We develop data assimilation techniques for non-linear dynamical systems modelled by moving mesh methods. Such techniques are valuable for explicitly tracking interfaces and boundaries in evolving systems. The unique aspect of these assimilation techniques is that both the states of the system and the positions of the mesh points are updated simultaneously using physical observations. Covariances between states and mesh points are generated either by a correlation structure function in a variational context or by ensemble methods. The application of the techniques is demonstrated on a one-dimensional model of a grounded shallow ice sheet. It is shown, using observations of surface elevation and/or surface ice velocities, that the techniques predict the evolution of the ice sheet margin and the ice thickness accurately and efficiently. This approach also allows the straightforward assimilation of observations of the position of the ice sheet margin.

From lava flows to tumour growth to water flooding, many time-evolving
processes can be mathematically modelled as moving boundary problems.
Predicting their evolution accurately requires not only the estimation of the
state variables of the system over a moving domain but also the estimation
of the location of the moving domain itself. In this paper, we propose to
combine data assimilation with a moving mesh numerical model to estimate both
the domain and the states of a moving boundary problem. Genuine moving mesh
methods use a fixed number of mesh points whose movement can be generated by
various techniques

Data assimilation (DA) aims to combine available observations of a
dynamical system with model predictions in order to provide optimal estimates
of the state of the system and an estimation of the uncertainty in these
estimates. DA has been applied successfully in various contexts and is
routinely used in operational systems such as numerical weather prediction
systems

Our approach is particularly relevant to the prediction of the dynamics of
ice sheets and glaciers. Future evolution of ice sheet boundaries is closely
linked with sea level rise

We adapt here two popular DA schemes, a 3-D variational scheme (or 3D-Var;
see, e.g.

We consider a single-phase, radially symmetric, grounded ice sheet (no
floating ice), centred on the origin

The geometry of the grounded ice sheet is described by its surface altitude,

Section of a grounded radially symmetrical ice sheet.

The evolution of an ice sheet is governed by the balance between the mass
exchanges at the surface (snow precipitation and surface melting) and the ice
flow that carries the ice from the interior of the ice sheet towards its
margins. This is summarised by the mass balance equation

The velocity of the ice is derived using the shallow ice approximation

Parameters involved in the computation of the vertically averaged
horizontal component of the ice velocity (Eq.

The moving point numerical method we use in this paper relies on the
computation of point velocities and point locations. This type of method
belongs to the family of velocity-based (or Lagrangian) methods

From the equations detailed in Sect.

The user provides the initial mesh and the ice thickness at mesh points in
order to initialise the numerical model. From these quantities, the total
mass and the mass fractions at the initial time are calculated by
discretising Eqs. (

We now recall the basics of data assimilation before explaining how to adapt the 3D-Var and the ETKF methods to our context. We then clarify the form of the observation operator for various types of observations that we assimilate.

We consider data assimilation in a discrete dynamical system evolving in
time. We denote by

The objective of DA is to provide an optimal estimate

The 3D-Var method (see, e.g.

We take the observation operator

In theory, the true background error covariance matrix

The ensemble Kalman filter (EnKF) introduced by

The forecast step propagates the ensemble from time

The ETKF may generate ensembles of analyses with underestimated spread, which
can lead to the divergence of the filter. We use an inflation procedure

In the twin experiments performed in Sects.

Traditionally, in a data assimilation scheme, the state vector includes all
the physical variables of the given dynamical system. For a fixed-grid
numerical method, the model variables are defined at fixed spatial positions.
For example, for a grounded ice sheet modelled with a fixed-grid method (and
assuming every parameter is perfectly known), the unknown variables are the
ice thicknesses located at known positions (see, e.g.

In contrast, the primary characteristic of a moving point method is that the
numerical domain evolves in time. The positions of the nodes evolve jointly
with the model variables (such as ice thickness) according to the dynamical
system equations and can be updated using the assimilation scheme. We
therefore include the positions of the nodes in the state vector. As a
consequence, we define the state vector

In particular, for an ice sheet model, this approach gives us a direct estimation of the position of the ice sheet margin that cannot be obtained
in fixed-grid methods without interpolation. In this case, we do not include in

This can be achieved with the 3D-Var method if the specified background
covariance matrix

The most difficult step with this form of analysis is, in general, to set
appropriately the cross-covariances in

For the moving point ice sheet model, the DA analysis step updates both ice
thickness variables and node positions, but the total mass and mass fractions
have to be updated as well, since they are not preserved by the analysis (and
there is no reason to preserve them). Therefore, these quantities need to be
“reset” from the analysed state vector. This is easily done by using
Eqs. (

calculate a forecast of the state vector

use the analysis scheme (Eqs.

from

evolve the analysis solution using the numerical moving point model to the next time where observations are available and

repeat steps 2–5.

The adapted ETKF roughly follows the same path as 3D-Var except that, at step
1, we calculate the forecast for each member of the ensemble and, at step 3,
the total mass and mass fractions have to be updated for each member of the
ensemble (they are different for each ensemble member). The background error
covariance is also updated using the ensemble statistics. The strict
positivity of ice thickness variables and the order required in Eq. (

We remark that observations outside the domain of the background state at the time of the update cannot be assimilated. This is a limitation on both methods, but the ETKF has the advantage that it can take into account such observations if the domain of the background of any member of the ensemble is large enough to include the reference domain.

In the twin experiments performed in Sects.

We may also assimilate observations of the position of the ice sheet margin.
Using a moving point method allows the movement of boundaries to be tracked
explicitly. In our context, the position of the ice sheet margin is
represented by

To demonstrate the efficiency of our DA approach, we perform twin experiments
with two different configurations. In this section, we consider experiments
using an idealised system with a flat bedrock and the EISMINT surface mass
balance detailed in Eq. (

We first generate a model run with the moving point numerical model from known initial conditions. From this simulation, observations are created with added error sampled from a Gaussian distribution. This run is used as a reference to measure the quality of the DA estimates.

We define the reference initial ice thickness profile by the function

Ice thickness profile from the reference run in a simple case (flat
bedrock, EISMINT surface mass balance from Eq.

From the reference run, we generate observations of ice thickness and the
position of the ice sheet margin at times

To evaluate the performance of our DA approaches, we compare the estimated
ice thickness profiles with their reference counterparts. This is mostly done
graphically. We also study the quality of the estimates of two variables: the
ice thickness at the ice divide at

We begin by studying the performance of the DA schemes in the idealised configuration where we assimilate observations of ice thickness only. We start with an experiment using the 3D-Var algorithm in which only the ice thickness is updated at the assimilation times and the mesh point positions are not updated.

The background state is defined as follows:

At initial time, the background ice thickness profile is set using the same profile as the reference (Eq.

The background mesh consists of

The model time step is

As we are using a 3D-Var scheme in this experiment, the background error
covariance matrix

This definition of

Covariance matrices for ice thickness variables

The formulation of

We now evaluate the quality of the estimates. Figure

We now use 3D-Var to update both ice thickness variables and node locations.
The definitions of

Results for the ice thickness profile are shown in Fig.

Standard deviations and correlation matrix

The 3D-Var method provides information on the analysis covariance structures
for ice thickness variables and mesh point positions. In
Fig.

In these experiments, we have specified a fixed form for the background error covariance matrices, which are defined in terms of the positions of the nodes. We next show, using an ETKF, how the covariances are expected to evolve in time with the model dynamics and the effects of this on the assimilation.

Standard deviations and correlation matrix

We now perform the same experiment as before except that we now use an ETKF.
The key question is how to generate the initial ensemble composed of

In this experiment, we generate an initial ensemble of

the same background state used in the experiments detailed in Sect.

with

We do not use any inflation in this experiment (

Results are summarised in Fig.

Evolution of the absolute error of the estimated ice thickness at

The background error covariance matrices used by the 3D-Var and ETKF
methods to produce the analysis at time

The ETKF provides information on the covariance structures for ice thickness
variables and mesh point positions. We display estimated standard deviations
and an estimate of the correlation matrix

We now compare the results from applying the 3D-Var and ETKF assimilation
schemes in the case where we observe only the ice thickness. We focus on the
accuracy of the estimated ice thickness at

Figure

For 3D-Var without node updates, the analysis at the second time of
assimilation (

Using the ETKF assimilation scheme, where the covariance matrix fully evolves in time, is seen to improve the overall estimates. At each assimilation time, the errors in the estimated ice thickness and the position of the margin are decreased. Notably, we do not observe any degrading of the estimates at the second time of assimilation. This improvement can be attributed to the better background forecast produced by the ETKF at each assimilation time.

Evolution of the absolute error of the estimated ice thickness at

In Fig.

In this section, we perform the same experiments as previously, but we now assimilate not only the same observations of ice thickness as before but also observations of the position of the margin. We consider only the case of 3D-Var with grid update and the ETKF.

Absolute errors for the estimates of the ice thickness at

Adding observations of the position of the margin in the data assimilation
system reduces the estimated standard deviations obtained with the ETKF for
variables close to the margin. For example, the estimated standard deviation
for the position of the margin is now

In this section, we consider experiments using a more realistic configuration
with a non-flat bedrock and an advanced surface mass balance, detailed in
Appendix

We generate observations from a new reference run. We use a non-flat fixed
bedrock whose elevation is defined by the equation

Start with an ice sheet profile following Eq. (

Run the numerical model with a fixed climate forcing, as defined in
Eq. (

From this steady state, run the numerical model with a linearly warming climate forcing from

Initial state used to obtain a 20-year reference run under a warming
climate as detailed in Sect.

The reference is obtained by running the model over 20 years from the initial
state with a time step

ETKF results for the advanced configuration where observations of
surface elevation are assimilated over the first 10 years and a forecast is
made for 10 further years.

We generate observations of surface elevation, surface ice velocity and the
position of the ice sheet margin at times

We compare the influence of the observations on the quality of the DA
estimates and the subsequent forecasts for the 3D-Var and ETKF methods.
Again, we focus on the two variables: the ice thickness at the ice divide at

Standard deviations and correlation matrix

We begin by studying the performance of the DA schemes where we assimilate only observations of surface elevations.

For 3D-Var, the estimates are obtained using an initial background state
defined as

The ETKF uses an ensemble with 200 members. The initial ensemble is generated
by adding to

Evolution of the absolute error of the estimated ice thickness at

ETKF results for the advanced configuration where observations of
surface ice velocity and the position of the margin are assimilated over the
first

We first study the results obtained with the ETKF. At the end of the data
assimilation window,

With respect to the covariance matrix, the estimates seem to show a similar
behaviour to those of the experiment detailed in Sect.

We now compare the ETKF with results obtained with 3D-Var. Absolute errors in
the ice thickness at

We now consider assimilating observations of surface ice velocity and the position of the margin (if we only assimilate observations of surface ice velocity, the problem is undetermined).

Standard deviations and correlation matrix

Evolution of the absolute error of the estimated ice thickness at

Again, we want to compare the accuracy of 3D-Var and the ETKF using this new
set of observations. We use the same background state, the same structure for

We first study the results obtained with the ETKF. At the end of the DA
window,

Estimates of the standard deviations and covariances, as shown in Fig.

We finally compare the ETKF with results obtained with 3D-Var. Absolute
errors in the ice thickness at

In this paper, we have adapted standard data assimilation techniques (a 3D-Var
scheme and an ETKF) to estimate the state of a 1-D ice sheet model using a
moving point method. This is done by including both ice thickness variables
and the location of mesh nodes in the state vector. The only requirement is
to ensure that the update does not produce a non-ordered moving mesh. This
can be achieved empirically either by using an appropriate flow-dependent
background covariance matrix with large correlations between adjacent mesh
points or by using an ensemble with the same properties. This combination has
been validated with various twin experiments assimilating classical available
observations for an ice sheet (ice thickness, surface elevation and surface
ice velocity) and also observations of the position of the boundary. These
twin experiments show the following:

The form of the state vector allows the explicit tracking of boundary positions for moving boundary problems.

This form also allows a straightforward and efficient assimilation of boundary positions (in this paper, the position of the margin).

Assimilating spatially distributed observations gives better estimates if node locations are updated in the analysis step.

3D-Var can have issues with assimilating observations if they are located outside the forecast domain; the ETKF can overcome these issues if at least one member of the ensemble has its numerical domain large enough to include the location of these observations.

ETKF tends to provide better estimates than 3D-Var, mainly because of its capacity to provide flow-dependent statistical estimates of the background error covariances, but 3D-Var still provides satisfactory estimates.

ETKF provides not only good state estimates but also interesting information on the structure of the covariances; these are expected to be dominated by the statistics of the initial ensemble and the type of observations that are assimilated.

Moving mesh approaches are also suitable for modelling the evolution of 2-D
moving boundary phenomena

Data have been generated using the ice sheet model as described in the text and can be reproduced by the reader using the algorithm given in Bonan et al. (2016). The ETKF used in this paper follows the implementation of Hunt et al. (2007). The 3D-Var method and the design of each experiment are fully detailed in the paper.

For the twin experiments performed in Sect.

For the twin experiments performed in Sect.

List of parameter values used for the parameterised surface mass balance.

The authors declare that they have no conflict of interest.

This research was funded in part by the Natural Environmental Research Council National Centre for Earth Observation (NCEO) and the European Space Agency (ESA). Edited by: Olivier Talagrand Reviewed by: two anonymous referees