Yelmo is a hybrid ice sheet model which heuristically sums the shallow-ice and shallow-shelf approximations (SIAs and SSAs, respectively) to obtain the ice velocity at a given location . It is thermomechanically coupled, and it employs Glen's flow law with an exponent of n=3. The model has been run at a horizontal resolution of 32 km with 20 vertical layers. Additional details of the model can be found elsewhere ; however here we will describe the key model properties that are relevant to this study.

The climate is calculated following a classical snapshot method, in which the present-day climate (PD) and that of the last glacial maximum (LGM) are known (e.g., ). The climatic forcing for other times is an interpolation of the two snapshots with weighting following a glacial index. The near-surface air temperature field T is thus calculated as 7T=Tpd+αcTlgm-Tpre, where Tpd is the present-day climatology obtained from a regional climate model simulation, Tlgm and Tpre are results from climate model simulations of the LGM and preindustrial period, respectively, and αc is the glacial climate index shown in Fig. . Precipitation is calculated analogously, 8P=PpdαcPlgmPpre-1+1+αpΔPhol, although the anomaly scaling is applied as a ratio rather than a sum to avoid negative values. An additional index, αp (Fig. 1c), is used to impose a spatially constant precipitation anomaly only during the Holocene using the parameter ΔPhol. This term adds a degree of freedom to the precipitation, which represents atmospheric changes not captured by the available snapshots. The temperature precipitation fields are additionally scaled to be consistent with the dynamic ice sheet topography via a fixed lapse rate of -6.5 K km-1.

The glacial climate index αc used here is a hybrid of several reconstructions of the Greenland temperature anomaly that span the time period of interest . The individual reconstructions are scaled to match temperature anomalies reconstructed for the NGRIP ice core , and then a low-pass filter is applied to remove climate variability at timescales under 10 kyr. Finally, the time series is normalized to give αc(PD)=0 and αc(LGM)=1. The glacial index is applied to monthly climate data. Figures  and show the mean annual near-surface air temperature and precipitation for PD and LGM. The PD snapshot was obtained as the 1981–2010 climatic average of a simulation of the regional climate model MAR forced at the boundaries by the ECMWF ERA-Interim reanalysis . In the case of temperature, the LGM snapshot is the ensemble mean of simulations contributed to the Paleoclimate Modelling Intercomparison Project Phase III (PMIP3) from several participating models (CCSM4, CNRM-CM5, FGOALS-G2, GISS-E2-R, IPSL-CM5A-LR, MIROC-ESM, MPI-ESM-P, MRI-CGCM3) . Given the large uncertainty associated with precipitation, we sample the mean P¯lgm and standard deviation σP of the PMIP3 simulations to obtain the LGM precipitation snapshot for a given simulation as 9Plgm=P¯lgm+fLGMσP, where fLGM is a free parameter used to scale the uncertainty around the mean. The mean and standard deviation of the LGM precipitation fields are shown in Fig. .

Once the monthly temperature and precipitation fields are known for a given time, the surface mass balance is calculated using the positive degree day (PDD) method . Temperatures above the freezing point are converted into melting of snow and ice via the parameters βs and βi, respectively. Here βi=7 mmK-1d-1, and βs is kept as a free parameter. The ice surface temperature is calculated as Ts=minTann+0.0266m˙s,273.15K, where Tann is the mean annual near-surface air temperature, and m˙s is the net melt rate at the surface. m˙s=0 myr-1 when no melting occurs or all melt is refrozen in the snowpack.

Marine-shelf melting is calculated following previous work , with the basal mass balance of floating ice calculated as 10b˙shlf=b˙ref-κΔTshlf.

Here b˙ref=-10 myr-1 is a spatially constant shelf basal mass balance representing the PD rate, and κ is a heat-flux coefficient that translates oceanic temperature anomalies into basal melting. Values of κ=10 myr-1K-1 and κ=1 myr-1K-1 are used for shelf ice near the grounding line and the broader shelf, respectively. The oceanic temperature anomaly relative to PD, ΔTshlf, is calculated as a fraction of the atmospheric temperature anomaly: ΔTshlf=0.25T-Tpd ; b˙shlf is limited to negative values (i.e., melting) to prevent unrealistic rates of ice accretion during cold climates.

At the lower boundary, the geothermal heat flux Qgeo is imposed with a spatially constant value. It too is a free parameter. Isostatic rebound is calculated dynamically using an elastic lithosphere relaxing asthenosphere (ELRA) isostasy model with a spatially constant time constant set to 3 ka . Sea level is spatially constant and varies in time following the global glacial cycle simulation of (Fig. ).

The spin-up procedure consists of two steps: a temperature spin-up and optimization of the basal friction coefficient field. First, the ice sheet is simulated under constant LGM climatic conditions for 20 kyr using the SIA ice dynamics solver alone, which is mainly intended to spin-up the ice temperature field. Next, the ice sheet is run for another 10 kyr with the full hybrid ice dynamics active. The cb field used here initially has been tuned to give good results in ice thickness for a steady-state present-day simulation. By the end of this 30 kyr spin-up, the ice sheet extends to near the continental shelf break, in good agreement with the reconstructed LGM ice extent .

Second, transient simulations from the LGM to PD are performed iteratively, with the basal friction field cb modified at each iteration to improve the simulated present-day ice thickness following . When the simulation reaches PD, the ice thickness error is calculated (Herr=Hsim-Hobs). For each grid point, we then use the upstream Herr to calculate a tuning factor: 11ε=HerrH0, where H0=1000 m is a scaling parameter controlling the magnitude of changes to cb for a given iteration. For a given location, Herr is defined as the velocity-weighted average of the upstream values in both lateral dimensions. Also, we limit ε∈-1.5,1.5 in order to avoid scaling cb too rapidly for large values of Herr. The tuning factor is then used to modify the local basal friction coefficient as 12cbn+1=10-εcbn, where the indices n and n+1 indicate the current and next iteration. Thus, a positive bias in upstream ice thickness Herr results in a reduction of the basal friction coefficient β, which in turn leads to a lower basal velocity for the same basal shear stress (Eq. ). Once the cb for the next iteration has been calculated, the model state is reset to the LGM spin-up, and the transient simulation is repeated. We performed 10 iterations, after which the error in simulated PD tends not to reduce further, and the cb solution is rather stable. The above procedure results in a reasonable internal temperature distribution in the ice sheet that is consistent with the optimized basal friction configuration. This spin-up procedure is applied to each model version in the ensemble method described below.

The tracing of isochronals is based on . Key to the successful simulation of the isochronal surfaces is a vertical grid that is defined in time, not space, to avoid vertical advection across grid interfaces and therefore eliminate numerical diffusion. The spacing of this isochronal grid is arbitrary and can in principle be variable, but here we use a constant resolution of 200 years. This means that the domain is virtually empty at the beginning of a simulation, aside from 10 initialization layers. They are inconsequential for the analysis below as they can be regarded as older than 160 kyr and become very thin in the present day. Over the course of the simulation, one additional layer of ice is added every 200 years until the grid is full at the end of the simulation period. Thus, at the end of the simulation, the isochronal grid contains a total of 810 layers. The horizontal grid of the layer tracing scheme is the same as in Yelmo.

A major difference from the original implementation is that the calculation of the ice physics and all boundary conditions are now being handled by Yelmo. The host model provides the two horizontal velocity components, the total ice thickness, and the mass fluxes at the ice surface and the bed to the tracer scheme at every tracer time step, Δt = 5 yr (Fig. ). These relatively modest requirements make it in principle possible to replace Yelmo with virtually any three-dimensional ice sheet model. The ice velocities are vertically interpolated to the isochronal grid using the vertically integrated layer thicknesses of the latter as reference. This can be done safely with a simple linear interpolation scheme because velocities vary smoothly in the vertical dimension. Inside the isochronal layer scheme, layer thickness is a passive tracer variable that is advected using an implicit upstream scheme and the interpolated velocities. Advection is strictly two-dimensional within each isochrone so that mass is never exchanged along the vertical axis of the isochronal grid. In summary, the isochronal scheme defines depositional age as the grid and layer thickness as an advected property, which is exactly the opposite from Eulerian age tracers that have a grid defined in space and age as a tracer. Our approach therefore eliminates numerical diffusion on the important vertical axis by design.

While interpolation and advection are performed at the Yelmo time step, mass fluxes at the upper and lower boundaries are applied to the tracer scheme at the temporal resolution of the isochronal grid. Between these times, Yelmo mass fluxes are integrated to a temporary buffer. Where the cumulative surface mass balance of the previous 200 years is positive, it defines the thickness of a new layer. Where the mass balance is negative, at the surface and the bed, the appropriate amount is removed from the existing layers. The newly added layer has a thickness of zero in regions of negative mass balance so that older layers naturally outcrop at the surface near the margins. Empty layers below the one that at the current simulation time is the youngest can only be filled by lateral advection within each isochrone even if the local mass balance turns positive.

The less-frequent application of mass fluxes on the isochronal grid causes small discrepancies in the total ice thickness. This effect is very small but with a tendency toward either consistently positive or negative anomalies in certain regions, so it may accumulate over time. To counteract drifting and ensure continuous consistency between Yelmo and the simulated isochrones, we finalize the time step with a normalization to the Yelmo ice thickness.

An ensemble of 300 simulations were performed to investigate the impact of model and experimental choices on the simulation of isochronal layers. Each simulation consists of the spin-up and cb optimization procedure described above, followed by simulations of the ice sheet from 160 kyr ago to PD with fully transient boundary conditions. Four model parameters – the PDD factor for snow βs, the geothermal heat flux Qgeo, and the flow enhancement factors for shearing Eshr and glacial ice Eglac – and two boundary-forcing parameters – the Holocene precipitation anomaly ΔPhol and the glacial precipitation anomaly fLGM – were perturbed using a Latin hypercube sampling approach.

The simulated ice thickness and isochrone depths are in qualitatively good agreement with observations in both BESTice and BESTall, but closer inspection reveals important differences. BESTice agrees with the observations of ice thickness within a few tens of meters but also shows large disagreements in the depth of the isochronal surfaces (Fig. ). The contrary is true for BESTall (Fig.  ). The anomaly pattern of ice thickness of both simulations shows the expected deviations near the ice margin and a mostly homogeneous pattern in the ice sheet interior which do not point to any systematic bias. The anomalies in isochrone depths do show a more detailed pattern of positive and negative anomalies in BESTall. The positive anomaly in the northwest is visible in all four isochrones and may indicate too much accumulation in the simulation, a too rapid ice flow in this region that includes the ice divide, or both. Similarly, the negative anomaly in the northeast may be interpreted as too little accumulation or a dynamic bias in the model potentially associated with the Northeast Greenland Ice Stream.

A section through the summit confirms that BESTice more closely captures the correct ice thickness but that the englacial stratigraphy is in worse agreement with observations than in BESTall (Fig. ). However, looking at the absolute elevation of the isochrones instead of their depth below the surface, it is clear that the relatively good agreement in isochrone depth in BESTall is largely due to an underestimated ice thickness. This is an interesting result because it suggests that future improvements should focus on the early part of the simulation. We also observe that despite these shortcomings, BESTall is among the best simulations in simulating the total ice thickness, while BESTice shows a low skill in the combined RMSE (Fig. ). Similarly, the four simulations with the best results for the individual isochrones (indeed four different ones) have reasonably low values for the RMSE of total ice thickness. In contrast to this, the simulation with the best ice thickness performs poorly in all isochrone depths.

Not all of the six tuning parameters contribute equally to the ensemble spread (Fig. ). The Holocene precipitation anomaly ΔPhol appears to have the largest impact in terms of RMSE for all evaluation surfaces, followed by the scaling factor of the glacial precipitation anomaly fLGM. The optimal ice thickness is simulated with relatively high values for ΔPhol, but high Holocene precipitation deteriorates the simulated depth of the isochrones. This appears to be the primary reason behind the large differences between BESTice and BESTall. Lower values of ΔPhol also worsen the RMSE of isochrone depth, so the optimal value overall is close to 0 myr-1. All isochrones respond similarly to ΔPhol because the anomalous precipitation increases their depth similarly. The clear minimum of RMSE in the middle of the parameter range suggests that it was chosen appropriately, although the use of a constant offset for all locations has its shortcomings as will be discussed below.

The scaling of the glacial precipitation uncertainty fLGM has the strongest impact on the 57 and 115 ka isochrone depths in that both benefit from lower values. The same tendency, albeit weaker, is visible for the 29 ka isochrone depth. The reduced importance is probably due to the relatively short period during which the anomalous precipitation can impact this isochrone. Glacial precipitation has only negligible influence on the 11.7 ka isochrone depth and the total ice thickness. This is to be expected because glacial ice makes up a rather small portion of the total ice thickness.

The influence of the PDD melt factor for snow βs is mostly limited to the 115 ka isochrone because the preceding period, the last interglacial, is the only time when significant melting occurs in the region above 1000 m. The geothermal heat flux Qgeo and the two enhancement factors Eshr and Eglac have only minor effects. One possible explanation is that these three parameters are most important for ice dynamics that primarily take place near the bed and the ice margins. The deliberate exclusion of the margins but more importantly the very limited coverage of observational data for the deeper isochrones may bias the results to regions where dynamic thinning is least pronounced. In addition, the basal friction optimization may counteract the effects of changing the dynamic parameters because it is carried out for every combination individually. Lastly, the relatively strong impact of the precipitation parameters ΔPhol and fLGM can potentially dominate and therefore mask the effects of the other parameters.

The relatively low sensitivity of some of the parameters and the large scatter around the trend lines mean that several simulations across the parameter range produce RMSEs that are comparable to BESTice and BESTall. As a corollary to this finding, although these two sets of parameters produce the optimal results in our ensemble, the parameters themselves are not incontrovertibly optimal (Fig. ). The best 10 % of simulations agree well on ΔPhol and fLGM but otherwise span a large part of the tested parameters range.

Additional information can be obtained from the vertical distance between isochrones, not just their depth below the surface (Fig. ). ΔPhol does not have a direct effect on these differences, and so they allow inferences on the dynamic response. A higher Holocene precipitation generally has a slight positive effect, in particular on the deepest isochrones, suggesting that they benefit from stronger dynamic thinning. Interestingly, all isochrones agree that a high value for ΔPhol would be detrimental to their simulated depth below the surface (Fig. ). We therefore conclude that the imperfect thickness of the layer between the 115 and 57 ka isochrones is due to an unrelated model bias such as an unrealistic surface mass balance (SMB) during that time interval or shortcomings in the simulated dynamics. ΔPhol may partially counteract these by enhancing dynamic thinning but at the risk of inadvertently combining two errors into one seemingly correct outcome. A similar conclusion can be drawn from the decrease in RMSE of the 115–57 ka difference with higher fLGM. However, the thickness of the 29–11.7 ka layer, which is directly impacted by the change in glacial precipitation, clearly favors a small value for this parameter.

The RMSE does not provide information on the sign or the spatial pattern of the disagreements. To address this, we also analyze composites for how the evaluation surfaces depend on the high and low values of ΔPhol and fLGM. A full discussion is found in Appendix , but we summarize the main findings here. Even precipitation rates near the upper end of the tested range are barely enough to achieve a sufficiently thick ice sheet. The depth of the internal layers clearly shows the effect of this direct thickening for both ΔPhol and fLGM, as well as the dynamic thinning of additional accumulation. The thinning effect is secondary and therefore only visible in layers that do not receive more mass. This also applies to layers that accumulate after the precipitation anomaly, as is clearly seen in the notable shallowing of the 11.7 ka isochrone with high values of fLGM. The high accumulation before 11.7 ka creates a thicker ice sheet with steeper surface gradients. We find that the precipitation anomalies that result from ΔPhol and fLGM are not well suited to improve upon the data–model mismatch found in the ensemble average. However, comparison with reconstructed data may be a way of addressing this issue in future studies.

While most of the above analysis concerned a spatially comprehensive view of only four isochronal surfaces, the high resolution of our model in the temporal domain allows for a different and complementary analysis using the full age profile at certain locations to constrain model performance (Fig. ).

All locations show progressive thinning toward the bed, which is more pronounced for regions of high accumulation, as expected. Ice is too young in most of the ensemble simulations probably due to excessive accumulation, in particular at the more southerly locations of Dye-3 and Summit. A less severe mismatch is found for NGRIP, and the simulations agree much better with the reconstructed age profile at NEEM and EGRIP. The most plausible explanation for the difference in simulation quality between north and south is deficiencies in the surface mass balance due to the uncertain climate of the past or the relatively simple parameterization of mass balance used here. However, it is conceivable that the lack of observational isochrone data in the south accentuates these issues or adds to them. For example, the better coverage of radiostratigraphy data in the north biases the PDD melt factor βs toward this region and its climatic conditions, which likely is a poor choice for the climatically very different south. A similar effect is possible for the parameters that control ice dynamics that may be different for the fast-flowing ice of the south.

BESTice has much younger ice everywhere in the ice column due to its unrealistically high accumulation during the Holocene and glacial period. BESTall achieves a good simulation of the age profiles at NEEM and EGRIP, including periods of rapid age increase around 29 ka and between 60 and 70 ka that agree well with the radiostratigraphy data at the EGRIP site and less so at NEEM.

Comparison of the Eulerian age tracer with the age profile simulated by the isochronal tracing scheme in the same simulation BESTall (Fig. , dashed and solid blue, respectively) shows a clear disagreement, although the two diagnostics should ideally be identical because they simulate the same variable in the same simulation using two different methods. We observe that the Eulerian tracer data generally show a weaker curvature and almost invariably older ages, and they do not capture the variations in accumulation on shorter timescales, all of which indicate that it is subject to numerical diffusion in the vertical dimension. In the Eulerian age tracer old ages diffuse upward. The isochronal layer tracing scheme circumvents this problem by eliminating flow across layer boundaries and by using a numerically much less complex implementation that only requires advection in the two horizontal dimensions.

The spurious bias toward older ice with the Eulerian scheme has a noticeable effect on model behavior when the calibration is based on its results (Fig. , orange). Here we show the ages calculated with the isochronal scheme but derived from output of a simulation using parameters chosen to optimize the quality of fit for isochrones generated with the Eulerian age tracer. Note that this is a different simulation from the one shown in blue (BESTall) which uses parameters that optimize the fit for isochrones generated by the isochronal scheme. To counteract the upward diffusion of older ages, higher values for Holocene precipitation, ΔPhol, may seem advantageous (Fig. , orange). Following the same explanation, less glacial precipitation and so lower values of fLGM are necessary to obtain a good match with the reconstructed stratigraphy. Consequently, the simulation with the lowest RMSE based on the Eulerian age tracer produces an ice sheet whose true ages, according to the more accurate isochronal scheme, are younger than observations (Fig. , orange). The other ensemble parameters are less well constrained both for the isochronal and the Eulerian age schemes, and differences are negligible.

The tendency toward higher Holocene and lower glacial precipitation in the Eulerian-calibrated simulation is also evident in the total simulated ice volume (Fig. ). Simulations BESTice and BESTall show a similar ice volume for most of the simulation period, with notable exceptions in the relatively warm Holocene and Eemian. This suggests that the type of calibration has a significant impact on model sensitivity. The simulated ice volume of all simulations is low when compared to earlier reconstructions e.g.,. This is the result of relatively low basal friction in the marine sectors surrounding Greenland and the use of a hybrid ice sheet model, which leads to thinner grounded ice on the continental shelf. Previous ice volume estimates were based on SIA models that generally simulate thicker ice when constrained with the same ice extent. More recent simulations using hybrid models have shown lower volumes on the order of those shown here to be equally possible . Regardless of this open question, the ice thickness on the continental shelf is of negligible consequence for our comparison with the observed isochrones because they are far inland and not directly influenced by the remote ice margin during the glacial period.