcke et al., 2013), although we stress that the framework is not limited to this class of solutions. The mascon approach employed here uses the GRACE K-band inter-satellite range-rate (KBRR) data which are then binned and regularised using smoothness constraints. The fourth release (RL4) of the atmosphere/ocean model correction, which utilises the European Centre for Medium-Range Weather Forecasts atmospheric data and the Ocean Model for Circulation and Tides, was used (Dobslaw and Thomas, 2007). Some concerns with this correction have been reported (Barletta et al., 2012), but a release of the mascon data using the corrected version (Dobslaw et al., 2013) was not available for this study. Contributions to degree-one coefficients were provided using the approach by Swenson et al. (2008). Our mascon approach does not call for a replacement of C20 coefficients. We assume that GRACE does not observe SMB or ice mass changes over the floating ice shelves as they are in hydrostatic equilibrium. Hence, all observed gravity changes over the ice shelves are assumed to be caused by GIA.

Although the mascons are provided at a resolution of about 110 km, their fundamental resolution is nearer to that of the original KBRR data at about 300 km (Luthcke et al., 2013). For the statistical framework, it is important to quantify the correlation among the mascons so that it is taken into account when inferring both the processes and associated posterior uncertainties. We quantify the spatial correlation by determining an averaging model such that the diffused signal is able to loosely reconstruct the mass loss obtained using only altimetry (and assuming that all height change occurs at the density of ice). The averaging strength between mascon neighbours is also estimated during the inference (Zammit-Mangion et al., 2014). The error on the mascon rates is assumed to be a factor of the regression residuals on the trends, in a similar manner to the altimeter data (Zammit-Mangion et al., 2014). The a priori errors, after these two steps, are shown in Fig. 1, which also indicates the length scale over which we estimated the GRACE mascons to be uncorrelated.

GPS stations with vertical rate and errors, modelled elastic correction, and adjusted rates. The latter are used for inference.

The GPS trends used in this work were taken from Thomas and King (2011). Not all of the trends were suitable for our analysis, as the length of record did not always coincide with the 2003–2009 ICESat period. We only used stations with contemporaneous data, as well as those where we could access the original time series to confirm that the trend had remained constant, within the error bounds, for our observation period. For the northern Antarctic Peninsula, we followed the approach suggested in Thomas and King (2011) and used the pre-2003 trends, ignoring the later trend estimates, which are strongly influenced by elastic signals. All other stations were corrected for elastic rebound as in Thomas and King (2011) and subsequently assumed to be measuring GIA only (the published rates were used). A more advanced approach where the estimated ice loss is fed back into a dynamic estimate of the elastic rebound, is being implemented for a spatiotemporal extension of the Bayesian framework. The GPS data used in this study are detailed in Table 1.

Our framework makes use of several recent improvements in statistical modelling which can be exploited for geophysical purposes. Complete details regarding the mathematical methods employed are given in Zammit-Mangion et al. (2014), and here we provide a conceptual overview of the approach. A description of the software implementation can also be found in Zammit et al. (2015). The statistical framework hinges on the use of a hierarchical model where the hierarchy consists of three layers: the observation layer (which describes the relation of the observations to the measured fields), the process layer (which contains prior beliefs of the fields using auxiliary data sets), and the parameter layer (where prior beliefs over unknown parameters are described).

The “observation model” is the probabilistic relationship between the
observed values and the height change of each of the processes. For
point-wise observations, such as altimetry and GPS, the observations were
assumed to be measuring the height trend at a specific location. GRACE
mascons, however, were assumed to represent integrated mass change
over a given area. These mass changes were translated into height changes via
density assumptions: upper mantle density was fixed at 3800 kg m

Prior information and soft constraints applied to length scales and amplitudes based on expert judgement and analysis of the forward models discussed in Sect. 2.4.

In the “process model”, four fields (or latent processes) are modelled: ice dynamics, SMB, GIA, and a field which combines the processes that result in height changes but no mass changes, i.e. firn compaction and elastic rebound. We model the height changes due to these as spatial Gaussian processes, i.e. we assume that they can be fully characterised by a mean function and a covariance function. For each field we assume that the mean function is 0 (we do not use numerical models to inform the overall mean) and that the covariance function, which describes how points in space covary, is highly informed by numerical models and expert knowledge as described next. The relationship between the observations, priors, and the latent process, defined by the process model, is shown schematically in Fig. 2. Those processes that are influenced by an observation are linked by a solid arrow and it is evident that the problem is underdetermined as there are less independent observations than there are latent processes. This is why the use of priors is important to improve source separation (i.e. for partitioning elevation change between the four latent processes shown in Fig. 2). It should also be noted that SMB and firn compaction have been assumed, in this implementation of the framework, to covary a priori, as discussed later.

Schematic diagram showing the relationship between the observations, the process model defining the latent processes, and the priors employed.

The practical spatial range of surface processes – this describes the
distance beyond which the correlation drops to under 10 % – was
estimated from RACMO2.1 as described in Sect. 2.4. This analysis revealed,
for example, that locations at 100 km are virtually uncorrelated in the
Antarctic Peninsula but highly correlated inland from Thwaites Glacier.
Similarly, GIA was found to have a large practical range (

Length scales and prior soft constraints are easily defined for Gaussian
processes (or Gaussian fields) which nonetheless on the other hand, are also
computationally challenging to use. Gaussian fields can however be
re-expressed as Gaussian–Markov random fields by recognising that
Gaussian fields are in fact solutions to a class of Stochastic partial
differential equations (SPDEs, Lindgren et al., 2011). Numerical methods for
partial differential equations, namely finite element methods, can thus
be applied to the SPDEs in order to obtain a computationally efficient
formulation of a complex statistical problem (Zammit-Mangion et al., 2014).
Spatially varying triangulations (meshes) are used for the different
processes reflecting the assumption that, for example, ice loss is more
likely to occur at smaller scales near the margins of the Ice Sheet where
fast, narrow ice streams are more prevalent than in the interior. We thus use a
fine mesh at the margins (25 km) and a coarse mesh in the interior for this
field. GIA, however, is assumed to be smooth. This allows us to use a
relatively coarse mesh for this process (

We note that our methodology differs from others in that it is not an unweighted average of estimates with markedly different errors (Shepherd et al., 2012) or a sum of corrected data sources (Riva et al., 2009) but a process-based estimate. For each of the four fields (noting that elastic rebound and firn compaction covary in this implementation), we infer a probability distribution and standard deviation for every point in space. By relating pre-inference and post-inference variances, it is possible to assess the influence of different kinds of observation at each point on the resulting fields.

Inferential results are available for the four processes shown in Fig. 2 in isolation. In this section we report the results for each of the processes in turn but emphasise that these are presented to demonstrate the methodology rather than provide final estimates. This is because, as stated in Sect. 2, improvements are planned both to the framework and the data sets that we use in it. In all the examples shown, green stippling indicates where the signal is greater than the marginal standard deviation.

The gap in altimeter data around the pole results in spurious estimates for
that region, and the black shaded area south of 86

Height changes from firn compaction and elastic uplift of the crust
for 2003–2009 in m yr

Basin definitions used for West Antarctica (adapted from Sasgen et al., 2013).

Combined Ice and SMB loss trends for West Antarctica using RATES (pink), results from King et al. (2013) (blue), and results from Sasgen et al. (2013) (green). Basin definitions for King et al. (2012) differ for basins 1 and 24, so they are given in Table 3 instead. Our basin 25 is equal to the sum of basins 25 and 26 in King et al. (2012); this is given here as basin 25 for the King estimate.

Height changes from firn compaction and elastic rebound are estimated
together in a single field. Because they take place on similar length scales,
and there is no temporal evolution in our time-invariant solution presented
here, they are confounded in this study. Since firn compaction occurs at
relatively large rates (cm yr

Ice and SMB mass trends from RATES, Sasgen et al. (2013), and King
et al. (2012) in Gt yr.

Toy example illustrating the sensitivity of combination methods to
differing SMB estimates. The blue lines represent the set of equations that
solve for ice loss and GIA when SMB

Basin 23, which connects the ASE to the Southern Peninsula, also yields a
small uplift rate (0.4 mm yr

In Fig. 8 and Table 3 we present the basin-scale combined ice and SMB mass balance in comparison with two recent studies using GRACE (King et al., 2012; Sasgen et al., 2013). The latter study spans the ICESat period and the rates were taken from the publication. The former study, however, spans the 2002–2010 period. Basin definitions are the same as those in Sasgen et al. (2013) but differ from King et al. (2012): the sum of our basins 1 and 24 match the sum of their basins 1, 24, and 27. Our basin 25 matches the sum of their basins 25 and 26. Consequently, comparisons for these basins are not shown in Fig. 8 but provided in Table 3.

Overall, we obtain good agreement with Sasgen et al. (2013). Our mean
time-averaged ice loss rate of

We also compare our basin-scale results to ice loss rates from King et
al. (2012). Here, the observation periods are not identical, and the GIA
estimates differ. Still, there is generally good agreement at the
basin scale, in particular, where their GIA estimates (Whitehouse et al.,
2012) lie within our error ranges (basins 18, 19); the agreement is worst where their GIA
uplift rate is a multiple of ours (sum of basins 1 and 24). Overall, their
ice loss rate of

Mass trend values for each basin shown in Fig. 8 for different
values of the GIA length scale, SMB length scale, and ice surface velocity
threshold. All values in columns 2–4 are in Gt yr

Integrated over the domain studied, our loss estimate is smaller than other
recent estimates: Shepherd et al. (2012) arrive at

The results for SMB are more challenging to interpret because the trend, over
this time period, is relatively small (a few cm yr

Methods that combine altimetry and gravimetry such as Gunter et al. (2014)
and also the framework presented here are sensitive to the SMB anomaly used.
We illustrate this sensitivity through a simple calculation. Let the
unobserved processes on a 1 m

It is also worth examining the sensitivity of the solution to the prior
distributions that were derived from the forward models, auxiliary data sets,
such as surface ice velocity, and expert knowledge. To do this, we changed
the original amplitude and length-scale constraints as detailed in Table 4.
The table also lists the original mass trend (using constraints detailed in
Table 2) alongside the new estimates using the revised constraints. Changes
in the characteristic length scale for GIA and SMB have a rather small effect
on the integrated mass trend. However, the velocity threshold that
is used to determine whether the signal is likely to be associated with ice
dynamics appears to have a significant effect for the three basins that
comprise the Antarctic Peninsula: 23, 24, 25. This is because, for the
Peninsula, observed and balance velocities are missing in a number of places.
Where this is the case, they were set to 5 m yr

The GIA estimates from our study agree well with a recent GRACE-based estimate (Sasgen et al., 2013) and also compare well with a recent forward model (Ivins et al., 2013). Compared to AGE-1, the spatial pattern of our uplift maximum is shifted away from the Peninsula and towards the Ronne Ice Shelf. The spatial pattern is closer to that of W12a and ICE-5G models, with a bimodal uplift maximum centred underneath the Ronne and Ross Ice Shelves (Fig. 6a). This spatial structure is likely to have resulted from the use of GPS uplift rates, which were also used in the calibration of the most recent forward models (Whitehouse et al., 2012; Ivins et al., 2013). The W12a model yields slightly higher estimates for most basins but shows good agreement on the southern Antarctic Peninsula. Whitehouse et al. (2012) remark that the uplift rates using the W12 de-glaciation history – which are already substantially lower than the ICE-5G (Peltier, 2004) model rates – can be viewed as an upper bound. Separating secular and present-day viscous and elastic signals from the trends in this area remains a challenging task and will be treated in greater detail in the spatiotemporal version of our framework.

For this proof-of-concept study, our focus lies mainly on ice dynamics, SMB, and GIA estimates, neglecting to a certain extent the influence of mass-invariant height changes (due to firn compaction and elastic deformation of the bedrock). At this stage, the framework solves for a single process that combines both these processes. In this time-invariant framework, the two are confounded and cannot be separated, as they are not distinguishable by different densities or length scales. A better approach to solve for the elastic rebound of the crust would be to integrate a dynamic estimate that depends on the ice load changes. This approach is being implemented in the spatiotemporal version of the framework. Firn compaction is currently linked with SMB through a simple correlation model (Zammit-Mangion et al., 2014). This approach could be further improved by adding a temperature dependence, along the lines of a simple firn compaction model (Helsen et al., 2008). Finally, another open question concerns the extent of present-day viscoelastic rebound in the ASE.

Our proof-of-concept study shows that hierarchical modelling is a powerful tool in separating ice mass balance, SMB, and GIA processes when combining satellite altimetry, GPS, and gravimetry. We demonstrate that, using only smoothness criteria derived from forward models, it can provide an accurate estimate of the different processes. A time-varying version of the framework is currently being developed, which includes a number of improvements mentioned earlier. In particular, estimation of elastic rebound in the GPS time series, and more robust partitioning of ice dynamics and SMB will provide substantial improvements in source separation, error reduction, and GIA estimation. A central advantage of the framework is that new data – which need be neither regular or gridded – can be added at any point. For example, it is possible to extend the observation period forward or back in time using data from ERS-2, Cryosat2, or any other data set that contains information about one of the processes being solved for. This could include, for example, accumulation radar data or shallow ice cores for SMB variability or additional GPS sites as they become available. Preliminary tests have shown that the inference can also be performed without GRACE data.

J. L. Bamber conceived the study, co-wrote the paper, and made all revisions during review. N. Schoen and A. Zammit-Mangion prepared the data, implemented the framework, and co-wrote the paper. J. C. Rougier advised on the statistical inference framework and methodology. T. Flament and F. Rémy provided the Envisat data; S. Luthcke provided the GRACE data. All authors commented on the manuscript.

The authors would like to thank the following colleagues for helpful discussions: Volker Klemann, Ingo Sasgen, Matt King, Liz Petrie, Pete Clarke, Martin Horwath, Finn Lindgren, and Valentina Barletta. This work was funded by UK NERC grant NE/I027401/1.

Also, the following colleagues provided additional data without which the project would not have been possible: J. M. Lenaerts, S. Ligtenberg, Erik Ivins, Ricardo Riva, Brian Gunter, Pippa Whitehouse, Ingo Sasgen, Rory Bingham, Grace Nield, Liz Thomas. Edited by: E. Larour