Conversion of altimetry-derived ice-sheet volume change
to mass requires an understanding of the evolution of the combined ice and
air content within the firn column. In the absence of suitable techniques to
observe the changes to the firn column across the entirety of an ice sheet,
the firn column processes are typically modeled. Here, we present new
simulations of firn processes over the Greenland and Antarctic ice sheets (GrIS and AIS)
using the Community Firn Model and atmospheric reanalysis variables for more
than four decades. A data set of more than 250 measured depth–density
profiles from both ice sheets provides the basis of the calibration of the
dry-snow densification scheme. The resulting scheme results in a reduction
in the rate of densification, relative to a commonly used semi-empirical
model, through a decreased dependence on the accumulation rate, a proxy for
overburden stress. The 1980–2020 modeled firn column runoff, when combined
with atmospheric variables from MERRA-2, generates realistic mean integrated
surface mass balance values for the Greenland (
One of the most robust methods for measuring ice-sheet mass balance uses
satellite altimetry (Shepherd et al., 2012, 2018) to measure
changes in surface height through time and ultimately provide ice-sheet-wide
volume change estimates (Helm et
al., 2014; Paolo et al., 2015; Pritchard et al., 2009; Zwally et al., 2005,
2015). Interpretation of volume changes, however, requires ancillary
information because there are several processes that generate height changes
observable by satellite altimeters (Ligtenberg et al.,
2011; Smith et al., 2020). The measured surface height change is a
combination of signals, which reflect processes that involve ice or
solid-earth mass change, or even no mass change at all. Even if we remove
the solid-earth processes, partitioning the remaining ice-sheet-volume
change to the appropriate material densities remains a challenge.
Specifically, volume change due to ice dynamics represents a change at the
density of ice (917 kg m
In our modeling, we divide the ice sheets into two vertical layers of
different material density, referred to hereinafter as the firn and ice
columns. Typically extending tens of meters to over 100 m down from the
surface (Ligtenberg et al., 2011), the firn column
represents snow that has fallen, was subsequently buried, and is undergoing
densification yet remains less dense than ice. The rate at which firn
compacts varies and is dependent on its age, the weight of snow pressing
down on it from above, temperature, and meltwater infiltration and
refreezing. The ice column begins at a depth where material density becomes
approximately constant (917 kg m
The firn column is constantly evolving due to a changing climate, across all
timescales, and the deviations in snow accumulation, meltwater production,
and temperature from steady-state conditions drive changes in the firn layer
thickness. The goal of this work is to simulate these changes in the firn
column over the past 40
Changes in ice-sheet surface height reflect the integrated signal of several processes, some of which are related to ice or solid-earth mass change and others that reflect no mass change at all. Thus, we must decompose the full signal into various components in order to derive the quantity of interest; here, we are focusing on ice-mass change.
Observed height change (
Firn column changes, however, have a complicated relationship with mass change. Height changes due to variable rates of compaction of the firn column do not reflect a change in mass but impact the observed ice-sheet height variations through changes in volume and density. Meltwater production is more ambiguous: when it can infiltrate the firn and refreeze, there is no resulting mass change, but when infiltration is impeded and meltwater runs off, there is mass change. The effect of net snow accumulation always reflects a change in mass and can be positive or negative. As a result, the conversion between height, volume, and ultimately mass change requires understanding the material density of each component, which is neither constant in time nor space.
Rather than partition firn column changes by its individual components (see
above), we divide total firn column height change into changes in the
thickness of ice and the air thickness: surface mass balance (SMB) and firn
air content (FAC), respectively. Specifically, we define
The SMB is the summation of mass fluxes at the surface, including
precipitation (solid and liquid), evaporation/sublimation, and runoff
(Lenaerts et al., 2019). Here, we do not account for blowing snow
processes that likely impact local-scale SMB; however, these processes
comprise an overall small percentage of total SMB (Van Wessem et
al., 2018). Specifically,
The FAC or depth-integrated porosity represents the integrated volume of air
within the entire firn column and is defined as
We simulated firn column processes over both the GrIS and AIS using the Community Firn Model (CFM) framework (Stevens et al., 2020), forced by reanalysis climate variables. These simulations are referred to as the Goddard Space Flight Center firn densification model (GSFC-FDM, v1.2.1). First, we provide specifics relating to the CFM as well as our methodology for calibration, spin-up, and implementation. We then describe our selected climate forcing from NASA's Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2) used in our simulations. Third, we discuss the differences between GSFC-FDMv1.2 and its earlier versions, v1 and v0, the latter of which was used in Smith et al. (2020) and Adusumilli et al. (2020). Finally, we provide details regarding our uncertainty assessment as well as our SMB evaluation approach.
The Community Firn Model was built as a resource to the glaciology community, consisting of a modular, open-source framework for Lagrangian modeling of several firn- and firn-air-related processes (Stevens et al., 2020). The CFM allows the user to select the processes and/or physics of each simulation. The core CFM modules contain physics for firn density and temperature evolution; however, there are several modules for additional processes that the user can implement. For the GSFC-FDMv1.2.1 simulations, we use modules for grain-size evolution, meltwater percolation and refreezing, and sublimation. Grain-size evolution is simulated for testing purposes and not considered realistic. The user also has several options of firn densification physics from which to choose. Several of the models are calibrated using climate forcing from a regional climate model (RCM), atmospheric reanalysis, or even satellite-derived products, which means that any biases in these climate variables will bias the calibration coefficients in the firn densification model. Thus, it is necessary to have consistent climate forcing between the calibration and actual model runs, so we perform our own densification model calibration (Sect. 2.1.3). Finally, we use a simple bucket scheme for simulating meltwater percolation and refreezing; while the CFM contains a choice of physics of varying complexity, recent work by Verjans et al. (2019) suggests there is currently no evidence that the higher-order models perform better. Here, we use CFM v1.1.6 (Stevens et al., 2020, 2021).
To ensure that we do not impose any unwanted transients in our simulations,
we must have a sufficiently long spin-up interval during which most of the
firn column is refreshed. Due to variable snow accumulation rates across the
ice sheets, the time required to fully refresh the firn column can vary
significantly. Thus, we impose a variable spin-up time that is dependent on
the long-term mean climate. Specifically, we use the Herron and Langway (1980) densification model to approximate the depth to the bottom
of the firn column (delineated at a density of 910 kg m
The CFM has the option to impose a dry-snow spin-up; however, this solution
would build a firn column that is in dynamic equilibrium under dry
conditions only. If melt were then imposed, meltwater processes would create
large negative
We use a subset of 256 published firn depth–density profiles from both the
GrIS and AIS as the basis of our calibration procedure and perform a single
calibration that is representative of both ice sheets. The density profile
data set is described in Appendix A. The Arthern et al. (2010)
dry-snow densification model provides the physical basis for our
GSFC-FDMv1.2.1 simulations. Specifically, modeled dry-snow densification
rates are separated into two stages during which the parcels experience
different compaction processes and that are defined by the density of the
parcel:
Modeled time-invariant initial density for the
To begin the calibration procedure, we first run the model in its original form at 226 calibration sites across Greenland and Antarctica (Fig. 1). The number of model calibration runs is less than the actual number of observations (256) as some fall within the same grid cell (e.g., several observations from the vicinity of Summit, Greenland). All 256 observations are used. Unlike other calibration efforts (Kuipers Munneke et al., 2015; Li and Zwally, 2004, 2011; Ligtenberg et al., 2011), the calibration procedure presented here treats dry-firn densification from both ice sheets together, forming a single calibration parameterization, which benefits from a much wider range of climate conditions than if each ice sheet was treated individually.
The logarithm of the firn density profile with depth is approximately
linear, largely for stage 2 (Herron and Langway, 1980). More
discussion on the use of a logarithmic density profile is in Appendix B. For
each calibration site, we compare the slopes of the logarithmic density
versus depth for the two stages of densification between observations
(
Rather than develop a new physical form for calibration, we optimize two
parameters within the Arthern et al. (2010) model: the
exponential dependence on the mean annual accumulation rate since the parcel
was deposited and the activation energy for creep. Arthern et al. (2010) found evidence that the activation energy is not well
constrained for the sites investigated, suggesting that the physical
processes at play under various conditions are not fully understood.
Similarly, Ligtenberg et al. (2011) and Kuipers Munneke et al. (2015) found that the Arthern et al. (2010) model required additional dependence on snow accumulation
to best fit observations. Thus, we elect to calibrate the parameters
relating to variations in snow accumulation (
We define our calibration coefficients for the two stages of densification
(
The dry-snow densification calibration coefficients for
We finally iteratively solve for
Combining Eqs. (9)–(12), (15), we define our densification rate coefficients as
For Greenland, we define the ice boundary using the Greenland Mapping
Project (GIMP) ice mask posted at 90 m spatial resolution
(Howat et al., 2014). We identified approximately 13 200 of
the 12.5 km GSFC-FDMv1.2.1 pixels as ice if any of the GIMP pixels that fell
within were flagged as ice. For integrated SMB determination, we scale each
pixel by the area of ice within based on the GIMP ice mask; the total ice-sheet area along with the peripheral ice not connected to the main ice sheet
is
For Antarctica, we use the drainage basins at 1 km resolution defined
by Zwally et al. (2012). We identified any of its 12.5 km pixels that contain an ice-flagged pixel from Zwally et al. (2012) as ice, resulting in
just over 88 300 ice-covered pixels. We assume all the pixels are 100 %
ice-covered, which is equivalent in area to
The Antarctic GSFC-FDM simulation locations colored by the representative size of their neighborhood. Darker colors (larger neighborhoods) with more white space (redundant simulations) indicate that the gradients in mean annual climate variables do not vary significantly over short length scales. Paler colors suggest stronger gradients with fewer redundant simulations.
Because of the low accumulation rates over the ice sheets and the coarse (5 d) time resolution of our simulations, we anticipate significant reworking of the initial, low-density surface snowpack. Ideally, the imposed initial density would vary in time based on the ambient climate conditions; however, there are few studies that focus on the temporal evolution of freshly fallen snow over the ice sheets (e.g., Groot Zwaaftink et al., 2013). Thus, we focus rather on improving the bulk (or time-invariant) initial density for each grid cell based on the mean annual climate conditions as done by Helsen et al. (2008) and Kuipers Munneke et al. (2015). This approach means that, on average, we will approximate the surface density well, but we accept that there might be significant deviations from this bulk density over shorter timescales.
To build a model of initial density (
Comparison of observed and modeled initial densities. Solid circles indicate that they were not used in the final model development and represent an independent testing data set, whereas open circles represent the training partition used to build the Gaussian process regression model (see Sect. 2.1.5). The statistics in black are in reference to the solid circles only (testing partition), and those in gray are in reference to the open circles (training partition).
MERRA-2 is a global atmospheric reanalysis developed at the Global Modeling
and Assimilation Office (GMAO) at the NASA Goddard Space Flight Center
(Gelaro et al., 2017). Atmospheric variables are provided at
MERRA-2 employs the incremental analysis update (IAU) scheme of Bloom et al. (1996). The IAU uses predictor and corrector model forward integrations where differences with observations are
first computed in the predictor segment and then added as an additional
forcing term in the corrector run. It may be noted that an entirely
different global model may employ the IAU scheme to correct to MERRA-2
innovation variables every 6 h, a process referred to as “replay”
(e.g., Mapes and Bacmeister,
2012). The MERRA-2 12 km replay integration (M2R12K) was produced as part of
the NASA Downscaling Project (Tian et
al., 2017) and covers the period December 1999 to November 2015. A
non-hydrostatic version of the Goddard Earth Observing System (GEOS) model
was used in the replay integration with an output grid spacing of
Mean annual net accumulation (snowfall minus sublimation) for the
Greenland
Mean annual skin temperature for the Greenland
The high-resolution M2R12K only spans 15 years, so it cannot be used as direct forcing of the firn densification model. Rather, we retain the seasonal magnitudes in the atmospheric variables from the M2R12K to provide hybridized MERRA-2 output. First, the MERRA-2 output is oversampled to the M2R12K grid. We then determine the 2000–2014 monthly means in MERRA-2 and remove them from the full MERRA-2 record (1980–2021). The 2000–2014 M2R12K monthly means are then added to the MERRA-2 residuals to form the hybridized MERRA-2 atmospheric variables. In such a manner, the magnitude of the gradients in precipitation and temperature from the high-resolution M2R12K are transferred to the coarse MERRA-2 output. Figures 5 and 6 show the mean annual net accumulation (snowfall minus sublimation) and skin temperature, respectively, for the GrIS and AIS. For simplicity, we hereinafter refer to the hybridized MERRA-2 as MERRA-2.
While the variables are provided at hourly resolution, to maximize computational efficiency, we perform the firn simulations at a resolution of 5 d. The 5 d MERRA-2 time series are built by averaging the hourly data over 5 d intervals. Although MERRA-2 includes meltwater processes, only net runoff is retained. Thus, we use a degree-day approach to build gridded meltwater time series, which is described in Sect. 2.2.1.
For both ice sheets, we used a simple model to generate meltwater fluxes for
input into the CFM. Specifically, meltwater production (
Degree-day model evaluation for
Mean degree-day factors over the Greenland Ice Sheet binned at 250 m elevation intervals for a temperature threshold,
We calibrated our melt model for Antarctica using a calibration data set of
surface meltwater fluxes (Trusel et al., 2013) that span the
1999 to 2009 melt seasons, which are linearly interpolated to our 12.5 km
grid. Rather than calibrate our model to 5 d meltwater fluxes, we
optimized correspondence of annual meltwater production between the model
and calibration data and set
We estimated a temperature threshold over the GrIS using a similar approach.
While we used an observation-based calibration data set over Antarctica, a
similar data set does not exist for Greenland, so we instead used
independent model output as the basis of our calibration. Specifically, we
used the 1980–2014 annual meltwater rates from the MARv3.5.2 regional
climate model (RCM) (Fettweis et al., 2017). Although this product
provides sub-annual resolution, we opted to calibrate to annual meltwater
production once more. In such a manner, the short-timescale meltwater fluxes
were driven by MERRA-2, but the calibration to annual RCM output ensured
that the simple model remains aligned with realistic annual magnitudes from
MAR. For Greenland, we found a threshold, identical to the Antarctic, of
Mean annual meltwater fluxes for the
For both ice sheets, the temperature threshold is below freezing, which
suggests either (1) a cold bias in MERRA-2 or (2) too strong a melt within the
calibration data sets. The former has been found over Greenland
(Hearty III et al., 2018) and Antarctica
(Gossart et al., 2019; Huai et al., 2019) for summer
months, but we cannot eliminate the latter as a contributor to the
sub-freezing threshold as well, which we discuss more in Sect. 4. We assess
the realism of the calibrated GrIS DDF by plotting the mean values over
250 m elevation bins. Moving into the interior, we would expect lower DDFs
as the surface is typically bright snow, whereas lower elevations are more
likely to exhibit bare ice and lower albedos, which would yield higher
DDFs. For Greenland, the relationship between elevation and DDF exhibits
high values at lower elevations, which drop off to a near-stable value
around 1500 m, above which the values rapidly increase
(Fig. 8). We assume that the stable values around
0.13 kg m
The results presented here build off prior simulations, GSFC-FDMv0 and v1,
detailed in a previous publication (Smith et al., 2020;
Medley et al., 2020). We have since incorporated major improvements to the
GSFC-FDMv1.2.1, which we outline below. Version 1.2 is obsolete as there was
a bug in the CFM that excluded time steps with net sublimation. The CFM bug
was fixed for v1.2.1 runs.
GSFC-FDMv1.2.1 includes a spatially variable initial density ( GSFC-FDMv1.2.1 includes the calibration of the dry-snow/firn compaction model that limits the inclusion
of observations based on the ratio of mean annual meltwater production to
snowfall (see Sect. 2.1.3). The calibration approach for v0 did not discard
observations based on their exposure to liquid water processes. This change
in v1 and v1.2.1 should lead to an improvement in the representation of dry
compaction, but we note that this calibrated dry-snow/firn compaction model
is still used in regions of meltwater percolation. GSFC-FDMv1.2.1 includes a more robust approach to handling mass fluxes at the surface. The CFM underwent a significant update between v0 and v1, including allowing the
explicit removal of mass via sublimation and also inclusion of rainfall. For
v0, sublimation was handled by aggregating the accumulation from neighboring
time steps until positive, thereby still accounting for sublimation but at
the cost of smoothing out the accumulation signal. Rainfall was not included
in v0. For v1 and v1.2.1, mass via rainfall can now be added to the total
liquid volume present and become subject to liquid water processes. GSFC-FDMv1.2.1 includes an improved meltwater model. The degree-day approach for both v0 and v1 is
the same; however, the v0 model was built using skin temperature, which
cannot exceed 273.15 K and will not capture the large temperature deviations
above freezing, especially in Greenland. For v1, we use 2 m air
temperature (see Sect. 2.2.1), which is a more robust approach; however,
extreme DDF values, largely in the interior, resulted in unrealistic melt
rates. Thus, for v1.2.1, we capped DDFs based on realistic dry-snow
values, which should improve meltwater fluxes in the nearly dry interior of
the GrIS. GSFC-FDMv1.2.1 includes runoff as an output. The older CFM version used for v0 did allow for melt,
percolation, and refreezing but did not provide runoff as an output. Thus,
we are now able to calculate surface mass balance using v1 and v1.2.1. GSFC-FDMv1.2.1 includes an uncertainty analysis of the dry-snow calibration coefficients, which was not completed in v0 or v1. This exercise provides part of the basis for
estimating total uncertainty in FAC and its evolution in time as well as
total height and volume change. GSFC-FDMv1.2.1 includes a time resolution of 5 d for both the GrIS and AIS. The prior versions
(v0 and v1) ran subsets of the AIS at 5, 10, and 20 d, depending on their
mean climate. Within v1.2.1, the entire AIS is run at 5 d resolution.
Comparison of observed and modeled slopes of the logarithmic
density with depth for
To evaluate the model improvement through our calibration procedure (Sect. 2.1.3), we evaluate the uncalibrated and calibrated model abilities to
capture the slopes of the logarithmic density versus depth for both stages
against the calibration data set. We found that the mean absolute error
(MAE) in modeled slopes for both stages was reduced by nearly one-half after
calibration, and the explained variance between observed and modeled was
significantly increased in stage 2 (Fig. 10). The
mean observed slope is 0.067 m
We would ideally prefer to perform an evaluation of modeled firn
densification rates, but a substantial number of published observations are
lacking. Here, we further evaluate the ability of GSFC-FDMv1.2 to reproduce
the observed densities in our full data set of sites that are in both dry
and wet conditions. Most of these observations were used in the calibration;
however, those with significant melt were excluded (see Sect. 2.1.3). Thus,
we break out our evaluation into sites exhibiting zero, moderate, and high
melt rates, quantified by their ratio to net snowfall, and there are at least
two observations within a stage. Specifically, these are respectively
defined as 0 %, less than 10 %, and more than 10 % of the mean annual
snowfall, and we evaluate the modeled mean absolute error in reproducing
depth–density observations (Fig. 11). The error
increases with larger melt fractions, especially for stage 1, where the
impact of melt is stronger. Because most of these observations are included
in the calibration, we report them as interquartile ranges and assume the
upper bounds are more representative of a realistic error for each group.
For stage 1, we expect density errors of 15.2 to 29.9 kg m
If we evaluate the bias in our model-derived density profiles for each
stage, we find that with increasing melt, the modeled profiles exhibit a
more positive bias (bias
We estimated the uncertainty in the total FAC and its variability through time through ensemble perturbation runs of the CFM at select locations over each ice sheet. Specifically, we completed principal component analysis (PCA) on the 5 d climate time series of variables of critical importance to our simulations: SMB and temperature. We then found the principal components that account for 95 % of the variability for both SMB and temperature. This selection yielded 41 principal components (PCs) for SMB and 4 for temperature for AIS and 14 and 4 for GrIS, respectively. We then correlated each individual PC time series with the equivalent time series at every grid cell over the respective ice sheet. The grid cell with the largest correlation with the PC was selected as a perturbation site. As such, we had 45 sites for AIS and 18 sites for GrIS. We used these locations because they are the most representative of the forcing time series across the entire ice sheet. PCA analysis of melt was not performed because it is determined by the temperature (Sect. 2.2.1).
Boxplot of the mean absolute errors in density for stages
The 1
For each of the calibration sites, we ran the CFM 100 times, each time
applying 11 perturbations to the climate forcing variables, CFM parameters,
and the reference climate interval. Each of the perturbations sampled from a
Gaussian distribution with a mean of zero and a standard deviation based on
observations or model performance with specifics found in the second
column of Table 1. The choice of reference climate interval and the
parameterization for the thermal conductivity of ice were exceptions: we
assumed a uniform distribution of each of the various scenarios in Table 1,
which details each perturbation, their sampling window, and any references.
For each of the 100 perturbations, we sampled each of the aforementioned
Gaussian distribution of uncertainties for the modeled initial density
(
Comparisons of the estimated 2
We assessed uncertainties by taking the standard deviation of the mean FAC
for each of the 100 perturbations over the entire time series for a given
site. We next used mean annual climate parameters (snow accumulation, rain,
melt, and temperature) for each site (the original, non-perturbed MERRA-2
mean values) to predict the standard deviations in FAC. We broke the
regression into two groups based on the ratio of the mean annual liquid
water content (melt
To quantify the uncertainty in FAC variability through time, we used the
same set of perturbations and estimated the standard deviation in FAC change
for each of the 100 perturbation runs over every 5 d time step, producing
a time series of standard deviations. We then scaled the standard deviations
in 5 d FAC change by dividing them by the absolute value of the mean 5 d FAC
change, yielding a time series of standard deviations relative to the
absolute value of the mean FAC change. Finally, we calculated the median
scaled standard deviation over the entire time series to approximate the
typical uncertainties in FAC change, which was done for each of the
perturbation sites. We were unable to quantify a relationship between the
relative error in FAC change and the mean climate forcing even when
separating between sites that experience melt and those that do not. Rather,
the relative uncertainty in 5 d FAC change (
We evaluated our SMB estimates through comparison with in situ measurements from across both ice sheets. For the AIS, we attempted to replicate the analyses as presented by Mottram et al. (2021) to ease comparisons of our performance against a suite of state-of-the-art SMB models. We used a new compilation of SMB observations from Wang et al. (2021), excluding those from Dattler et al. (2019) and Medley et al. (2013). The former study generated SMB using airborne shallow radar; however, because of the lack of an age constraint of the observed radar horizons, the layers were dated in a way to allow the derived SMB estimates to match the large-scale MERRA-2 mean. Thus, the Dattler et al. (2019) data set is dependent on the MERRA-2 SMB and is excluded. The Medley et al. (2013) data set was not used because it was excluded in the Mottram et al. (2021) evaluation, which cited the challenge in evaluating a coarsely resolved SMB data set against finely resolved radar-derived measurements. We performed a separate analysis that includes the Medley et al. (2013) data set.
After filtering the observations as described in Mottram et al. (2021) by limiting observations to the 1950–2018 interval, we arrived at a total number used in the evaluation of 16 427. We used a reference interval of 1987–2015 to match Mottram et al. (2021). For SMB observations that fall entirely within the reference interval, we compared the observation against the model mean SMB over the contemporaneous period. For the observations that cover years outside of the reference interval, we used those that span more than 5 years and compare the mean against the mean SMB over the reference interval. We also used the same aggregation approach by (1) interpolating the modeled SMB values to the location of the SMB observation and (2) averaging all the interpolated model values and observations that fall within the same grid cell. We do not do the comparison on the same common grid as Mottram et al. (2021) but rather use the 12.5 km grid used in this analysis. The final number of aggregated observations for comparison against modeled SMB was 1037 as many of the observations fall within the same grid cell (1207 if the Medley et al., 2013, data set is included).
For the GrIS, we performed a similar analysis as with the AIS using ice core observations of SMB compiled by Fettweis et al. (2020) and PROMICE (v2020) SMB observations compiled by Machguth et al. (2016b), filtering the latter to observations of greater than 3 months with a start date after 1980. We also used an ensemble mean of 13 SMB models (GrSMBMIP) to add context to the evaluation (Fettweis et al., 2020). For each observation, we linearly interpolated the model SMB to the observation location, repeating for both the GSFC and GrSMBMIP models. To minimize bias imparted by poor spatial sampling, we averaged all the observations and their associated model values into the 12.5 km grid used in this study, as done in Mottram et al. (2021). We compared the observations to the models in three ways. First, we determined the mean GSFC SMB over the exact observation interval. Second, to ease comparison with GrSMBMIP, we calculated the annual GSFC mean SMB and took the mean GSFC annual SMB of the years the observation interval covered (referred to as GSFC/ANN). Third, we perform the same as the latter with the GrSMBMIP ensemble (referred to as GrSMBMIP/ANN). The final number of aggregated observations for comparison against GSFC, GSFC/ANN, and GrSMBMIP/ANN was 312. Results from the SMB evaluation follow in Sect. 3.4 and provide the basis of our SMB uncertainty analysis in Sect. 2.5.1.
The mean firn air content (FAC) for the
During the RCI, the average firn air content over the GrIS was 15.7 m
(the mean 2
Because of the much colder conditions, the AIS firn column contains, on
average, substantially more air than the GrIS. The average FAC during the
RCI for the AIS was 24.0 m (the mean 2
The net mass flux at the surface of an ice sheet is referred to as the
surface mass balance (SMB; Eq. 5) and is typically presented in units of
mass per unit time. Here, we use gigatons per year (Gt yr
Over the RCI (1980–1995), the mean annual SMB of the GrIS was
After 2003, the mean annual SMB for the GrIS was
The SMB of Greenland peripheral ice was never positive over the entire
1980–2021 period with a mean of
The SMB of the AIS is nearly entirely controlled by snowfall
(Fig. 15). Of the
Most mass gains over the AIS occur in the form of net accumulation over the
grounded ice sheet (
The combined fluctuations in SMB and FAC drive the total ice-sheet volume changes due to surface processes, yet only the former constitutes an actual mass change. We evaluate the relative contributions of mass (SMB) and air (FAC) at seasonal and multi-annual timescales. When propagating errors, we account for the variable correlation in time and space.
Height and volume change of the Greenland Ice Sheet.
The seasonal amplitudes of the SMB and FAC components of ice-sheet-wide
volume change averaged over the RCI are 143 and 236 km
Rate of height change resulting from changes in firn air content over the Greenland Ice Sheet between 1 September 2003 and 1 September 2021. Note the asymmetric color bar.
Height and volume change of the Antarctic Ice Sheet (grounded and
floating ice).
The seasonal amplitude of height change due to surface processes alone
averages to 6 cm over the entire AIS (Fig. 18c),
which is one-fourth that of the GrIS (23 cm). Due to its large area,
however, the seasonal volume change amounts to 808 km
To contextualize the SMB values derived here from MERRA-2 and the CFM, we perform SMB evaluation against observations inspired by two recent SMB model intercomparison exercises for the AIS (Mottram et al., 2021) and GrIS (Fettweis et al., 2020).
Modeled GrIS SMB performance statistics against
The comparison between observations and modeled SMB for the GrIS indicates
that our model performs similar to several of the models within the GrSMBMIP
exercise. Figure 20 shows the performance of the
GSFC/ANN comparison against the GrSMBMIP/ANN, and
Table 2 provides the statistical comparison with the
observations. We note here that the GrSMBMIP ensemble mean resolved SMB
better than any individual model within the ensemble, so we expect the GSFC
model to have lower performance metrics than the ensemble mean. The GSFC
model reproduces observed SMB under near-equal performance as GrSMBMIP for
observations with SMB
Rate of height change resulting from changes in firn air content over the Antarctic Ice Sheet between 31 March 2003 and 31 March 2021.
Comparison of SMB observations and the GSFC/ANN (purple) and GrSMBMIP/ANN (green) data sets for the GrIS (see Sect. 2.5.2 for the method).
Using the
The difference in mean annual SMB between GSFC and the GrSMBMIP ensemble mean.
Comparison of annual melt from the MERRA-2 degree-day model (Sect. 2.2.1) and MARv3.5.2, the model used to train the degree-day model (Fettweis et al., 2017).
We also directly compare the GrSMBMIP ensemble mean annual SMB with our GSFC
results in Fig. 21 over the common 1980–2012
interval, interpolating our model results onto the GrSMBMIP grid. The GSFC
model exhibits elevated SMB over the interior relative to the GrSMBMIP
ensemble mean with variable differences in sign around the periphery (i.e.,
exhibits positive and negative differences). The statistical summary in
Table 2 suggests that integrated over the entire ice
sheet, the GSFC model has a slightly higher SMB. The annual mean SMB from
the GrSMBMIP of 347 Gt yr
Finally, we compare our degree-day model annual melt rates with those used
to train our model (i.e., MARv3.5.2; Fig. 22). The
time series have a high correlation (
A breakdown of GSFC SMB performance against observations separated
into different elevation bands to match Mottram et al. (2021)
analysis.
Replicating the analysis within Mottram et al. (2021) was more
straightforward, so we present analysis that allows for direct comparison
with their results. Figure 23 compares all SMB
observations with the GSFC modeled SMB, and statistics of the evaluation
are presented in Table 3, broken down into different
categories as done by Mottram et al. (2021). Considering the AIS as
a whole, GSFC SMB has a very small positive mean bias (6 kg m
Comparison of
Difference in GSFC SMB and observations over the AIS, including the results from Medley et al. (2013) (see Sect. 2.5.2 for method).
The 1980–2010 mean annual GSFC SMB is 2620 Gt yr
Comparison of annual melt from the MERRA-2 degree-day model (Sect. 2.2.1) and Trusel et al. (2013) QuikSCAT-derived surface meltwater fluxes used to calibrate the degree-day model, as well as two regional climate models, RACMO2.3p2 (Van Wessem et al., 2018) and MARv3.6.4 (Agosta et al., 2019).
Finally, we compare our degree-day model annual melt rates with those used
to train our model (i.e., Trusel et al., 2013;
Fig. 25). We also compare our annual melt fluxes
against two regional climate models (Van Wessem et al.,
2018; Agosta et al., 2019) to provide a longer context because the QuikSCAT
observations cover only a decade. By design, our degree-day model best
matched the magnitude of the observations from Trusel et al. (2013). The contemporaneous (1981–2016) mean annual melt rates
from our degree-day model, RACMO2.3p2, and MARv3.6.4 are 99, 107, and 83 Gt yr
We present simulations of GrIS and AIS firn processes using the CFM forced by MERRA-2 atmospheric reanalysis data spanning more than 40 years. Specifically, we calibrate the Arthern et al. (2010) firn densification model through modification of its dependence on overburden and temperature. The resulting model reduces the rates of densification, largely due to reduced sensitivity to increasing overburden, which is approximated by the mean accumulation rate. Modification to the temperature dependence was necessary for the second stage of densification, which is in line with other studies that found the accumulation rate as a key parameter in model calibration (Kuipers Munneke et al., 2015; Ligtenberg et al., 2011). Our calibration differs and is comparable to the approach by Verjans et al. (2020), as we derive the form of our calibration using the original form of the Arthern et al. (2010) densification equation, which provides adjusted model parameters that best fit observed depth–density profiles and the MERRA-2 climate conditions. Additionally, we calibrate the model using observations from both ice sheets, resulting in one set of adjusted parameters. It is important to note that the adjustments to the densification model parameters reflect missing physical processes as well as persistent biases within the climate forcing (e.g., if the forcing exhibited a cold bias). Thus, application of these adjustments when using a different climate forcing is not recommended. Future work will investigate use of alternative calibration equations to assess its impact on the resulting volume changes.
The surface density parameterization is also dependent on the mean annual
climate conditions derived from MERRA-2, so any biases will manifest in the
derived coefficients. We note that while the model does a satisfactory job
of reproducing moderate to high surface densities (325–415 kg m
We next review other limitations of the work we have presented, which will be the focus of future work. The choice of running the model at 5 d time steps was a subjective choice, based on the need for computational efficiency. The firn is subject to diurnal changes in temperature and melt that our model is not capable of resolving; however, we attempt to capture much of the signal at 5 d windows through accumulating fluxes at hourly resolution such as melt and snow accumulation. In the prior simulations, we used an effective mean temperature to try to capture the non-linear impact of the large diurnal fluctuations in temperature and their resulting impact on the densification rate. We abandoned that effort (see Sect. 2.1.3 and Appendix C) given its degraded performance when compared against simulations performed at 1 d time steps. Future work preserving both the physical and effective temperature means through time will help us better understand if we can adequately capture the sub-time-step temperature impact on densification moving forward.
While we indicate that our choice of RCI was our best attempt at capturing the long-term conditions, it remains a partly subjective choice that does have an ultimate impact on our results and interpretation. The challenge for all firn modeling efforts is that the firn column was built of tens to thousands of years of snow accumulation, yet we only have a spatiotemporally complete understanding of polar climate conditions arguably since the beginning of the satellite era (1979 and onwards). Thus, we make assumptions regarding how that firn column will respond to modern conditions without knowledge of the past prevailing conditions. Studies suggest variable spatial trends in both snow accumulation rates (Medley and Thomas, 2019; Thomas et al., 2017) and air temperatures (Steig et al., 2009; Nicolas and Bromwich, 2014; Bromwich et al., 2013) over the AIS, which are not considered in this work. Similarly, the reconstructed SMB from a 20th century reanalysis found significant trends over GrIS since 1870, which this work does not capture (Hanna et al., 2011). Thus, any deviation of the RCI atmospheric conditions from reality will bias the trends in firn column evolution. Future work investigating the impact of these reconstructed trends would help to quantify the resulting uncertainty in height changes due to long-term climate change. Deviations from observed height changes thus reflect both errors in firn modeling efforts and unknown trends due to a lack of constraint on recent climate, impacting results over both ice sheets.
Meltwater fluxes as well as their ultimate fate remain the largest source of uncertainty in our firn modeling effort. Our simple degree-day model of melt was employed due to the absence of MERRA-2 meltwater flux output. At present, the CFM does not have an energy balance model subroutine, although it is in preparation, so future versions of GSFC-FDM will use a physically based melt model. Comparisons against the degree-day model training data, as well as other RCM results, suggest that we are capturing a significant portion of the annual signal (Figs. 22 and 25). The total magnitude of melt is less than the training data set for the GrIS, which might be due to (1) an overestimation of melt within the RCM used to train our model, (2) a cold bias in the MERRA-2 air temperatures, (3) capping DDFs above 1500 m, or (4) a combination of the aforementioned. Thus, the runoff produced by GSFC-FDMv1.2.1 is on the lower end of several existing SMB models for the GrIS and exhibits a smaller increase in runoff through time. We note that a recent study by Smith et al. (2022) found that the older melt model used for GSFC-FDMv1.1, as well as a more recent version of MAR than used in this study (i.e., MARv3.11.5; Amory et al., 2021), systematically overpredicted the height changes within the high-elevation parts of the ice sheet, particularly in association with melt events. After capping the unrealistic melt factors above 1500 m, the melt model in v1.2.1 yields a better match of the firn height changes with satellite altimetry. Because this comparison only covers two melt seasons, the evaluation suggests improvement, but comparison against more melt event/seasons is necessary to fully evaluate this improvement, rule out possible compensating errors, and highlight other potential future improvements.
While not the focus of the work, one important output from the
GSFC-FDMv1.2.1 simulation is surface runoff, which allows us to estimate ice-sheet SMB. The mean annual GrIS SMB is comparable to SMB estimates from an
ensemble of models of varying complexity
(Fettweis et al.,
2020). Our estimates of the 1980–2012 GrIS mean annual SMB (
Deviations in SMB from its mean over the RCI result in ice-sheet height and volume fluctuations; however, these SMB deviations along with changes in temperature also modulate the total air content within the firn column, amplifying the mass-related height and volume fluctuations. Thus, the SMB impact on height change is twofold: imposing both a change in mass and a change in air (i.e., fresh snowfall is a matrix of ice and air), which means that the fluctuations in SMB and FAC change are strongly correlated. We keep height changes due to mass separated from those due to air because of the relevance to interpretation of satellite-derived height changes. While the SMB and FAC contributions to total firn volume change over multiannual timescales are somewhat comparable, the seasonal signal is dominated by FAC for both ice sheets. This difference suggests that 62 % for the GrIS and 71 % for the AIS of sub-annual volume fluctuations are in response to a change in the air content rather than actual mass change. Thus, determination of seasonal mass change using satellite altimetry requires a substantial FAC correction, highlighting the importance of firn densification and the atmospheric models that force the FDMs, especially when investigating shorter intervals of change as not being mindful of the seasonal cycles of SMB and FAC can generate large biases.
Finally, we briefly note the differences between the GrIS GSFC results for v1.1 and v1.2.1 (differences were negligible over the AIS). The largest difference is the muted FAC change through time integrated over the GrIS (Fig. 16b). This change is partly due to the improved surface density model that yields lower densities over the interior and higher densities around the periphery, which led to a larger increase in firn air over the interior in response to additional snowfall and a smaller decrease in firn air in the percolation zone (Fig. 17). Other factors include the modification of the melt regime at high elevations, which acted to reduce total meltwater fluxes in the interior, reducing the FAC losses due to melt. Finally, the overestimation of density (or underestimation of FAC) at sites with high melt would potentially generate FAC change biased low as there is less air to lose when melt occurs. Thus, substantial FAC loss occurs along the periphery of the GrIS, but those losses are partly balanced by gains in the interior. Small changes in the surface density and liquid water processes yield measurable changes in FAC and SMB, and their uncertainty limits our ability to constrain mass balance estimates from satellite altimetry. Thus, future work constraining melt, its routing, and the initial density and their spatiotemporal evolution is necessary and should be a priority.
The time series of firn height and volume change, split into its respective SMB (ice) and FAC (air) components, provide the data necessary to isolate the ice-dynamical change from the changes observed using airborne and satellite altimeters. Future work improving the representation of the near-surface climate, initial density, and especially liquid water processes within the firn column should improve future iterations of GSFC-FDM-modeled firn volume changes. Because of the challenges in measuring firn processes in situ, future evaluations of firn densification model representation will likely rely on direct comparisons with altimetry-derived volume changes.
The calibration depth–density data were compiled through combination of the SUMup data sets (Koenig and Montgomery, 2018; Montgomery et al., 2018) and other compiled sources that are listed in Table A1.
Locations and sources of the depth–density profiles used in model calibration.
Continued.
Continued.
Continued.
Continued.
We compared the fit statistics when making a linear fit to the logarithmic
density profile versus a linear fit to the actual density profile for each
stage of densification. Table B1 summarizes the results taken from the
Fit statistics. Bold values indicate where the fit to logarithmic density profile is significantly better than using the actual density data based on a two-sample
We tested how the model results are affected by the surface-temperature averaging scheme, which is needed to upscale the forcing data from its native 1 h resolution to the desired 5 d resolution for the CFM runs.
To do so, we performed three types of model runs. In the first, we ran the
CFM with 1 d time steps, using the 1 h MERRA-2 fields (labeled in
Fig. C1 as “1 d”). In the second, we ran the CFM with 5 d time steps,
and the surface temperature was calculated by taking the mean temperature
for each 5 d period (labeled as “15 d, mean T”). In the third, we also
ran the CFM with 5 d time steps, but we calculated the 5 d “effective”
mean temperature, given by
We ran the CFM with the three types of runs for two different sites (South Pole and Summit, Greenland). Figure C1 shows the firn air content (FAC) change from 1980 to 2021 predicted for the two sites for each of the three model run types. Table C1 shows, for each site, the mean FAC for the entirety of each model run (mean FAC row), the change in FAC from the start of the model run to the final time step (FAC change), and the mean modeled FAC in 2020 minus the mean modeled FAC in 1980.
In both cases, the effective mean runs produce a lower total FAC than the 1 and 5 d mean runs. The FAC change using the 5 d mean setting gives a FAC change that is closer to the 1 d value, whereas the effective mean runs predict a smaller FAC change than the 1 d runs. Thus, the use of an effective mean was abandoned; however, future work on the CFM might allow for tracking of both effective mean and physical mean of the firn parcels, which might resolve these discrepancies.
The mean FAC, change in FAC, and 2020 mean FAC minus 1980 mean FAC
predicted for each of the three model run types, for each site. The 5 d
mean
The change in FAC for the duration of the model run for the South
Pole
The NASA GSFC MERRA-2 data are available at
BM led the GSFC-FDM model development including calibration, processing the MERRA-2 climate forcing, and analyzing the output. BM, TAN, HJZ, and BES designed the study and contributed to the manuscript. CMS wrote code for the CFM, ran model simulations, and contributed to the manuscript.
The contact author has declared that none of the authors has any competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors would like to acknowledge all who have contributed to the Community Firn Model effort and made it a useful resource for the community. We would also like to acknowledge Richard Cullather and Lauren Andrews, who provided significant insight into MERRA-2 and the NASA GMAO for the M2R12K data. Finally, we would like to thank Tyler Sutterley and Susheel Adusumilli for providing feedback on the GSFC-FDMv0 and v1 output.
This research was supported by the ICESat-2 Project Science Office at the NASA Goddard Space Flight Center.
This paper was edited by Nicolas Jourdain and reviewed by Vincent Verjans and one anonymous referee.