Mass change over Greenland can be caused by either changes in the glacial
dynamic mass balance (DMB) or the surface mass balance (SMB). The GRACE
satellite gravity mission cannot directly separate the two physical causes
because it measures the sum of the entire mass column with limited spatial
resolution. We demonstrate one theoretical way to indirectly separate
cumulative SMB from DMB with GRACE, using a least squares inversion
technique with knowledge of the location of the glaciers. However, we find
that the limited 60

Mass change occurring over the ice sheets can be divided into two parts: changes due to dynamical responses of glaciers (thinning and calving) and changes due to large-scale patterns in surface melting, runoff, sublimation, and precipitation. The glacial response is known as dynamic mass balance (DMB), while the atmospherically forced signal is the surface mass balance (SMB). These two types of mass change are typically modeled or measured separately. One exception to this rule is when the ice sheet mass balance is measured by satellite gravity, such as the Gravity Recovery And Climate Experiment (GRACE); these measurements are sensitive to the sum of all mass changes, without the direct ability to separate one cause from another. In this paper, we demonstrate one theoretical way to separate cumulative SMB from DMB using GRACE, based on a priori knowledge of glacier locations on the ice sheet. Using simulations, we determine the GRACE spatial resolution needed to separate cumulative DMB and SMB around large glaciers within acceptable error limits.

Impact of spatial resolution on the apparent shape and amplitude of
a 1 cm signal over Helheim Glacier, given the a priori weight distribution
in

In recent years, inverse least squares estimation techniques have been used to localize the smoothed signal observed by GRACE into more precise, geophysically relevant regions (Schrama and Wouters, 2011; Jacob et al., 2012; Sasgen et al., 2012; Bonin and Chambers, 2013; Luthcke et al., 2013; Wouters et al., 2013). Most often, these techniques have focused on the mass change over all of Greenland, or else within 8–16 large drainage basins covering the island. We expand this technique to include regions designed to contain the mass signal of the largest of Greenland's glaciers: Kangerdlugssuaq, Helheim, and Jakobshavn. These glacial regions experience two different physical processes atop each other: the localized DMB signal and the broader-scale SMB signal. Unlike most places in Greenland, the DMB signals in Kangerdlugssuaq, Helheim, and Jakobshavn glaciers are expected to be larger than the local SMB signal. That fact allows us to potentially separate the dynamical effects from the SMB effects in these regions, by making a pair of assumptions. First, since SMB is correlated over fairly large regions, we assume that the SMB signal across each of the large glaciers is similar to the SMB just outside the glacier. Second, we assume that any local signal within the glacier region which is not defined by the broader SMB signal is caused by glacial dynamics. The latter is a reasonable assumption in the case of these three glaciers, due to the relatively large size of the expected DMB signal compared to discrepancies in local SMB relative to nearby SMB. This allows us to use two overlapping basins to separate the two independent signals: first, a large SMB basin, similar to those used in previous studies; and second, a small glacial basin covering only the area just around the glacier. The smaller basin is designed to trap the localized signal, which we know to be mostly caused by the DMB, while the larger basin will trap the underlying larger-scale signal, which we know to be mostly caused by the SMB.

From a purely mathematical perspective, the least squares approach should be
able to separate a localized signal (DMB) from a wider-spread signal (SMB).
However, Bonin and Chambers (2013) found out via simulation that
estimating mass change via an inversion modeling method, even over
relatively large SMB basins, can result in trend errors of

A significant reason inversion techniques give weak results in very small areas is due to the innate limited spatial resolution of the GRACE Release-05 (RL05) data. At GRACE's typical maximum degree/order of only 60, a strong spatially localized signal is effectively indistinguishable from a weaker, more spread-out signal. However, at higher maximum degrees, such signals become distinct (Fig. 1) – and thus, should become separable by the least squares inversion process. However, this benefit must be balanced with the cost of greater satellite errors at higher degrees. We thus aim to answer two questions. First, how high of a maximum degree/order of gravity coefficients is needed to separate the localized, large-magnitude DMB from the broader-scale, smaller-magnitude SMB? Second, what level of satellite errors is required for current or future satellite gravity missions to separate the signals with reasonable uncertainty? In this paper, we design a series of GRACE-like simulation sets with known “truth” values to test this.

Throughout this paper, we use a modified version of the least squares
inversion method described mathematically in Bonin and Chambers (2013).
This technique uses a set of pre-defined “basin” shapes on a
1

Mathematically, this can be written as a set of models for each latitude and
longitude (

SMB and glacial basins for Greenland. Glacial basin (J)akobshavn overlaps with SMB basin 7, while (H)elheim and (K)angerdlugssuaq overlap basin 4. White grid cells show the central glacier cell, while black are lesser-weight glacier cells.

In Bonin and Chambers (2013), we assumed that mass was distributed evenly
within each individual basin. However, that assumption was only accurate to
first order, since the SMB is dominated by higher losses near the coast.
Here, we instead weight the 8 external Greenland SMB basins (1–8), Iceland,
Ellesmere Island, and Baffin Island to accentuate coastal mass change. We
compute the weights using data from the RACMO2 regional climate model (Ettema
et al., 2009). By summing RACMO2's SMB data from 2002 to 2012, then removing
the mean at each location, we get grids of cumulative SMB anomaly, similar to
GRACE's monthly mass anomalies. We use the RMS of this RACMO2 cumulative SMB
data as the weights for our external Greenland basins. The internal Greenland
SMB basins and other external basins are still assumed to have uniform mass
distribution. The glacial basins are each dominated by a single
1

Although in Bonin and Chambers (2013) we determined that a diagonal constraint matrix assisted in the optimization, experimentation since has demonstrated that when using non-uniform basin weights, such “process noise” does not improve accuracy. As such, our least squares inversion technique computes the set of optimal basin multipliers using no additional constraints or regularization.

Our primary goal is to quantify the accuracy of the least squares inversion
method, given a fixed set of pre-defined basins and basin weights. We do
this by creating multiple 1

Each simulation contains three parts: a cumulative SMB signal (Sect. 3.1), a cumulative DMB signal (Sect. 3.2), and an estimate of GRACE stripe errors (Sect. 3.3). The combination of these three pieces results in as full a simulation to the truth as we can create. By varying the SMB and DMB signals in the next two sections, we can determine the impact that misfits in the spatial distribution of the basin weights and the two ice mass signals have on the least squares results. The variation in satellite errors allows us a better statistical handle on the likely effect of the GRACE stripes. Summed together, we can determine if the combined errors are small enough to create a meaningful estimate of the truth signal – and therefore learn if this inversion technique can be used to correctly separate the SMB from DMB signals in this region.

The “SMB-only” simulations actually include the land hydrology and
oceanography signals as well as the SMB. (We call them SMB-“only” since,
over Greenland, the signal is “only” SMB, not DMB or stripes.) The
hydrology model used is the average of the GLDAS-Noah
(Rodell et al., 2004) and WGHM (Döll et
al., 2003) models. Over the oceans, we use the JPL_ECCO ocean
model, run at the Jet Propulsion Laboratory (JPL) as a contribution to the
Estimating the Circulation and Climate of the Ocean (ECCO), and available at

To this, we add an SMB simulation. Since we had already used RACMO2 to compute the SMB basin weights, we could not directly use it to test the errors caused by misfits of those weights. So we chose to simulate plausible cumulative SMB signals using RACMO2 as a baseline. We separated the actual 2002–2012 RACMO2 signal into a long-term trend, a 12-month climatology, and the remaining residual. The long-term trend and monthly climatology together make up 83 % of the RACMO2 cumulative SMB variability across Greenland, including over 95 % of the coastal signal, making them the dominant terms in need of careful reproduction. The residual part contains both sub-annual variability and interannual variability, the latter of which is especially important in mass estimates over Greenland due to its connection with long-term climate change. For the SMB part of our simulations, we sought to mimic the trend, monthly climatology, and residual parts of the cumulative SMB signal by creating semi-randomized truth simulations which vary realistically but randomly from the mass distribution used in our basin weights, using the following two-part method.

Example components of the simulation-building process:

We created altered versions of the trends and monthly climatology maps, by
varying the cumulative SMB signals away from the RACMO2 trends and
climatology in a random but physically meaningful manner. To do so, we began
by estimated typical correlative length scales,

We then created generic randomized maps,

We then took each actual RACMO2 trend/climatology map,

For both trends and climatology, we are probably creating a conservative
estimate, since RACMO2 has been determined to have much less than 50 %
error (Ettema et al., 2009). However, error estimates in such studies have
focused on the errors in the total mass change over all of Greenland, not the
mass change in a far smaller area like a single grid cell. Since positive and
negative errors will tend to average out over large areas, we presume that
local 1

While the trends and climatology describe the strongest parts of the RACMO2
cumulative SMB estimate, 17 % of its variance is driven at other
frequencies, including significant interannual variability. To simulate both
higher- and lower-frequency variability in the simulated data, we used a
random walk process. We first created a series of the random,
locally correlated maps described previously, one for each desired month of
simulated data. We then used an autoregressive process such that the
simulation at month

Each cumulative SMB simulation series is made from the summation of trend, climatology, and random-walk pieces, for each month. We created 50 simulations of 11 years of cumulative SMB simulation, designed to represent the GRACE years 2002–2012. To these, we added the modeled hydrology and oceanography series, to form the final “SMB-only” simulation truth series. We transformed these into spherical harmonic representations of maximum degree/order 60, 75, 90, 120, and 180 for use in the least squares inversion process. The difference between the inverted results of the SMB-only simulations will estimate the sensitivity of the least squares process to imperfect SMB basin definition and weights.

In comparison, the set of simulated cumulative DMB signal is artificially simple. We considered using a random walk process, similar to that used in the residual SMB simulation, but decided to avoid such unnecessary complexity. Firstly, we did not have access to good, monthly measurements of the mass signal in any of the three glaciers we were looking at, so we had no clear estimate of the expected variability, particularly at sub-annual frequencies. Secondly, the glacial basins are only 2–4 grid cells in size, and are each dominated by a single central grid cell, so there is minimal concern about signal overlap from nearby glacial bins with vastly different temporal signals. Instead, we kept things simple and manufactured a piecewise linear truth signal for each glacial basin (Fig. 4c). The simulated DMB signal is of roughly comparable magnitude to modeled estimates (Howat et al., 2011) and is thus much larger than the cumulative SMB signal is, though across a far smaller area. Everything outside the near-glacier regions in Fig. 4a is set to zero (since the signals there are already included in the “SMB-only” simulations).

We expect misfit errors from the cumulative DMB to arise from the imperfect basin weightings we gave to the non-central glacier cells. To test how large an effect that has, we created an ensemble of 50 simulated cumulative DMB series, each to maximum degree/order 60, 75, 90, 120, and 180. Each run has the same total DMB signal per glacier, but we altered the spatial distribution of that signal slightly each time (for example, Fig. 4b vs. Fig. 4a), via the following method.

96

We first computed the average weight originally given to the non-central
grid cells (

Since north–south stripe errors dominate any individual map made from unconstrained, unsmoothed GRACE data, we have created a series of simulated stripes to approximate their impact on the least squares inversion results. The stripe simulation technique is based on an observation by Swenson and Wahr (2006) that due to the north–south stripes, same-order odd-degree harmonics tend to correlate, as do same-order even-degree harmonics. Bonin and Chambers (2013) demonstrated that, given the real GRACE variances at each spherical harmonic as well as correlations with other harmonics, one can make randomized sets of simulated “GRACE-like” stripes.

We use the variances and correlations from the standard RL05 GRACE solutions
from the Center for Space Research (CSR), with the AOD1B ocean dealiasing
monthly averages added back. We create stripe-only simulations from harmonic
cases 60

To best simulate stripe errors, we remove as much of the geophysical signal as possible, to end with what we hope is mostly errors in GRACE. Thus, we removed the ocean and hydrology models used previously, as well as the RACMO2 model over Greenland and Antarctica. None of these models are perfect, so we fit a mean, trend, annual, and semiannual signal to what remained. We know that much of the remaining trend and annual signal is important geophysical signal, but some stripes also fall into those categories. To further separate that, we pulled aside only the trend/annual components of the harmonics which explained at least 50 % of that harmonic's full variability. That fraction is added to the “model” and removed from the “residual”. The result is a set of “model” maps that do not visibly show stripes, and a set of “residual” maps that are heavily dominated by stripes (Fig. 5a and b).

We calculated the actual variance and harmonic cross-correlations from these
residual GRACE series, then used the technique in Bonin and Chambers (2013)
to make randomized sets of north–south stripes with approximately the
same spatial distribution as what is actually seen in GRACE (Fig. 5c). We
created 50 randomized variations of the stripes for each GRACE series
(degrees 60, 96, and 120). The stripe simulation technique begins to break
down at high degrees/orders, overweighting the stripe amplitude within

We chose to create simulated stripes, rather than directly use the residual signal as the GRACE errors because a close look at the residuals reveals that some probably real interannual signal remains in several of the coastal Greenland basins, even after the trend and/or annual fit and removal. This is caused by an imperfect SMB/glacial model and the fact that not all remaining signal is perfectly linear or annual. In terms of the simulated stripes, it implies that our stripe estimates will tend to somewhat overstate the true north–south stripes, since the variance of the remaining interannual signal will go into simulated stripes. This makes our stripe simulation a slightly conservative estimate of the expected GRACE errors.

Numerous techniques exist to reduce these stripe errors, including a variety of spatial smoothings, correlation-based destriping methods, and spatial and temporal constraints; however each one necessarily impacts the signal along with the error. More critical to our interest here, they effectively reduce the spatial resolution of the GRACE data, by damping both error and signal at higher degrees/orders. To use any such post-processing method would undo the benefits of inverting a high-resolution series, making it more difficult to reach the needed resolution to separate SMB from DMB signals. As such, we choose to use no spatially based stripe-reduction method.

However, we do use one simple technique to reduce the errors at no spatial
cost: applying a year-long temporal moving window to the data. This is
useful since a majority of the stripe RMS occurs at periods of less than 1 year. For example, in the 120

We thus have three sets of simulations, to test the three likely types of error in the least squares inversion process. The SMB-only simulation set will be used to test the impact that imperfect definitions of SMB basins and basin weights will have on the inverted results. The DMB-only simulation set will be used to test the impact that imperfect glacial basin weights will have on the inverted results. And lastly, the stripe-only simulation set will be used to test the impact that satellite errors have on the inverted results. Since our least squares inversion routine is perfectly linear, the errors of the inversion of a summed version of the three simulation pieces are the same as the sum of the inverted errors of the three individual pieces. However, by separating the simulation into three known pieces, we can determine from which part different errors arise, and thus learn which is the most limiting factor to getting accurate results using the least squares inversion technique over Greenland.

To do so, we used the least squares inversion method on each of the 50
SMB-only, 50 DMB-only, and 50 stripe-only simulation sets, for smoothed
versions of maximum degrees 60, 75, 90, 120, and 180 each. For each
inversion, we fit to the full set of SMB, glacial, and external basins. We
then subtracted each simulation's estimated inverted basin amplitudes
(

In Sect. 4.1, we compare each SMB truth input to its inverted response, to determine the errors caused by using imperfect SMB basins in the least squares method. Sect. 4.2 similarly calculates the errors due to the imperfect glacier basins, and Sect. 4.3 shows a visualization of the sum of both types of basin misfit errors. In Sect. 4.4, we determine how large the satellite errors can be, when combined with the total basin misfit errors, to allow for a signal-to-noise ratio of 2. We then compute the RMS of GRACE's satellite errors, to determine if either the current GRACE or a future probable satellite gravity mission might be able to accurately separate the glacier signal from the SMB signal.

Average RMS difference from truth per basin for the SMB-only simulations, for data of increasing maximum degree, in SMB basins and glacial basins. Yearly windowing applied.

Figure 6 shows the average basin error from the SMB-only simulation set, for each of the 13 SMB basins and the three glacial basins, using yearly averaged data. The effect of spatial resolution is seen clearly: with increasing maximum degree/order, the errors decrease. This demonstrates the ability of the least squares inversion technique to correctly partition SMB signal, so long as the basins it is trying to fit to are sufficiently resolvable by the limited set of spherical harmonics.

As the maximum degree is lowered, the biggest degradations are seen in basin
7 (which overlaps with Jakobshavn Glacier) and basin 4 (which overlaps with
the other two glaciers), with particularly big changes seen as the maximum
degree drops from 90 to 75 to 60. In the case of basin 7 and Jakobshavn
Glacier, the two overlapping basins have large and consistently
anti-correlated error time series, particularly in the 60

Even near Jakobshavn, however, the strength of this error is highly
sensitive to the spatial resolution used. For example, the basin 7 and
Jakobshavn SMB-misfit errors are cut in half merely by increasing the
spatial resolution from 60

Average RMS difference from truth per basin for the DMB-only simulations, for data of increasing maximum degree, in SMB basins and glacial basins. Yearly windowing applied.

Visualization of the spread caused by the combined SMB and glacial
basin misfit errors, at the three glaciers, for maximum degrees

Figure 7 shows the average basin errors from the DMB-only simulation set, for
each of the 13 SMB basins and the three glacial basins, using yearly averaged
data. In most basins, increasing the maximum degree/order from 60 to 90 (or
above) reduces the errors. However, in the critical basins 4 and 7 and the
glaciers themselves, the situation is less simple. Basin 4 shows highest
errors for the 75

To visualize the relative size of the above misfit errors compared to the truth geophysical signals, we have plotted the inverted glacial signals from the 50 combinations of SMB and DMB simulations in Fig. 8. In the dark solid lines, we show the truth signal from each glacier basin, for comparison. The majority of the errors are driven by misfits between the SMB data and the pre-defined SMB basins, with only a small effect due to the misfit between the DMB data and the pre-defined glacial basins.

Average RMS error per basin for the noise-only simulations, for data of increasing maximum degree, in SMB basins and glacial basins. Yearly windowing applied.

Comparison of maximum allowed stripes (green boxes) based on SNR
>2, and the actual estimated stripes per basin (red lines) for
the

Figure 8 demonstrates that the two types of basin weight misfit errors do not
cause an insurmountable hurdle to our ability to separate the cumulative SMB
from DMB signal. Error-free 60

Unfortunately, GRACE observations do contain satellite errors. The RMS errors caused by inverting the stripe-only simulations, after yearly smoothing, can be in Fig. 9. As expected, the errors increase as the maximum degree increases. The errors in the glacier basins are significant, and the errors in the SMB basins overlapping those glaciers grow very large, in comparison to the basin misfit errors (Figs. 6 and 7).

Figure 10a shows the SMB-misfit and glacial-basin-misfit errors from the
previous sections combined in quadrature. If GRACE had no satellite errors,
Fig. 10b would be the signal-to-noise ratio (SNR) of the inversion
technique, computed by dividing the basin RMS of the ideal truth
SMB

Most of the coastal basins have SNRs greater than 5 at all maximum degrees.
However, basin 7 has the lowest SNR of the coastal basins: 0.7 at maximum
degree and only 2.3 by degree 90. Basin 4 gives nearly as poor a showing,
with noise-free SNRs of 1.2 at 60

Now we consider the situation if GRACE satellite stripe errors are also
included. To call the cumulative SMB from DMB signals separable, we require
a minimum desired stripe-inclusive SNR of 2.0 – that is, the signal RMS
must be at least twice the total error RMS of the stripes and basin misfit
errors combined. In Fig. 10c, we show the maximum stripe errors which meet
this SNR>2 goal, given the known basin misfit errors and
truth signals. We compute this using

Unfortunately, the actual yearly windowed inverted errors from the
stripe-only simulations are large and grow larger quickly with increasing
maximum degree/order (Fig. 9). Figure 11 shows a direct comparison of the
possible ranges of stripe errors which allow a stripe-inclusive SNR of at
least 2 (green bars), relative to the actual RMS errors found from the
stripes-only simulation. The non-glacier-overlapping SMB coastal basins of
the 60

A basin-based least squares inversion technique can theoretically be used to
separate the cumulative SMB signal from the cumulative DMB signal in
Greenland, assuming sufficient spatial resolution of the input data. We found
that a maximum degree of 60

Unfortunately, this is true in theory only. Realistically, when current GRACE noise estimates are included, a SNR > 2 is never achievable for the SMB basins where the three targeted glaciers are located. Since GRACE errors increase far faster with degree than the inversion method's basin-misfit errors decline, this problem becomes worse as the maximum degree of GRACE increases. There is no point where the misfit errors in the inversion method (highest at low degrees) balance with the satellite errors (lowest at low degrees) to allow a good SNR. If SNR levels higher than 2 are desired, the GRACE errors would need to be brought down even further, as they depend on the inverse square of the target SNR.

Significant stripe reduction could potentially allow for cumulative SMB and
DMB to be separated using the least squares inversion method, particularly if
errors are also reduced via temporal smoothing, as we have done here. Taking
into account yearly averaging, the GRACE noises would need to be reduced by
approximately a factor of 10 at 90

The authors want to express great thanks to Himanshu Save at the Center for
Space Research in Austin, TX, for the use of his 120

Support for this research was funded by the NASA GRACE Science Team program and by the NASA New (Early Career) Investigator Program in Earth Science.Edited by: E. Larour