Introduction
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 . The moving mesh method used
here is based on conservation of local mass fractions
. The major
advantage of our moving mesh method is that only a small number of mesh steps
are needed to accurately determine the positions of the boundaries, unlike
fixed or adaptive mesh methods . Our moving mesh method has been successfully
applied to a number of moving boundary problems, including one- and
two-dimensional models of ice sheet flow, tumour growth and chemical
spreading .
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 . In particular, DA has already been used
with fixed and adaptive grid models in the context of moving boundary
problems. In these cases, estimates outside the moving domain are generally
non-physical and need to be reanalysed .
Furthermore, with fixed or adaptive grids, DA does not provide an explicit
estimate of the extent of the domain; this can be only done by interpolation.
By combining DA with our moving mesh numerical model, we show here that the
explicit extent of the domain can be estimated efficiently and accurately and
that non-physical estimates do not appear.
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 and ice sheets are now
relatively well-observed bodies . Our moving mesh numerical
method for ice flow has already been validated for both 1-D and 2-D models of
ice sheets . In this paper, we describe the
application of data assimilation to the moving mesh method and demonstrate
the combined techniques using a one-dimensional moving mesh model of a
grounded shallow ice sheet as described in . Although the
model is relatively simple, there is no reason that these techniques cannot
be extended to much more complex problems.
We adapt here two popular DA schemes, a 3-D variational scheme (or 3D-Var;
see, e.g. ) and an ensemble transform Kalman filter
(ETKF; see ), to estimate the state of an ice
sheet modelled by our moving mesh method . The approach is
validated by twin experiments using available classical surface observations
(surface elevation and surface velocity; see ).
Observations of the position of the moving boundary (see, e.g.
for observations of continental margins in
palaeoglaciology) are also assimilated using a straightforward observation
operator. The paper is organised as follows: in Sect. 2 we recall the key
points of the moving point ice sheet model, in Sect. 3 we describe how to
apply the 3D-Var and the ETKF methods for our state estimation problem and
in Sects. 4 and 5 we validate our approach by performing several twin
experiments before concluding in Sect. 6.
Moving point ice sheet model
Ice sheet dynamics
We consider a single-phase, radially symmetric, grounded ice sheet (no
floating ice), centred on the origin r=0 of the radial coordinates. The
origin is called the ice divide.
The geometry of the grounded ice sheet is described by its surface altitude,
s(t,r), the ice thickness, h(t,r) and the altitude, b(r), of the fixed
bedrock on which the ice sheet lies (see Fig. ). These
quantities are linked through the relation
s=b+h.
The position of the edge of the ice sheet rl(t), also known as the ice
sheet margin, is implicitly determined by the Dirichlet boundary condition
h(t,rl(t))=0.
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
∂h∂t=m(t,r)-1r∂rhU∂r,
where m(t,r) is the surface mass balance and U(t,r) is the vertically
averaged horizontal component of the ice velocity in the sheet. In the
numerical experiments (see Sects. and ), we use
two different surface mass balances: a function that only depends on the
radius r and a more complex surface mass balance which depends on the
atmospheric temperature that evolves with the geometry of the ice sheet. Both
surface mass balances are described in detail in Appendix .
The velocity of the ice is derived using the shallow ice approximation
, which leads to the following analytical formulation of the
vertically averaged horizontal component of the ice velocity U(t,r):
U=-2n+2A(ρig)nhn+1∂s∂rn-1∂s∂r,
where s is given by Eq. () and the parameters involved in
the shallow ice approximation (SIA) are summarised in
Table . Since we consider only radially symmetrical ice
sheets, a symmetry condition also holds at r=0
U(t,0)=0and∂s∂r(t,0)=0.
Parameters involved in the computation of the vertically averaged
horizontal component of the ice velocity (Eq. ).
Parameter
Value
n
exponent of the creep relation
3
A
coefficient of the creep relation
10-16 Pa-3 yr-1
ρi
density of ice
910 kg m-3
g
gravitational acceleration
9.81 m s-2
Moving point method
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
. Here, the velocity of mesh points is obtained by conserving
local mass fractions . To calculate the velocity,
we first define the total volume of the ice sheet θ(t) as
θ(t)=2π∫0rl(t)rh(t,r)dr.
Assuming that the flux of ice through the ice sheet margin is zero, its rate
of change θ˙ depends only on the surface mass balance,
θ˙(t)=2π∫0rl(t)rm(t,r)dr.
We now define the relative mass fraction μ(r^) relative to the
moving point r^(t). Since the density of ice ρi is assumed
constant, volume fractions and mass fractions are equivalent and
μ(r^)=2πθ(t)∫0r^(t)rh(t,r)dr.
The velocity of the moving point r^(t) is defined implicitly by
keeping μ(r^) constant in time, that is, dμ(r^)dt=0. By differentiating Eq. () with respect to time using the Leibniz
integral rule, we obtain the velocity of every interior point
dr^dt=U(t,r^(t))+1r^(t)h(t,r^(t))μ(r^)∫0rl(t)rm(t,r)dr-∫0r^(t)rm(t,r)dr.
One of the points is dedicated to the static ice divide at r=0, while
another point tracks the position of the margin rl(t), which moves at the
velocity
drldt=U(t,rl(t))-m(t,rl(t))∂h∂r-1.
Once the velocity of each moving point has been obtained from Eq. () or (), the moving points are moved
in a Lagrangian manner using the explicit Euler scheme:
r^(t+Δt)=r^(t)+Δtdr^dt.
The total mass θ(t) is updated in the same way using θ˙(t)
from Eq. (). Finally, the ice thickness profile is updated
by differentiating Eq. () with respect to r^, giving
h(t,r^(t))=θ(t)πdμ(r^)d(r^2).
Numerical model
From the equations detailed in Sect. , a finite difference
algorithm is derived (see for the full algorithm). The
mesh consists of nr moving nodes with the positions
0=r^1<r^2<…<r^nr-1<r^nr=rl(t).
No further assumption is made on the spatial distribution of the moving
nodes. At each node r^i there is an associated ice thickness hi
and a fixed mass fraction μi. By construction, μ1=0, μnr=1 and the ice thickness at the ice sheet margin hnr=0.
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. () and () using the following
composite trapezoidal rule:
θ=π2∑i=1nr-1(hi+hi+1)(r^i+12-r^i2),μ1=0,μi+1=μi+π2θ(hi+hi+1)(r^i+12-r^i2),i=1,…,nr-1.
The mesh points are then evolved using a discrete form of Eq. (9) and
the ice thickness is determined using a discrete form of Eq. (12), with the mass
fractions {μi} kept constant over a time step. Full details are given
in Bonan et al. (2016).
State estimation of a system modelled with a moving mesh
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.
Data assimilation
We consider data assimilation in a discrete dynamical system evolving in
time. We denote by xk
the vector of size nx
describing the state of the system at time tk. For example, in our
numerical ice sheet model, ice thicknesses at mesh points are elements of the
state vector. The state xk is propagated forward in time to a
time tk+1 by the non-linear model Mk,k+1. Assuming the
model is perfect, we have
xk+1=Mk,k+1xk.
Observations are available at times tk and are related to xk
through the equation
yk=Hkxk+εk,
where yk is a vector of pk observations taken at time tk,
Hk is the (possibly non-linear) observation operator and
εk is the observation error vector, which is
assumed to be unbiased (zero mean) with covariance matrix Rk.
The objective of DA is to provide an optimal estimate xka of the
system, called the analysis, by combining observations with information
derived from the model. We consider in this paper two different DA schemes: a
3D-Var scheme and an ETKF.
3D-Var
The 3D-Var method (see, e.g. ) aims to provide
the optimal estimate xka by minimising the cost function
J(x)=12x-xkbTBk-1x-xkb+12yk-HkxTRk-1yk-Hkx,
where xkb is a prior, or background, estimate of the state of the system (generally obtained by propagating forward in
time the previous analysis xk-1a with Eq. ). The error in the prior estimate is assumed to be unbiased with
covariance matrix Bk and to be uncorrelated to errors in the observations.
We take the observation operator Hk to be linear around
xkb, meaning that
Hk(x)≈Hk(xkb)+Hkx-xkb,
where Hk is the linearisation of the observation operator about
the background xkb. Under this assumption, the cost function has
an explicit minimum
xka=xkb+Kkyk-Hkxkb,
where
Kk=BkHkTHkBkHkT+Rk-1.
The analysis error covariance matrix can be estimated as
Pe,k=(I-KkHk)Bk.
In theory, the true background error covariance matrix Bk should
be updated at each time step. However, this process is extremely expensive
for real-time applications and, instead, we use a matrix with a simplified
structure specified by the user. We will see in the numerical experiments
(Sects. and ) how setting Bk
appropriately is essential in order to obtain good estimates. Although the
assimilation scheme we propose here to use with the moving mesh method is a
variant of the traditional non-linear 3D-Var method, it is in essence a
variational method with a fixed form for the background covariance matrices
and we will refer to it as the 3D-Var method in the rest of the paper.
Ensemble transform Kalman filter
The ensemble Kalman filter (EnKF) introduced by
approximates a fully non-linear Monte Carlo filter. At each time step, the
state of the system is represented by an ensemble of Ne realisations
xk(i),i=1,…,Ne. The
state estimate is given by the ensemble mean
x‾k=1Ne∑i=1Nexk(i),
and the state error covariance matrix by the ensemble covariance matrix
Pe,k=1Ne-1XkXkT,
where Xk is the anomalies matrix defined as
Xk=xk(1)-x‾k,…,xk(Ne)-x‾k.
From the ensemble covariance matrix, we can define the matrix Corr
that contains an estimate of the correlation between the state variables to
be
[Corr]i,j=[Pe,k]i,j[Pe,k]i,i[Pe,k]j,j,
where [Corr]i,j and [Pe,k]i,j denote the entry
in the ith row and jth column of Corr and
Pe,k, respectively.
The forecast step propagates the ensemble from time tk to tk+1 with
the non-linear model Mk,k+1. For the analysis step, we use the
efficient ETKF introduced by
and follow the implementation of the algorithm given by
.
The ETKF may generate ensembles of analyses with underestimated spread, which
can lead to the divergence of the filter. We use an inflation procedure
here to avoid this potential degeneracy. In the rest of
the paper, the inflation factor is denoted by the parameter
λinfla.
In the twin experiments performed in Sects.
and ,
we use a large number of ensembles to avoid producing spurious correlations
in Pe,k. Therefore, no localisation has been employed in this
paper.
Form of the state vector in the moving mesh case
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 x as follows:
x=xhxrwithxh=h1⋮hnr-1andxr=r^2⋮r^nr.
Estimates obtained by combining DA with this formulation of x
using a moving point numerical model provide more information on the state of
the system than if we were using a fixed-grid method.
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 x the ice thickness at the margin hnr or the
position of the ice divide r^1 as both are fixed to zero. The DA schemes must, however, provide estimates with strictly positive ice
thicknesses hi, i=1,…,nr-1 and a preserved order
for node positions to respect the assumption of the moving mesh scheme.
This can be achieved with the 3D-Var method if the specified background
covariance matrix Bk in Eq. () is prescribed
carefully. At time tk, we decompose the background error covariance matrix
B and the tangent linear matrix of the observation operator
H (we drop the time index k for clarity) as
B=BhBrhTBrhBrandH=HhHr=∂H∂xh(xf)∂H∂xr(xf),
where Bh is the background error covariance matrix between the
model variables, Br is the error covariance between mesh point
locations and Brh includes the cross-covariances between errors
in point locations and errors in model variables. The different components of
the state vector are then updated by the following analysis step:
xha=xhb+BhHhT+BrhTHrTHBHT+R-1y-Hxbxra=xrb+BrhHhT+BrHrTHBHT+R-1y-Hxb.
The most difficult step with this form of analysis is, in general, to set
appropriately the cross-covariances in Brh that are needed for
the update stage. For example, if either Hh or Hr is
zero, a non-zero Brh matrix is the only way to correct
estimates of both xh and xr. However, we will see in
the next section that in our assimilation systems for the ice sheet model,
the observation operator depends explicitly on both ice thickness variables
and mesh node locations, and therefore by setting Brh to zero
we can still obtain good estimates.
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. () and (). The adapted 3D-Var scheme is
performed according to the following steps:
calculate a forecast of the state vector xb by using the previous analysis solution to initialise
the numerical moving point model,
use the analysis scheme (Eqs. and ) to produce the analysis
xa,
from xa, calculate the analysed total mass θa and update the mass fractions μa using
Eqs. () and (),
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. () for node positions are ensured by appropriately setting
the initial ensemble in the ETKF.
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.
Type of observations assimilated
In the twin experiments performed in Sects. and ,
we use three different conventional types of observations of an ice sheet
system that are available in reality (see, e.g. ). The
first is direct observations of the ice thickness. Assuming that we have an
observation of the ice thickness located at position ro, we define the
associated observation operator as
H(x)=hi+ro-r^ir^i+1-r^ihi+1-hiifr^i≤ro≤r^i+10elsewhere,
which is merely a piecewise linear interpolation operator. Note that
H depends on both ice thickness variables hi and node
locations r^i. We also assimilate observations of surface elevation
and surface ice velocity. We again use a piecewise linear interpolation
operator as in Eq. (). For observations of surface elevation,
we have
H(x)=si+ro-r^ir^i+1-r^isi+1-siifr^i≤ro≤r^i+1b(ri)elsewhere,
with
si=hi+b(ri).
For observations of surface ice velocity, from a discretisation of
Eq. () (see Appendix B2 in ), we have
H(x)=us,i+ro-r^ir^i+1-r^ius,i+1-us,iifr^i≤ro≤r^i+10elsewhere,
with
us,i=12A(ρig)3sgnsnr-snr-1hi4∂b∂r(ri)3+35hi5-hi-15r^i-r^i-1∂b∂r(ri)2+13hi3-hi-13r^i-r^i-12∂b∂r(ri)+27343hi7/3-hi-17/3r^i-r^i-13,
except for us,1=0.
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 r^nr. As a consequence, the observation operator for
such an observation is defined by
H(x)=r^nr.
The operator is continuous and linear. This makes the assimilation of the
position of the margin straightforward in comparison with the same
assimilation with a fixed-grid model (see, e.g. ).
Numerical experiments with an idealised model
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. ().
Experimental design
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
h(0,r)=h01-rrmax23/70≤r≤rmax
where h0=2000 m and rmax=450 km. This function gives a
smooth interior profile with a steep snout at the ice sheet margin
rmax. This is in compliance with the physics involved in the ice
sheet model and provides an initial state with a margin that is immediately
in motion. The reference run is obtained with an initial mesh of nr=28
points evenly spaced between r^1=0 and r^nr=450 km.
The model time step is Δt=0.02 years, the bed elevation b is fixed
to zero and the surface mass balance used is from the EISMINT benchmark (Eq. ). The experiment starts at time t=0 years and ends at
t=2000 years. The evolution of the reference ice thickness profile can be
seen in Fig. .
Ice thickness profile from the reference run in a simple case (flat
bedrock, EISMINT surface mass balance from Eq. ). The
initial state follows the profile of Eq. () with h0=2000 m and rmax=450 km. The reference run is obtained with an
initial mesh of nr=28 points evenly spaced between r^1=0 and
r^nr=450 km.
From the reference run, we generate observations of ice thickness and the
position of the ice sheet margin at times t1=500 and t2=1500 years.
Observations of thickness are taken at each point except at the margin (a
total of 27 observations) with added random noise from the Gaussian
distribution N(0,σho2), σho=100 m. For the
position of the margin, the observational noise is sampled from
N(0,σro2), σro=10 km.
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 r=0 and the position of the ice sheet
margin.
Updating the ice thickness only
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. )
but with h0=2100 m (+5 % error from the reference) and rmax=472.5 km (also +5 %
error).
The background mesh consists of nr=28 points, evenly spaced between r^1=0 and r^nr=472.5 km at initial
time.
The model time step is Δt=0.02 years.
As we are using a 3D-Var scheme in this experiment, the background error
covariance matrix B needs to be prescribed at both times of
assimilation (t1=500 and t2=1500 years). In this first experiment,
we only update ice thickness variables, so we set the background error
covariance matrix for point positions Br and the cross-covariance
matrix Brh to zero. We define Bh the covariance
matrix for ice thickness variables as
Bh=Dh1/2ChDh1/2,
with Dh the diagonal variance matrix and Ch the
correlation matrix. Dh is simply set to σhb2Inr-1 with σhb=100 m. The background error
correlation structure follows a second-order autoregressive (SOAR)
distribution with
Chi,j=1+|r^ib-r^jb|Lhexp-|r^ib-r^jb|Lhi,j=1,…,nr-1,
where Chi,j denotes the entry in the ith row
and jth column of Ch, r^ib the location of the
ith mesh point of the background state at the time of assimilation and
Lh is some correlation length scale to be fixed. The SOAR function is
preferred to a Gaussian structure as the matrix Ch is better
conditioned for inversion in that case . We set Lh to
100 km.
This definition of B takes into account the flow dependency of the
moving point locations, making our approach adaptive. Figure displays Bh at assimilation times t1=500 and t2=1500 years. As the distance between grid points increases in
time in the experiment, the covariances tend to reduce between the two
assimilation times. For example, the covariance between the location of points
r^1b and r^nr-1b is reduced from
[Bh]1,nr-1=530.7 at t1=500 years to
[Bh]1,nr-1=446.6 at t2=1500 years. In addition, we note
decreased correlations for points around the centre of the mesh due to a
greater distance between adjacent nodes in the centre of the grid than at the
boundaries.
Covariance matrices for ice thickness variables Bh used
by the 3D-Var at assimilation times t1=500 and t2=1500 years.
Covariances between variables at distant locations tend to reduce between the
two assimilation times. The distance between adjacent nodes also tends to be
greater in the centre of the mesh than at the boundaries, leading to a
decreasing covariance at t2=1500 years in this area.
The formulation of B forces the recomputation of the matrix at
every assimilation time. This is a limiting factor of our 3D-Var approach,
especially for high-dimensional systems, making it cost more than traditional
3D-Var for fixed-grid models in which B is only computed once.
Nevertheless, our experiments demonstrate that this formulation of the
background error covariance matrix ensures that the moving point framework
produces positive estimates of ice thickness variables and a smooth interior
profile in accordance with the physics of the system.
(a) 3D-Var analysis at time t=500 years compared with
the forecast and the reference when we update only ice thickness variables.
The ice thickness profile is improved, especially between r=100 and r=400 km. (b) Evolution of the position of the margin with
time. Even if the position of the margin is not directly updated, the
trajectory of the margin is corrected as a result of the ice thickness
update. (c) Evolution of the position of grid points with time.
The trajectory of each grid node is corrected after each analysis, as is the
margin.
We now evaluate the quality of the estimates. Figure
(left) displays the analysed ice thickness profile compared to its background
and reference counterparts at the first time of assimilation t1=500 years.
The picture shows that the ice thickness profile in the interior of the ice
sheet is substantially improved by DA. For example, the absolute error in ice
thickness at the ice divide (r=0) is reduced from 100 to 58.3 m by
the 3D-Var analysis. Results are even better between r=100 and
400 km. Since we only update xh in this experiment, the
position of the margin is not modified by our update. Nevertheless, by
correcting the interior of the ice sheet, the forecast of the migration of
the margin is improved (see the central and right pictures after t=500 years;
Fig. ), and at the second assimilation time, t=1500 years, the absolute difference between the position of the margin before
analysis and its reference position is only 5.6 km (instead of 15.9 km
without DA).
Updating ice thickness variables and node positions
We now use 3D-Var to update both ice thickness variables and node locations.
The definitions of Bh and Brh remain the same as in
the previous experiment, but we set the covariance matrix for node positions
Br to be Br=Dr1/2CrDr1/2 with Dr the diagonal variance matrix and
Cr a correlation matrix. The matrix Dr is set to
σrb2Inr-1 with σrb=22.5 km and Cr follows a SOAR distribution with
Cri,j=1+|r^i+1b-r^j+1b|Lrexp-|r^i+1b-r^j+1b|Lr,i,j=1,…,nr-1,
where Lr is a correlation length scale fixed to 100 km. The correlation
matrix Br constrains the movement of the assimilated mesh points and the
correlation function used in the formulation of Br is selected to ensure
that the order of the points
defined by Eq. () is preserved by the 3D-Var algorithm.
Since the distance between nodes evolves in time, it is even more important
than in the previous case to use a flow-dependent background error covariance
matrix B.
(a) 3D-Var analysis at time t=500 years compared with
the forecast and the reference when we update ice thickness variables and
node locations. In contrast to the results shown in Fig. , the ice thickness profile is substantially improved
close to the margin. (b) Evolution of the position of the margin
with time. The estimates are of very good quality even if the margin is not
observed directly. (c) Evolution of the position of mesh points
with time. The trajectory of each node is corrected by each analysis, as is
the margin.
Results for the ice thickness profile are shown in Fig. .
Overall estimates obtained with updating both ice thickness variables and
node positions are better than when we update only ice thickness variables.
The absolute error in ice thickness at the ice divide (r=0) is reduced
from 100 to 60.2 m by the 3D-Var analysis at time t1=500 years, which
is similar to the previous experiment. However, we now obtain at t1=500 years a very accurate ice thickness profile close to the margin and its
estimated position has an absolute error of only 0.2 km. This shows that
the estimated position of the ice sheet margin can be accurately corrected by
only using standard observations (no observation of the position of the
margin is involved in this experiment). At the second time of assimilation at
t2=1500 years, the estimate is degraded, however, as a result of using
fixed variances in the matrix B. This behaviour is discussed
further in Sect. .
Standard deviations and correlation matrix Corr estimated
from the 3D-Var analysis at time t=500 years when we use only observations of
ice thickness. Auto-correlations between ice thicknesses are located in the
top left corner of Corr; auto-correlations between node positions
are in the bottom right corner. The rest of the matrix depicts the
cross-correlations.
The 3D-Var method provides information on the analysis covariance structures
for ice thickness variables and mesh point positions. In
Fig. , we display the estimated standard deviations
and the error correlation matrix Corr (see Eq. )
obtained at time t=500 years using the estimated analysis error covariance
matrix Pe,k given by Eq. (). We see that
the 3D-Var method produces decreased standard deviations and correlation
length scales for ice thickness variables close to the ice divide and
decreased standard deviations and correlation length scales for node
locations close to the margin. The 3D-Var method also produces strong
anti-correlations between ice thickness variables and node positions, meaning
that in order to fit the observations where the ice thickness variables
become larger, the associated nodes need to retreat.
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.
(a) ETKF analysis at time t=500 years compared with the
forecast and the reference. The ice thickness profile is improved over the
whole domain and the reference profile is within the ensemble spread.
(b) Evolution of the ice thickness at r=0 with time. The
estimates are of very good quality and the estimates seem to converge towards
the reference value at the end of the study. (c) Evolution of the
position of the margin with time. The ETKF provides consistent estimates and
the reference value is always within the ensemble spread.
Standard deviations and correlation matrix Corr estimated
from the ETKF analysis ensemble at time t=500 years when we use only
observations of ice thickness. Auto-correlations between ice thicknesses are
located in the top left corner of Corr; auto-correlations between
node positions are in the bottom right corner. The rest of the matrix depicts the
cross-correlations.
Using an ETKF
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 Ne
members. The easiest way is to add noise to a background state sampled from a
Gaussian law N(0,B) with B as the
background error covariance matrix defined in Eq. ().
In this experiment, we generate an initial ensemble of Ne=200 members
using:
the same background state used in the experiments detailed in Sect. and ,
Bh defined by Eq. () with the diagonal matrix Dh=σh2Inr-1,
σhb=100 m, Ch defined by Eq. (), Lh=100 km,
Br taken as Dr1/2CrDr1/2 with Cr defined by
Eq. ()
with Lh=100 km and the diagonal matrix Dr defined as
[Dr]ii=minσrb,αr^ii=1,…,nr-1
with σrb=22.5 km and α=0.2, and
Brh set to zero.
Note that the definition of B is slightly different from the
previous experiment as we choose different diagonal variances. The change is
because of the high probability of generating useless initial meshes with
negative radii using Dr=σrb2Inr-1,
as the background standard deviation σrb is larger than the
background position of the first points (for example, r^2b=17.5
km). To avoid this problem, we have decided just to reduce the variance for
the position of points near the ice divide using Eq. (). The
new ensemble mean has, at the initial time, an estimated position of the
margin of 472.9 km with an estimated standard deviation of 22.8 km (where
the true value at t=0 is 450 km).
We do not use any inflation in this experiment (λinfla=1).
Results are summarised in Fig. . At the first time of assimilation
t1=500 years, the analysis step corrects the ice thickness profile well.
The estimate of the ice thickness at r=0 is of the same quality as in the
previous experiments (absolute error of 46.9 m) and the estimate of the
position of the margin is reduced from 483.1 km (forecast mean with
estimated standard deviation 18.9 km) to 468.8 km (analysis mean with
estimated standard deviation 7.1 km). The estimate obtained by the ETKF is
in accordance with the true value (which is within the ensemble spread) and
the absolute error of 7.5 km is of the same order as the estimated standard
deviation. The rest of the experiment exhibits the same quality in terms of
recovering the ice thickness profile.
Evolution of the absolute error of the estimated ice thickness at
r=0 and the estimated position of the margin when we observe only the ice
thickness. We compare the absolute errors obtained when we use 3D-Var without
and with correction of the position of grid nodes and when we use an ETKF.
The background error covariance matrices used by the 3D-Var and ETKF
methods to produce the analysis at time t=1500 years.
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 Corr (see Eq. ) in Fig. for the analysis ensemble
at time t=500 years. The ETKF produces decreased standard deviations and
correlation length scales for ice thickness variables close to the ice
divide. For example, the standard deviation of the ice thickness at the ice
divide is more than halved by the analysis, from 97.4 m before analysis to
41.6 m. Decreased standard deviations and correlation length scales are
also obtained for node locations but close to the margin in this case. The
standard deviation for the position of the margin is reduced from 18.9 km to
7.1 km by the analysis. The ETKF also produces strong anti-correlations
between ice thickness variables and node positions, meaning that where ice
thickness variables become larger associated nodes need to retreat to fit the
observations of ice thickness.
Comparing 3D-Var and the ETKF
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 r=0 and the position of the
margin.
Figure shows the evolution of the absolute errors in the estimates
of the ice thickness at r=0 and in the position of the margin for the
ETKF and for 3D-Var, with and without node updates. All three methods provide
improved estimates at the first analysis time (t1=500 years), leading to
good forecasts up to the next assimilation time. We find that the ETKF tends
to perform better than the variational approach and that for 3D-Var the
estimates obtained by updating both ice thickness variables and node
positions are generally better than those where only ice thickness variables
are updated.
For 3D-Var without node updates, the analysis at the second time of
assimilation (t2=1500 years) of the ice thickness at r=0 is
unfortunately degraded relative to the forecast, but the estimated position
of the margin is still improved by the second analysis. In the case where ice
thickness and nodes are updated, the estimates of both ice thickness at r=0
and the position of the margin are degraded at the second time of
assimilation. This weakens the confidence in the forecast and we partially
lose what we had gained from the previous analysis. The experiment shows the
sensitivity of 3D-Var to current observations resulting from the dependence
of the prescribed covariance matrix B on the positions of the mesh
nodes.
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
r=0 and the estimated position of the margin when we observe the ice
thickness and the position of the margin. We compare the absolute errors
obtained when we use 3D-Var with correction of the position of grid nodes and
when we use an ETKF. In both experiments, the results are improved with
respect to the position of the margin (compared to results detailed in Fig. ). No improvement (nor degradation) is observed for the ice
thickness at r=0.
In Fig. , we display the background error covariance matrices
used by the 3D-Var and ETKF methods to produce the analysis at time t=1500 years. At the previous assimilation time t=500 years, the analysis covariances
produced by both methods are very similar, as seen in
Figs. and . However,
because the 3D-Var error covariance matrix has a fixed form, the background
covariance matrix used by 3D-Var at the assimilation time t=1500 years has not
changed significantly. In contrast, it can be seen that the ETKF background
error covariance has fully evolved and contains much more information than
the 3D-Var error covariance matrix. This explains the better ability of the
ETKF to provide accurate estimates in the context of the moving point model.
Propagating the background error covariances using the ensemble statistics
ensures that the ETKF is a more reliable scheme than 3D-Var. This improvement
has a computational cost, however, as we now need to run the model Ne
times instead of once for 3D-Var.
Assimilating observations of the position of the margin
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 r=0 and the
position of the margin are shown in Fig. . In both cases,
assimilating observations of the position of the margin is beneficial to our
estimates of the margin and of the ice thickness profile close to the margin.
For example, the estimated position of the margin at time t=500 years has an
absolute error of 4.2 km for the ETKF (compared to 7.5 km previously).
Not surprisingly, it does not change the results for the ice thickness at r=0.
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 5.8 km instead of 7.1 km. Not
surprisingly it has no influence on the standard deviation for variables
close to the ice divide. The estimated correlation structure (not shown) is
also not modified by adding observations of the position of the margin in the
DA system.
Numerical experiments with an advanced configuration
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 investigate the case of a rapidly warming
climate over a short timescale.
Experimental Design
We generate observations from a new reference run. We use a non-flat fixed
bedrock whose elevation is defined by the equation
b(r)=1000m-1400m⋅r1000km2+700m⋅r1000km4-120m⋅r1000km6.
The reference run is generated from a realistic initial state obtained with
the following steps:
Start with an ice sheet profile following Eq. () with h0=2000 m, rmax=300 km
and nr=21 computational mesh points evenly spaced between r^1=0 and r^nr=300 km.
Run the numerical model with a fixed climate forcing, as defined in
Eq. (), where Tclim=4 ∘C until it reaches the
steady state (a 30000-year run with a Δt=0.01-year time step).
From this steady state, run the numerical model with a linearly warming climate forcing from Tclim= 4 ∘C with
dTclim/dt=0.02 ∘C yr-1 for an extra
100 years (Δt=0.01 years). The state obtained at the end of the run is
the initial state (see Fig. ).
Initial state used to obtain a 20-year reference run under a warming
climate as detailed in Sect. with nr=21 grid points and a
non-flat bed.
The reference is obtained by running the model over 20 years from the initial
state with a time step Δt=0.01 years and the same linearly warming
climate forcing as defined in Appendix A.2, with Tclim= 6 ∘C at initial time t=0 years, and
Tclim=6.4 ∘C at t=20 years (that is, Tclim=6+0.2t). Over
the 20-year run, the geometry of the ice sheet stays relatively similar to
the geometry of the initial state due to the slow dynamics of the model. The
ice sheet margin retreats from 1160.9 to 1158.6 km and the ice
thickness at the ice divide increases by 1.5 m.
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. (a) ETKF analysis at time t=10 years
compared with the reference. (b) Evolution of the ice thickness
at r=0 with time. (c) Evolution of the position of the margin
with time.
We generate observations of surface elevation, surface ice velocity and the
position of the ice sheet margin at times t=1,2,…,10 years from the reference run. The observations of the surface are taken at each
point including the margin with an added Gaussian noise (uncorrelated with
standard deviation σso=200 m). The observations of the surface ice
velocity are located at the midpoints between mesh points (so we have 20
observations of surface velocity). Observations are noised using a Gaussian
law (standard deviation σuso=30myr-1,
uncorrelated). For the position of the margin, the observational noise is
sampled from N(0,σro2) with σro=50 km.
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 r=0
and the position of the ice sheet margin.
Standard deviations and correlation matrix Corr estimated
from the analysis ensemble at time t=10 years in the advanced configuration
where we observe surface elevation. The matrix Corr has the same
structure as B defined by Eq. (). Both standard
deviations and correlation structures are similar to
Fig. .
Assimilating observations of surface elevation
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 xb=0.95xref(0) with a 5 %
smaller extent than the reference state. The flow-dependent background error
covariance matrix B is defined as in Eq. (). The
matrix Bh is defined as in Eq. () with a SOAR matrix
for Ch (σhb=200 m, Lh=240 km) and Br
is defined with a SOAR matrix for Cr (σrb=60 km, Lr=240 km). The matrix Brh is set to 0.
The ETKF uses an ensemble with 200 members. The initial ensemble is generated
by adding to xb a random noise drawn from the Gaussian law
N(0,B). The background covariance matrix
B is defined as previously, except for Br for which we
still use a SOAR matrix for Cr (Lr=240 km) but with
variances decreased near the ice divide following Eq. ()
(σrb=60 km and α=0.2). We tested different values for the
inflation parameter λinfla; the best results were obtained
with λinfla=1.01.
Evolution of the absolute error of the estimated ice thickness at
r=0 and the estimated position of the margin in the advanced configuration
where we assimilate surface elevations over the first 10 years. We compare
the absolute errors obtained when we use 3D-Var with the correction of the
position of grid nodes and when we use an ETKF. The ETKF performs better than
the 3D-Var for both variables.
ETKF results for the advanced configuration where observations of
surface ice velocity and the position of the margin are assimilated over the
first 10 years and a forecast is made for the following 10 years.
(a) ETKF analysis at time t=10 years compared with the reference.
(b) Evolution of the ice thickness at r=0 with time.
(c) Evolution of the position of the margin with time.
We first study the results obtained with the ETKF. At the end of the data
assimilation window, t=10 years, the ice thickness profile is retrieved well
everywhere by the mean of the ensemble and the reference profile is within
the ensemble spread (see Fig. ). We note that the estimate of the
ice thickness at the ice divide is improved by the first analysis. After time
t=7 years, however, the estimate is worsened by the analysis. This is
because the ensemble spread is too small from that time onwards. This can be
fixed by taking a larger inflation parameter λinfla, but
the estimates of other variables are then degraded. The estimated position
(mean) of the margin at t=10 years is 1158.0 km with an ensemble standard
deviation of 3.1 km. In comparison to the reference value at that time, r=1159.9 km, we see that the ETKF with a large ensemble performs well. The
quality of the estimates is also kept high during the forecast (from t=10 to t=20 years). For example, the absolute error on the position of the
margin is kept below 2.5 km over this time window.
With respect to the covariance matrix, the estimates seem to show a similar
behaviour to those of the experiment detailed in Sect. using
the ETKF where observations of ice thickness are assimilated (see Fig. ), but with a larger correlation length scale. The
similarity can be explained by the similarity of the construction of the
initial ensemble (the same structure for the background covariance matrix
B used to sample the Gaussian noise added to the background state)
and by the similarity of the observation operators for ice thickness and
surface elevation.
We now compare the ETKF with results obtained with 3D-Var. Absolute errors in
the ice thickness at r=0 and in the position of the margins are displayed
for both cases in Fig. . As in previous experiments, the ETKF
performs better than 3D-Var. For example, the absolute error for the ice
thickness at the ice divide stays below 60 m after t=1 year for the ETKF.
By contrast, the absolute error for 3D-Var can be up to 125 m. The same
statement remains valid for the absolute error in the position of the margin,
which stays below 8 km for the ETKF after t=2 years, yet can be up to 20 km for 3D-Var. We remark that, since the domain of the background state is
smaller than that of the reference state, 3D-Var does not assimilate all
available data. Indeed, the algorithm cannot incorporate observations outside
the background domain because of the form of the observation operator (see
Eq. ). This is not, however, the case for the ETKF, even if
the ensemble mean has a smaller domain than the reference domain, since in
this case there is at least one member of the ensemble with a bigger domain
than that of the reference. At the end, both approaches show a similar
accuracy in the forecast state after time t=10 years, showing again the
efficiency of both DA schemes.
Assimilating observations of surface velocity and position of the margin
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 Corr estimated
from the analysis ensemble at time t=10 years in the advanced configuration
where we observe surface ice velocity and the position of the margin. The
matrix Corr has the same structure as B defined by
Eq. (). Both standard deviations and cross-correlation
structures are different from those shown in Fig. .
Evolution of the absolute error of the estimated ice thickness at
r=0 and the estimated position of the margin when we observe surface ice
velocities and the position of the margin in the advanced configuration. We
compare the absolute errors obtained when we use 3D-Var with the correction
of the grid-node positions and when we use an ETKF. The ETKF performs better
than the 3D-Var with respect to the position of the margin, but 3D-Var gives
better results for the ice thickness at r=0 in this case.
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
B and the same initial ensemble as before. The observation
operator for surface velocities is non-linear (see Eq. ) and,
even though the ensemble is large, inflation is necessary in this case. We
take an inflation of λinfla=1.10. If the inflation is
taken any larger in this example, the ETKF analysis produces ensemble members
with a non-ordered grid and the experiment cannot be pursued.
We first study the results obtained with the ETKF. At the end of the DA
window, t=10 years, the ice thickness profile is retrieved well everywhere by
the mean of the ensemble, except near the ice divide at r=0 (see Fig. ). This is due to the relatively large uncertainty of surface
velocity observations near the ice divide compared to the reference value at
the same point (here σuso=30myr-1 and the
reference surface velocity near the ice divide is below
0.1myr-1). The estimated (mean) position of the margin at t=10 years is 1144.7 km with an ensemble standard deviation of 12.1 km.
This is an absolute error of 15 km, so it is worse than in the case where we
observed the surface elevation, but assimilating these data still provides
better estimates than those obtained with no assimilation. This comment
remains valid for the forecasts obtained after t=10 years since estimates of
the position of the margin are not degraded over the time window [10,
20 years].
Estimates of the standard deviations and covariances, as shown in Fig. , differ from those of the previous experiment (see
Fig. for comparison).
The reduction in the standard deviation for ice thickness variables close to
the ice divide is less significant than in the previous experiment. This is
due to the relatively large uncertainty of surface velocity observations near
the ice divide compared to the reference value at the same point. We remark
that assimilating observations of surface ice velocity together with the
position of the margin leads to an increased correlation length scale for ice
thickness variables and to a smaller correlation length scale for node
positions compared to the previous experiment. Finally, the cross-covariances
have smaller anti-correlations and positive correlations appear between ice
thickness variables in the interior of the ice sheet and between node
positions close to the margin. These differ significantly from the case where
we assimilate observations of surface elevation as a result of the difference
in observation operators.
We finally compare the ETKF with results obtained with 3D-Var. Absolute
errors in the ice thickness at r=0 and in the position of the margins are
displayed for both cases in Fig. . As in previous experiments, the
ETKF performs better than 3D-Var for the position of the margin,
but 3D-Var gives better results for the ice thickness at r=0 and performs
reasonably well overall in this non-linear context. The forecast trajectory of
the margin after t=10 is improved by DA in both cases. This
demonstrates again the robustness of our DA approach in the context of an ice
sheet modelled with a moving point numerical model.