The gravity-driven free surface flow of ice is solved using the finite-element ice flow model Elmer/Ice .

For the force balance, we solve the shelfy stream approximation (SSA) equation in one horizontal dimension. This is a vertically integrated model that derives from the Stokes equations for small aspect ratio and basal friction. In 1D, this leads to the following non-linear partial differential equation for the horizontal velocity field u: 1∂∂x4η‾H∂u∂x-τb=ρigH∂zs∂x, with ρi the ice density, g the gravity norm, H=zs–zb the ice thickness, and zs and zb the top and bottom surface elevations, respectively. Using Glen's constitutive flow law, the vertically averaged effective viscosity η‾ is given by 2η‾=1H∫zbzs12A-1/nDe(1-n)/ndz, where De is the second invariant of the strain-rate tensor, here equal to De2=(∂u/∂x)2, A is the rate factor and n is the creep exponent, taken equal to the usual value n=3 in the following. The basal friction τb is null under floating ice and is represented by the non-linear Weertman friction law for grounded ice: 3τb=Cum, with C and m the friction coefficient and exponent, respectively. In the following, we use the classical power law with m=1/n=1/3. When in contact with the ocean, the ice is assumed to be in hydrostatic equilibrium. The floating condition is evaluated directly at the integration points, and τb in Eq. () is set to 0 wherever ice is floating .

The time dependency is introduced by the evolution of the top and bottom free surfaces. Because of the hydrostatic equilibrium, the ice sheet topography is fully defined by the bed elevation b and only one prognostic variable. Equation () is then coupled with the vertically integrated mass conservation equation for the evolution of the ice thickness H: 4∂H∂t+∂(uH)∂x=as-ab, with as the surface accumulation rate and ab the basal melt rate. The free surfaces zs and zb are obtained from the floating condition which, for zs, using a constant sea level zsl=0, gives 5zs=b+HforH≥-bρwρizs=H1-ρiρwotherwise with ρw the sea water density.

We start by building an initial steady marine ice sheet. The domain extends from x=0km, where we apply a symmetry condition, u=0 in Eq. (), to x=800km where we have a fixed calving front. We use 1D linear elements with a uniform mesh resolution of 200m, leading to 4001 mesh nodes.

Following , we generate a synthetic bed geometry that reproduces a typical large-scale overdeepening with some small-scale roughness. The bed b=btrend+br is the sum of a general trend btrend defined as 20btrend=-1100+xfor ×≤450km-650-5(x-450)for ×>450km, and a roughness signal br that is computed at 200m resolution using a random midpoint displacement method . This is a classical algorithm for artificial landscape generation. In 1D, the algorithm recursively subdivides a segment, and a random value drawn from a normal distribution N(0,σ2) is added to the elevation of the midpoint. The standard deviation σ is decreased by a factor of 2h between two recursions. Here we have used 12 recursions using an initial standard deviation σ=500m and a roughness h=0.7. The resulting bed is shown in Fig. .

For the basal friction, we use a synthetic sinusoidal function with two wavelengths for C (MPam-1/3a1/3) 21C=0.020+0.015sin(52πxL)sin(1002πxL), with L=800km (Fig. ).

While not tuned to match any specific glacier, this synthetic design compares relatively well to the conditions found in Thwaites Glacier (Antarctica). Thwaites has been the focus of many recent studies as it is undergoing rapid ice loss and, connected to deep marine-based basins, its retreat could trigger a large-scale collapse of the West Antarctic Ice Sheet over the next centuries . In Fig. , C and b are compared with model results from along three streamlines. In , C has been inferred from the observed surface velocities using a time-independent control inverse method and a SSA model. We can see that our synthetic design is realistic in terms of both amplitude and spatial variations. As the other characteristics (geometry, small flow divergence and convergence) are also similar, the model velocities have a good order of magnitude.

Using a uniform ice rigidity B=(2A)-1/n=0.4MPaa1/3, we grow an ice sheet to steady-state using a uniform surface accumulation as=0.5ma-1 and no basal melting ab=0. The steady state GL is located at x=440km, just downstream of the region of overdeepening (Fig. ).

In , observed ice flow accelerations in the Amundsen Sea sector have been attributed to the decadal oceanic variability, where warm phases associated with increased basal melt induce a thinning of the ice shelves, reducing their buttressing effect and initiating short-lived periods of unstable retreat of the most vulnerable GLs. In a flow line experiment the ice shelf does not exert any buttressing effect. Using a suite of melting and calving perturbation experiments for Pine Island Glacier, have shown that, when initiated, the dynamics of the unstable retreat are fairly independent of the type and magnitude of the perturbation. Here, to trigger the initial acceleration, we instantaneously decrease the ice rigidity to B=0.3MPaa1/3 at t=0, keeping all the other parameters constant.

This initial perturbation induces an acceleration, a thinning and a retreat of the GL. The model is then run for 200 years with a time step dt=510-3a-1. After a short stabilisation at x=437.2km between t=13a and t=32a, the GL retreats at a rate of approximately 1kma-1 during the following 100 years, and then the rate decreases as the GL enters an area of down-slopping bed (Fig. ). The retreat rate shows small variations associated with spatial variations in the topography and basal friction.

From the reference run, we generate synthetic noisy observations that are typical of the resolution and performance of actual observing systems.

For the bed, we mimic an airborne radar survey conducted perpendicular to the ice flow with an along-flow resolution of approximately 15km. For this, we randomly select 54 locations between x=0 and x=800km and then linearly interpolate the true bed and add a random uncorrelated Gaussian noise with a standard deviation σbobs=20m (Fig. ).

We assume that the surface elevation and velocities are observed at an annual resolution at each mesh node. We then add an uncorrelated Gaussian noise with a standard deviation σzsobs=10m for the surface elevation and σuobs=20ma-1 for the velocity. The most recent velocity products are now posted with a monthly to annual resolution . The reported uncertainty for individual velocity estimates using the 6 and 12 d image pairs from the Sentinel-1A/B satellites is 6.2 and 17.5ma-1 for the two horizontal velocity components in stable conditions; however this could be underestimated in the coastal areas. For the surface elevation, the spatial and temporal resolution as well as the coverage and uncertainty will depend on the sensors. The ArcticDEM (http://arcticdem.org, last access: 31 January 2019) is a collection of openly available digital surface models derived from satellite imagery and posted at 2 m spatial resolution. After co-registration, a standard deviation ranging from 2 to 4m has been reported for the uncertainty of elevation difference between two individual models of static surfaces . Using the same satellites, Greenland digital elevation models are now posted with a 3-month temporal resolution (https://nsidc.org/data/nsidc-0715, last access: 28 August 2019).

We recall that our aim is to initialise the model using the DA framework to estimate the state together with the basal conditions. As a simplification to realistic experiments, we assume in the following that the ice rheological properties (represented by the Glen flow law and its parameters) and the forcing (represented by the surface and basal mass balances in Eq. ) are perfectly known. In addition, we assume that the form of the basal friction follows Eq. () with m=1/3, so that only the spatially varying friction coefficient C is uncertain.

In our model, as the force balance Eq. () contains no time derivative, the velocity is a diagnostic variable. Because of the flotation condition, the topography can be represented by only one prognostic variable. The state vector x is then given by the free surface elevation zs at every mesh node, and we use the floatation Eq. () for the mapping between the ice thickness H and zs. The state vector is augmented by the two parameters to be estimated, the bedrock topography b and the basal friction coefficient C. For the parameters we assume a persistence model, i.e. no time evolution, during the forecast step (Eq. ). Because the velocities are insensitive to the basal conditions where ice is floating, these two parameters are included in the state vector only for the nodes where at least one member is grounded. In addition, to insure that C remains positive, we use the following change of variable for the assimilation C=α2. Although it does not insure uniqueness of the estimation as α and -α would lead to the same C, this change of variable is classical and was chosen as the reference friction coefficient spans only 1 order of magnitude. Similar performances were found using the other classical change of variable C=10α as in .

Because both zs and b are included in the state vector, the analysis does not conserve the ice sheet volume, for either the ensemble mean or the individual members. However, as illustrated in Fig. , the estimation of zs and b, and thus of the ice thickness, should be improved at each analysis as more data are assimilated, and the final state is the best estimation provided by the filter knowing the model, all the observations during the assimilation window and their uncertainties. As mentioned in the introduction, if the main interest is an analysis of past volume changes, a smoother or a variational method might be more appropriate. The smoother extension of the ESTKF can be found in . Note however that, interestingly, if we expect that the filter will improve the estimation of the ice thickness, there is no guaranty in general that it will provide a better estimate of the total volume as an a priori state with a totally different thickness distribution could lead, by compensation of the errors, to a perfect estimate of the true volume.

Kalman-based filters are based on the hypothesis of the independence between the background, i.e. the initial ensemble, and the observations that are used during the assimilation. As the synthetic bed observations will be used to construct the initial ensemble (see next section), we assimilate only the surface elevation and velocity observations, every year from t=1a up to t=35a. The observation operator H is a simple mapping for the surface elevation and is given by the non-linear SSA equation (Eq. ) for the surface velocities.

Finally, to illustrate the effect of the transient assimilation on model projections on timescales relevant for sea level projections, the analysed states at t=20a and t=35a are used to run deterministic and ensemble forecasts up to t=200a. The deterministic forecast uses the ensemble mean produced by the analysis while the ensemble forecast propagates the full ensemble.

For atmosphere and ocean models, the initial state is usually sampled from a climatology, either observed or from a model run. This method can not be used for the parameters and the initial ensemble must reflect the background and the estimation of its uncertainty, available a priori before the assimilation. Following previous studies , we assume that the initial distributions for b and C are Gaussian with a given mean and a prescribed covariance model. Furthermore we assume no cross-correlation between the initial b, C and zs, and we draw the initial ensembles independently.

For b and C, the initial samples are drawn using the R package gstat . As is classical in geostatistics, the covariance model is prescribed using a variogram γ(d) that is half the variance of the difference between field values as a function of their separation d. It is usually defined by two parameters, the sill s that defines the semi-variance at large distances and the range ra which, for asymptotic functions, is defined as the distance where the γ(ra)=0.95s. The package gstat allows one to directly draw simulations, i.e. random realisations of the field, from the prescribed spatial moments .

For the bed we use an exponential function, 22γ(d)=s(1-e-3dra), with ra=50km and s=4000m2. We also add a nugget model defined by 23γ(d)=0d=0nugd>0, with nug=200m2. This model is meant to represent the bed measurement error. To draw the initial ensemble, the simulations are conditioned with the bed observations. This procedure gives an initial ensemble that is drawn from the posterior probability distribution that would be obtained using ordinary kriging with the same observations and variograms. The ensemble mean and spread for a 50-member ensemble are shown in Fig.  and the first three members are shown in Fig. . As expected, the ensemble spread increases with the distance from the observations. At the observation locations, the spread is controlled by the nugget. For the individual members, the nugget controls the small-scale variability, resulting in a roughness larger than the reference. When averaged this roughness disappears, and the ensemble mean has a much smoother topography.

For the friction coefficient, we assume that we know the mean value Cmean=0.020MPam-1/3a1/3 and draw unconditional simulations. For the spatial dependence, we use a Gaussian function γ(d)=s(1-e-3(dra)2) for the variogram using a range ra=2.5km and a sill s=8.10-5MPa2m-2/3a2/3. This results in initial ensemble members that have approximately the same maximal amplitude as the reference, as shown in Fig. .

For the free surface, we initialise all the members using the observed (noisy) free surface at t=0. Doing so, we implicitly assume that the spread of the ensemble induced by the uncertain initial conditions at the first analysis is small compared to the spread induced by the uncertain parameters. This is motivated by the fact that divergence anomalies induced by uncertainties in model parameters can typically reach tens to hundreds of metres per year in fast-flowing areas .

To assess the performance of the DA in retrieving the basal conditions, we compute the root-mean-square error (RMSE) between the analysed ensemble mean and the reference for both the bed and the friction coefficient, RMSEb and RMSEC. After each analysis, the RMSE is computed using all the nodes where the basal conditions have been updated by the assimilation, i.e. at least one member is grounded, and where x≥ 300km. The last value mentioned is close to the position reached by the grounding line after 200 years in the reference simulation; moreover, during the assimilation window, the reference velocity at this location is close to 80ma-1 (Fig. ), so that the relative noise is ∼25% and we do not expect too much improvement from the DA upstream as the velocity tends to 0.

Here the size of the state vector x, Nx, is approximately 8400, i.e. zs at every node and the basal conditions, b and C, in the grounded part. To test the performances of DA in conditions that would be numerically affordable for real applications, we run the assimilation with relatively small ensemble sizes Ne=30, Ne=50 and Ne=100. In this case, inflation and localisation are required to counteract the effects of undersampling and we test a range of forgetting factors ρ and localisation radius r. The errors obtained at t=20a relative to the errors from the initial ensemble mean are shown in Fig. . The performances of the assimilation for Ne=50 and Ne=100 are very similar. The filter diverges and produces errors larger than the initial errors for a localisation radius r≤4km. However, for larger localisation radii, the assimilation is relatively robust for a wide range of r and ρ, with errors reduced by ∼30% for b and ∼40% for C. Decreasing the ensemble sizes reduces the filter performance but there is still a reduction of the errors by ∼20% and ∼30%, respectively, with Ne=30. For the two smallest ensembles, there is an optimal value for r, and increasing r above this value decreases the filter performance. In general, this optimal value for r increases as ρ decreases, because the ensemble spread reduction induced by assimilating more observations is counterbalanced by the inflation.

In the sequel we discuss the results obtained with an ensemble size Ne=50. As a compromise between the performances in retrieving b and C, we choose a forgetting factor ρ=0.92 and a localisation radius r=8km. The evolution of the RMSEs as a function of assimilation time together with the initial and final ensembles are shown in Fig. . RMSEb decreases steadily from ∼25m for the initial ensemble at t=0 to ∼12m at t=35a. For the basal friction, RMSEC is decreased by a factor of 1.75 during the first 10 years; then there is still a slight but much smaller improvement as new observations are assimilated.

At the end of the assimilation, for both fields, the spatial variations are well reproduced by the ensemble mean, and, compared to the initial ensemble, the difference from the reference is decreased everywhere except between 300 and 325km for C. The reduction in the error is also accompanied by a diminution of the ensemble spread, represented by the minimum and maximum values in Fig. . This reduction is the most important just upstream of the grounding line where the relative noise for the velocity is the smallest. For the first 100km upstream of the grounding line, the ensemble standard deviation increases by a factor of 4, from approximately 4 to 17m for b and from 1×10-3 to 4×10-3MPam-1/3a1/3 for C. Downstream of the GL where all members are floating, the model is insensitive to the basal conditions and the initial ensemble is unchanged.

We expect that uncertainties in the ice sheet interior should not affect the short-term forecast of the coastal regions ; however for completeness we also show the results for the first 300km in Fig. . For the bed there is only a small improvement of the ensemble mean with an RMSE decreasing from 50 to 45m after 35 years. Because the relative observation error on the velocity is very high in the first kilometres, the reduction of the ensemble spread due to the assimilation of new observations is very small and eventually outperformed by the inflation, leading to an ensemble spread that becomes larger than before the assimilation. The model seems more sensitive to the basal friction and this effect is less pronounced for C with a continuous decrease in the RMSE and a small reduction of the ensemble spread everywhere.

Figure  shows that some members undergo a fast GL retreat of a few kilometres before assimilation at the end of the first year. Interestingly, as the assimilation updates both the thickness and the bed, it also corrects the GL position, which never departs by more than a few nodes from the reference for the rest of the assimilation period.

As in realistic simulations the true bed and friction are not available to assess the performance of the DA, we also look at the variables assimilated by the model. Figure  shows the RMSEs between the ensemble mean and the reference for the velocity u (RMSEu) and the free surface zs (RMSEzs), computed for the entire domain (0≤x≤800km). We also report the evolution of the ensemble spread, computed as the square root of the averaged ensemble variance. The velocities before and after the analysis at t=1 and t=35a are shown in Fig. . The RMSEs are largely decreased during the first few years, especially for the velocity with an error of more than 300ma-1 before the first assimilation to approximately the noise level 20ma-1, at t=20a. For zs, RMSEzs is already below the noise level before the first analysis and decreases relatively steadily to reach ∼2m after 35 years. RMSEu increases at the end of the period when the reference GL leaves the stable region. As can be shown in Fig. , the error is dominated by the larger difference over the ice shelf due to the few members that still have their GL at the stable location, largely affecting the ensemble mean.

In general, during the first and last years of the assimilation period, the error and the ensemble spread increase during the forecast step. The analysis step reduces both the error and the ensemble spread (Fig. ). With the stabilisation of the grounding line, both the error and the spread remain relatively stable during the forecast, and as RMSEu and RMSEzs have already reached levels comparable to the observation noise, there is no much improvement during the analysis. After a few assimilation steps, as expected for a reliable ensemble, the error and the spread have similar values.

Similar conclusions are drawn if the assimilation is pursued up to t=50a. Because of the sensitivity of the ice shelf velocities to the grounding line position, RMSEu shows a higher variability, but, with a few exceptions, stays close to the noise level. RMSEb and RMSEc stagnate as the reconstruction continually improves mostly in the first few tens of kilometres upstream of the GL where the relative noise on u is the smallest.

To assess the influence of the observation uncertainties in the performance of the DA, we repeat the experiment with the same localisation and inflation but different levels for the uncertainties on the observed surface velocity (σuobs) and surface elevation (σzsobs) (see Sect. ). We recall that these uncertainties are not correlated spatially and temporally. As shown in Fig. , the performance of the DA to retrieve both b and C increases when the uncertainty on the velocity observation σuobs decreases. However, when looking at the model velocities and surface elevation, this improvement is not significant as the RMSEs were already below the noise level. As shown in Fig. , as expected, decreasing σzsobs improves the analysis for the surface elevation. However, it does not necessarily reflect on the basal conditions and, on the contrary, reducing σzsobs below 10m leads to an increase in RMSEC from 0.004 to 0.005MPam-1/3a1/3. However, again, this effect does not reflect on the model velocities that are retrieved with the same accuracy.

We now discuss model projections from the initial state to t=200a.

Without assimilation, the deterministic forecast, i.e. using the ensemble mean basal conditions, rapidly leads to the fastest GL retreat, and after a few years the GL position is no longer included within the previsions from the ensemble (Fig. b). This is due to the fact that the ensemble mean is smoother than the reference and any of the ensemble members. The reference GL position is included in the ensemble, and at the end of the simulations most of the members are within ±25km from the reference. However, for a few members the GL remains very stable near its initial position for tens to hundreds of years, eventually never switching to an unstable regime during the duration of the simulation. Retreat rates are relatively variable from one member to the other, depending on the basal conditions.

With assimilation, the ensemble mean is improved and the difference from the reference reduced. The deterministic forecast cannot be distinguished from the ensemble members any more (Fig. c–d). Retreat rates are closer to the reference, with the previsions from all the ensemble members being more or less parallel to the reference. We note however that when the forecast starts after an assimilation window of 20 years, i.e. during a period of stable GL position for the reference, the deterministic forecast leaves the stable position with a delay of approximately 25 years, and a few members remain stable for the entire simulation. On average between t=13 and t=32a, the thinning rate at the GL in the reference simulation is approximately 0.6 m a-1, decreasing to 0.25 m a-1 during the last 2 years. The total thinning between two analyses is then much lower than the noise in the observed surface elevation and cannot be captured accurately by the DA. In addition, at the GL, the difference between the minimum and maximum bed elevation given by the ensemble is approximately 20 m. This remaining uncertainty induces a difference of more than 2 m for the floatation surface and combined with the small thinning rates explains the delays in the initiation of the instability.

Extending the assimilation window up to t=35a when the reference has switched in a fast retreat allows the forcing of all the members in the unstable retreat. There is a very good agreement between the reference and the deterministic forecast up to t=110a. This is also true for the ensemble, and after that the spread is larger and the predicted GLs are less retreated than for the reference.

These results can be summarised by looking at the distribution of the ensemble forecasts for the grounding line position and volume above floatation (VAF) at t=100a in Fig.  where the relative VAF change is computed as (VAFt=100-VAFt=0ref)/VAFt=0ref, with VAFt=0ref the reference VAF at t=0. As expected there is a clear correlation between grounding line retreat and mass loss, with higher retreat leading to higher mass loss. The distributions are clearly non-Gaussian; however, even without assimilation there is already a mode close to the reference. The mode is more pronounced, with more members close to the reference as observations are assimilated. As discussed before, with no assimilation or a short assimilation up to t=20a before the unstable retreat, the deterministic forecast can be very different from the mode of the ensemble forecast. However they are very similar if the assimilation is pursued up to t=35a, within 1% for the relative volume loss or 5 km for the GL position.