Marine-based sectors of the Antarctic Ice Sheet are increasingly contributing to sea level rise. The basal conditions exert an important control on the ice dynamics and can be propitious to instabilities in the grounding line position. Because the force balance is non-inertial, most ice flow models are now equipped with time-independent inverse methods to constrain the basal conditions from observed surface velocities. However, transient simulations starting from this initial state usually suffer from inconsistencies and are not able to reproduce observed trends. Here, using a synthetic flow line experiment, we assess the performance of an ensemble Kalman filter for the assimilation of transient observations of surface elevation and velocities in a marine ice sheet model. The model solves the shallow shelf equation for the force balance and the continuity equation for ice thickness evolution. The position of the grounding line is determined by the floatation criterion. The filter analysis estimates both the state of the model, represented by the surface elevation, and the basal conditions, with the simultaneous inversion of the basal friction and topography. The idealised experiment reproduces a marine ice sheet that is in the early stage of an unstable retreat. Using observation frequencies and uncertainties consistent with current observing systems, we find that the filter allows the accurate recovery of both the basal friction and topography after few assimilation cycles with relatively small ensemble sizes. In addition it is found that assimilating the surface observations has a positive impact on constraining the evolution of the grounding line during the assimilation window. Using the initialised state to perform century-scale forecast simulations, we show that grounding line retreat rates are in agreement with the reference; however remaining uncertainties in the basal conditions may lead to significant delays in the initiation of the unstable retreat. These results are encouraging for the application to real glacial systems.

Despite recent significant improvements in ice sheet models, the projected magnitude and rate of the Antarctic and Greenland ice sheets' contribution to 21st century sea-level rise (SLR) remains poorly constrained

Improving SLR estimates requires, amongst other things, correctly modelling the dynamics of the grounding line (GL), i.e. the location where the ice detaches from its underlying bed and goes afloat on the ocean

For obvious reasons of inaccessibility, the basal conditions (topography and friction) are an important source of uncertainties. Because of the intrinsic instability of marine ice sheets resting over a seaward up‐sloping bed, the resolution of the bed topography in the coastal regions can significantly affect short-term ice sheet forecasts

Uncertainties in the model state and parameters can be reduced by data assimilation (DA). The objective of formal DA methods is to update the model using observations in a framework consistent with the model, the data and their associated uncertainties

Because historic remote sensing data collections are spatially incomplete as well as temporally sparse, most distributed maps are mosaicked, stacked or averaged to maximise the spatial coverage at the expense of the temporal information

Ensemble DA methods, based on the ensemble Kalman filter (EnKF), have been successful in solving DA problems with large and non-linear geophysical models. Comparative discussions of the performances and advantages of variational and ensemble DA methods can be found in, e.g.

EnKF approximates the state and the error covariance matrix of a system using an ensemble that is propagated forward in time with the model, avoiding the computation of the covariance matrices and the use of linearised or adjoint models. Contrary to time-dependent variational methods where the objective is to find the model trajectory that minimises the difference with all the observations within an assimilation window, EnKF assimilates the observations sequentially in time as they become available using the analysis step of the Kalman filter, as illustrated in Fig.

Principle of data assimilation (adapted from

As Monte Carlo methods, EnKFs suffer from under-sampling issues as often the size of the ensemble is much smaller than the size of the system to estimate. Localisation and inflation are popular methods to counteract these issues and to increase the stability of the filtering. Because they are based on the original Kalman filter equations, EnKFs are optimal only for Gaussian distributions and linear models. However, the many applications in geoscience with large and non-linear models have shown that the method remains robust in general and EnKFs are used in several operational centres with atmosphere, ocean and hydrology models

In the context of ice sheet modelling, encouraging results have been obtained by

The purpose of this paper is to explore the performance of ensemble Kalman filtering for the initialisation of a marine ice sheet model that includes GL migration. In particular, we want to address (i) the quality of the analysis for the simultaneous estimation of the basal topography and friction in the context of a marine ice sheet that is undergoing an unstable GL retreat and (ii) the effects of the remaining uncertainties for the predictability of GL retreat. The ice flow model and the EnKF used in this study are described in Sect.

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

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

For the assimilation, we use the error subspace ensemble transform Kalman filter

As an EnKF, ESTKF approximates the state

The algorithm can be decomposed in two steps, the

The analysis provides a new estimation of the system state by combining the information from the forecast and the observations. In the following we will omit the time index

In practice, with large models (

After some algebra using Eqs. (

Finally, the update step is obtained as a single equation for the transformation of the forecast ensemble

Finally, the analysed ensemble

We draw attention to several remarks on the algorithm.

To compute the innovation

Several ensembles can have the same mean and covariance matrix, which is why several EnKFs exactly satisfy Eq. (

As written here, the ESTKF leads to the same ensemble transformation as the ETKF. However, as the computations are not performed in the same subspace, tiny differences due to the finite precision of the computations may grow, leading to slight differences at the end of the assimilation window

The leading computational cost of the ensemble transformation in ESTKF is

In practice for large-scale problems, EnKFs as Monte Carlo methods suffer from under-sampling issues.
First, because of the rank deficiency of the covariance matrix

Second, the rank deficiency of

Here, the forgetting factor

To evaluate the performance of the DA framework we perform a twin experiment. In this section we first describe the synthetic reference simulation that will be used to assess the performance of the DA framework. From this reference, we generate a set of synthetic noisy observations that will be used by the assimilation scheme. Finally, we describe the initial ensemble constructed using a priori or background information.

We start by building an initial steady marine ice sheet. The domain extends from

Following

For the basal friction, we use a synthetic sinusoidal function with two wavelengths for

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

Thwaites Glacier (Antarctica). Model results from

Using a uniform ice rigidity

In

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 d

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

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

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.

In our model, as the force balance Eq. (

Because both

Kalman-based filters are based on the hypothesis of the independence between the background, i.e.

Finally, to illustrate the effect of the transient assimilation on model projections on timescales relevant for sea level projections, the analysed states at

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

For

For the bed we use an exponential function,

For the initial ensemble for the bed and friction coefficient, the ensemble mean is the dashed blue curve, and the shading shows the ensemble spread. Coloured solid lines show the first three members. The reference is shown in black and the synthetic bed measurements are shown as green triangles.

For the friction coefficient, we assume that we know the mean value

For the free surface, we initialise all the members using the observed (noisy) free surface at

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,

Velocity,

Here the size of the state vector

RMSE at

In the sequel we discuss the results obtained with an ensemble size

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

We expect that uncertainties in the ice sheet interior should not affect the short-term forecast of the coastal regions

Same as Fig.

Figure

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

RMSE (solid lines) and square root of the averaged ensemble variance (dashed lines) during the assimilation window for

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.

Similar conclusions are drawn if the assimilation is pursued up to

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 (

Sensitivity to the surface velocity observation error

Sensitivity to the surface elevation observation error

We now discuss model projections from the initial state to

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.

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.

Extending the assimilation window up to

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

Ensemble forecast at

Here, we have tested an ensemble Kalman filter to assimilate annually observed surface velocities and surface elevation in a marine ice sheet model.
Similar to previous studies, we have shown that, in fast-flowing regions, it is possible to accurately separate and recover both the basal topography and basal friction from surface observations

Using a scheme that assimilates time-dependent observations provides a model state consistent with transient changes and that can directly serve as an optimal initial condition to run forecast simulations without the need of an additional relaxation

Good results have been obtained with relatively small ensembles (50 to 100 members) for a state vector of size

We have used inflation and localisation to stabilise the filter. The inflation giving the best results in

In the experiments presented above, we have used a depth-integrated model for the force balance equations where GL migration is implemented through a hydrostatic floatation condition. This allows a full description of the ice topography with only one prognostic variable. Adaptation of the framework to a full-Stokes model requires minimum adaptations as these models do not rely on the floatation condition and solve a proper contact problem for the grounding line migration

Before generalising such methods to real glacial systems, several points must be taken into consideration. They are independent of the DA method but they will eventually be treated differently in a variational or in an ensemble framework.

First, if the implementation is not an issue, the computational cost implied by running a full-Stokes model might remain a limiting factor. Compared to the Stokes solution, the SSA is known to overestimate the effects of bed topography perturbations on the surface profile for wavelengths less than a few ice thicknesses

Second, the quality of the analysis and the accuracy of the error estimates depends on the observation error covariance matrix

A review paper by

Third, the results depend on prior assumptions on the control variables and their variability, represented here by the initial ensemble. For the basal topography, current reference maps provide local error estimates

Finally, in our synthetic applications, we have not accounted for all potential sources of uncertainty which are, for example, as follows.

Developing model initialisation strategies that properly reproduce the ice sheet dynamical mass losses observed over the last decades requires developing transient assimilation frameworks that are able to account for the growing availability of dense time series, especially from space observations. Here, we presented a synthetic twin experiment demonstrating the possibility of calibrating a marine ice model using an ensemble Kalman filter which requires fewer numerical developments than variational methods.

Using resolutions and noise levels consistent with current observing systems, good performances are obtained to recover both the basal friction and basal topography with an ensemble of at least 50 members. Localisation and inflation have been tuned manually; however the results are consistent over relatively wide ranges. Future studies should investigate how these values can be transposed to realistic applications. Nevertheless, there is an abundant and growing literature in other geophysical fields to overcome problems that we might be facing in future studies.

Once the GL enters an unstable region, retreat rates largely depend on the basal conditions; thus using DA to reduce the associated uncertainties largely increases the skill of the model to predict rates and magnitude of GL retreat for timescales relevant for sea level rise projections. In our simplified application, the assimilation of the surface observations was sufficient to capture the GL migration during the assimilation window, without explicitly assimilating the observed position. However, for the GL to enter an irreversible retreat, the thickness must reach a tipping point, i.e. the thickness at the GL must reach floatation. This can seriously impact the predictability of the system as, for small perturbations, remaining uncertainties on the basal conditions can lead to an uncertainty on the residence time of the GL on stabilisation points, which can be similar to the simulation timescale. However, if the assimilation is pursued up to a time when the glacier is engaged in unstable retreat, all the members exhibit instability, and the spread of centennial-scale model projections, in terms of volume and grounding line position, is largely reduced.

Finally, we have discussed the main challenges to tackle before generalising transient DA in ice sheet modelling. This includes a better assessment of the uncertainties in the model and in the observations used for the background and for the assimilation.

Elmer/Ice code is publicly available through GitHub (

Notations and values used in this study associated with the ice flow model.

Notations and values used in this study associated with the ensemble filter.

FGC designed the experiments and wrote the paper.

The author declares that there is no conflict of interest.

The author thanks Gael Durand, Olivier Gagliardini and Jérémie Mouginot for valuable comments on the first drafts of the manuscript.

This research has been supported by the French National Research Agency (ANR) through the TROIS-AS project (grant no. ANR-15-CE01-0005-01).

This paper was edited by Carlos Martin and reviewed by Dan Goldberg and two anonymous referees.