In this study, we use satellite gravimetry data from the Gravity Recovery
and Climate Experiment (GRACE) to estimate regional mass change of the
Greenland ice sheet (GrIS) and neighboring glaciated regions using a least
squares inversion approach. We also consider results from the input–output
method (IOM). The IOM quantifies the difference between the mass input and
output of the GrIS by studying the surface mass balance (SMB) and the ice
discharge (

We use a simulation model to quantify and correct for GRACE approximation errors in mass change between different subregions of the GrIS, and investigate the reliability of pre-1990s ice discharge estimates, which are based on the modeled runoff. We find that the difference between the IOM and our improved GRACE mass change estimates is reduced in terms of the long-term mass change when using a reference discharge derived from runoff estimates in several subareas. In most regions our GRACE and IOM solutions are consistent with other studies, but differences remain in the northwestern GrIS. We validate the GRACE mass balance in that region by considering several different GIA models and mass change estimates derived from data obtained by the Ice, Cloud and land Elevation Satellite (ICESat). We conclude that the approximated mass balance between GRACE and IOM is consistent in most GrIS regions. The difference in the northwest is likely due to underestimated uncertainties in the IOM solutions.

During the last decade, the ice mass loss from the Greenland ice sheet (GrIS)
has become one of the most significant mass change events on Earth.
Because of its ongoing and potentially large future contribution to sea
level rise, it is critical to understand the mass balance of the GrIS in
detail. As a result of increasing runoff and solid ice discharge (

To quantify the recent changes in the GrIS mass balance, three methods are used: satellite altimetry, satellite gravimetry and the input–output method (IOM) (Andersen et al., 2015; Colgan et al., 2015b; Sasgen et al., 2012; Shepherd et al., 2012; Velicogna et al., 2014; Wouters et al., 2013). We will concentrate on the latter two methods in this study, using results from satellite altimetry for validation purposes.

The IOM is used to evaluate the difference between
mass input and output for a certain region. It considers two major mass
change entities: surface mass balance (SMB) and solid ice discharge. SMB
is commonly estimated using climate models (Ettema et al., 2009; Fettweis,
2007; Tedesco et al., 2013; van Angelen et al., 2012), whereas

The satellite gravity observations from GRACE (Gravity Recovery and Climate Experiment) provide snapshots of the global gravity field at monthly time intervals, which can be converted to mass variations. Mass variation solutions of a given area that are obtained from GRACE observations are, however, influenced by measurement noise and leakage of signals caused by mass change in neighboring areas. Furthermore, the GRACE monthly gravity fields contain north–south oriented stripes as a result of measurement noise and mis-modeled high-frequency signal aliasing. Therefore, to estimate the mass balance for GrIS subregions from GRACE data, we apply the least squares inversion method (Schrama and Wouters, 2011) in this study with an improved approach (Xu et al., 2015). However, as shown by Bonin and Chambers (2013) in a simulation study, the least squares inversion method introduces additional approximation errors.

Previous studies have compared regional GrIS mass change from different
independent methods. In Sasgen et al. (2012), the mass balance in seven major
GrIS drainage areas was derived from the IOM and GRACE data using a forward
modeling approach developed by Sasgen et al. (2010). When separating out
the IOM components and comparing them with the seasonal variability in the
derived GRACE solution, the relative contributions of SMB and

The GrIS mascon layout, based on the drainage system definition by
Zwally (2012). A mascon with the same digits refers to a region belonging to
the same drainage system. The “a” and “b” terms indicate the GrIS margin
(

In this study, we aim to investigate the two aforementioned sources of uncertainties in GRACE and IOM mass balance estimations: (i) we present a way to reduce the error from the inversion approach and (ii) we investigate different discharge estimates. We then evaluate our results by comparing the GRACE and IOM estimates both with each other and with published estimates from satellite altimetry.

The GrIS drainage system (DS) definition of Zwally (2012) is employed here to investigate the mass balance in GrIS subregions. This definition divides the whole GrIS into eight major drainage areas, and each drainage area is further separated by the 2000 m elevation contour line, creating interior and coastal regions for each drainage area. This GrIS DS definition is employed by several other studies (Andersen et al., 2015; Barletta et al., 2013; Colgan et al., 2015a; Luthcke et al., 2013; Sasgen et al., 2012). The regional GrIS mass change estimated with GRACE are influenced by mass change from areas outside the ice sheet, i.e., from Ellesmere Island, Baffin Island, Iceland and Svalbard (EBIS) (Wouters et al., 2008). Therefore, we include four additional DSs to reduce the leakage from these regions. The overall mascon definition used in this study is shown in Fig. 1.

The main topic of this study is to provide improved GrIS regional mass balance estimates from GRACE and the IOM. We show that the improved GRACE solution reduces the regional differences between two mass change estimates, especially in the southeastern GrIS region. Furthermore, we compare the GRACE solution with the IOM, which employs different reference discharge estimates, showing that the uncertainties in the reference discharge can result in an underestimated mass loss rate in the IOM regional solution, in particular in the northwestern GrIS region.

In Sect. 2, we present SMB mass change from a recently improved regional
atmospheric climate model (RACMO2.3) (Noël et al., 2015) and discharge
estimates of Enderlin et al. (2014), which are based on a near-complete
survey of the ice thickness and velocity of Greenland marine-terminating
glaciers. In Sect. 3, we introduce the least squares inversion approach. In
Sect. 4, we start by investigating different methods to calculate mass
change in GrIS drainage areas using the modeled SMB and

For the GrIS, precipitation (

For both the whole GrIS and a complete drainage area from ice sheet maximum
height to the coast, the total mass balance is

In order to fit the temporal resolution of the modeled SMB data, we
interpolate the yearly

In a previous study of mass balance with the IOM, for which estimates of

For the reference period we defined the period for the integral in Eq. (5)
to be from years 1961 to 1990. For the subsequent period the lower and upper
bounds of the integral are 1991 and

As explained before, when Eq. (6) is used to compute the mass balance for
the regions below and above 2000 m separately, the ice flux across the 2000 m
contour (

Based on these two assumptions, we apply Eq. (6) for the interior and
coastal GrIS regions, yielding

In this study, we use the GRACE release 5 level 2 monthly spherical
harmonics coefficients

GRACE observations of mass change within a subregion of the GrIS are
affected by the mass change in neighboring areas, a phenomenon known as
leakage (Wahr et al., 1998). Furthermore, GRACE data should be corrected for
known oceanic and atmospheric mass motions, continental hydrology and
glacial isostatic adjustment (GIA). The oceanic and atmospheric mass change
is already removed from the coefficients provided by CSR. The Global Land
Data Assimilation System (GLDAS) model (Rodell et al., 2004), specifically
the 1

We correct for the GIA effect in the GRACE data of the GrIS by using the
output of the model by Paulson et al. (2007), which is based on the ICE-5G
ice loading history and the VM2 Earth model (Peltier, 2004). Hereafter, we
refer to this model by Paulson-07. In addition to this approach, three GIA
models with a total of 11 variations are employed based on different ice
histories and viscosity models to determine the uncertainty in the GIA
correction. For instance, the models of van der Wal et al. (2013) include 3-D
changes in viscosity and the model of Simpson et al. (2009) uses a different
ice loading history. A summary of the GIA models used in this study is given
in Table A3. An isotropic Gaussian filter is employed to reduce the noise in
GRACE data (Wahr et al., 1998), with a half width of

To estimate the regional mass balance in separate GrIS drainage area, we use
a constrained least squares inversion approach (Bonin and Chambers, 2013;
Schrama and Wouters, 2011):

In this study, the total error in SMB

The systematic error is the uncertainty in the SMB derived from model
output, whereas the averaging error is related to the variability of the
reference SMB

Correlation between the anomaly of the discharge

We also investigate the uncertainties of the 1961–1990 reference
discharge. In this study we employ D-14 as the

Another way to obtain historic discharge estimates is by using the presumed
correlation between discharge and SMB or runoff (Rignot et al., 2008; Sasgen
et al., 2012). This approach assumes that the anomaly of the discharge with
respect to a reference SMB (

Comparison between cumulative TMB (2000–2012) obtained with three
different methods. method 1, using no reference TMB, is shown with a green
curve. For method 2 (red markers and curve), the reference discharge is
based on the estimation from D-08, while the discharge estimation from D-14
is used (

Hereafter, the regional cumulative discharge anomaly (

To evaluate the SMB

In the solution of

Correlation between the linear trend in the simulations

The simulation model

Next we apply Eq. (9) to yield approximated regional mass change

The simulated trends in regional mass changes

Comparison of the 2003 to 2013 regional mass change rates between the GRACE solution and the IOM solutions. The first column on the left refers to the entire GrIS and the following ones to the right indicate the complete drainage area according to Zwally et al. (2012). The regional mass change rates from GRACE before correcting for the approximation error are represented by the light blue hollow squares; the filled dark blue squares indicate the mass change rates after implementing the correction. The numbers show the mass change rates in blue and red color which indicate the GRACE solution and IOM solution, respectively. The dashed line separates the solutions from the interior regions (above the dashed line) and from the coastal regions (below the dashed line). The error bars are estimated in Sect. A4.

For the coastal regions, there is a linear relationship between the
simulations

Contrary to the coastal regions, the linear relation between

We compare the regional mass change rate from GRACE with the IOM (Fig. 5)
before and after applying the approximation error correction to GRACE, and
with different discharge estimations implemented in the IOM, separately for
coastal and interior regions. Note that in this figure, the time interval is
January 2003 to December 2013; we extrapolate the 2013 ice discharge from Enderlin-14

We only consider TMB from the IOM in order to reduce the influence of the
individual uncertainties in SMB and

Figure 5 shows that the agreement between GRACE and the IOM improves after
correcting the GRACE approximation errors and applying the runoff-based
discharge estimations in DS3a, DS5a, DS6a and DS7a. The difference between
GRACE and IOM estimates is also reduced in DS1a and DS2a, where the
remaining difference falls within the uncertainty margins. The corrected
GRACE solution in DS4a is only

One reason for the discrepancies could be the discharge from peripheral
glaciers, which is not included in the IOM but which does affect GRACE
estimates. Previous studies, e.g., Bolch et al. (2013) and Gardner et al. (2013),
show that a mass rate of approximately 40 Gt yr

For the regions above 2000 m altitude, GRACE-inferred regional mass change
rates agree with the estimations from IOM within their uncertainties (see
Fig. 5). A noticeable mass increase appears in both the GRACE and IOM
solutions in DS2b (northeastern interior). A second observation is that in the
IOM the runoff dominates the regional mass balance on the edge of the
southern GrIS interior, which results in a mass loss of

We also compare our GRACE and IOM solutions to (1) GRACE, (2) IOM and (3) ICESat
altimetry estimates from different studies, as shown in Table 1. All listed
GRACE solutions agree within the uncertainty levels in DS1, DS2, DS3, DS5
and DS8. In the southeastern region DS4, there is a deceleration of the mass
change after 2007, when the regional acceleration of mass loss becomes
negligible (

The rates of mass changes in GrIS regions based on satellite
gravity data (GRACE), IOM output and altimetry data (ICESat), in
Gt yr

The IOM is also relatively uncertain in DS4 (Sasgen et al., 2012). Even
though the mass-change rates between GRACE and the IOM in this region show a
relatively large difference, agreement is obtained within the large
uncertainties. For ICESat-based mass loss estimates, the retrieved long-term
mass loss can be very different, e.g.,

Another area where GRACE and the IOM do not agree is the northwest (region
DS8). We find that our GRACE solution shows

We have reduced the approximation error in the GRACE solution for this
region, although by a small amount (

The uncertainties of the GIA effect are included as part of the
uncertainties of the GRACE solution for DS8 as well (see Table A3), but
adding these still cannot bridge the gap between GRACE and IOM. The ICESat-
and Operation IceBridge-based mass change estimates by Kjeldsen et al. (2013)
yield a mass loss rates of 55

In this study, we implement a simulation of the GrIS mass change and show
that the approximation errors caused by the least squares inversion approach
can be quantified and reduced in the GRACE solution. When using the IOM, we
also improve the reference discharge estimate by utilizing the modeled
runoff. We show that the regional differences between our GRACE and IOM
solutions are reduced and agree within their calculated confidence
intervals. This is confirmed by an intercomparison with ICESat-based
regional mass change rates. In the southeast, the corrections for the
approximation errors in GRACE data products are especially important. We
find that the IOM solutions underestimate the mass loss in the northwest
compared to the GRACE and ICESat solutions, which we attribute to incorrect
estimates in the reference

The GrIS ice discharge

To determine the linear relationship between the simulated regional mass
balances with the associated approximations after applying the least squares
inversion, the linear-fit parameters

The GrIS monthly mass balance simulations that are used in Sect. 4.2 are
based on the RACMO2.3 model and the discharges estimates from Enderlin et
al. (2014). Note that the discharge estimates are given in the form of
lumped mass change for 178 different geographical locations. To get SMB and

A summary of the uncertainties in the regional mass balance (linear trend)
is shown in Table A2. In our GRACE-inferred mass balance, the uncertainties
are associated with (1a) the standard deviations of the CSR RL05 GRACE
spherical harmonics coefficients (including the standard deviations of the
external degree

We apply the GIA correction to the GRACE data using three GIA models with a total of 11 different parametrizations before estimating the associated regional mass change in 20 GrIS and surrounding Arctic regions (see the mascon definition in Sect. 3). By comparing with one without applying GIA correction, we assume the differences are the regional GIA effects. In addition to the Paulson-07 GIA model, we use a GIA model with lateral changes in viscosity and the ICE-5G loading history (van der Wal et al., 2013).

Moreover, we use another GIA model based on the ice history model from
Simpson et al. (2009), provided by Glenn Milne within the scope of the IMBIE
project. The upper mantle viscosity ranges from 0.3

In Table A3, the GIA related mass change can vary from

In order to quantify the uncertainties of the regional GIA in the Paulson-07, since it is the GIA model we used to derive our GRACE solution, we estimate the standard deviation of all models with respect to Paulson-07. The uncertainties are summarized in Table A2.

The linear-fit parameters

The uncertainties associated with the regional mass change rates.
For the GRACE-inferred regional solutions, “coef. SD” refers to the errors
due to the standard deviations in the CSR RL05 spherical coefficients,
“GIA” refers to the errors obtained from comparing 11 GIA models. Note
that the GIA uncertainties in the interior GrIS are all close to 0 and are
therefore negligible. In the column with the header “cor” we show the
uncertainties which are caused by the approximation error correction. For
SMB and

The GIA effects on mass balance in different GrIS regions based on
11 different GIA models. The unit is Gt yr

The equivalent water thickness of the linear trend

This research is funded by means of scholarship GO-AO/27 provided by the Netherlands Organization of Scientific Research, NWO. We are grateful to Ian Joughin for the suggestions of estimating the ice flux at high elevation. Furthermore, the authors acknowledge the thoughtful comments by Etienne Berthier and three anonymous referees on the manuscript. Edited by: E. Berthier