Ice flow models are now routinely used to forecast the ice sheets' contribution to 21st century sea-level rise. For such short term simulations, the model response is greatly affected by the initial conditions. Data assimilation algorithms have been developed to invert for the friction of the ice on its bedrock using observed surface velocities. A drawback of these methods is that remaining uncertainties, especially in the bedrock elevation, lead to non-physical ice flux divergence anomalies resulting in undesirable transient effects. In this study, we compare two different assimilation algorithms based on adjoints and nudging to constrain both bedrock friction and elevation. Using synthetic twin experiments with realistic observation errors, we show that the two algorithms lead to similar performances in reconstructing both variables and allow the flux divergence anomalies to be significantly reduced.

Robustly reproducing the responsible mechanisms and forecasting the
ice sheets' contribution to 21st century sea-level rise is one of the major
challenges in ice sheet and ice flow modelling as highlighted by
community-organised efforts such as SeaRISE (Sea-level Response to Ice Sheet
Evolution)

Such projections on decadal timescales are sensitive to the model initial
state which can account for an important source of uncertainty in the model
response

However, remaining uncertainties lead to non-physical ice flux divergence
anomalies

Among the remaining uncertainties, one of the most important is the
uncertainty related to the bedrock elevation. The basal topography is derived
from ice thickness measurements, mostly obtained from airborne ice-penetrating
radars. These measurements can have large uncertainties and are
usually at a lower resolution than required model grids

Because of theses large uncertainties, several methods have been proposed to
consider the bedrock elevation as an optimisation variable. For example,

In their work,

Several methods have been explored to construct model states where both the
basal friction and the basal topography are treated as optimisation
variables. In a pioneer work,

In this paper, we explore two different algorithms to infer both the basal
friction and the basal topography and initialise the model state using
simultaneous observations at a given time. The first algorithm is in line
with

For the force balance, we use the standard vertically integrated shallow
shelf approximation (SSA) equations

Natural boundaries are the calving fronts where the Neumann condition results
from the difference between the ice pressure and the sea water pressure:

The objective of the methods is to produce a model state from
Eq. (

The misfit between the model and the corresponding observations is evaluated
using cost functions. The first cost function measures the difference between
modelled (

The second cost function measures the misfit between modelled and observed
thickness rates of change:

The modelled rate of change of ice thickness

In general, both Eqs. (

The objective is then to find the parameter vector

The two cost functions have an implicit dependence on the parameter vector

The implementation in Elmer/Ice is carried out in a way that stays as close
as possible to the differentiation of the discrete implementation of the
direct equations. This method should lead to a better accuracy on the
gradient computation than the discretisation of the continuous equations.
Elmer/Ice uses programming features that are not supported by automatic
differentiation tools and the differentiation of the crucial parts of the
discrete source code (e.g. cost function computation, matrix assembly) has
been done manually. If the problem is non-linear, as here due to the
dependence of the viscosity to the strain rate, and the non-linearity solved
using a Picard iterative scheme, the iterations should be reversed (at least
partially) in the adjoint code to achieve a good accuracy of the computed
gradient

Inverse problems are often ill-posed, leading to instabilities. It is then
necessary to add regularisation terms to the cost function to avoid
overfitting of data. This can be done in the form of a Tikhonov
regularisation. Here, we define two different regularisations. The first one
measures the norm of the first spatial derivative of the component

The computation of the gradients of these two functionals with respect to

This minimisation is achieved using the quasi-Newton routine M1QN3

By definition, the steady-state solution of Eq. (

From the methods presented in the previous section we design two algorithms
to infer simultaneously the friction coefficient

This algorithm uses the gradients of the cost functions derived using the
adjoint method to optimise both

In this algorithm, the adjoint method is first used to optimise

Reference (solid lines) and initial (dashed lines) state for

These two steps are then repeated iteratively until changes in

A twin experiment is designed to investigate the ability of the two methods to reproduce simultaneously good estimates of the basal friction coefficient and the bedrock elevation. A flowline geometry is preferred to reduce the computational cost and easily test the method, however all the algorithms can be applied to 2-D plane view simulations. A reference experiment for which all the model parameters are prescribed is produced to generate synthetic observations. These observations are then used to test the performances of the two algorithms.

A flowline of Jakobshavn Glacier, Greenland, is used to test the two
algorithms with realistic conditions. Jakobshavn Isbrae is one of Greenland's
three largest outlet glaciers and has one of the largest drainage basin on
the ice sheet's western margin

The geometry is discretised through a mesh of 500 linear elements,
increasingly refined to the front of the glacier. The element size decreases
from

Results will only be presented on the first 100

Synthetic observations are generated by sampling and/or adding noise to the
reference simulation. Details for each required field are given below. These
synthetic observations and initial fields for the inverse methods are
compared to the reference in Fig.

Mean error on the thickness rate of change (rms

Surface velocities are assumed to be observed at the same resolution as the
reference simulation but with a white Gaussian noise with a mean

The surface mass balance,

The surface elevation is assumed to be perfectly observed. For the bedrock
elevation

The ice viscosity is assumed to be perfectly known and corresponds to the viscosity used in the reference experiment.

Assuming that no observation of the friction coefficient is available, an
initial solution has to be postulated. A good first guess for

The rms errors on the surface velocities and the rate of change of ice
thickness between the initial state and the synthetic observations are,
respectively, 761 and 357 m a

The average relative error on the basal shear stress is measured as

Results of the ATP algorithm with (orange) and without (red)
optimisation of

A set of 255 pairs (

The optimisation of both

Results of the ANC algorithm (purple):

In order to assess the influence of accounting for

Introduction of a Gaussian noise on

The steps for the optimisation of

In addition to the regularisation parameters of Eq. (

The five new references build from a 5-year perturbation of the
initial reference by an increase of the friction parameter:

The model is in good agreement with observations with a rms misfit of
46.1

As for ATP, introduction of a Gaussian noise in the observed thickness rate
of change

Range of values for ATP algorithm for the five perturbations of the
friction coefficient

Same as Fig. 5 but for the ANC algorithm.

In order to evaluate the efficiency of both algorithms in transient states,
we construct new reference cases where

The time period for the glacier to come back to equilibrium, after this
change of friction parameter, depends on the amplitude of the perturbation.
Here, the perturbation is only applied during 5 years in order to keep the
five cases in disequilibrium. Resulting thickness rates of change

The results of the optimisations for the five cases of perturbation are shown
in Fig.

Evolution of

In this section, we assess the impact of our initialisation algorithms on the
prognostic response of the model forward in time assuming the same constant
forcing used to build the reference state. By doing so, if the initialisation
was perfect, one would expect no change of the geometry and ice flow during
this prognostic simulation. The experiment is performed from ATP and ANC
initial states. A third initialisation state is constructed for which only
the friction coefficient has been optimised, keeping

Ice surface elevation

The prognostic simulations are conducted during a 10-year period in order to see
how the initial thickness rate evolves during this time and how it impacts
the final ice thickness and ice surface. The thickness rates of change after 1
and 10 years of simulation are shown in
Fig.

ANC and ATP initial states involve thickness rates of change much closer to
zero than the optimisation of “

In that way, the two algorithms implemented in this study show substantial
improvements compared to the optimisation of “

The presented algorithms allow the reconstruction of two poorly known
parameters: the bedrock topography

The optimisation of these two parameters mainly relies on the knowledge of
some other data that are easier to measure: ice surface velocities and thickness rates
of change. Some local measurements of bedrock elevation and associated errors
are necessary in order to define a background

The two algorithms are based on the optimisation of the friction coefficient

We have shown that the ATP algorithm is capable to well reproduce

Furthermore, the transient simulations over 10 years from initial states reconstructed with the two algorithms developed give very encouraging results. The model divergence is clearly decreased with respect to usual inversion methods of the friction coefficient only. The integration of observations like thickness rates variation through an optimisation of the divergence during inversion or nudging steps, allows to regularise the solution in a physical way and also clearly improves the results.

Finally, the sensitivity experiments shows that the different algorithms can take into account the disequilibrium of mass balance, which is particularly interesting considering that a large amount of outlet glaciers in both Greenland and Antarctica present this feature.

The construction of the twin experiment presented in this article is
partially based on real data. Surface velocities come from Joughin et
al. (2010), while surface and bedrock geometries come from Bamber et
al. (2013). Notice that surface topography slightly differs from Bamber et
al. (2013) in order to reach steady state. The simulations were performed
using the Elmer/Ice finite element model
(

We would like to thank the editor, A. Le Brocq, as well as the two referees, S. L. Cornford and R. Arthern, for their positive and constructive comments which greatly improved the initial version of the manuscript. This work was supported by the French National Research Agency (ANR) under the SUMER (Blanc SIMI 6) 2012 project ANR-12-BS06-0018. LGGE is part of Labex OSUG@2020 (ANR10 LABX56). Edited by: A. Le Brocq Reviewed by: R. Arthern and S. L. Cornford