Increased sea-level contribution from northwestern Greenland for models that reproduce observations

Significance

The Greenland Ice Sheet has been thinning over the past several decades and is expected to contribute significantly to sea-level rise over the coming century. Ice flow models that make these projections, however, tend to underestimate the amount of mass lost from the ice sheet compared to observations, which complicates adaptation and mitigation planning in coastal regions. Here, we constrain a model of northwestern Greenland with a time series of satellite-derived surface velocity data and time-dependent physics to infer unknown ice properties. The model reproduces observed mass loss over the past 13 y within uncertainty. This model—constrained by more data—leads to about 8 to 17% greater sea-level rise contribution from this region by 2100.

Abstract

State-of-the-art ice sheet model simulations used in the Ice Sheet Model Intercomparison Project (ISMIP) that informs the Intergovernmental Panel on Climate Change tend to underestimate observed mass loss from the Greenland Ice Sheet, leading to the question of whether future sea-level rise may be larger than projected. We use one of these models, the Ice-sheet and Sea-level System Model, to investigate how transient calibration impacts historical and projection simulations. Transient calibration is an emerging capability in ice flow models; it uses time series of surface observations and time-dependent physics to constrain uncertain model parameters—in this case, the basal friction coefficient in the sliding law. With more constraints than the common snapshot inversion method, transient calibration has been shown to better capture trends in ice dynamics. Here, we apply both methods to northwestern Greenland, a region undergoing rapid changes. For simulations initialized with the snapshot inversion, we find that subsequent modeled velocities are generally too slow, leading to an underestimation of the mass loss. With transient calibration, however, our simulation better matches a time series of observed velocities, bringing it within observational error for mass loss; however, the fit to observed surface elevation is slightly reduced. Together with the ISMIP results, our simulations show that reproducing the high rates of historical mass loss leads to greater projected sea-level contribution from this region over the coming century. Finally, we suggest a path forward for making transient calibration scalable to the entire Greenland Ice Sheet.

The Greenland Ice Sheet is projected to contribute an additional 1.5 to 14 cm to sea-level rise between 2015 and 2100 on top of the committed contribution, according to the Ice Sheet Model Intercomparison Project for the Coupled Model Intercomparison Project phase 6 (ISMIP6) (1). These ISMIP6 projections are a primary source of information for the Intergovernmental Panel on Climate Change Sixth Assessment Report (2). The numerical models used to estimate these sea-level contributions, however, systematically underestimate mass loss over the last several decades (3), leading to the question of whether their projections are too low.

Observations and projections of the Greenland Ice Sheet indicate that increasing surface melt and ice discharge through the grounding line contribute about equally to ongoing and future mass loss, with a much smaller contribution from basal melt (4–6). Ice sheet models simulate ice discharge, whereas the surface mass balance (SMB) is commonly an externally estimated forcing that has weak or no feedbacks with ice sheet geometry on decadal or shorter timescales (7, 8). When model simulations are compared to observation-based mass balance reconstructions derived from the input–output method (e.g., refs. 4, 5, and 9), the same SMB estimates can be used, thus isolating the difference between observed and simulated ice discharge. Similarly, model projections can be forced with the same climate conditions and use different dynamical parameterizations, thus isolating the uncertainties in simulating future ice discharge.

Changes in ice discharge are a function of the evolving ice velocity and thickness, which, on the subannual to multidecadal timescales of mass balance observations and near-term projections, are controlled by changes in SMB, basal motion, and, for the many marine-terminating outlet glaciers, terminus position. Similarly to SMB, historical terminus positions can be forced because there are now sufficient observations to prescribe subannual ice front positions (e.g., refs. 10 and 11, and references therein), thereby reducing the uncertainty of simulating their evolution with calving laws and frontal melt parameterizations. For projections, terminus position forcings, including calving rates and melt rates, are unknown, and assumptions must be made, for example, based on modern observations and climate model projections (e.g., refs. 1, 12, and 13). Basal motion cannot be forced like SMB and terminus positions during the historical period because it is difficult to observe and depends on properties such as bed material, bed roughness, and water pressure at the ice base, which remain all largely unknown.

Basal sliding and sediment deformation are the dominant modes of ice motion along the ice sheet margins, where Greenland’s fast-flowing outlet glaciers slide over thawed beds (e.g., refs. 14 and 15). Sliding laws parameterize this motion, but their parameters remain uncertain, leading to the common practice of inverting for a basal friction coefficient from satellite-derived ice surface velocity. The most common inversion method is called a “snapshot inversion,” which minimizes the model-observation misfit for the model at one time step (commonly the initial time step) in a manner that is consistent with the conservation of momentum (e.g., refs. 16–18). While this method has proved valuable for initializing models for over 30 y, subsequent forward simulations do not always capture trends in observations (19–23). An emerging method in the field of ice flow modeling is “transient calibration,” which allows for the assimilation of a time series of observations into a time-dependent model to optimize the basal friction coefficient (or any other uncertain model parameters) (21–23) (Materials and Methods). This adjoint data assimilation method is one of several methods that are currently being developed to calibrate ice sheet models with time series of observations (e.g., refs. 24–28). Transient calibration has been shown to reproduce past trends and improve forecasts of ice surface velocity—even when only a short time series of observations are assimilated (23). In addition, Goldberg et al. (22) show that transient calibration can improve the data-model misfit for variables withheld from the assimilation.

We use the Ice-sheet and Sea-level System Model (ISSM) (29) to simulate northwestern Greenland (Fig. 1A, Inset) as a test case to quantify the impact of transiently calibrating basal friction with surface velocity observations on 1) the model’s ability to reproduce observations of surface velocity, surface elevation, and mass change in this region from January 2008 through December 2020 and 2) projections of mass change during the 21st century. This region includes many tidewater glaciers that display a range of subannual to multidecadal behaviors (e.g., refs. 11 and 30–32), is undergoing rapid changes (e.g., refs. 4, 31, and 33), and has high-resolution, high-accuracy datasets available for bed topography, ice surface velocity, ice front positions, and ice surface elevation (Materials and Methods).

Fig. 1.

Model mean errors for surface velocity and surface elevation along the northwestern coast of Greenland. Panel (A) Inset shows the model domains (red outlines) for the Northwest (NW) and Central West (CW) regions (4) and the coastal area (black box) shown in panels (A–H). The basemap is made with Natural Earth. Panel (A) shows the average observed surface velocities from 2007 to 2021 (35–38) masked to the observation locations and initial model ice area with a log color scale. The black line is the coastline from Natural Earth. Panels (B–D) show the difference between modeled and observed surface velocity averaged over time with panel (B) showing results for Snapshot-2007-region, panel (C) for Snapshot-2007-2022-region, and panel (D) for Transient-2007-2022-catchment (labeled “Transient calibration”). Snapshot-2007-2022-catchment is similar to the Snapshot-2007-2022-region result. Panels (E–H) are similar to (A–D); however, they are for surface elevation observations from 2018 to 2021 (39). Mean absolute errors (MAE) for surface velocity and surface elevation are shown for all simulations in SI Appendix, Fig. S3.

Results

Ice Flow Simulations and Calibration.

We run two sets of forward simulations of ice sheet evolution for the Northwest (NW) and Central West (CW) regions of Greenland (4) (together referred to as “northwestern” Greenland). The “historical” set is run from 2007 to 2022, and interpreted from 2008 to 2021. The “projection” set is run from 2015 to 2100, and interpreted from 2021 to 2099 in order to overlap with the majority of the ISMIP6 results (1).

The two primary historical simulations are the Snapshot-2007-region simulation with the basal friction coefficient inferred from a snapshot inversion for 2007 and the Transient-2007-2022-catchment simulation with this coefficient inferred using transient calibration from 2007 to 2022. All basal friction coefficients in this study are spatially variable and temporally constant. The Snapshot-2007-region calibration and simulation are run separately for the NW and CW regions, while transient calibration is run separately for each glacier catchment as defined by Mouginot et al. (4). The catchment results are compiled into a single basal friction coefficient map for each region before running the historical simulations.

Two additional sensitivity historical simulations are run to understand the source of the differences between the Snapshot-2007-region and Transient-2007-2022-catchment simulations. First, for the Snapshot-2007-2022-region simulation, the snapshot inversion is run using the full time series of observations used for transient calibration. Second, for the Snapshot-2007-2022-catchment simulation, the snapshot inversion is run using the observation time series and separately for each glacier catchment, as for transient calibration. Together, the four historical simulations show the influence of adding more observations to the snapshot inversion, completing the inversion separately for each glacier catchment, and adding in time-dependent physics (i.e., accounting for both conservation of momentum and mass at every time step). Though the sensitivity simulations are useful for this purpose, we caution their use for actual ice sheet model calibration because of the known inconsistencies between the time of observations and geometry of the model is inappropriately incorporated into the calibrated parameter during the snapshot inversion.

For the projections initialized from each historical simulation, we use the ISMIP6 protocol (8, 13) and apply two climate forcing scenarios (34)—MIROC5 RCP2.6 and RCP8.5 with medium ocean sensitivity—which impact both the SMB and ice front retreat. We prescribe the same ice front retreat as for the JPL-ISSM1 ensemble member of ISMIP6 (1), but begin with the initial ice sheet geometry from January 2015 of each of our historical simulations to reduce shocks to the system.

Inferred Basal Conditions.

The inferred basal friction coefficients from the snapshot inversions and transient calibration are broadly similar, with lower values in fast flowing areas and higher values in slow flowing areas (SI Appendix, Figs. S1A and S2 B, F, and J). The domain, amount of data, and magnitude of the model-data misfit impact the resulting calibration. Regional and catchment scale variations in the basal friction coefficients (SI Appendix, Figs. S1 and S2) are due to the domain over which each simulation is calibrated: some are calibrated for each region, while others are calibrated separately for each glacier catchment (see above). Despite these regional and catchment scale variations, there is continuity across the entire model domain. For example, calibration with the data time series tends to decrease the friction coefficient near the divide, increase it about 100 km downstream, and decrease it close to the margin and in fast flowing areas (SI Appendix, Figs. S1 and S2). For individual glaciers, the Snapshot-2007-region, Snapshot-2007-2022-region, and Transient-2007-2022-catchment lead to especially different basal friction coefficients, while the Snapshot-2007-2022-catchment coefficient pattern displays similarities with both the Snapshot-2007-2022-region and Transient-2007-2022-catchment results (SI Appendix, Fig. S2). This indicates that the choices of observations and physics most strongly influence the calibration, with a lesser degree of influence from the choice of calibration domain.

Capturing Spatiotemporal Observations.

Surface velocity.

To compare the modeled and observed surface velocity, we compute the MAE for locations and time steps for which there are observations in the fast flowing areas (here defined as greater than 200 m/y as in ref. 40). The Snapshot-2007-region simulation has a MAE of 84 m/y averaged spatially and temporally, which is 13.6% of the average observed fast flowing velocities. This error arises from the ice flowing too slowly for most locations and at most times, with the largest differences being in the trunks of outlet glaciers (Fig. 1B). The Snapshot-2007-region simulation performs better than all the other historical simulations in 2007 (Fig. 2A) because it is calibrated for that time and surface velocity dataset only. After 2007, the Snapshot-2007-region simulation performs worse than the other ones, which could be caused by the snapshot inversion overfitting to 2007 or by the issue of equifinality due to the underconstrained nature of the inversion.

Fig. 2.

Model MAE for surface velocity and surface elevation through time. Panel (A) shows the difference between modeled and observed surface velocity averaged over the fast flowing areas (defined as any observation that is 200 m/y or greater) for times and locations with observations. Panel (B) is the same except for surface elevation. Here, the fast flowing areas are defined by the average over observed velocities from 2007 to 2021. For both panels, it is the difference between the four simulations that is of particular note, whereas the change in MAEs through time is a function of both model performance and changes in observation density. The 2008 to 2014 period has the lowest density of surface velocity observations and they are focused on the fastest flowing areas, which together lead to the highest errors for all simulations.

Because transient calibration assimilates a time series of surface velocity observations from 2007 through 2022 and takes into account the complete time-dependent physics of the model (conservation of momentum and mass for each time step), the Transient-2007-2022-catchment simulation will, by design, better fit the complete record of surface velocity observations than the Snapshot-2007-region simulation, though it likely will not perform as well in 2007. As expected, the Transient-2007-2022-catchment MAE is highest in 2007 compared to the other historical simulations, but far lower for 2008 through 2020 (Fig. 2A). Averaged spatially and temporally, we find a MAE of 59 m/y (9.7% of the average observed surface velocity) for Transient-2007-2022-catchment, which is the lowest of all the simulations. The improvement from Snapshot-2007-region to Transient-2007-2022-catchment is spatially heterogeneous, with some locations showing larger reductions in the misfit and some reversing the sign of the misfit (e.g., now flowing slightly faster than observations instead of consistently too slowly) (Fig. 1 B and D). The greatest improvements are along the margin, while some locations in the slower flowing interior show larger MAEs for Transient-2007-2022-catchment (SI Appendix, Fig. S3). Overall, the Transient-2007-2022-catchment simulation has smaller MAEs and a lower bias (Figs. 1 and 2). In the fast flowing areas this bias is slightly positive (2.4 m/y mean error) rather than the large slow bias of $-$52 m/y for Snapshot-2007-region.

The Snapshot-2007-2022-region and Snapshot-2007-2022-catchment inversions assimilate the same data as Transient-2007-2022-catchment but only account for the conservation of momentum with the geometry of the year 2007 and do not account for conservation of mass. Their spatially and temporally averaged MAEs are 75 and 76 m, respectively (12% of the averaged observed surface velocity), suggesting that calibrating by region versus catchment makes little overall difference. Some larger errors in the interior that only show up for Snapshot-2007-2022-catchment and Transient-2007-2022-catchment; however, suggest that calibrating by catchment may cause some higher MAEs in the interior (SI Appendix, Fig. S3). The overall MAEs for the two sensitivity simulations are smaller and less biased (though still biased slow) than for Snapshot-2007-region, except for in 2007 when they perform worse (Fig. 2B). The MAEs, however, are larger than for Transient-2007-2022-catchment (Figs. 1 and 2 and SI Appendix, Fig. S3), with only a few exceptions, mostly in the interior and in 2007. This suggests that, though calibrating with an observation time series helps match those same observations, the most improvement is achieved when the time-dependent physics is also included.

Surface elevation.

To compare the modeled and observed surface elevation (Fig. 1E), we again compute the MAE for locations and time steps for which there are observations in the fast flowing areas (now defined as greater than 200 m/y for the temporal average of the 2007 to 2021 surface velocity observations). The Snapshot-2007-region simulation has a MAE of 21 m (2.2% of the average observations in the fast flowing areas) and a bias of 12 m (elevations tending to be too high in the model) (Fig. 1F). This surface elevation error is comparable to the two sensitivity experiments (Figs. 1 F and G and 2B), but slightly smaller than for the Transient-2007-2022-catchment simulation, which shows a MAE of 34 m (3.4% of the average observations) and a bias of 13 m. Spatially, the Transient-2007-2022-catchment shows larger errors than the other three simulations around Sermeq Kujalleq (Jakobshavn Isbræ) in the southernmost part of the CW region and upstream of some other fast flowing outlet glaciers (Fig. 1F–H and SI Appendix, Fig. S3).

Mass balance.

The Snapshot-2007-region simulation underestimates the Mankoff et al. (5) observation-based reconstruction of mass change for the combined region of NW and CW Greenland for several of the years from 2008 to 2021 (Fig. 3A). This leads to a cumulative underestimate of 182 Gt (11% of the total observed mass loss for this period), which is about 0.55 mm sea-level equivalent (SLE) (Fig. 3B). This error is 110% that of the observational discharge uncertainty (5) and is larger than the difference between the Regional Atmospheric Climate MOdel—RACMO (41) and the Modèle Atmosphérique Régional—MAR (42) SMB reconstructions (Fig. 3 B and C). By using transient calibration instead of the standard-practice snapshot inversion (as for Snapshot-2007-region), the modeled velocities are faster, leading to greater dynamical thinning (SI Appendix, Fig. S4), and a larger mass loss that reduces the error to 66 Gt (4% of the total observed mass loss), or about 0.20 mm SLE. The slight high bias in both the velocities and surface elevations lead to a mass loss that is larger than the observations. Transient calibration thus brings the simulation to within observed mass loss uncertainty (the model-data difference being 41% of the observational discharge uncertainty). Though there are differences between the NW and CW regions, the Transient-2007-2022-catchment simulation consistently shows improvement over the Snapshot-2007-region simulation (SI Appendix, section A and Fig. S5). The two sensitivity experiments show the lowest cumulative mass balance errors, with an underestimate of 48 Gt for Snapshot-2007-2022-region and 11 Gt for Snapshot-2007-2022-catchment. The slow bias in surface velocity and high bias in surface elevation for these simulations may lead to a good fit to mass loss, in part, through compensating errors.

Fig. 3.

Historical and projected mass balance. Panel (A) shows annual mass change for the four historical simulations (colored lines) and an observation-based reconstruction (5) using a surface mass balance (SMB) estimate from the RACMO (41) (Obs-RACMO, black points). Observation uncertainties are too small to see on this plot. Panel (B) shows the same, but for cumulative mass change with uncertainty shading for the observations and with results from ISMIP6 historical and MIROC5 RCP2.6 (light blue) and RCP8.5 (dark blue) forcings with medium ocean sensitivity (1). Panel (C) shows the historical (2008 to 2021) mass change versus the projected (2021 to 2099) mass change for each model simulation from ISMIP6 (blues) and this study (purples and orange). Circles and stars show results from RCP2.6 and RCP8.5 projections, respectively, and the black lines are observed historical mass loss using different SMB estimates: dashed line for RACMO and dash-dotted line for MAR (42). Panels (D) and (E) show the projected cumulative mass change results for all of the simulations under RCP2.6 and RCP8.5 forcings, respectively.

Projected Mass Loss.

Projecting the mass change to 2100 continues to show that using transient calibration instead of the standard-practice snapshot inversion (as for Snapshot-2007-region) causes the model to lose more mass (Fig. 3C–E and SI Appendix, Figs. S6 and S7). Under the MIROC5 RCP2.6 scenario conditions converted into SMB and ice front forcings by ISMIP6, the Transient-2007-2022-catchment projection loses 8,336 Gt, which is 1,183 Gt more mass than the Snapshot-2007-region projection from 2021 to 2099, equivalent to about 3.5 mm of additional sea-level rise. Under the MIROC5 RCP8.5 forcings, the difference between the calibration methods is slightly greater: The Transient-2007-2022-catchment projection loses 16,132 Gt, which is 1,338 Gt more mass lost (4 mm SLE). For comparison, the differences between the mass loss for the 2099 projections forced by RCP2.6 and RCP8.5 are 7,642 and 7,796 Gt (both about 23 mm SLE) for the Snapshot-2007-region and Transient-2007-2022-catchment projections, respectively. For the sensitivity experiments, their projected mass loss falls between Snapshot-2007-region and Transient-2007-2022-catchment, just as it does for the historical mass loss. We note that projected mass loss for Snapshot-2007-2022-catchment is nearly indistinguishable from Transient-2007-2022-catchment; however, we caution the use of the sensitivity experiments for projections because of the known time offset between the observations and model geometry during calibration.

For our small sampling of the climate forcing and calibration uncertainty, we find that the impact of the calibration method is about 15 to 17.5% that of the climate forcing uncertainty at the year 2099. The choice of calibration method, however, is a larger share of the uncertainty prior to 2099. At 2060, the projections show a transition from the uncertainty being dominated by the choice of calibration method to being dominated by the choice of climate forcing. Prior to 2060, the choice of calibration method is most important, especially from 2035 to 2059. These findings are similar for the NW and CW regions individually (SI Appendix, section A and Fig. S7).

Discussion

Sea-Level Projections.

The future contribution of the Greenland Ice Sheet to sea-level rise remains an open question. Here, we compare projections of northwestern Greenland mass loss to results from ISMIP6 for the same region (1). The results, however, should only be compared under the following considerations. First, our results include peripheral ice masses, unlike the observations and the ISMIP6 simulations. Therefore, our simulations show an additional amount of mass loss; however, it is likely insufficient to fully explain the differences (Materials and Methods). Second, the SMB forcings for the historical period vary. We use a recent RACMO SMB forcing (version 2.3p2), while most ISMIP6 results use an earlier version of RACMO, MAR, or a different SMB forcing. For ISMIP6 models that use RACMO2.3, we expect the mass balance comparison with our results to be representative of differences in ice discharge, whereas ISMIP6 models that use MAR, for example, would be biased toward more mass loss when compared to our results (Fig. 3C). Third, we stitch together regional models which have a higher spatial and temporal resolution than many continent-wide models, but do not capture the interaction between the regions (e.g., thickness changes, divide migration, and cross-region ice flow). Our regional models are also calibrated at the regional to catchment scale, likely allowing for a more accurate calibration of smaller glaciers than a continent-wide calibration. Finally, the ISMIP6 ensemble is usually reported with a control run subtracted because the models are not necessarily designed to capture historical mass changes (1, 43). Here, we keep the control run in the ISMIP6 results for direct comparison to our results. Together, these considerations suggest that there may be compensating influences on a comparison between this study and the ISMIP6 ensemble, which could be explored or corrected for in future work.

Similar to Aschwanden et al. (3), we find that the ISMIP6 ensemble consistently loses less mass than both our simulations and the observations (5) during the historical period of 2008 to 2021 (Fig. 3B). ISMIP6 also has a lower ensemble mean projection of sea-level rise by 2099 (Fig. 3C–E). This suggests a strong connection between simulations that lose more mass during the historical period (thereby better capturing the observed historical mass trends) and a tendency to lose more mass in projections and contribute more to future sea-level rise (Fig. 3C).

Evaluating Models with Historical Observations.

Models that better match historical observation-based reconstructions of mass loss may or may not lead to more accurate projections. Choi et al. (23), showed that bringing a model into alignment with observations in a manner consistent with time-dependent model physics (e.g., with transient calibration) should increase confidence in its projections, at least over decadal timescales. Models calibrated during a historical period, however, may still result in tuned parameters or parameterization choices that are not necessarily representative of future ice dynamics, especially for longer projections as the environmental conditions and ice sheet geometry change.

Even for near-term forecasts, historically calibrated models must be carefully evaluated because they can fit observations for reasons other than improved physical representation of the system. For example, in this study, there may be influences of overfitting observations, inconsistencies between datasets inappropriately incorporating uncertainties from one parameter or parameterization into a different parameter, and compensating errors for scalar values. For the Transient-2007-2022-catchment simulation, surface velocity errors are reduced while surface elevation errors are slightly increased compared to the three snapshot inversion simulations (Figs. 1 and 2), which may result from overfitting the assimilated surface velocity observations, suggesting that assimilating the elevation observations or strengthening the regularization may be necessary. All of the simulations, however, may overfit the observation-based mass balance reconstruction (5) because the model and reconstruction rely on many of the same datasets for velocity and ice thickness. The Snapshot-2007-2022-region and Snapshot-2007-2022-catchment simulations, may additionally suffer more from inconsistent datasets and compensating errors than for the other simulations. The calibrations for these two simulations are done for an ice sheet geometry from 2007 and surface velocity observations from 2008 to 2022. The snapshot inversion will try to correct for possible inconsistencies between these two datasets through the basal friction coefficient, resulting in a calibrated parameter that is impacted by known (and fixable) errors. With absolute surface elevation changes averaging 28 m from 2007 to 2021 in the fast flowing areas, as seen in the comparison of BedMachine v5 (44) and ICESat-2 data (39) (SI Appendix, Fig. S4A) and for each of the historical model simulations (SI Appendix, Fig. S4 B–E), the slope changes, and therefore the corresponding balance velocities, could be significant. Therefore, the influence of inconsistent datasets on the calibrated basal friction coefficient may also be significant (SI Appendix, Figs. S1 and S2). Additionally, these two sensitivity simulations show a stronger influence of compensating errors than for Snapshot-2007-region and Transient-2007-2022-catchment: They provide the best fit to the observed mass balance (Fig. 3), but the relatively large surface velocity errors are spatially distributed between both positive and negative values (Fig. 1) and the small slow bias in the velocities may be compensated by the positive surface elevation bias.

Scaling Transient Calibration to Larger Domains.

Our results indicate that transient calibration is a useful tool for capturing ice dynamics and mass loss in Greenland, though more work remains to ensure it is not overfitting the surface velocity observations. Because this method captures trends with greater fidelity than the standard-practice snapshot inversion method (i.e., calibrating with a single dataset from as close to the same time as the initial model geometry), it may be well-suited for calibrating and initializing models used to project Greenland contribution to sea-level rise; for example, it could be used for some models in the upcoming ISMIP7 effort. With the difficulty of implementing transient calibration and its added computational expense, however, community adoption of this method is unlikely without making it more accessible across models and more computationally scalable, for example, to models with larger domains or higher resolutions than this study. This study, combined with previous work, suggests several paths forward on this front, including applying transient calibration to a subset of a model domain (this study), calibrating with shorter time series of observations (23), or transferring calibrated parameters between models (45) or from a coarser to finer resolution mesh (23). It may also be possible to combine several of these approaches without significantly impacting the results. Though it may be tempting to take a shortcut by instead using the snapshot inversion with a different surface velocity dataset, for example, one that has faster flow speeds like for Snapshot-2007-2022-region, the resulting basal friction coefficients will be compensating for errors in the conservation of momentum rather than being representative of the ice–bed interface, making this a less accurate calibration method given the current available knowledge.

Though the calibrated basal friction coefficients in this study have similar patterns at large scale, small differences greatly impact the simulated ice flow. We investigate which of these differences are most important for capturing the observed mass change. To do this, we run five additional historical simulations that isolate the transient calibration updates to basal friction in fast flowing and near margin areas (SI Appendix, section B and Fig. S8). We find that only the changes closest to the margin impact discharge, where the cutoff distance can be estimated from an average surface velocity and the time period of the simulation. For example, with an average surface velocity of 1,500 m/y and a simulation length of 100 y, we may only need to apply transient calibration to the areas within 150 km of the coast. Even after expanding this distance to account for the fastest flowing glaciers, the area for which it is necessary to run transient calibration would still be much smaller than the entire domain, especially when considering a model for the full Greenland Ice Sheet. A downside of running transient calibration on only a subset of the mesh is that fewer observations would be available for the calibration and smoothing may be needed to avoid abrupt transitions between the transiently calibrated area and upstream. We suspect this would not significantly impact the results, however, future work is needed to verify this approach.

As scalability is further explored, it is also important to note that transient calibration, like most calibration methods, will be impacted by the choices made in the method, model, and assimilated observations. For example, we adjust the basal friction coefficient in the sliding law, despite the presence of other uncertain model parameters. We find that some of the most impactful updates to the basal friction coefficient are in the fast flowing trunks of each outlet glacier where sliding dominates ice flow. This may suggest that adjusting basal friction in fast flowing areas is key for capturing observed mass change; however, additional work is needed to determine whether other uncertain variables, such as ice rheology, bed topography, flow law formulation, and frontal forcings, could have a similar impact (e.g., softening the shear margins of outlet glaciers would also increase the trunk velocity). Additionally, changes in the basal friction coefficient may be compensating for the choice of sliding law; the linear Budd sliding law and sliding laws without effective pressure have been shown to result in slower velocities in Northwest Greenland compared to other sliding laws (46, 47). Such uncertainties could begin to be quantified, though research into this is ongoing (3, 48).

Key Takeaways.

Calibrating our model of the northwestern Greenland Ice Sheet with time series of ice surface velocity data and time-dependent physics (transient calibration) leads to simulations that better match observed surface velocity and mass loss over the last 13 y than the standard-practice snapshot inversion approach, though the fit to observed surface elevation is slightly below that of the snapshot inversion. When the model reproduces mass loss observations, projections show greater contribution to future sea-level rise, suggesting that underestimates from models in the historical period carry forward. Reproducing total historical mass loss only, however, does not necessarily mean that the model is appropriately calibrated. We suggest instead that calibration and evaluation should be based on spatiotemporal datasets and a physical understanding of the system. Though transient calibration has not yet been applied to models at continental scale, the computational expense can likely be reduced without significantly impacting the benefits of this method.

Materials and Methods

Ice Flow Model.

We use the ISSM, a finite-element solver for ice flow equations (29), to simulate the Northwest (NW) and Central West (CW) regions (4) of Greenland (Fig. 1A, Inset). We create two unstructured triangular meshes of 465,007 and 230,186 elements for the NW and CW regions, respectively. The model resolution is 400 m near the ice margin, in fast flowing areas, and along shear margins, and gradually transitions to coarser resolution inland with a cap at 10 km. The initial bed and surface topography are interpolated onto the mesh from BedMachine v5 (44) and GIMP DEM (49), respectively, and are nominally for the year 2007. Grounded and floating ice extents are computed from this geometry using a hydrostatic assumption.

For conservation of momentum, we use the shallow-shelf approximation with an ice rheology parameter estimated from the depth-averaged temperature field of the JPL-ISSM simulation for ISMIP6 (1) and converted to a prefactor using the table of Cuffey et al. (50). We soften the shear margins of Sermeq Kujalleq (Jakobshavn Isbræ) by a factor of three, where the shear margins are defined by locations with modeled effective strain rates that are greater than 0.5 y⁻¹ and velocities that are in the lower 90th percentile for the catchment. This allows us to capture the large velocity gradient caused by ice softening in shear margins, similar to what was done in previous studies (51, 52). For basal motion, we use a Budd sliding law (53):

$${\tau }_b=C^{2}N{u}_b^{\frac{1}{m}},$$ [1]

where ${\tau }_b$ is the norm of the basal shear stress, $C$ is the basal friction coefficient, $N$ is the effective pressure, $u_b$ is the magnitude of the basal velocity, and $m$ is a constant exponent. The Budd sliding law has been shown to be appropriate to capture recent changes in Northwest Greenland (46). For simplicity and stability, we set $m=1$, which results in a linear approximation of sliding, and we assume that the effective pressure is such that the bed is fully connected to the ocean, similar to previous studies (e.g., ref. 46). To initialize the basal friction coefficient in the sliding law, we perform a snapshot inversion (16) using a Greenland composite surface velocity dataset with a nominal year of 2005 as the target observations (35, 36). Importantly, the surface velocity observations are for a similar year as the nominal year of the initial geometry.

At the ice–atmosphere interface, we prescribe a surface mass balance forcing from the RACMO version 2.3p2 downscaled to a 1 km spatial resolution for each month from January 2007 to August 2021 (41). At the ice–ocean interface, we prescribe monthly ice front positions from a data compilation (11) using the level-set method (54) to capture (within data uncertainty) the combined influence of frontal melting and calving. For all forcings, we linearly interpolate conditions for model time steps that fall between the forcing timestamps.

For each historical simulation, we treat the first year as a relaxation period, though most of the abrupt velocity changes resulting from the initial conditions resolve after 0.1 y. For each projection, because the ice extent in January 2015 for our model differs slightly from the January 2015 extent in the ice front forcing taken from the JPL-ISSM ensemble member of ISMIP6 (1), we ensure that the glacier ice fronts do not artificially advance by always choosing the most inland ice front from our model and the forcing.

Snapshot Inversion and Transient Calibration.

Snapshot inversion and transient calibration are adjoint data assimilation methods, which minimize a cost function, $\mathcal{J}$, by first computing the sensitivity (gradient) of $\mathcal{J}$ with respect to the parameter(s) of interest for a given model. Here, this parameter is the spatially variable (but time invariant) basal friction coefficient ($\mathcal{C}$, Eq. 1). We compute an initial estimate of $\mathcal{C}$ from the sliding law assuming the basal shear stress is equal to the driving stress. This coefficient is then adjusted using a gradient descent method until $\mathcal{J}$ reaches a minimum. Here, the cost function quantifies the absolute misfit between the modeled and observed velocities in the $x$ and $y$ directions, the misfit between the logarithm of the modeled and observed velocities, and the gradient of $\mathcal{C}$. For the snapshot inversion, the model physics are the conservation of momentum at the initial state, so the modeled and observed velocities are compared at one point in time, resulting in the following $\mathcal{J}$:

$$\begin{array}{cc}\mathcal{J}\left(\mathcal{C}\right)& ={\gamma }_1\underset{\text{Ω}}{{\displaystyle \int }}\frac{1}{2}\left({\left(v_x(\mathcal{C})-v_x^{\text{obs}}\right)}^2+{\left(v_y(\mathcal{C})-v_y^{\text{obs}}\right)}^2\right)d\text{Ω}\\ & +{\gamma }_2\underset{\text{Ω}}{{\displaystyle \int }}\frac{1}{2}\ln {\left(\frac{v(\mathcal{C})}{v^{\text{obs}}}\right)}^{2}d\text{Ω}+{\gamma }_3\underset{\text{Ω}}{{\displaystyle \int }}\frac{1}{2}\Vert \nabla \mathcal{C}{\Vert }^{2}d\text{Ω},\end{array}$$ [2]

where ${\gamma }_i$ are the weights given to each term, $v$ is velocity with the subscript indicating a direction and superscript indicating whether it is an observation, and $\mathrm{\Omega }$ is the model domain. The first two terms quantify the model-data misfit, while the last term is a regularization that constrains the variability of $\mathcal{C}$ to try to avoid overfitting the data. We select the weights (${\gamma }_i$) for each term based on standard practices: the absolute velocity misfit is about five times larger than the log velocity misfit, which puts more weight on fitting the fast flowing areas most relevant to capturing ice discharge. Additionally, we aim for the data misfit to be about ten times larger than the regularization term. Future work should incorporate a physical interpretation of the basal friction coefficient to select the regularization weight (48). However, our results are robust to small changes in the weights, including order of magnitude adjustments to ${\gamma }_3$.

Unlike the snapshot inversion, transient calibration can assimilate a time series of observations into a time-dependent model (which accounts for both conservation of momentum and mass for all time steps). This makes the gradient of the cost function more difficult to compute, often requiring automatic differentiation (AD). AD applies the chain rule to the model operations to calculate partial derivatives, such that it can estimate the gradient of a cost function or any other quantity of interest with respect to any parameter of interest (e.g., refs. 23 and 55). The current implementation of AD in ISSM is described by Morlighem et al. (55). To ensure the most straightforward comparison between the two calibration methods, we use AD for both snapshot inversions and transient calibration in this study instead of the built-in snapshot inversion in ISSM; however, this should have little impact on the results since the two implementations are equivalent.

AD is computationally expensive for transient simulations, so we calibrate the basal friction coefficient field for each glacier catchment individually within the NW and CW regions as defined by Mouginot et al. (4). We then assemble the calibrated fields to produce a complete map of basal friction across the NW and CW regions. As for the snapshot inversion, we calibrate a spatially variable coefficient that is constant in time, though transient calibration does make it possible to calibrate parameters that change through time (21–23).

Using procedures similar to Morlighem et al. (55) and Choi et al. (23), we assimilate a time series of ice surface velocity data (described below) with the following cost function over the January 2007 to January 2022 simulation:

$$\begin{array}{cc}& \mathcal{J}\left(\mathcal{C}\right)\\ & ={\gamma }_1{\displaystyle {\displaystyle \sum }_{t=2007}^{2022}}\underset{\text{Ω}}{{\displaystyle \int }}\frac{1}{2}\left({\left(v_x(\mathcal{C},t)-v_x^{\text{obs}}(t)\right)}^2+{\left(v_y(\mathcal{C},t)-v_y^{\text{obs}}(t)\right)}^2\right)d\text{Ω}\\ & +{\gamma }_2{\displaystyle {\displaystyle \sum }_{t=2007}^{2022}}\underset{\text{Ω}}{{\displaystyle \int }}\frac{1}{2}\ln {\left(\frac{v(\mathcal{C},t)}{v{(t)}^{\text{obs}}}\right)}^{2}d\text{Ω}+{\gamma }_3\underset{\text{Ω}}{{\displaystyle \int }}\frac{1}{2}\Vert \nabla \mathcal{C}{\Vert }^{2}d\text{Ω},\end{array}$$ [3]

where $t$ are the times for which we have surface velocity data, even if only for part of the domain. We use the same weights as for the snapshot inversion, but for our initial guess of $\mathcal{C}$, we use the results of the snapshot inversion with an a priori constraint of $\pm$50%.

As for the snapshot inversion, the transient calibration is considered complete when either fifty gradient descent iterations are complete or the basal friction coefficient changes by less than 1e⁻⁵ (s/m)^1/2 between two iterations, whichever occurs first. We note, however, that during our investigation we found that running even just a few iterations of transient calibration was enough to significantly improve the simulation.

For the two sensitivity snapshot inversions (to examine the relative influence of the quantity of data, the spatial area over which the calibration is completed, and the time-dependent physics), we use the same setup as for transient calibration (but account for just the conservation of momentum at the initial state) and vary the area over which the calibration is completed (i.e., regions versus catchments).

Datasets for Model Calibration, Evaluation, and Analysis.

For ice surface velocity in the snapshot inversion, we use a 20-y mosaic nominally for 2005 (35, 36) both to be consistent with the other initialization datasets (nominally for 2007) and to obtain a spatially complete calibration dataset. Throughout the main text, this dataset is referred to as the 2007 surface velocity observation. For transient calibration and the two snapshot inversion sensitivity experiments, we assimilate monthly gridded observations from the MEaSUREs’ Greenland Ice sheet Mapping Project (GrIMP) for January 2015 to January 2022 (35, 36, 38). Prior to 2015, we assimilate the gridded SAR product (35, 37) whenever available; however, we do not assimilate the 2005 surface velocity mosaic.

We evaluate our results against surface elevation and mass balance data products. For surface elevation, we use the combined ICESat-2 ATL14 and ATL15 altimetry data products (39), which overlap the interpretation period for our model simulations from September 2018 through December 2020. For mass balance, we compare our results to the observation-based reconstruction from Mankoff et al. (5), where total mass balance here is defined as SMB minus ice discharge. This reconstruction is made using the input–output method, which combines ice geometry, surface velocity observations, and an SMB reconstruction. We chose the RACMO SMB (solid black line) for direct comparison to our model (41), but also show the impact of choosing the MAR SMB (42) (Fig. 3C). This total mass balance reconstruction agrees well with others using both similar and different methods (5, 56, 57) and has a subannual resolution, making it an optimal comparison dataset for our model results. Unlike this reconstruction, we do not subtract peripheral glaciers and ice caps from our domain, which may cause our simulations to lose about 2 to 6 Gt/y more than for just the main ice sheet (58). Holding this rate constant, this is about 26 to 78 additional gigatonnes of loss for our historical simulations and 165 to 468 gigatonnes for our projections.

We compare our results to results from the Ice Sheet Model Intercomparison Project for the Coupled Model Intercomparison Project phase 6 (ISMIP6) (1) historical simulations and projections that use the MIROC5 RCP2.6 and RCP8.5 with medium ocean sensitivity forcings (34). We exclude the two UAF-PISM ensemble members due to an issue with the time dimension in the results reported from Goelzer et al. (1).

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Model script, figure scripts, and model output data have been deposited in Zenodo (https://doi.org/10.5281/zenodo.15079083). Previously published data were used for this work (1, 5, 11, 29, 33, 35–39, 41, 44, 49).