Introduction
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) or ice2sea
e.g..
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 . Improving the reliability of the
model projections requires the model initial state to be better constrained
from observations. The problem is that observations are often uncertain,
sparse in time and space, and indirect, so that the model state depends on
many poorly determined physical parameters and boundary conditions.
Gradient-based optimisation methods, such as the control method
or the Robin inverse method
, are efficient means to constrain such
model parameters and boundary conditions. These methods have been implemented
and applied with success in ice flow models of different complexity in order
to infer the basal drag, one of the most uncertain model parameters
e.g..
However, remaining uncertainties lead to non-physical ice flux divergence
anomalies resulting in undesirable transient
effects in the free surface evolution. A solution to dissipate these
transients is to conduct a surface relaxation step prior to the projections
. This allows admissible flux divergence
rates to be reached but at the expense of the accuracy of the modelled
surface elevation and surface velocities which can then depart significantly
from observations after the relaxation step.
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
. Standard bedrock elevation maps for Antarctica
and Greenland are then produced by interpolation or Kriging, and report
standard errors ranging from a few tens of metres to several hundreds of metres
depending on the distance to observations and local topographic variability
. For comparison, the
uncertainty on the surface elevation is usually 1 order of magnitude lower
.
Because of theses large uncertainties, several methods have been proposed to
consider the bedrock elevation as an optimisation variable. For example,
derived the adjoint of the continuity equation for
the ice thickness. The depth-averaged velocities and surface mass balance are
then optimised to minimise the mismatch between modelled and measured ice
thicknesses. Surface velocity measurements are used as initial guess for
depth-averaged velocities, and (by construction) the flux divergence produced
by this approach is in equilibrium with the prescribed surface mass balance.
However, there is no constraint that the optimised velocities are a solution
of the stress equilibrium equations, so that, in general, the above method
does not guarantee that the flow divergence anomalies resulting from an ice
flow model initialised with the optimised fields will be reduced.
In their work, developed an iterative algorithm where the
discrepancy between the surface elevation predicted by the model and the
observations is used to correct the bedrock elevation. Thus, the method does
not rely on the accurate computation of the derivative of a cost function, as
in a control method, and is then more similar to nudging methods that have
been widely studied in the past decades in meteorology
e.g. and later in oceanography
. However, the method
proposed by does not use observed surface
velocities to control the model parameters.
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, developed a
least-squares inversion using analytical solutions for the transmission of
small-scale basal perturbations to the ice surface. This method has been
extended in a non-linear Bayesian framework by
and applied to an Antarctic ice stream by
. have tested the
performances of an ensemble Kalman filter on twin experiments using a
shallow-ice flowline model. The adjoint method has been tested by
and with models
of different complexity. All these methods usually show good performance in
reconstructing both basal friction and basal topography when using
observations of both surface elevation and surface velocities, so that mixing
between the two variables does not seem to be too problematic for realistic
applications . In addition,
and show better
performance when the rates of surface elevation change are also constrained
from observations.
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 and since it
uses the adjoint solution of the force balance equation. We use the shallow
shelf approximation to facilitate the derivation of the adjoint. Indeed, in
this case the ice thickness appears as a state variable, while it changes the
geometry of the domain for higher order approximations
. In its simplest formulation, the algorithm
minimises the misfit between model and observed surface velocities, but an
additional constraint where the flux divergence is close to a given surface
mass balance can be added. The second method is an algorithm combining
inversion of basal friction using the adjoint method and nudging of the
bedrock topography. The control from the surface velocity observations is
imposed by the adjoint step while the nudging step allows to decrease the
discrepancy between the flux divergence and the surface mass balance. The
main motivation of this second algorithm is its ease of implementation as no
inversion of the model with respect to the ice thickness is required. Our
objective is then to illustrate its ability to reconstruct the bedrock
topography by comparison with the results of the more mathematically founded
first algorithm. Both algorithms are implemented in the finite element
ice sheet/ice flow model Elmer/Ice . The
methods and algorithms are described in details in Sect. . To
test their performances, we design a twin experiment in Sect. . The
results are discussed in Sect. .
Methods
Direct model
For the force balance, we use the standard vertically integrated shallow
shelf approximation (SSA) equations . This
approximation neglects the effects of vertical shearing and is, hence, more
adapted to model the flow in areas where the friction is low, resulting in an
ice motion dominated by sliding. The horizontal velocity field (u,v) is a
solution of
∂∂x2Hν2∂u∂x+∂v∂y+∂∂yHν∂v∂x+∂u∂y-βu=ρgH∂zs∂x∂∂xHν∂v∂x+∂u∂y+∂∂y2Hν∂u∂x+2∂v∂y-βv=ρigH∂zs∂y,
where β is the friction coefficient, ν the vertically averaged
effective viscosity, ρi the ice density, g the gravity, and H=zs-zb the thickness, with zs and zb the top and bottom surface
elevations, respectively.
Natural boundaries are the calving fronts where the Neumann condition results
from the difference between the ice pressure and the sea water pressure:
4Hν∂u∂xnx+2Hν∂v∂ynx+Hν(∂u∂x+∂v∂x)ny=(ρigH-ρwgH0)nx4Hν∂v∂yny+2Hν∂v∂xny+Hν(∂u∂x+∂v∂x)nx=(ρigH-ρwgH0)ny,
where ρw is the water density, H0 the ice thickness below sea
level, and nx and ny the two components of the horizontal unit vector normal
to the calving front. Dirichlet boundary conditions are prescribed for other
non-natural boundaries. The continuity equation for the ice thickness is
given by
∂H∂t+∂(uH)∂x+∂(vH)∂y=a,
where a is the surface mass balance and accumulation/ablation at
the bedrock interface is neglected.
Inverse methods
The objective of the methods is to produce a model state from
Eq. () that best fits the observations of surface velocities and
the rates of change of ice thickness. To minimise the discrepancy between the
model and the observations, the optimisation parameter vector p
contains both the basal friction coefficient β and the bedrock
elevation zb (p=(β,zb)).
Cost functions
The misfit between the model and the corresponding observations is evaluated
using cost functions. The first cost function measures the difference between
modelled (u) and observed (uobs) surface velocities:
Jv=∫Γ12(|u-uobs|)2dΓ,
where Γ is the model domain.
The second cost function measures the misfit between modelled and observed
thickness rates of change:
Jdiv=∫Γ12∂H∂t-∂H∂tobs2dΓ.
The modelled rate of change of ice thickness ∂H/∂t is
evaluated from Eq. () as the difference between the flux
divergence solution of Eq. () and the prescribed surface mass
balance. Observed rate of change of ice thickness (∂H/∂t)obs can be estimated from surface elevation trends extracted from
radar altimetry measurements .
In general, both Eqs. () and () could be weighted with
error covariance estimates such as the one of .
However, this information is not often available. In this paper, observed
ice surface velocities (uobs) and observed rate of change of
ice thickness (∂H/∂t)obs are considered
perfectly known or perturbed with a Gaussian noise which would make
unnecessary the addition of a covariance term.
The objective is then to find the parameter vector p that minimises
Jv and Jdiv. This can be achieved in different ways as
illustrated in the following sections.
Adjoint method
The two cost functions have an implicit dependence on the parameter vector p
through the model surface velocities u which are solutions of
Eq. (). The gradient of the cost functions with respect to p can
be computed efficiently using the adjoint equations of Eq. (). The
derivation of the continuous adjoint equations and the gradient of
Jv with respect to the friction coefficient β can be found in
. This can be easily extended for the
computation of the gradient with respect to H.
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 . However, as the present direct solver
is equipped with a Newton linearisation of the ice viscosity so that it
remains self-adjoint , the Newton iterations are
not reversed in the adjoint code and we only keep the last iteration. The
adjoint code has been validated on standard tests by comparing the gradients
with those obtained from a finite difference evaluation. The agreement is
usually better than 0.1 %.
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 pi of
p, thus allowing to give preference to smooth solutions:
Jregi=12∫Γ∂pi∂x2+∂pi∂y2dΓ.
The second forces the optimisation variables to stay close to a certain prior
or background information pb. This background can be based on
observations or on empirical knowledge. This second regularisation term is
written as
Jbi=12∫Γ1σp2(pi-pi,b)2dΓ,
where σp is a spatial parameter allowing to give more or
less weight to the prior information pi,b.
The computation of the gradients of these two functionals with respect to
p is trivial. How these regularisation terms are weighted with
respect to the model–data misfit functionals Eqs. () and
() is described in more details with the description of the
algorithms in Sec. .
This minimisation is achieved using the quasi-Newton routine M1QN3
implemented in Elmer/Ice. This method uses an
approximation of the second derivatives of the cost function and is therefore
more efficient than a fixed-step gradient descent.
Nudging method
By definition, the steady-state solution of Eq. () where a
is replaced by the apparent mass balance a-(∂H/∂t)obs is the minimum for Jdiv. Running the model forward in
time with a constant forcing is then a simple way to minimise Jdiv,
equivalent to a relaxation step. Here, we assume that the surface elevation is
known so that computed changes in ice thickness are used to correct the
bedrock elevation zb. During this process, the ice thickness can
substantially deviate from observations. Nudging methods, also called
Newtonian relaxation, can remedy to this problem by constraining the
thickness to fit observations through an additional callback term in
Eq. (), which now writes
∂H∂t+∂(uH)∂x+∂(vH)∂y=a-∂H∂tobs-k(H-Hobs),
where the coefficient k defines the amplitude of the callback at each node
of the model. These methods imply a trade-off, adjustable through k,
between model physics and observations. The callback term can depend on many
different criteria such as observation accuracy or distance to observation
. Here, we take k as a Gaussian function of
the distance to the closest observation so that the callback is maximum
where an observation is available and decreases to zero far from all
observations. The choice of the variance for the Gaussian function is
discussed in Sect. .
Algorithms
From the methods presented in the previous section we design two algorithms
to infer simultaneously the friction coefficient β and the bedrock
elevation zb. To ensure that the friction coefficient remains
positive during the inversion, we use the following change of variable
β=10α.
Adjoint method with two parameters (ATP)
This algorithm uses the gradients of the cost functions derived using the
adjoint method to optimise both α and zb. For the
regularisation, a constraint on the smoothness is imposed for α using
Eq. () while a constraint on the background information is imposed
for zb using Eq. (). The total cost function then
writes
JATP(α,zb)=Jv+γJdiv+λαJregα+λzbJbzb,
where γ is a constant fixed to give a similar weight to Jv
and Jdiv while λα and λzb are
two constants allowing to adjust the weight given to the regularisation
terms. Following , several pairs
(λα, λzb) are tested using a L-curve
approach, and optimal values are taken from the combinations that avoid two
extremes: overfitting of the observations or excessive regularisation.
Adjoint-nudging coupling (ANC)
In this algorithm, the adjoint method is first used to optimise α only
by minimising the following total cost function
JANC(α)=Jv+γJdiv+λαJregα.
The bedrock elevation is then updated using the nudging method by solving
Eq. () for a given time period T. T should be neither too
short nor too long to allow to reduce Jdiv without overfitting
observations. The sensitivity of the method to T is discussed in the
results section.
Reference (solid lines) and initial (dashed lines) state for
(a) the bedrock elevations zb, (b) the estimated
basal traction τb=βu, and (c) the surface velocities. In
(a), synthetic observations every 10 km are the plain black
circles. In (c) the observed velocities are depicted by the
circles and the shaded green curve is the absolute difference between
observed and reference surface velocities (right axis).
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These two steps are then repeated iteratively until changes in Jv
and Jdiv between two iterations are less than 1 %.
Manufactured data sets
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.
Reference experiment
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 . It is
also the fastest Greenland glacier with a terminus velocity greater than
13 kma-1 . The
flowline is 550 km long and runs from the ice divide to the ice
front. The surface and bedrock elevations are taken from available digital
elevation models . The basal friction coefficient
field is first adjusted so that the model velocities fit observed velocities
. To have realistic thickness rates of change,
the free surface is relaxed to steady state. The surface mass balance a in
Eq. () has been calibrated so that the steady state is close
to the initial geometry, and is meant to take into account the flow
convergence or divergence along the flowline. The steady-state solution is
used as the reference of the twin experiment.
The geometry is discretised through a mesh of 500 linear elements,
increasingly refined to the front of the glacier. The element size decreases
from ∼ 2 km in the upper part of the glacier to ∼ 400 m down to
the front.
Results will only be presented on the first 100 km upstream of the
glacier front where velocities are above 100 ma-1 and where the
SSA is more appropriate but the inversion is done all along the flowline up
to the ridge.
Synthetic observations
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 (rmsdiv) as a
function of the mean error on velocity (rmsu) for the 255 pairs of
regularisation parameters (λα, λzb).
Colour scales show the normalised regularisation terms
(a) Jregα and (b) Jzb
(0 corresponds with the lowest value and 1 with the highest value obtained
with the 255 pairs). The chosen value (λα=1011,
λzb=107) is shown with a black circle.
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Surface velocities
Surface velocities are assumed to be observed at the same resolution as the
reference simulation but with a white Gaussian noise with a mean μ=0
and a standard deviation σ=50 ma-1. This corresponds to
a root mean squared (rms) error of 47.8 ma-1 for the entire
flowline. The reference and noisy observed surface velocities are shown in
Fig. c together with their absolute difference.
Surface mass balance and thickness rate of change
The surface mass balance, a, and thickness rate of change, (∂H/∂t)obs, in Eq. () are assumed to be perfectly
observed. As the reference simulation corresponds to a steady state, (∂H/∂t)obs=0. However, the methods are also tested in
Sect. for cases where (∂H/∂t)obs≠0 to show their ability to initialise the model when the
flux divergence is not in equilibrium with the surface mass balance.
Surface and bedrock elevations
The surface elevation is assumed to be perfectly observed. For the bedrock
elevation zb, we simulate observations representing airborne radar
measurements crossing the flowline. Bedrock elevations are sampled every
10 km with a Gaussian noise centred on zero and with a standard deviation of
σ=50 m. This leads to a rms error of 62.4 m on the
55 observation points of the entire flowline. This error is similar to the
errors given in practice on recent bedrock elevation maps
. For the mesh nodes between
the observations, the bedrock is linearly interpolated as shown in
Fig. a. This is used as the first guess for the
inverse methods and as the background information for the regularisation in
Eq. ().
Model parameters
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 β is
provided by using the driving stress to estimate the basal shear stress:
βini(x)=ρigH(x)|θ(x)||u(x)|,
where H(x), θ(x), and u(x) are, respectively, the ice thickness, the
surface slope and the surface velocity at position x. The reference and
initial values are shown in Fig. b.
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-1.
The average relative error on the basal shear stress is measured as
ετb=1L∫Γ|τb|-|τb,ref||τb,ref|dΓ,
where τb,ref is the basal shear stress in the reference
experiment and L the length of the flowline. The relative error on the
basal shear stress with our initial estimate of the basal friction
βini is 394 %. The performances of the two algorithms in
reducing these initial errors are presented in the following section and will
be compared to these initial errors.
Results of the ATP algorithm with (orange) and without (red)
optimisation of Jdiv, i.e. γ=1 or γ=0, respectively,
in Eq. (): (a) absolute difference between observed and
model velocities, (b) estimated basal traction, and
(c) estimated bedrock elevation. The green shaded area is the
difference between the noisy reference velocities and the true velocities.
The green solid lines are the reference values and the black dashed line is
the initial guess for the bedrock elevation.
![]()
Results
Adjoint method with two parameters (ATP)
A set of 255 pairs (λα, λzb) is tested to
adjust the weighting of the regularisation terms of Eq. (). The
misfits on the different cost functions of Eq. () for the
different pairs (λα, λzb) is given in
Fig. . Both graphs show that most of the pairs
fitting well the observed velocities can also adequately reproduce the
observed rate of change of the ice thickness.
Figure b also shows that smaller misfits on
Jzb clearly involve higher rms misfits on the ice surface
velocities (rmsu) and on the rate of change of the ice thickness
(rmsdiv). On the contrary, Fig. a does
not show a clear relation between the magnitude of Jregα
and the magnitude of rmsu and rmsdiv. Both graphs also show a
high density of pairs for small rmsu and rmsdiv. However, the
pair (λα=1011, λzb=107) seems to
come off the others, giving a good trade off between data fitting and
regularisation. Notice that the constant γ is fixed to 1 since
Jv and Jdiv have the same order of magnitude.
The optimisation of both α and zb simultaneously allows a
rms misfit of 49.7 m a-1 on velocities to be reached, very similar to
the observation rms error, showing no overfitting of velocity data. The rate
of change of ice thickness misfit is also largely decreased with a rms value
of 19.2 m a-1. The resulting basal traction τb and
zb as well as the misfit for the surface velocities are given in
Fig. . The basal traction variability is accurately reproduced
with a corresponding average relative misfit of only 25 % along the entire
flowline with respect to the reference basal shear stress
τb,ref, i.e. more than a 10-fold decrease of the initial misfit.
We only notice local overestimations of slipperiness in bedrock pits without
significant impacts on the flow velocities. Indeed, under a defined value of
β corresponding to a nearly perfectly sliding case, an additional
reduction in friction has no impact on the flow. The same reasoning applies
to a nearly perfectly sticky case, where an increased friction would not
involve more decrease of the velocity. The bedrock elevation zb is
well reconstructed in the first 50 km upstream of the glacier front. The
discrepancy with respect to the reference bedrock is larger upstream where
the cost function Jv is less sensitive because of lower velocities.
This could possibly be improved by using a cost function measuring the
logarithm of the misfit as in , but with a
greater risk of fitting noise since the relative observation error is higher
in these regions.
Results of the ANC algorithm (purple): (a) absolute
difference between observed and model velocities, (b) estimated
basal traction, and (c) estimated bedrock elevation. The green shaded
area is the difference between the noisy reference velocities and the true
velocities. The green solid lines are the reference values and the black
dashed line is the initial guess for the bedrock
elevation.
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In order to assess the influence of accounting for Jdiv on the
method, the optimisation is repeated without the Jdiv term in the
total cost function Eq. (). The pair (λα,
λzb) is kept equal to the previous case since the optimum
is hardly affected by the absence of Jdiv in the total cost
function. The result is given and compared to the previous one in
Fig. . The friction coefficient β is again pretty well
reconstructed, with a corresponding relative average misfit of 31 % on
basal shear stress τb to be compared to the 25 % obtained with
the optimisation of the Jdiv term. However, zb shows
non-consistent high frequencies involving a higher discrepancy with respect
to the reference bedrock elevation in the case of optimising
Jdiv. Therefore, the optimisation Jdiv has a clear
regularisation effect on the parameter zb, by reducing the
non-consistent high-frequency oscillations of the solution.
Introduction of a Gaussian noise on (∂H/∂t)obs
has been investigated in order to assess its effect on the optimisation of
Jdiv. Different levels of standard deviation σ have been
tested. Results show that the optimisation is little affected by this noise
even for standard deviations σ going up to the same order of magnitude
as the surface accumulation a. Introduction of systematic bias on
(∂H/∂t)obs in a physically acceptable range, i.e.
of the same order of magnitude as surface accumulation a, also have
few consequences on the optimisation.
Adjoint-nudging coupling (ANC)
The steps for the optimisation of α only are conducted with a value
λα=5×109, which allows a good agreement between the
different cost functions and a value γ=1.
In addition to the regularisation parameters of Eq. (), ANC
algorithm depends on the time period for the nudging steps T and the
variance of the Gaussian k in Eq. (). The nudging period T
impacts the convergence on Jv and Jdiv after each cycle.
The convergence is substantially similar for T from 1 to 4 years. Longer
periods mainly involve a worse minimisation of Jv since there is no
control on velocities during nudging. Shorter relaxation times do not involve
sufficient change of zb inducing a lower minimisation of
Jdiv for a given number of cycles. Therefore, a relaxation time T=1 year is adopted, which seems sufficient to allow significant changes of
zb without too much adaptation to the previous intermediate value
of the friction coefficient. The algorithm is stopped after 10 cycles,
corresponding to the stopping criterion of Sect. . For a
given T period, tests show that variance values of the Gaussian k in
Eq. () larger than 1 km are excessive and induce non-physical
callback amplitudes when departing from observations. After a few cycles,
the resulting bedrock induces an increase between modelled and observed
velocities that cannot be overcome by the basal drag inversion. Variance
smaller than 1 km has little impact on the final result in terms of cost
functions. However, among the acceptable values, the 1 km variance gives the
best agreement between misfit on the surface velocities and misfit on the
rate of change of the ice thickness.
The five new references build from a 5-year perturbation of the
initial reference by an increase of the friction parameter: β=2βref (green), β=3βref (red), β=4βref (blue lines), β=5βref (purple), and
β=10βref (orange). New references for (a) the
thickness rate of change for the different perturbations,
(b) velocities (without observation noise), and
(c) friction coefficients β.
![]()
The model is in good agreement with observations with a rms misfit of
46.1 ma-1 in the range of observation noise for velocities, and
15.8 ma-1 for thickness rates of change. The basal shear stress
τb is close to the reference one despite exacerbated variations
at some locations. The corresponding relative average misfit with respect to
the reference τb,ref is 30 % for the entire flowline. The
reconstructed bedrock elevation zb is also close to the reference
on almost 100 km upstream of the front of the glacier. This reflects,
especially in fast flowing region, a real improvement of the basal knowledge
with respect to the first guess. Moreover, the use of nudging, instead of the
adjoint method, does not show the same problem of non-sensitivity in regions
of slower flow velocities, as mentioned in Sect. . Note,
however, that zb significantly departs from the reference bedrock
elevation from 80 to 100 km to the front, strongly linked to the
poorer fit of β (see Fig. ).
As for ATP, introduction of a Gaussian noise in the observed thickness rate
of change (∂H/∂t)obs has also been tested.
Results show no significant impacts on the optimisation. Nevertheless,
introduction of systematic bias in (∂H/∂t)obs
has direct consequences on the nudging steps inducing an offset of
zb of the range of the systematic bias cumulated on the nudging
period T. ANC is therefore more sensitive to systematic bias than ATP.
Range of values for ATP algorithm for the five perturbations of the
friction coefficient β. (a, b) Minimum (dark orange shade) and
maximum (light orange shade) of absolute difference between observed and
model velocities and relative error for τb, respectively.
(c) Range of values for bedrock elevation zb (orange
shade). The green solid line is the reference value and the black dashed line
is the initial guess for the bedrock elevation.
![]()
Same as Fig. 5 but for the ANC algorithm.
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Further sensitivity experiments
In order to evaluate the efficiency of both algorithms in transient states,
we construct new reference cases where (∂H/∂t)obs≠0. This is achieved by multiplying βref by a
factor of 2, 3, 4, 5, and 10. As a consequence, increasing the basal friction
involves a disequilibrium of the glacier, an ice thickening, and a decrease of
ice flow velocities.
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 (∂H/∂t)obs≠0 are in the same order of magnitude as the
tuned surface mass balance a. The five new reference cases are presented in
Fig. .
The results of the optimisations for the five cases of perturbation are shown
in Fig. for ATP and
Fig. for ANC. The velocity misfit for ATP
increases with the amplitude of perturbation with rms values between 47.8 and
52.3 ma-1 while the rms misfit for thickness rate of change
increases from 12.7 to 21.8 ma-1. ANC reaches rms misfits from
45.9 to 47.2 ma-1 for velocities and 12.5 to
21.5 ma-1 for thickness rate of change. The friction coefficient
β is well reconstructed for both methods. The corresponding average
relative error (with respect to each reference) on basal shear stress
varies from 22 to 30 % for ANC and 20 to 28 % for ATP, still according to
the amplitude of the perturbation. Both algorithms also allow to improve the
knowledge of the bedrock elevation zb with regard to the first
guess. We notice a tendency to overestimate the amplitude of bumps and pits
in some locations which generally corresponds to an underestimation in the
amplitude of variations of β. This latter behaviour highlights the
limits of the algorithms and the difficulty of distinguishing the effects of
two basal parameters as closely linked as the friction and the bedrock
topography. This behaviour had been already highlighted in
and where a
higher ice thickness with respect to the reference is compensated by a higher
basal friction, and conversely as well.
Evolution of ∂H/∂t after 1 year (a) and
10 years (b) of prognostic simulation and the resulting mismatch
after 10 years between surfaces obtained with three different initial states
and reference surface (c). The orange and purple lines give the
results for ATP and ANC. The red line gives the result for inversion of
β only.
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Flow divergence in transient model
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 zb equals to
the a priori zbb. This third initialisation, called
“β only” involves a rms misfit on velocities of
43.3 ma-1 and an average relative error
ετb,ref of 36 % on basal shear stress, similar to
the ATP and ANC initial states. However, the rms misfit on the thickness rate
of change is significantly higher, 147.8 m a-1.
Ice surface elevation zs after 10 years of prognostic
simulation for three different initial states: initialisation with ATP
algorithm (orange line), with ANC algorithm (purple line), and with the
inversion of β only (red line). The green line is the reference surface
elevation. The figure focuses on the first 50 km next to the front of the
glacier.
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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. a and b, respectively, while the mismatch
on the surface elevation after 10 years is shown in
Fig. c.
ANC and ATP initial states involve thickness rates of change much closer to
zero than the optimisation of “β only”. This also leads to a lower
mismatch εs on the surface elevation with respect to the
reference after 10 years of simulation. Indeed, this mismatch is well below
20 m for both ANC and ATP, except on a few kilometres in the upstream
region, whereas the optimisation of β only gives rise to a mismatch
globally above 20 m with some regions exceeding 50 m.
In that way, the two algorithms implemented in this study show substantial
improvements compared to the optimisation of “β only”. We especially
notice a better reproduction of low-scale variations of the surface elevation
due to the transfer of similar variations from the bedrock elevation
zb (Fig. ). These variations tend to
disappear with the optimisation of the friction only, giving rise to a lower
resolution of the surface. However, we should point out that this direct
transfer of bedrock variations to the surface is a consequence of the SSA
ice flow model used and that a full Stokes model would produce a more
diffusive transfer response.
Conclusions
The presented algorithms allow the reconstruction of two poorly known
parameters: the bedrock topography zb and the friction coefficient
β at the same time.
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 zb. The two
algorithms aim to infer the set of parameters which minimises the misfit
between the model and the corresponding observations of ice surface
velocities and thickness rates of change. If the optimisation of ice surface
velocities is usually sufficient to infer β, the inference of a second
parameter requires more information to distinguish the effects of each
parameters on the flow. Observations of rates of change of ice thickness are
necessary to allow optimising zb as well.
The two algorithms are based on the optimisation of the friction coefficient
β with the adjoint method. The bedrock geometry zb is
reconstructed in two different ways, again with the adjoint method for the first
algorithm (ATP) and with a nudging method based on mass conservation equation
for the second one (ANC).
We have shown that the ATP algorithm is capable to well reproduce β and
the corresponding basal shear stresses, while the bedrock elevation
zb is only well reproduced in high velocities regions. The lower
the velocity, the harder for zb to depart from its initial
background value. The iterative algorithm coupling adjoint method and nudging
(ANC) gives results that are just as good. Moreover, ANC allows a better reconstruction of
the bedrock geometry zb in most regions. This is a very good sign
for an adaptation of the method to non-depth-integrated flow models such as
full Stokes models where the bedrock topography is no more a state variable
but affects the domain geometry making the derivation of the adjoint even
more demanding . Indeed, there is no need to
inverse a shape variable like bedrock topography which is a usual obstacle to
adjoint-based methods.
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.