Recent observations provide evidence that the rate of mass loss of the Greenland ice sheet (GrIS) is continuously increasing . Simulating the GrIS response under future warm periods is therefore crucial to establish reliable projections of future sea-level rise at decade to century timescales but also to investigate the effects of ice sheet changes on the climate system . As a result, better constraining the GrIS evolution has become a key objective of the climate and ice sheet modelling communities.

Reliable simulations of the GrIS require a proper ice sheet model initialisation procedure to avoid an unphysical model drift which can be caused by inconsistencies between the initial conditions of the ice sheet model and the boundary conditions (external forcing fields). These initialisation procedures consist of finding the initial physical state of the ice sheet (such as the internal temperature), the model parameters and sometimes the boundary conditions that best reproduce the observations with a minimal model drift. Recent observations, such as surface and bedrock topographies and horizontal surface velocity offer only a partial description of the GrIS current state and a major source of uncertainty lies in the poor knowledge of the basal properties (e.g. water content in the sediment or basal dragging) and of the internal thermo-mechanical conditions (e.g. temperature and deformation profile). Indeed, both the basal properties and the internal conditions have a strong impact on the ice motion and thus on the simulated GrIS state . Optimising the initialisation procedure of ice sheet models is therefore an active area of research and a multidisciplinary effort. The ice sheet model initialisation experiments intercomparison project (initMIP) gives a recent example of this effort. Its goal is to compare different initialisation techniques and to assess their impact on the dynamic responses of the models.

The goal of ice sheet model initialisation is to infer internal properties (e.g. temperature), some boundary conditions (e.g. basal drag) and model parameter values. To this aim, different techniques have been developed. One approach is to allow the ice sheet model to evolve freely over a long enough time (ice sheet spin-up). This approach has long been the most commonly used technique to initialise ice sheet models (, and other references in ). It consists of simulating the ice sheet during one or more glacial–interglacial cycles to account for the long-term ice sheet history and thereby to obtain internal consistency between the simulated ice sheet and the climate forcing evolution derived from ice core records. Even if model parameters can be chosen to reduce the mismatch between modelled and observed present-day ice sheet state (e.g. topography, velocity), this approach may lead to important errors. In addition, due to the long integrations needed (>10000–100 000-year long), such spin-up methods can only be used with low computational-cost models, which are often unable to properly capture fast ice flow processes. To compute the internal properties, an alternative approach is to keep the topography fixed, while vertical temperature fields, and possibly velocity fields, are allowed to freely evolve e.g.. In this case, because the simulated ice flux divergence is generally far from being balanced by the net mass balance (i.e. surface and basal mass balance), an artificial drift arises when free evolving topography is restored .

A second category of initialisation methods relies on data assimilation techniques, whose goal is to infer model parameters or poorly known boundary conditions and which are also used to minimise the mismatch between model variables (most often surface velocities) and observations e.g.. However, this approach may lead to internal inconsistencies between the simulated internal conditions (temperature and velocities) or between the simulated ice velocities and the observational datasets, such as surface and bedrock topography. These inconsistencies may have a strong impact on the results of forward simulations. To circumvent this drawback, other authors e.g. developed a multi-parameter inversion technique to optimise both the sliding velocities and the bedrock topography in such a way that the modelled surface ice velocities match with the observed ones. This allows the set of initial conditions to be self-consistent. However, if not constrained by observed ice thickness, these methods may lead to unrealistic simulated topography. An alternative approach, which avoids the previously mentioned shortcomings, consists of considering only the observed ice sheet geometry as the final target by finding appropriate basal conditions (generally the basal drag coefficient; see Sect. ) that minimise the differences between observed and simulated ice thickness . However, methods that choose to invert the basal drag coefficient only are not able to correct ice thickness errors in regions where there is no sliding (i.e. where the bed is frozen). Moreover, while inverse methods are designed to produce an ice sheet state close to observations, the inferred basal drag coefficient may cancel errors coming from erroneous simulated basal temperatures and/or model physics shortcomings. Yet, as outlined by , the risk of cancelling errors is of less importance compared to those related to inconsistencies between internal conditions and surface properties that will likely to be considerably reduced with expected future improvements in ice sheet models and better observations of basal conditions.

Here, we present a new iterative initialisation procedure that relies on the same basic principles as those developed by (referred to as PDC12 in the following) and applied by for the Antarctic ice sheet using linear and non-linear sliding laws. Similarly to PDC12, we compute the basal drag coefficient that minimises the error in the simulated ice thickness and relates basal stresses to basal velocities. However, while PDC12 requires long (multi-millennial) integrations for the method to converge, we suggest instead an iterative method of short (decadal to centennial) integrations starting from the observed ice thickness. Our iterative method ensures a more rapid convergence and is thus suitable for computationally expensive models.

The paper is organised as follows. In Sect.  we present the main characteristics of the ice sheet model used in this study. Section  describes the iterative minimisation procedure in detail. The main results are presented in Sect.  and sensitivity experiments in Sect. . These sections are followed by a discussion and the main conclusions of the present study (Sect. ).

The GRISLI (GRenoble Ice Shelf and Land Ice) ice sheet model was first designed to describe the Antarctic ice sheet and further adapted to the Northern Hemisphere ice sheets . The version used in this study has been specifically developed for Greenland with a horizontal resolution of 5 km ×5 km (301×561 grid points) and 21 evenly spaced vertical levels. GRISLI accounts for the coupled behaviour of temperature and velocity fields. It relies on basic principles of mass, heat and momentum conservation. The evolution of ice sheet geometry is a function of surface mass balance, ice dynamics and bedrock altitude. Since this study only deals with present-day steady-state simulations, the module describing the isostatic adjustment is not activated here. The evolution of the ice thickness is governed by the mass balance equation: 1∂H∂t=-∇(UG‾H)+SMB-bmelt, where H is the ice thickness, UG‾ is the depth-averaged velocity (2-D vector), SMB is the surface mass balance and bmelt is the basal melting.

The ice flow velocity is derived from a simplified formulation of the Stokes equations (i.e. the stress balance) using the shallow-ice and shallow-shelf approximations. The shallow-ice approximation (SIA) assumes that, owing to the small ratio of vertical to horizontal dimensions of the ice sheet, longitudinal stresses can be neglected with respect to vertical shearing along the steepest slope. Conversely, in the shallow-shelf approximation (SSA), the horizontal strain rates become dominant and the horizontal velocities do not vary with depth. In the model, the velocities are computed as the heuristic sum of the SSA and the SIA components, as in but with a no-weighting function . In this case, the SSA velocity is used as the sliding velocity. We assume no-slip conditions for a frozen bed (i.e. basal temperature below the melting point), and in these conditions, the SSA velocity is set to 0. In the model version used in this study, we assume a linear viscous till with a uniform thickness, in which the basal shear stress (τb) and basal velocity (ub) are related via the following expression: 2τb=-βub, where β is the basal drag coefficient and varies with space.

To describe the effect of ice rheology, the deformation rate and stresses are related via Glen's flow law . As in other large-scale ice sheet models, GRISLI uses a flow enhancement factor (Ef) in Glen's flow law to artificially account for the impact of ice anisotropy on the deformation rate. This enhancement factor depends on the stress regime e.g.. Lower enhancement factors lead to lower deformation rates and as such to slower ice velocities. The grounding line position is defined according to a flotation criterion and floating points are treated following the SSA only. Calving physics is not explicitly computed, but if a grid point at the ice-shelf front fails at maintaining a thickness threshold, it is automatically calved . The ice thickness cut-off threshold is set to 250 m.

Since GRISLI is thermo-mechanically coupled, the ice temperature influences the ice velocity via the viscosity. The temperature is computed both in the ice and in the bedrock by solving a time-dependent heat equation. The temperature signal itself depends on ice deformation, surface temperature forcing and geothermal heat flux.

The basic principle of inverse modelling approaches for the ice sheet initialisation procedure is to adjust the basal drag coefficient (β) which varies spatially, in order to reduce the mismatch between the simulated surface ice velocities and/or the ice sheet geometry and the observed ones.

While numerous studies are based on fitting the modelled ice velocities e.g. or both surface velocities and basal topography , only few authors have opted for fitting ice surface elevation . Here, we decided to adjust the basal sliding velocities via the adjustment of the β coefficient to fit the GrIS thickness to the observed one. Similarly to , our choice is motivated by the need to refine the estimates of GrIS contribution to future sea-level rise without the sea-level rise signal being contaminated by unphysical transients from the initial condition. However, while adopted a formal minimisation approach (i.e. adjoint-based model), we suggest instead an ad hoc method potentially applicable to any ice sheet model.

The GRISLI climate forcing, i.e. surface mass balance and surface air temperature (Fig. ), is provided by the regional atmospheric model MAR forced at its boundary by the ERA-Interim reanalyses . Both forcing fields are averaged over the 1979–2005 period (Fig. a and b). They are interpolated on the GRISLI grid (5 km ×5 km) and corrected for surface elevation differences between MAR and GRISLI by applying the method developed by . For the geothermal heat flux we use the data generated for the SeaRISE (Sea-level Response to Ice Sheet Evolution) project . The initial geometry consists of the present-day observed ice thickness and bedrock elevation taken from . To compute initial conditions consistent with the boundary conditions, we run a 30 000-year integration of the model imposing a fixed topography. For this long experiment, similar to the fixed topography spin-up method, we assumed a perpetual present-day climate forcing (Fig. a and b) and we used a basal drag coefficient (Fig. a) coming from a previous simulation carried out within the Ice2Sea project . The resulting basal temperature after this long integration, presented as a difference with respect to the pressure melting point, is shown in Fig. c. It shows areas with temperature largely below the pressure melting point, associated with frozen bed, and areas with temperature at the pressure melting point (red colours), associated with thawed bed. Compared to the recent synthesis of GrIS basal temperatures (see Fig. 11 in ), our initial basal temperature generally agrees well with the reconstructions in the north-western and north-eastern parts of the GrIS but is probably overestimated, with too large a thawed bed area, in the eastern and central parts of the GrIS (not shown). The impact of ice temperature on the minimisation procedure is discussed in Sect. .

In order to avoid inconsistencies between the different datasets used as boundary and initial conditions, GRISLI is first run forward (free-evolving surface elevation and temperature) for 5 years (relaxation step, blue box in Fig. ). After this short relaxation period, we start the iterative minimisation procedure (red box in Fig. ). This procedure is based on an iterative process set up to adjust the basal drag coefficient in such a way that the mismatch between observed and simulated ice thickness is reduced. Instead of optimising the basal drag coefficient every 5000 years as in PDC12, here the optimisation is done at every time step (which is set to 1 year for the present study), using an ice thickness ratio to correct the simulated sliding velocity with the help of a modification of the basal drag coefficient.

The iterative minimisation procedure itself consists of repeated cycles, each cycle being divided into two main steps (red box in Fig. ): first step:

The first step consists of a free-evolving simulation (thickness and temperature) during which we adjust, at each model time step, the basal drag coefficient so that the ice thickness difference with respect to the observations becomes minimal. To this end, from the simulated vertically averaged velocity (UG‾) computed from the previous time step (or from the values obtained after the relaxation for the first iteration), we compute a corrected vertically averaged velocity field (Ucorr‾) as a function of the computed (HG) and observed ice thickness (Hobs): 3Ucorr‾=UG‾×HGHobs. As seen before (Sect. ), the mean velocity field UG‾ is the sum of two velocity components: the sliding velocity Usli and the vertically averaged velocity Udef‾ due to vertical ice deformation: 4UG‾=Usli+Udef‾. Assuming that the differences between UG‾, the simulated vertically averaged velocity field, and Ucorr‾, the idealised vertically averaged velocity field, are only due to changes in the sliding velocity Usli, we can write 5Ucorr‾=Ucorrsli+Udef‾. Following Eqs. () and (), we can deduce the corrected sliding velocity (Ucorrsli): 6Ucorrsli=Ucorr‾-UG‾+Usli. Ucorrsli represents the corrected sliding velocity whose difference with Usli indicates how the simulated sliding velocity must change to reduce the mismatch between HG and Hobs.

As such, we use the ratio between the simulated and the corrected sliding velocities UsliUcorrsli to compute a new basal drag coefficient (βnew). This results in slowing down or speeding up the simulated sliding velocity and acts to reduce the gap between HG and Hobs: 7βnew=βold×UsliUcorrsli. Equation () is essentially identical to what is done in except that they use observed and modelled velocities rather than observed and modelled ice thickness to adjust the basal drag coefficient. It should be noted that Ucorrsli can be lower or equal to 0, leading to infinite or negative basal drag coefficient. This can happen when the velocity due to vertical shearing Udef‾ is greater or equal to Ucorr‾. In this case we artificially impose a no-slip condition by assigning to the basal drag coefficient a maximum value set to 5×105 Pa yr m-1. On the other hand, in case of too small a Udef‾ velocity, β may be as low as 1 Pa yr m-1 to facilitate ice sliding. Owing to its design, the method is only able to correct for the ice thickness mismatch where sliding occurs, i.e. where the base of the ice sheet is at the pressure melting point. Throughout this step, the basal drag coefficient is updated at each time step for each model grid point. In the following, the duration of this step is referred to as Nbinv and has a typical value of a few decades.

Note that, using Eqs. () and (), we can show that Eq. () can be rewritten as 8βoldβnew=rH+Udef‾Usli(rH-1)whererH=HGHobs. As such, the adjustment of the basal drag coefficient is stronger in regions dominated by ice deformation.

In summary, our iterative minimisation procedure consists of

i.

adjustment of the basal drag coefficient at each time step (each year) for Nbinv years (first step, Eqs.  to ).

In addition, to assess the performance of the minimisation procedure (i.e. the quality of the inferred basal drag coefficient), we compute some quality metrics at the end of each cycle (green box in Fig. ). The metrics are computed at the year 200 of the free-evolving simulation of the second step, independently of its duration (i.e. Nbfree). If Nbfree is shorter than 200 years, we simply extend the simulation for 200 years. The quality metrics discussed in Sect.  include in particular the root mean square error (RMSE) of the simulated ice thickness with respect to the observations and the drift in geometry (integrated ice thickness changes). These metrics help to decide whether an additional cycle is required or not. In the following, we also discuss the spatial patterns of ice thickness and ice velocity mismatches with respect to observations. Our method does not use the observed surface velocity as a constraint. However, at the end of the minimisation procedure (e.g. minimal thickness error and minimal drift), the simulated velocity tends nonetheless to approximate the balance velocity, that is the depth-averaged velocity required to maintain the steady-state of the ice sheet.

Once the optimal basal drag coefficient is found, it can be used to run prognostic forward simulations such as in and .

In order to improve the reliability of Greenland ice sheet simulations under a future transient climate, an accurate evaluation of the present-day trend of ice flow dynamics is required. One of the major difficulties in addressing this need lies in the poorly constrained observational data of the basal conditions that strongly control the ice motion in the entire ice sheet. Here, we present an inverse method to infer the spatial distribution of the basal drag coefficient in such a way that the mismatch between simulated and observed GrIS thickness is minimised. As such, our target criteria are defined for the sets of minimisation procedure parameters providing minimum values of ice thickness RMSE (with respect to observations) and ice thickness trend, which are respectively as low as ∼50 m and 15 cm yr-1 for our best fit. This remains in the range of PDC12 results. The great advantage of the method is its rapid convergence (i.e. 1320 years) making it suitable for more computationally expensive models. Moreover, we have also shown that it only weakly depends on the initial guess of the spatial distribution of the basal drag coefficient and the initial temperature profile.

Our method, based on the adjustment of the basal sliding, cannot be applied in regions of frozen bed and is only effective in thawed bed areas where basal sliding may occur. However, in case of too large a deformation rate in these regions, the basal drag coefficient is set to its maximum value to counteract the overly fast ice flow. The limit of applicability of the method led us to investigate the impact of the enhancement factor, which is expected to have a large influence on the deformation rate and, thus, on the ice flow and subsequently on the simulated ice thickness. We performed a series of simulations with a range of various values of the enhancement factor (from 0.5 to 5) and showed that the mismatch between the simulated and the observed GrIS topography is reduced with an appropriate tuning of the enhancement factor. This highlights that the overall performance of the method is critically dependent on the basal thermal state and highlights that the finding of appropriate initial conditions with a simple adjustment procedure remains an undetermined issue. Actually, multiple combinations of the enhancement factor and the basal drag coefficient can produce a simulated ice thickness close to the observed one, but this cannot discard the possibility of errors in modelled basal and vertical temperatures. A logical next step could lie in the adjustment of the basal drag coefficient combined with a similar approach for the adjustment of the enhancement factor in frozen bed areas. However, we have shown that the minimisation procedure presented in this paper is able to reduce the ice thickness mismatch regardless of the initial temperature profile. This offers the possibility to tune the thermal state to be as close as possible to the observations (inferred basal temperature as in or vertical profiles at ice core locations) before running the iterative minimisation procedure. Increasing our confidence in the vertical temperature profile would therefore increase our confidence in the choice of Ef and β values.

Finally, we have shown in this paper that the iterative adjustment of β produces modelled surface velocities that compare well with the observed ones. This suggests that future work could include an additional metric related to surface ice velocities so as to further reduce the uncertainties associated with the choice of model parameters and variables.

Another limitation of the method may come from the model resolution. The succession of higher/lower ice thickness due to the succession of valleys/ridges in mountain areas may be poorly resolved. Owing to the insulation effect of the ice, this may lead to an erroneous representation of the basal temperature patterns, and SSA regions may be erroneously interpreted as frozen bed regions and vice versa . This drawback is clearly illustrated in our study in Fig.  (Ef =1). Indeed, the simulated ice thickness obtained with the inversion procedure is generally less than 50 m in most GrIS areas but can be greater than several hundred metres in coastal mountain ranges such the central eastern margin area where ice flow occurs in deep valleys. An alternative solution consists of correcting the basal temperature to account for bedrock roughness, similarly to what was done in PDC12 to improve their inversion procedure in the Transantarctic Mountains. On the other hand, higher-resolution models can also better account for the dynamics of small-scale outlet glaciers and for their interactions with floating ice that strongly influence the ice sheet mass balance e.g.. However, due to the elliptic character of the SSA equation e.g., the local adjustment of the basal drag coefficient has an impact on the ice velocity of neighbouring points. As a result, increased resolution may increase the noise, unless a smoothing function that filters the high-frequency noise is introduced .

The reliability of the method also depends on the quality of observation data and of climate forcing. Errors in observed surface or bedrock topography or in SMB patterns different from those associated with the observed ice thickness would give rise to errors in the present-day estimated ice thickness and thus to an erroneous choice of the best spin-up parameters. In the same way, large uncertainties remain in the reconstruction of the geothermal heat flux that strongly impacts the basal temperature. Finally, we would like to stress that in our simulations, the spatial distribution of the basal drag coefficient does not change through time. However, changes in basal hydrological conditions along with changes in ice surface elevation and ice extent are likely to occur in a changing climate. While a constant spatial distribution of the β coefficient may seem reasonable for short-term projections, it is more questionable at the century timescale, and future modelling efforts should therefore be undertaken to compute interactively the basal drag coefficient as a function of changes in basal conditions.