Introduction
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 (D), the GrIS
has been experiencing a considerable and increasing mass loss since the
mid-1990s (Hanna et al., 2005; Rignot and Kanagaratnam, 2006; van den Broeke
et al., 2009). The changes in mass loss rates are due to different
processes. For instance, mass loss acceleration in the northwestern GrIS is
linked to the rapidly increasing discharge in this region (Enderlin et al.,
2014; Andersen et al., 2015), while in the southeast the increase in mass
loss rate after 2003 is mainly due to enhanced melting and less snowfall
(Noël et al., 2015).
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 D
can be estimated from combined measurements of ice velocities and ice
thicknesses, e.g., Rignot and Kanagaratnam (2006), Enderlin et al. (2014) and
Andersen et al. (2015). To reduce the uncertainties in the mass change of
SMB and D, the SMB and D from 1961 to 1990 are sometimes used as a reference
when applying the IOM method (van den Broeke et al., 2009; Sasgen et al.,
2012). However, introducing SMB and D as reference may introduce new
uncertainties in the IOM. We will discuss the details of the IOM, as well as
the uncertainties of the reference SMB and D, in Sect. 2.
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 D to the
annual mass balances were revealed. In the northwestern GrIS, important
differences between IOM and GRACE were noted, which were attributed to the
uncertainty in the regional discharge component in this area, where detailed
surveys of ice thickness are lacking. The comparison between two approaches
shows a mass loss difference of 24 ± 13 Gt yr-1 in this
region, and as a result the uncertainty in the regional mass balance
estimate is estimated at ∼ 46 %. However, using new
discharge estimates and the corresponding IOM regional mass change in the
northwestern GrIS, Andersen et al. (2015) found that the difference between
GRACE and IOM mass loss estimates fell within the combined uncertainty
range. However, using the least squares based inversion approach of Schrama
and Wouters (2011), we find that the mass change differences between results
obtained from GRACE and the IOM in the southern GrIS increase and cannot be
explained by the assumed uncertainties. An example of the regional
differences between the GRACE data and the IOM solution can be seen in Fig. A1.
The details of this difference will be discussed in Sect. 4.
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
(< 2000 m) and GrIS interior (≥ 2000 m), respectively. There are
16 GrIS mascons and 4 neighbouring Arctic mascons. The locations of the
three largest discharge outlets are marked with a star, i.e., Jakobshavn
(green), Kangerdlugssuaq (red) and Koge Bugt (blue) glaciers. The glacier
surface area is defined in the RACMO2.3 model.
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 D estimates.
Subsequently, we identify the approximation errors in regional mass change
estimates from GRACE data, followed by a comparison of mass change estimates
from GRACE and the IOM and a discussion of the remaining differences. The
conclusions and recommendations are given in Sect. 5.
IOM method
SMB and D
For the GrIS, precipitation (P) in the form of snowfall is the main
contribution to the mass input, while mass loss is a combination of
sublimation (S), meltwater runoff (R) and D. The
SMB equals to P - S - R, and subtracting D from SMB
yields the total mass balance (TMB). In this study, we use the Regional
Atmospheric Climate Model, version 3 (RACMO2.3), to model the SMB of the
GrIS. RACMO2.3 (Ettema et al., 2009; Van Angelen et al., 2012; van den
Broeke et al., 2009) is developed and maintained at the Royal Netherlands
Meteorological Institute (KNMI) and has been adapted for the polar regions
at the Institute for Marine and Atmospheric Research, Utrecht University (UU/IMAU).
RACMO2.3 model output is currently available at
∼ 0.1∘ spatial resolution for January 1958 to December 2014. The differences
between a previous model version (RACMO2.1) and other SMB models are
discussed by Vernon et al. (2013). For RACMO2.3, we assume 18 %
uncertainties for the P and R components in each grid cell. Assuming both
components to be independent, the uncertainty of the SMB is the quadratic
sum of uncertainties of P and R. The magnitude of S is small and its
absolute uncertainty is negligible compared to those in P and R. The
RACMO2.3 model also provides estimates of the SMB in the peripheral glacier
areas, which we have included in this study.
D estimates from Enderlin et al. (2014) (hereafter
Enderlin-14, with the associated discharge estimates D-14) are used in this
study. In Enderlin-14, the ice thickness of 178 glaciers is estimated from
the difference in ice surface elevations from repeat digital elevation
models and bed elevations from NASA's Operation IceBridge airborne
ice-penetrating radar data. The ice surface velocity is obtained from
tracking the movement of surface features that are visible in repeat Landsat
7 Enhanced Thematic Mapper Plus (ETM+) and Advanced Spaceborne Thermal and
Reflectance Radiometer (ASTER) images. For glaciers with thickness transects
perpendicular to ice flow (i.e., flux gates), the ice flux is estimated by
summing the product of the ice thickness and surface speed across the
glacier width. For glaciers with no thickness estimates, we use empirical
scaling factors for the ice flux, as derived in Enderlin et al. (2014).
Because the ice fluxes are calculated within 5 km of the estimated grounding
line locations, the SMB gain or loss between the observations and the
grounding lines will be small and the ice discharge is estimated directly
from the fluxes (Enderlin et al., 2014). The resulting estimation of
discharge uncertainty of 1–5 % for each glacier is smaller
than in previous studies, e.g., Rignot et al. (2008) (hereafter Rignot-08,
and the associated estimates are denoted by D-08), which relied on interior
ice thickness estimates that were assumed to be constant in time.
Cumulative TMB anomaly
For both the whole GrIS and a complete drainage area from ice sheet maximum
height to the coast, the total mass balance is
TMB=SMB-D.
In this study, we further separate each GrIS drainage area in a downstream (I)
and upstream (II) region, separated by the 2000 m surface elevation
contour line. Thus, for the subdivided regions Eq. (1) becomes
TMB=TMBI+TMBII,
where
TMBII=SMBII-FII,
and
TMBI=SMBI+FII-FI,
in which FII
refers to the ice flux across the 2000 m elevation contour, and
FI refers to the ice flow across the flux gate. Note that FII is canceled if the study area includes
both the regions below and above the 2000 m contour but
FII has to be considered when the upstream and downstream regions are considered
separately. As described above, we assume that SMB changes downstream of the
Enderlin-14 flux gates are negligible and that FI = D.
In order to fit the temporal resolution of the modeled SMB data, we
interpolate the yearly D on a monthly basis. Significant seasonal variations
in ice velocity have been observed along Greenland's marine-terminating
outlet glaciers (Moon et al., 2014). However, since we focus mostly on
long-term mass change in this study, monthly variations in D should have a
negligible influence on our analysis, and we assume that D is approximately
constant throughout the year. Contrary to the total mass represented in the
GRACE data, the SMB, D and TMB are estimates of rates of mass change (i.e.,
mass flux) in “Gt per month” or “Gt per year”. Hence, in order to compare with GRACE, one has
to integrate the SMB and D from a certain month (or year) onwards, which yields
ΔTMBi=∫i0iSMBt-Dtdt,
where ΔTMBi is the cumulative mass change at
month i in the IOM (unit is Gt) and the integration time period is from a
certain initial month i0 to month i.
In a previous study of mass balance with the IOM, for which estimates of D
were not available for certain regions (Rignot et al., 2008), the 1961 to
1990 reference SMB is used to approximate the missing regional D (Sasgen et
al., 2012). Also, due to the uncertainties in the SMB model, accumulating
the TMB over decades may also lead to unrealistic mass gains or losses (van
den Broeke et al., 2009). By removing the reference, the influence of large
uncertainties and interannual variability in SMB and D can be reduced (van
den Broeke et al., 2009). For instance, the uncertainties due to model
configurations could be similar in every monthly SMB estimate, and
accumulating those over a long period may result in a large uncertainty. The
choice of reference period is based on the assumption that the mass gain
from the surface mass balance during that period is compensated by ice
discharge, i.e., the GrIS was in balance during that period (no mass change).
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 i, respectively. Since we assume the GrIS
was in balance during the reference period, ∫19611990(SMBt - Dt) dt = 0. By removing the reference SMB
and D (i.e., SMB0 and D0) Eq. (5) becomes
ΔTMBi=∫1991iδSMBt-δDtdt,
where δSMBt = SMBt - SMB0 and
δDt = Dt - D0. Note that SMB0 and δSMBt
are both rates of mass change (and similarly for the discharge).
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 (FII) has to be considered. Therefore we introduce two assumptions:
(1) FII is constant over time, which means FII = F0II
(F0II is the FII during the reference period), so that
∫1991iδFtII dt = 0,
and (2) the separate GrIS interior and coastal regions are all in balance
during the 1961–1990 reference period,
i.e., ∫19611990(SMB0II - F0II) dt = 0 and
∫19611990(SMB0I + F0II - D0) dt = 0. Assumption (1) is necessary since there
is a lack of yearly measurements of ice velocity across the 2000 m contour.
An estimate of decadal change by Howat et al. (2011) suggests it is
reasonable to assume a constant FII for the entire GrIS, except for a few glaciers, such
as the Jakobshavn glacier where, after 2000, the FII may be higher than
F0II. In Andersen et al. (2015), the mass balance of the interior GrIS (in their
study defined as the ice sheet above the 1700 m elevation contour) was
41 ± 61 Gt yr-1 during the 1961–1990 reference period
and in Colgan et al. (2015a) the ice flux across the 1700 m contour was
estimated to be 54 ± 46 Gt yr-1 for the same time
period, indicating the assumption of balance approximately holds within the
given uncertainties.
Based on these two assumptions, we apply Eq. (6) for the interior and
coastal GrIS regions, yielding
ΔTMBiII=∫1991iδSMBtIIdt,
and
ΔTMBiI=∫1991iδSMBtI-δDtdt.
We quantify the combined uncertainties of assumptions (1) and (2) by comparing
the results from Eq. (7) to the regional mass balance derived from GRACE by
Wouters et al. (2008), as well as those derived from Ice, Cloud and land Elevation Satellite (ICESat) by Zwally
et al. (2011), resulting in an uncertainty of ±15 Gt yr-1
for the entire interior GrIS. The regional uncertainties are
summarized in Table A2. Note that for each region, the same uncertainty is
applied to both the interior and coastal areas. For the whole drainage area,
the uncertainties associated with assumption (1) and (2) will vanish, because
these two assumptions are needed only when we separate the coastal and
interior regions.
Cross-validation
Reference SMB and D
In this study, the total error in SMB0, hereafter σSMB0, is
a result of the systematic error caused by the assumption of a reference
period and the averaging within the chosen reference period. Both components
will be explained hereafter.
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 SMB0 during 1961–1990. To quantify the latter, we perform a
Monte Carlo simulation to evaluate the standard deviations of the SMB0
that result from using different combinations of 20-year averages of the
SMB, following van den Broeke et al. (2009). The sampled combinations are
randomly chosen from the months between 1961 and 1990. For RACMO2.3, we find
20 Gt yr-1 averaging errors in σSMB0. The
SMB0 from RACMO2.3 yields 403 Gt yr-1; hence the
systematic error is approximately 73 Gt yr-1 (considering
18 % uncertainty in RACMO2.3). If we assume both errors are independent,
then σSMB0 = 75 Gt yr-1.
Correlation between the anomaly of the discharge δD with respect to a reference SMB (y axis) and the
4-year averaged runoff δR (x axis) in
GrIS regions. The symbols with different colors refer to different
estimations of D. The grey bars for both δD and δR indicate the
errors. The correlation coefficients R2 are also shown in each plot.
We also investigate the uncertainties of the 1961–1990 reference
discharge. In this study we employ D-14 as the D estimate in the IOM.
However, the D-14 time series starts from the year 2000, when the GrIS
already was significantly out of balance. To retrieve D0 for
D-14, we use D0 = 413 Gt yr-1 in 1996 from
D-08 (D0-08) for the entire GrIS and assume that the regional D
changes from 1990 to 2000 in D-08 are proportional to the changes in D-14 in
each region, i.e., that D-14 and D-08 are linearly correlated. The details of
the interpolation of regional D0 are given in Sect. A1. Note
that the averaging error in D0 is minimized via an iteration
process, the details of which can be found in Rignot et al. (2008). Due to
the lack of ice thickness information before 2000, the reference D0 in
Rignot-08 has high uncertainty, especially in the northwest of the GrIS.
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 (δD = SMB0 - D) is correlated with a reference
runoff (δR = R - R0), which is based on the anomaly
of the 5-year averaged runoff. Note that the lagging correlation is
discussed in Bamber et al. (2012) and Box and Colgan (2013). In this study
we choose to use the runoff output from the RACMO2.3 model. We consider
three estimates of D, i.e., by Rignot-08, Enderlin-14 and Andersen et al. (2015),
which use different types of measurements of the ice thickness and
flux velocity changes, integration areas (areas between the flux gate and
the grounding line), SMB and ice storage corrections. Additionally, they
differ in whether the peripheral glaciers are included or not. In this study
we provide runoff-based estimates for D0 only for those GrIS drainage
areas where the correlation coefficients between δD and
δR are equal or higher than 0.7 (Fig. 2), i.e., for DS1,
DS3, DS7 and DS8. For the entire GrIS, we obtain a high correlation (R2 = ∼ 0.86),
similar to the correlation found by Rignot et al. (2008). However, the
high correlation cannot be obtained in all the GrIS
drainage areas; the regional correlation coefficients vary from 0.19 to 0.94.
In DS7 and DS8, the discharge anomaly is obviously correlated with the
runoff anomaly (R2 > 0.9), while in other regions (i.e., in
DS2, DS4, DS5 and DS6), the correlation is low (R2 < 0.5). In
DS3a, when we consider only the D estimates from Enderlin et al. (2014) and Andersen et al. (2015), the correlation increases to R2 = 0.72. Note that, the
regions with high correlation are also those that have a large fraction of
marine-terminating glaciers. We derive the linear relation between
δD and δR for eight major GrIS DSs and calculate the
regional annual δD from 1961 to 2013 using this linear relation.
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 (δD-14) for the years after 2000. Method 3 (blue markers and
curve) interpolates the reference discharge using the modeled runoff data
(only in DS1, DS3, DS7 and DS8), is dented by δDR.
δD-08 refers to the discharge estimated using D-08. All the discharges are
shifted upward by 200 Gt for visualization purposes. The numbers in each
plot indicate the annual TMB change rates in Gt yr-1. The
x axis shows the last two digits of the years from 2000 to 2012.
Hereafter, the regional cumulative discharge anomaly (δD),
which is derived from the RACMO2.3 runoff, is denoted as δDR,
while δD-08 and δD-14 refer to δD based on
Rignot-08 and Enderlin-14, respectively. We compare δDR,
δD-08 and δD-14 in Fig. 3 for the time interval 2000 to
2007, which is common to both δD-08 and δD-14. In DS7 we
find a correlation of R2 = 0.94. In that region, δD-08 and
δD-14 are similar, i.e., 20.1 ± 1.9 and
17.6 ± 2.2 Gt yr-1, respectively, but δDR
is 8.9 ± 4.7 Gt yr-1. The difference between the
runoff-derived and flux gate D estimates may indicate that the reference
D0 for this region should be ∼ 9 Gt yr-1
lower than D0 estimated by Rignot-08. A similar difference can be seen
in DS4 where we obtain 36.2 ± 2.5 Gt yr-1 for
δD-14 and 37.9 ± 2.8 Gt yr-1 for δD-08, but
δDR is 8.4 ± 3.3 Gt yr-1. However, in DS4,
δDR is probably not reliable, since the runoff-to-discharge
correlation is weak in this region (R2 = 0.38). For the entire GrIS,
the reference discharge D0 is 427 ± 30 Gt yr-1 for
δD-08 and 414 ± 44 Gt yr-1 for δD-14.
When applying the runoff-based interpolated D0 only for DS1, DS3, DS7
and DS8, with the rest of drainage areas using δD-14, D0
becomes 410 ± 37 Gt yr-1 and all three versions of
reference discharge agree within their respective uncertainties.
To evaluate the SMB0 and D0 used in this study, we compare the IOM
regional mass balance in eight major drainage areas (interior and coastal
regions are combined) and apply both Eqs. (5) and (6). The latter
equation relies on the determination of the SMB0 and D0 while Eq. (5)
does not, so the comparison can provide an indication about the
reliability of the SMB0 and D0 for some drainage area. For the
application of Eq. (6) we use two methods. Method 2 uses δD-14 while
method 3 uses δDR in DS1, DS3, DS7 and DS8. As can be seen in
Fig. 3, the three methods agree for the whole GrIS, as well as for most of
the drainage areas, within their uncertainties. In DS4, DS7 and DS8,
however, methods 1 and 2 are significantly different, which may be caused by
underestimated cumulative errors in Eq. (5) or a less accurate reference
surface mass balance SMB0 and reference discharge D0. This is
further discussed in Sect. 4.3.
Approximation errors
In the solution of x^, two types of errors occur: (a) systematic
errors caused by measurement errors propagated through the least squares
approach and (b) an additional error that is introduced when applying Eq. (9).
For the type (b) error, Bonin and Chambers (2013) show that the use of
Eq. (9) results in a noticeable difference between the approximation
x^ and the “truth” (a GrIS mass change simulation), especially in
GrIS subregions, which we recognize as an error source; see also the
discussion in Tiwari et al. (2009). Hereafter the type (b) error is denoted
as “approximation error” or ε. We estimate
ε by using simulations of GrIS for
x, following Bonin and Chambers (2013), so that the
approximation error becomes ε = x - x^. The simulated regional mass change on the
mascons is x = [x1, x2,
x3, … , xn], where n is the total number of mascons. We will
show that there is a relation between x^ and x,
which can be used to correct for the approximation errors.
Correlation between the linear trend in the simulations
x′ (y axis) and the corresponding approximation
x^′ (x axis). The units are Gt yr-1. The colors are
associated with the changing range of x′ for a
standard deviation going from 1σ to 5σ. The numbers refer to
the R2 coefficients of the error bounds of the corresponding colors.
The simulation model y = f(θλ) is based on the
10-year linear trend (2003 to 2012) of mass change of SMB and D estimates
(see Sect. A3), with uncertainties of the simulation model written as
σ(θλ). We employ a Monte Carlo
approach to simulate a sample of 1000 randomly distributed observations,
according to yl′ = y + klσ with
kl = kl(θλ) a vector of random
scaling factors that vary from -1 to 1 and index l running from 1 to 1000.
It is important to note here that we assume that measurement errors do not
exist (i.e., the simulation model is assumed to be reality). In addition, we
assume that the generated samples in the simulation (σ) are normally distributed.
Next we apply Eq. (9) to yield approximated regional mass change
x^ = [x^m], in which m is the index of the mascons (see Fig. 1).
From the simulation we can derive the real regional mass change rate
x = [xm]. As mentioned above, the difference between
x^ and x equals the approximation error. In Fig. 4 we
show that the xi are linearly correlated with x^m. By
applying this correlation to the approximations derived from GRACE data, one
can reduce the approximation errors in the GRACE-based regional mass balance
approximations.
The simulated trends in regional mass changes x′ and the
corresponding approximation x^′ are shown in Fig. 4. It can be
noticed that the approximations are strongly correlated over time with the
simulation in the coastal regions, having an average correlation coefficient
R2 of ∼ 0.9. This means that the approximated regional
solutions are close to the simulation. The correlation in region DS1a is
weaker (∼ 0.6), which suggests that the approximation for
region DS1a is influenced more by mass change in neighboring regions such
as region DS8a. In the simulation, the inter-region correlation between DS1a
and DS8a is ∼ -0.1, while in the approximations, the
correlation rises to ∼ -0.5. By comparison, another neighbor
of DS8a, DS7a, has a very weak inter-region correlation with DS8a
(∼ 0.04), both in the simulation and in the approximation. The
inter-region correlation errors are a systematic result of the least squares
inversion (Bonin and Chambers, 2013; Schrama and Wouters, 2011). Previous
work shows that the regional approximation errors can be reduced by
specifying constraints for the GrIS coastal and inland regions. However, in
this study all the sub-drainage areas within the coastal region are
constrained by the same prior variance, resulting in the large remaining
correlation between DS1a and DS8a.
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 x and the approximation x^, as can be
seen in Fig. 4. We fit this relationship by x = α1x^ + α0,
with α1 and α0 given in
Table A1. The linear relationship between the simulated and the approximated
regional mass change rates is found to be stable; even when the simulation
uncertainties are multiplied with a factor of 5 (light green marks in
Fig. 4), the regression parameters (α1 and α0) vary by less
than ∼ 1 % for the coastal mascons. Therefore, it is
reasonable to assume α1 and α0 should also reflect the
relationship between reality and the approximation as derived from GRACE
observations. When the vector of observations y becomes
the GRACE observations, we can derive an improved GRACE regional solution by
applying the linear relationship to the corresponding approximation
x^. We will show in Sect. 4.3 that this correction yields better
agreement between GRACE and the IOM.
Contrary to the coastal regions, the linear relation between x and x^
is weak in the interior regions, where the mean correlation coefficient is
∼ 0.2. This may be due to the fact that interior regions show
smaller mass change rates than the coastal regions. For simulations created
within a 1σ range, the highest correlation coefficient is only 0.47
for DS7b. The strong constraint used for these regions, i.e., a priori
variance of 0.1 m2, may cause the approximation to be more strongly
determined by this constraint than in the simulation. However, even if we
apply a weaker constraint, i.e., λ = 106, the correlation
coefficients between x and x^ in these regions
remain below 0.5. This means that correcting the approximation errors using
the same method as for the coastal regions may create larger uncertainties.
Following Bonin and Chambers (2015), we choose to include the approximation
errors in the uncertainties, but only for the interior regions. The
uncertainties are shown in Table A2.
Results and discussions
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
D estimates. For the coastal regions, we find that the correction of the
approximation errors in the GRACE solutions adjusts the mass distributions
between adjacent mascons. For instance, the corrected mass loss rate in DS3a
increases by 10 Gt yr-1 while it reduces the mass loss rate
in the adjacent region DS4a by 15 Gt yr-1. In DS6a,
correcting for the approximation error causes a mass loss increase of
13 Gt yr-1. One may notice that the corrected GRACE mass loss
rates exceed the uncertainty range of the mass loss rates before correction;
e.g., in DS1a and DS3a. This is because the uncertainty before our correction
is estimated without considering the approximation error.
We only consider TMB from the IOM in order to reduce the influence of the
individual uncertainties in SMB and D. We obtain two IOM solutions, using
the reference D0 by Rignot-08 (method 2) and the interpolated
discharges based on the RACMO2.3 runoff (method 3). In DS1a and DS3a, we
obtain lower discharge changes rate from method 3 than from method 2. In
DS7a, which includes Jakobshavn glacier, method 3 results in smaller mass
change than method 2.
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 ∼ 3 Gt yr-1
lower than the IOM solution, while it was ∼ 10 Gt yr-1
higher before correcting for the approximation error. However,
regardless of whether the approximation errors are corrected, the GRACE-inferred regional mass balance agrees with IOM mass balance in DS4a due to
the large uncertainties in the GRACE solution and the RACMO2.3 model there,
i.e., ±17 Gt yr-1 (see Table A2). From Fig. 5 we can
also make an inference about the effect of using different methods to
estimate the reference discharge. That is, a change in the reference
discharge will accumulate over time and cause the same amount of change in
the mass change rates. Only in DS8a, the IOM and GRACE do not agree within
the uncertainties.
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-1
comes from the peripheral glaciers. However, this is not the reason for the
difference between GRACE and the IOM. In this study we are using RACMO2.3
SMB estimates for not only the GrIS but for all of Greenland, including the
majority of the mass loss from the peripheral glaciers and ice caps.
Discharge from the peripheral glaciers and ice caps is expected to be small,
because there are far less marine-terminating glaciers that drain the
glaciers and ice caps than those that drain the ice sheet. Less than half of
the glaciers and ice caps are marine-terminating in Greenland (Gardner et
al., 2013). Moreover, given the relationship in the discharge data found by
Enderlin et al. (2014) between glacier width and area for the ice sheet's
marine-terminating glaciers, we expect the discharge from these glaciers to
be small. Consequently, we expect the regional mass change in these glacier
areas to be dominated by changes in SMB, which are captured by RACMO2.3. The
GRACE–IOM difference due to the exclusion of discharge from peripheral
marine-terminating glaciers and ice caps will likely be negligible as long
as the SMB for the whole of Greenland is considered, not just the ice sheet.
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 -8 Gt yr-1.
The overall IOM uncertainties in the coastal regions are mainly
influenced by the uncertainties in the SMB and D estimates, but the
assumption on the flux across the 2000 m contour (F2000) contributes to
additional uncertainties in the GrIS interior regions. In the GRACE
solution, the uncertainties are due to the errors in the GRACE coefficients
which are not dependent on the altitude. Therefore, the uncertainty level is
similar for the interior regions and the coastal regions.
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 (∼ -0.1 Gt yr-2). However, different GRACE
solutions for the same time period do not result in the same mass change
rates. This suggests that a large approximation error, which is associated
with different approximation approaches, is likely present in the GRACE
solutions for this region. As shown in Fig. 5, if we consider the time
period of 2003–2013, the regional mass change is reduced by 29 % in
this region after applying the correction introduced in Sect. 4.2.
The rates of mass changes in GrIS regions based on satellite
gravity data (GRACE), IOM output and altimetry data (ICESat), in
Gt yr-1. The sources are Zwally et al. (2011), Sasgen et al. (2012),
Barletta et al. (2013), Colgan et al. (2015b), Groh et
al. (2014) and Sørensen et al. (2011). Note that the IOM solution from this
study is shown in brackets.
Drainage area
DS1
DS2
DS3
DS4
DS5
DS6
DS7
DS8
Zwally-11 (2003–2007)
1 ± 0
13 ± 0
-51 ± 1
-75 ± 2
-10 ± 0
-4 ± 0
-14 ± 0
-33 ± 1
ICESat
Sørensen (2003–2009)
-16 ± 1
-16 ± 3
-40 ± 18
-43 ± 11
-26 ± 5
-51 ± 7
-53 ± 3
ICESat
Colgan (2003–2010)
-21 ± 6
1 ± 6
-47 ± 13
-28 ± 7
-24 ± 4
-33 ± 7
-23 ± 9
-42 ± 12
GRACE
Sasgen (2003–2010)
-20 ± 4
-16 ± 5
-31 ± 8
-66 ± 21
-20 ± 7
-66 ± 20
-26 ± 12
IOM
Sasgen (2003–2010)
-16 ± 5
-12 ± 5
-38 ± 6
-42 ± 6
-24 ± 6
-56 ± 7
-53 ± 7
GRACE
Barletta (2003–2012)
-17 ± 2
-12 ± 2
-36 ± 4
-35 ± 3
-23 ± 2
-66 ± 4
-44 ± 4
GRACE
This study (IOM solution in brackets)
2003–2007
-14 ± 6
-1 ± 8
-49 ± 6
-60 ± 13
-18 ± 5
-6 ± 9
-25 ± 6
-32 ± 6
GRACE (IOM)
(-22 ± 7)
(-9 ± 4)
(-55 ± 12)
(-73 ± 21)
(-11 ± 6)
(-14 ± 8)
(-17 ± 3)
(-13 ± 3)
2003–2010
-19 ± 4
-9 ± 5
-33 ± 5
-50 ± 11
-19 ± 7
-24 ± 8
-34 ± 5
-46 ± 5
GRACE (IOM)
(-25 ± 8)
(-10 ± 4)
(-37 ± 7)
(-61 ± 20)
(-16 ± 7)
(-32 ± 12)
(-26 ± 5)
(-28 ± 6)
2003–2012
-20 ± 4
-10 ± 4
-35 ± 5
-51 ± 11
-20 ± 7
-38 ± 8
-37 ± 5
-49 ± 5
GRACE (IOM)
(-26 ± 8)
(-11 ± 6)
(-40 ± 8)
(-59 ± 18)
(-18 ± 7)
(-46 ± 14)
(-38 ± 6)
(-30 ± 8)
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., -75 ± 2 Gt yr-1
by Zwally et al. (2011) compared to -43 ± 11 Gt yr-1 by
Sørensen et al. (2011). This may be explained by the complicated regional
ice surface geometry in the coastal areas (Zwally et al., 2011) or
uncertainty resulting from the conversion of height changes to mass change,
e.g., different firn corrections and density conversions.
Another area where GRACE and the IOM do not agree is the northwest (region
DS8). We find that our GRACE solution shows -32 ± 6 Gt yr-1 mass
change during 2003–2007 and -46 ± 5 Gt yr-1 during 2003–2010. The
ICESat solutions show similar mass loss rates in this region (see Table 1),
while the IOM solution shows lower mass change rates, i.e., -13 ± 3 and
-28 ± 6 Gt yr-1 for the time period 2003–2007 and
2003-2010, respectively. Moreover, if we determine the mass change rates for
the time interval from 2007–2011, the rate is -57 ± 6 Gt yr-1
(GRACE) and -49 ± 11 Gt yr-1 (IOM). Both
results agree with the rate of -58 ± 14 Gt yr-1 from
Andersen et al. (2015).
We have reduced the approximation error in the GRACE solution for this
region, although by a small amount (-2.3 Gt yr-1). To assess
the influence of the remaining approximation errors, we compare the GRACE
and IOM solutions in the surrounding areas, i.e., DS1 and DS7. It can be seen in
Fig. 5 that mass change rates are consistent between IOM and GRACE
solutions, within their uncertainties. This suggests that the approximation
errors become negligible in the GRACE solution. The comparison for Ellesmere
Island is more difficult because discharge is not included in our IOM
solution. However, an IOM solution including the D estimates by Gardner et
al. (2011), showed that the mass change rate of the glaciers on this island
is 37 ± 7 Gt yr-1 between the years of 2004 and 2009.
This agrees (to within its uncertainty bound) with our solution for the same
time interval, i.e., -35 ± 7 and -29 ± 3 Gt yr-1
for IOM and GRACE solutions, respectively. Hence, it
is reasonable to believe the mass change estimated from IOM and GRACE
agree with each other in this region. After comparing the GRACE and IOM
solutions on all the neighbor regions of DS8, no significant differences
between the IOM and GRACE solutions are found. This suggests that the
remaining approximation error is not the major source of the difference in DS8.
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 ± 8.4 Gt yr-1
from 2003 to 2010, which is consistent with the GRACE solution obtained in
this study. A combination of all evidence indicates that the IOM method
underestimates the mass loss rate in this drainage area by
∼ 15 Gt yr-1. In Sasgen et al. (2012), the discharge
estimations from Rignot-08 are used, in which a portion of DS8 was
un-surveyed, to which they attributed the difference between GRACE and IOM
(24 ± 13 Gt yr-1). In this study, the discharge
estimation from Enderlin et al. (2014) covers the entire glacier area in
this region, but only for the years after 2000. Therefore, despite
observations of relatively stable terminus positions for the majority of the
marine-terminating glaciers in northwestern Greenland between 1985 and 2000 (Howat
and Eddy, 2011), we hypothesize that the estimated reference discharge
overestimates the regional D0. In deriving D0 from D-14, we have
used the assumption that D from 1990 to 2000 follows Rignot-08, which
contains high regional uncertainties. However, if we use the
runoff-based estimate of D0, uncertainties are influenced by the
uncertainty of the RACMO2.3 model. The SMB intercomparison study of Vernon
et al. (2013) shows that the 1961–1990 reference SMB0 of RACMO2.3 model
is larger than some other SMB models, e.g., MAR or PMM5. It is interesting to
see that when the cumulative TMB is calculated independently from the
reference SMB0 and D0 (using Eq. (5) and method 1), the mass change
rates agrees with the GRACE mass balance in this region within
uncertainties. This indicates that the modeled SMB (as well as SMB0)
could have uncertainties that are larger than 18 %.