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Calibration of a higher-order 3-D ice-flow model of the
Morteratsch glacier complex, Engadin, Switzerland
H. ZEKOLLARI,
1
P. HUYBRECHTS,
1
J.J. FU
¨RST,
1
O. RYBAK,
1
O. EISEN
2
1
Earth System Sciences & Departement Geografie, Vrije Universiteit Brussel (VUB), Brussels, Belgium
E-mail: harry.zekollari@vub.ac.be
2
Alfred-Wegener-Institut fu
¨r Polar- und Meeresforschung (AWI), Bremerhaven, Germany
ABSTRACT. We have reconstructed the ice thickness distribution of the Morteratsch glacier complex,
Switzerland, and used this to simulate its flow with a higher-order 3-D model. Ice thickness was
measured along transects with a ground-penetrating radar and further extended over the entire glacier
using the plastic flow assumption and a distance-weighted interpolation technique. We find a maximum
ice thickness of 350
52.5 m for the central trunk of Vadret da Morteratsch, resulting from a bedrock
overdeepening. The average thickness of the glacier complex is 72.2
18.0 m, which corresponds to a
total ice volume of 1.14
0.28 km
3
. The flow of the glacier is modelled by tuning the rate factor and the
sliding parameters taking into account higher-order terms in the force balance. The observed velocities
can be reproduced closely (root-mean-square error of 15.0m a
–1
,R
2
= 0.93) by adopting a sliding factor
of 12
10
–16
m
7
N
–3
a
–1
and a rate factor of 1.6
10
–16
Pa
–3
a
–1
. In this setting, ice deformation accounts
for 70% of the surface velocity and basal sliding for the remaining 30%. The modelled velocity field
reaches values up to 125 m a
–1
, but also indicates an almost stagnant front and confluence area, which
are crucial for understanding the ongoing glacier retreat.
INTRODUCTION
Recent worldwide glacier retreat is one of the clearest
consequences of global warming (Lemke and others, 2007).
Alpine glaciers in particular have experienced a strong retreat
since the end of the Little Ice Age (LIA), which has intensified
over the last few decades (Paul and others, 2004; Bauder and
others, 2007; Huss and others, 2010). The consequences of
this retreat are expected to be very large and dramatic,
extending far beyond mere touristic impacts (Elsasser and
Bu
¨rki, 2002). Over the next century, mountain and valley
glaciers are expected to make an important contribution to
sea-level rise (Kaser and others, 2006; Radic´ and Hock, 2011;
Jacob and others, 2012). They are also important water
suppliers (Lemke and others, 2007), as the annual glacier
melting cycle provides water in valleys during the warmest
and driest seasons, when it is needed most for consumption,
irrigation and cooling. In order to predict the future evolution
of glaciers, it is crucial to be able to correctly model and
understand their present-day behaviour.
A prerequisite for ice-flow modelling is adequate know-
ledge of ice geometry over the area of the glacier. Often this is
not well known, as there is a lack of direct field measure-
ments, related to difficult and labour-intensive working
conditions. However, several techniques exist to deduce
the ice thickness distribution without direct field measure-
ments, such as those developed by Li and others (2011, 2012)
or the one by Farinotti and others (2009), based on glacier
mass turnover and principles of ice-flow mechanics. These
techniques give good overall results, but are insufficient
when applying complex modelling methods at high spatial
resolution. In this paper, we reconstruct the thickness field
over the Morteratsch glacier complex, Switzerland, based on
an extensive field dataset, a high-resolution digital elevation
model (DEM) and theoretical relationships such as the plastic
flow assumption for central glacier regions.
Several modelling studies have attempted to reconstruct
the dynamics of alpine glaciers, extending from simple one-
dimensional (1-D) flowline models (Oerlemans, 1986;
Huybrechts and others, 1989) through two-dimensional
(2-D) models based on the widely used shallow-ice
approximation (SIA) (e.g. Le Meur and Vincent, 2003) to
complex three-dimensional (3-D) full-Stokes (FS) models
(e.g. Gudmundsson, 1999; Le Meur and others, 2004; Jouvet
and others, 2009, 2011). In the SIA, horizontal gradients in
stress are neglected, which reduces the complexity of the
force balance and the boundary conditions. However, this
assumption is based on a small aspect ratio between vertical
and horizontal characteristic dimension (0th-order approx-
imation), which does not apply to most traditional mountain
glaciers as ice thickness is (locally) often of the same order of
magnitude as glacier width. Le Meur and others (2004) have
shown that this becomes problematic (i.e. the SIA solution
diverges from the FS solution) for mountain glaciers with
bedrock slopes above 0.2 and when the maximum ice
thickness is of the same order as the glacier span. Another
concern of the SIA is that the flow solution is local, ignoring
the effect of membrane stresses, yielding unrealistic results in
a diagnostic model set-up.
To model the dynamics of alpine glaciers it is therefore
more favourable to include longitudinal and transverse stress
gradients in the force balance. Here we use a Blatter/Pattyn
type of higher-order (HO) 3-D model (Blatter, 1995; Pattyn,
2003; Fu
¨rst and others, 2011), which is applied to the
observed geometry in order to reproduce the flow field of
Morteratsch glacier. The model calibration occurs through
the tuning of flow and sliding parameters and is based on
measured surface velocities (e.g. Gudmundsson, 1999). In
other work this is often not the case and parameter values
are taken from the literature or the calibration is based on
indirect field evidence. Examples of the latter are the use of
past glacier extents (e.g. Le Meur and Vincent, 2003), which
adds a mass-balance uncertainty to the calibration. FS
modelling studies have been performed for mountain
glaciers elsewhere (e.g. Zwinger and others, 2007; Zwinger
Annals of Glaciology 54(63) 2013 doi:10.3189/2013AoG63A434 343
and Moore, 2009), but here also the calibration was not
based on direct field measurements.
In this paper we present a new 3-D HO model for the
Morteratsch glacier complex. The ice thickness field is
reconstructed from an extensive set of radar transects
collected from over a decade of field measurements. We
then use observations of surface velocity to calibrate
constitutive and basal sliding parameters in the model.
LOCATION, DATA AND FIELD MEASUREMENTS
The Morteratsch glacier complex is situated in the south-
eastern Swiss canton of Graubu
¨nden (Fig. 1). It consists of the
twin glaciers Vadret da Morteratsch and Vadret Pers. In 2010,
the Morteratsch glacier complex had a length of 6.3 km and
covered an area of 16 km
2
(Glaciological Reports, 1881–
2011). Vadret Pers flows into Vadret da Morteratsch at a
distance of about 1.5 km from the snout. The glacier front is
currently at 2050 m elevation, while the highest parts reach
4000 m, culminating at surrounding peaks such as Piz
Bernina (PB; 4049 m), Piz Zupo (PZ; 3996 m) and Piz Palu
¨
(PP; 3905 m). Like most alpine glaciers, the glacier has
undergone a strong retreat since the end of the LIA, with a
total retreat of 2.3 km and only 5 years of advance since
1878. Recently this trend has intensified and over the last
decade the glacier retreated more than 300 m (Glaciological
Reports, 1881–2011).
A 25 m resolution DEM from 2001 (acquired from the
Swiss Federal Office of Topography, swisstopo) is used to
obtain the ice mask and surface elevation. This resolution is
kept for the ice thickness field reconstruction and for the HO
modelling. The fact that the DEM dates from 2001 implies
that the thickness field and dynamics reconstruction also
represent the state of the glacier in the early 2000s.
Therefore, where possible, thickness and surface velocity
measurements from the beginning of the last decade (2001–
04) are used. However, except for the lowest parts of the
glacier, changes in ice thickness and surface velocity are
rather limited over the last 10 years.
Thickness measurements consist of transects and point
measurements obtained with a RAMAC monopulse bistatic
ground-penetrating radar (GPR) system (Mala
˚Geoscience)
in combination with a 5 MHz ice-penetrating point radar
system (Narod and Clarke, 1994). The RAMAC GPR data
were acquired in common-offset mode with nominal 12.5,
25.0 and 50.0 MHz unshielded antennas for Vadret da
Morteratsch and in addition a 100 MHz antenna for Vadret
Pers. For the point radar, a monopulse transmitter was used,
generating 1600 V pulses across a resistively loaded 10 m
antenna. The data acquisition and post-processing, such as
rectifications for migration and gain corrections, followed
the same methods as described in Eisen and others (2003).
Regions of the glacier where the ice thickness is known from
radar measurements are indicated in Figure 2. The un-
certainty in these measurements depends on the equipment
(highly frequency-dependent) and the bedrock characteris-
tics, but is usually on the order of 15–20% (Moran and
others, 2000).
Annual surface velocities are deduced from stake posi-
tions, which were measured each year at the end of the
ablation season (usually the beginning of October). A
Fig. 1. Map of the Morteratsch glacier complex showing glacier
elevation together with height contours of the surrounding area at
200 m intervals based on a DEM acquired from swisstopo repre-
senting the situation in 2001. The map uses the Swiss CH1903
coordinate system. Note that only ice that flows into the glacier
complex is considered in the mask and that isolated small glaciers,
like Vadret da la Fortezza (situated between Vadret Pers (P) and
Vadret da Morteratsch (M)), are not taken into account. The highest
surrounding peaks are Piz Bernina (PB; 4049 m), Piz Zupo (PZ;
3996 m) and Piz Palu
¨(PP; 3905 m). The inset shows the location of
the glacier in Switzerland.
Fig. 2. Ice thickness derived from radar measurements (coloured
points and lines) and locations of the 20 surface velocity points
used for model calibration (open circles). The latter correspond to
the average position of a network of mass-balance stakes. Grey
areas are either ice-free or contain local ice that does not contribute
to the flow of the glacier complex.
Zekollari and others: Calibration of higher-order 3-D ice-flow model344
Trimble GPS Pathfinder Pro XR was used to determine stake
locations. The positions were differentially corrected with
nearby Swiss reference stations (located at Samedan or
Zimmerwald), which typically reduces the horizontal
uncertainty to 0.5–1.0 m. Only measurements from the
period 2001–04 were retained for the ice model calibration,
except for a few locations where stakes were not available
during this reference period. For these, more recent
measurements are used, which are considered to be
representative of glacier flow in 2001. All the observed
surface velocities used for model calibration are summar-
ized in Table 1. Their locations are shown in Figure 2.
RECONSTRUCTION OF GLACIER THICKNESS AND
BEDROCK ELEVATION
To extend the ice thickness observations (Fig. 2) and obtain a
continuous thickness field for the entire glacier, several steps
are taken as indicated on the flow chart in Figure 3. First, the
25 m grid is filled with the direct field observations from the
radar measurements according to the nearest-neighbour
method. These are mostly located in the glacier centre. For
gridpoints adjacent to the margins we assume an ice
thickness of 5 m. This is realistic for gridpoints situated
12.5 m from the margin (half of the horizontal resolution).
Next, central flowlines are generated for all glacier
segments. To do this in an automated way, the discrete
Laplacian of the distance to the closest margin (d
i,j
)is
determined for each gridcell (index i,j) situated more than
125 m from the margin. On a transect, the minimum of the
discrete Laplacian corresponds to the point farthest away
from the margin, where we assume the ice to be the thickest.
The discrete Laplacian (l
i,j
) is calculated using a 2-D finite-
difference approximation of the Laplace operator:
li,j¼1
4diþ1, jþdi1, jþdi,jþ1þdi,j1
di,j:ð1Þ
Cells for which this value is lower than –5m are considered
to be located on the central flowline. For several locations
along this central flowline the ice thicknesses are known
from radar measurements. These measurements are used to
reconstruct the ice thickness for other locations along the
central flowline where the ice thickness was not measured.
Under plastic flow theory there is a relationship between ice
thickness and surface gradient, given by the yield stress
0
.
In the SIA this reduces to
0¼gHrh,ð2Þ
where is ice density (890 kg m
–3
for temperate glaciers; Le
Meur and others, 2004), gis the gravitational constant
(9.81 m s
–2
), His the ice thickness (m) and rhis the surface
elevation gradient (known everywhere from the DEM, taken
over a horizontal distance of 100m). The yield stress
0
is
first calculated for central regions where radar measure-
ments are available. We obtain a typical value of 150 kPa
(144.93 27.85 kPa), in good agreement with Cuffey and
Paterson (2010). For locations along central flowlines
situated more than 200 m from a radar measurement, yield
stresses are determined by interpolating the spatially vari-
able yield stresses obtained from radar measurements along
the central flowline. These are then inverted to obtain ice
thicknesses following Eqn (2).
In a subsequent step, an inverse quadratic distance
interpolation of ice thickness is performed between the
central flowlines and the margins at regular intervals, except
for regions where the ice thickness is known from the field
measurements. This ensures a parabolic cross profile as
already indicated by the available radio-echo sounding data.
Finally, any remaining gaps in the thickness field (because
the inverse quadratic distance interpolation was not applied
for lines between central regions and margins that cross
known measurements) are filled with a bilinear inter-
polation. From a rock outcropping that occurred in 2006,
it is known that ice thickness close to Isla Persa (western side
of lower Pers glacier) is overestimated in our reconstruction,
which was rectified with a minor manual correction. Finally,
a small-scale linear smoothing is applied with a window of
Table 1. Measured annual surface velocities and their respective
errors derived from stake positions. The stake locations given in the
second and third columns correspond to the average position of the
stakes over the respective years. In cases where the velocity of a
particular stake is measured over several years, the average is taken
and attributed to the midpoint location, except for stakes M56 and
M57. These are located at the foot of the Labyrinth icefall in a
region of strong spatial velocity gradients and are considered as
separate calibration points
Stake
label
CH1903
easting
CH1903
northing
Surface velocities
mm ma
–1
M11 791 751 144 723 6.2 0.6
(2001–02)
6.0 0.6
(2002–03)
4.8 0.7
(2003–04)
M12 791 738 144 280 16.1 0.9
(2001–02)
13.1 0.9
(2002–03)
14.9 0.5
(2003–04)
M20 791 429 142 212 45.3 0.6
(2002–03)
41.7 0.7
(2003–04)
M23 791 676 142 453 37.2 0.6
(2002–03)
34.5 0.8
(2003–04)
M44 790 279 139 085 4.1 0.8
(2002–03)
4.1 0.6
(2003–04)
M46 790 796 138 968 24.2 0.9
(2002–03)
M56(1) 791 236 141 670 84.4 0.9
(2002–03)
M56(2) 791 278 141 781 64.9 0.8
(2003–04)
M57(1) 791 228 141 314 141.0 0.5
(2003–04)
M57(2) 791 242 141 476 91.6 0.7
(2005–06)
M58 791 217 141 255 135.4 0.6
(2009–10)
M62 791 841 143 529 34.7 0.7
(2001–02)
51.0 0.6
(2002–03)
34.6 0.5
(2003–04)
P21 794 315 141 856 12.8 0.7
(2002–03)
12.2 0.6
(2003–04)
P22 794 012 141 394 41.9 0.8
(2001–02)
44.9 0.5
(2002–03)
44.5 0.5
(2003–04)
P23 793 790 140 641 51.4 0.6
(2001–02)
46.5 1.0
(2002–03)
P25 793 868 140 578 36.9 0.8
(2006–07)
P26 795 035 141 561 16.1 0.5
(2008–09)
P32 793 706 142 553 42.0 0.6
(2001–02)
42.0 0.9
(2002–03)
39.0 0.6
(2003–04)
P33 793 792 142 311 51.4 0.6
(2001–02)
46.2 0.6
(2002–03)
45.7 0.6
(2003–04)
P34 793 025 142 910 34.8 0.7
(2003–04)
36.1 0.4
(2004–05)
Zekollari and others: Calibration of higher-order 3-D ice-flow model 345
100 m. The bedrock elevation of the Morteratsch glacier
complex (Fig. 4b) is then derived by subtracting the
reconstructed ice thickness field (Fig. 4a) from the surface
DEM. In Figure 4b, we also determined the bedrock
elevation below Vadret da la Fortezza using the same
principles as described above with a yield stress of 150 kPa.
This gave ice thicknesses up to 30 m.
From the DEM, the 2001 area of the Morteratsch glacier
complex is 15.8 km
2
. The average ice thickness we
obtain is 72.2 18.0 m and the total glacier volume is
1.14 0.28 km
3
. The thickest ice is found in the central part
of Vadret da Morteratsch at 2500 m elevation, where radar
measurements indicate an ice thickness up to 350 52.5 m.
This is in line with the plastic flow assumption (Eqn (2)), as
this is the flattest part of the glacier. The high thickness
results in an overdeepening of the glacier bed (Fig. 4b),
which is commonly observed in large temperate alpine
glaciers (e.g. Glacier d’Argentie
`re, France: Hantz and
Lliboutry, 1983; Aletschgletscher, Switzerland: Hock and
others, 1999). At the glacier front and for Vadret Pers near
the confluence area, there is a strong horizontal gradient in
ice thickness resulting in thin ice. The measured low ice
thickness for the frontal regions is again in agreement with
the plastic flow assumption, as these regions are very steep.
The uncertainty on the reconstructed ice volume and ice
thickness distribution is estimated at 25%; 15% of that
uncertainty originates from the individual radar measure-
ments while another 20% is attributed to the interpolation
procedure. For the latter, uncertainties arise from estimating
the yield stress along central flowlines where there are no
observations and from any deviation from the assumed
parabolic cross section. If it is assumed that both error sources
are independent, they can be combined in quadrature.
To compare our methodology with other reconstruction
methods, we also applied the method developed by Farinotti
and others (2009) using the 2-D mass-balance field from
Nemec and others (2009). In general this gave comparable
results for ice thickness; in both cases the highest ice
thickness of Vadret da Morteratsch was found to be 50%
larger than that for Vadret Pers. This is due to the similarity
between the two methods. Farinotti and others (2009) first
determine the local ice thickness as the fifth-power root of
the mean specific ice volume flux (and other non-local
factors). Subsequently these are divided by a factor of sin rh
raised to the power 3/5. This implicitly makes ice thickness
inversely proportional to surface slope, as is the case under
the plastic flow assumption (see Eqn (2)). Locally, however,
the results obtained from Farinotti’s technique differed in
some places from our radar measurements. The ice thickness
of the lower glacier tongue tends to be overestimated in their
method due to the high estimated ice volume flux for these
regions.
Larger uncertainties in the geometry reconstruction exist
for the higher parts of the glacier, as direct thickness
measurements are only available for the saddle area
between Piz Bernina and Piz Zupo. Steep slopes and visual
observations of ice thickness of several hanging glaciers
suggest thin ice of the same magnitude as in our reproduced
thickness field. Overall, we have sufficient confidence in the
reconstructed ice thickness and bedrock elevation to use it
as input to model the glacier flow as detailed further below.
DESCRIPTION OF THE ICE-FLOW MODEL
We use a finite-difference HO model of Blatter/Pattyn type
(Blatter, 1995; Pattyn, 2003; Fu
¨rst and others, 2011) that
solves the 3-D momentum balance according to the LMLa
(Multilayer Longitudinal Stresses) approximation (Hind-
marsh, 2004). In contrast to SIA models, this model includes
longitudinal and transverse stress gradients. Compared to the
FS solution the HO solution assumes cryostatic equilibrium
in the vertical, neglecting the vertical resistive stresses, i.e.
bridging effects. A related assumption is that horizontal
gradients of the vertical velocity are small compared with
the vertical gradient of the horizontal velocity, so that the
model solves solely for the horizontal velocity components.
Fig. 3. Flow chart illustrating the main steps for reconstructing the ice thickness field of the glacier. Red boxes indicate the data sources of
input, while green boxes represent the main output.
Zekollari and others: Calibration of higher-order 3-D ice-flow model346
As the HO solution used here is only accurate to first order, it
is not expected to resolve the details of the flow pattern at
horizontal resolutions less than a few ice thicknesses.
Owing to a new discretization of the governing force-
balance equation that makes extensive use of information on
staggered gridpoints (Fu
¨rst and others, 2011), the model has
lower computational costs and is numerically more robust
(improved convergence) than older versions of the same
LMLa code (Pattyn, 2003). Here we use the same horizontal
resolution as the DEM (25 m) and consider 21 layers in the
vertical. The high horizontal resolution of 25 m is chosen
because it allows for a good numerical convergence and
because several steps in this paper rely on a nearest-
neighbour operation. This is the case for the measured ice
thicknesses for deriving the thickness field and for the
observed surface velocities used to calibrate the flow
parameters. In this way the observations can be located
close to their real positions. Nonetheless, the results should
not be interpreted at 25 m resolution but rather at a lower
horizontal resolution of 100–200 m. This is because the
25 m resolution is below the typical ice thickness and the
first-order momentum balance on which our model is based
does not provide any meaningful information at such a fine
resolution. Furthermore, the geometric input is not every-
where representative at a 25m resolution, as a part of the
thickness field is derived through inter- and extrapolation. In
addition, a final smoothing of this field was applied at the
100 m scale more in accordance with the approximations
underlying the flow model.
Unless stated otherwise, the theoretical and numerical
implementations are the same as described in Fu
¨rst and
others (2011). We assume the glacier is temperate, but
neglect the pressure-melting effects by assigning a tempera-
ture of 08C everywhere (Cuffey and Paterson, 2010).
As the constitutive equation for ice deformation, Nye’s
generalization of Glen’s flow equation is used (Glen, 1955;
Nye, 1957):
ij ¼2_
"ij,ð3Þ
¼1
2A01=n_
"eþ_
"0
ðÞ
1=n1,ð4Þ
where
ij
are deviatoric stresses and nis the power-law
exponent. A
0
is the rate factor and the first variable used for
model calibration. In this study we take nto be equal to 3,
which is its most common value found in the literature
(Cuffey and Paterson, 2010). A
0
is one of two variables that
are tuned to match the observed surface velocities. is the
effective viscosity, defined via the second invariant of the
strain rate _
"2
e¼1
2
_
"ij
_
"ij. The strain-rate tensor is defined in
terms of velocity gradients _
"ij ¼1
2@iujþ@jui
._
"0is a
negligible offset of 10
–30
that makes viscosity finite
(Pattyn, 2003).
As a lower boundary condition, a Weertman-type sliding
law (Weertman, 1964) is adopted in which the basal sliding
velocity (u
b
) is proportional to basal drag (
b
, basal shear
stress) raised to the third power, a common approach in
alpine glacier modelling (e.g. Le Meur and Vincent, 2003;
Jouvet and others, 2011):
ub¼Asl3
b:ð5Þ
The basal drag
b
is calculated following the HO approx-
imation and corresponds to the sum of all basal resistive
forces (Van der Veen and Whillans, 1989):
b, x¼xz bðÞ2xx bðÞþyy bðÞ
@b
@xxy bðÞ
@b
@y
b, y¼yz bðÞ2yy bðÞþxx bðÞ
@b
@yxy bðÞ
@b
@x
(,ð6Þ
in which bis the bedrock elevation. A
sl
is the sliding
parameter and is the second variable used for model
calibration. This is incorporated by a Picard iteration on
the basal boundary condition. The SIA is used for an initial
Fig. 4. (a) Reconstructed ice thickness field. The contours delineate
50 m ice thickness intervals. White areas are either ice-free or have
local ice that does not contribute to the flow of the glacier complex.
(b) Modelled bedrock elevation obtained by subtracting ice
thickness from surface elevation. The thick black line delineates
the Morteratsch glacier complex. The thin black lines are bedrock
elevation contours at 200 m intervals.
Zekollari and others: Calibration of higher-order 3-D ice-flow model 347
guess of the basal drag (
b,SIA
), after which the basal drag is
updated at each iteration with the basal drag calculated from
the HO model at the previous iteration:
b, SIA ¼gH @h
@x
2
þ@h
@y
2
!
1=2
:ð7Þ
This iterative set-up is similar to the implementation of a
Coulomb friction type of basal boundary condition by
Bueler and Brown (2009).
The iterative solver is the same as described by Fu
¨rst and
others (2011). It decomposes the momentum balance into a
linear system of equations and a nonlinear update. To reach
convergence, the solvers use a relative error for the linear
and nonlinear part, which has to fall below 10
–4
.
MODEL CALIBRATION
Calibration using observed surface velocities
By varying the rate factor A
0
and the sliding parameter A
sl
we
seek a best fit between the modelled surface velocity field
and the observed surface velocities (Table 1). An ensemble of
100 runs were done, each with a different combination of A
0
(from 0.6 10
–16
Pa
–3
a
–1
in steps of 0.2 10
–16
Pa
–3
a
–1
to
2.4 10
–16
Pa
–3
a
–1
) and A
sl
(from 0 10
–16
m
7
N
–3
a
–1
in
steps of 2 10
–16
m
7
N
–3
a
–1
to 18 10
–16
m
7
N
–3
a
–1
). For
each of these runs the root-mean-square error (RMSE)
between the modelled surface velocity (v
mod
) and the
observed surface velocity (v
obs
;n= 20) is calculated:
RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pn
i¼1vobs, ivmod, i
2
n
s:ð8Þ
The obtained RMSE for the different runs is summarized in
Figure 5. The best agreement between observed and mod-
elled velocities is obtained for the run with A
0
= 1.6
10
–16
Pa
–3
a
–1
and A
sl
=1210
–16
m
7
N
–3
a
–1
with an RMSE
equal to 15.0 ma
–1
. For this run (Fig. 6), internal deformation
accounts on average for 70% of the flow and basal sliding for
the remaining 30%. Locally, however, this ratio can vary
significantly. Overall, the model is able to reconstruct the
observed surface velocities with a large resultant correlation
and a high coefficient of determination R
2
of 0.93:
R2¼1Pn
i¼1vobs, ivmod, i
2
Pn
i¼1vobs, i1
nPn
i¼1vobs, i
2:ð9Þ
A
0
= 1.6 10
–16
Pa
–3
a
–1
is of the same order of magnitude as
values obtained in other alpine glacier modelling studies.
Gudmundsson (1999) found a value of 0.75 10
–16
Pa
–3
a
–1
for Unteraargletscher, Switzerland, Le Meur and Vincent
(2003) 0.63 10
–16
Pa
–3
a
–1
for Glacier de Saint-Sorlin,
France, and Iken and Truffer (1997) 1.6 10
–16
Pa
–3
a
–1
for
Findelengletscher, Switzerland. Our value is close to the
widely used value of 2.1 10
–16
Pa
–3
a
–1
(Paterson, 1994).
When multiplied with the average glacier thickness of
72.2 m, the sliding factor is A
sl
= 8.7 10
–14
m
8
N
–3
a
–1
(in
literature often a division by Hoccurs in the sliding law),
close to the value of 5 10
–14
m
8
N
–3
a
–1
obtained by Le
Meur and Vincent (2003).
Besides the ‘best-fit’ combination, other parameter
combinations also give low RMSEs. In fact, a diagonal with
minimum RMSE can be identified in Figure 5. This is
because A
0
and A
sl
are linearly correlated with, respectively,
the internal deformation and sliding. Internal deformation
and basal sliding produce a rather similar velocity pattern, as
both depend on shear stress raised to the third power. In
other words, there is a trade-off between sliding and internal
deformation. For instance, a model run with A
0
= 1.4
10
–16
Pa
–3
a
–1
and A
sl
=1810
–16
m
7
N
–3
a
–1
, in which slid-
ing accounts for almost half of the observed velocities, yields
an RMSE of 15.5 m a
–1
. Based on the RMSE only, it is
therefore difficult to determine with high confidence the
separate contribution of internal deformation and basal
sliding for the glacier complex.
For our ‘best-fit’ parameter selection, the largest differ-
ences between model and observations occur for four points
with rather low observed velocities (25–50 m a
–1
), with the
model underestimating the observations (see Fig. 6). These
low velocities, important to understanding the present
glacier retreat (see below), contribute comparatively less in
Fig. 5. RMSE between observed and modelled surface velocities for
runs with different combinations of A
0
and A
sl
.Fig. 6. Modelled versus observed surface velocities (1) for the
run with the lowest RMSE (A
0
= 1.6 10
–16
Pa
–3
a
–1
,A
sl
=12
10
–16
m
7
N
–3
a
–1
). The horizontal bars represent the error on the
observed velocity. The thick red diagonal line represents the 1: 1
ratio between observed and modelled velocities. Dashed red lines
represent the 1:1 ratio RMSE.
Zekollari and others: Calibration of higher-order 3-D ice-flow model348
the calculation of the total RMSE, as the RMSE is based on
absolute differences in velocities. To take this into account,
we used a weighted sum as an alternative and calculated a
relative RMSE (RRMSE):
RRMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pn
i¼1
vobs, ivmod, i
vobs, i
2
n
v
u
u
t:ð10Þ
Overall, the pattern in RRMSE is similar to that of the RMSE,
but the minimum is slightly shifted to A
0
= 2.0 10
–16
Pa
–3
a
–1
and A
sl
=410
–16
m
7
N
–3
a
–1
, corresponding to a larger
contribution of internal deformation. Furthermore the con-
trast along the ‘diagonal of minimal error’ is stronger. The
previously mentioned model run with A
0
= 1.4
10
–16
Pa
–3
a
–1
and A
sl
=1810
–16
m
7
N
–3
a
–1
(about half of
the velocity explained by sliding) now has an RRMSE that is
35% higher than for the run with the minimum RRMSE
(A
0
= 2.0 10
–16
Pa
–3
a
–1
,A
sl
=410
–16
m
7
N
–3
a
–1
). Con-
sidering both the RMSE and the RRMSE, this suggests that
the average contribution of internal deformation is larger than
that from basal sliding, but that the exact ratio is difficult to
establish from our limited number of direct observations.
Velocity field
The modelled surface velocity field with the lowest RMSE
(A
0
= 1.6 10
–16
Pa
–3
a
–1
and A
sl
=1210
–16
m
7
N
–3
a
–1
)is
represented in Figure 7, while Figure 8 shows the separate
contribution of internal deformation and sliding. The highest
surface velocities are found for central upper Vadret da
Morteratsch in the icefall (‘the Labyrinth’) near the equi-
librium-line altitude (ELA) and reach up to 125 m a
–1
. In the
ablation zone of the glacier the faster parts are mostly linked
to large ice thickness, while in the accumulation zone the fast
velocities are rather explained by steep surface slopes. For the
lowest parts of the glacier, the model is able to reproduce
the observed very low surface velocities. In the vicinity of the
glacier front, the ice is almost stagnant, indicating that there is
little to no mass supply from upstream. Because of this, here
the annual frontal thinning is almost equal to the local annual
surface mass balance, which is usually close to
–8 m w.e. a
–1
(Nemec and others, 2009). This is crucial for
explaining the present-day strong retreat of Vadret da Morter-
atsch and also generally applies to other alpine glaciers with
an extended valley tongue. Modelled surface velocities for
the lowest parts of Vadret Pers near the confluence with
Fig. 8. Modelled velocity due to (a) internal deformation and (b) basal sliding for the run with the lowest RMSE (A
0
= 1.6 10
–16
Pa
–3
a
–1
,
A
sl
=1210
–16
m
7
N
–3
a
–1
). For internal deformation the highest velocity is 95m a
–1
, while for sliding it is 62 m a
–1
.
Fig. 7. Modelled surface velocity field for the run with the lowest
RMSE (A
0
= 1.6 10
–16
Pa
–3
a
–1
,A
sl
=1210
–16
m
7
N
–3
a
–1
). The
highest modelled flow speed is 125 m a
–1
.
Zekollari and others: Calibration of higher-order 3-D ice-flow model 349
Vadret da Morteratsch are also very low (3–6m a
–1
). The
limited mass flux (from up-glacier) enhances the thinning of
this region and suggests the imminent separation of the two
glaciers. In 2001 the ice thickness was less than 50 m here,
which, in combination with an annual surface mass balance
of about –5 m w.e. a
–1
, indicates that the two glaciers will
probably separate in less than 5 years.
As internal deformation is the main contributor to the
flow, a large similarity exists with the modelled surface
velocity field. The basal sliding field is more fragmented and
reaches high values for the icefall of Vadret da Morteratsch.
This is due to the fact that the surface gradient is very high
here, combined with moderate ice thickness (Eqn (7)). For
the central trunk of Vadret da Morteratsch below the icefall,
the fraction of basal sliding is very small due to the low
surface slopes.
CONCLUSION
On the basis of radio-echo sounding data, theoretical
concepts and appropriate interpolation techniques, we
reconstructed the ice thickness and bedrock elevation of
the Morteratsch glacier complex. The thickest ice of up to
350 m was found for the central trunk of Vadret da
Morteratsch just below the Labyrinth icefall, while for Vadret
Pers the maximum ice thickness is 250m. The area of highest
ice thickness for Vadret da Morteratsch corresponds to an
overdeepened bed. Depending on drainage conditions, this
may eventually give rise to a proglacial lake if a glacier
retreat continues. For this specific glacier, ice thickness
reconstruction techniques based on estimates of the mean
specific ice volume flux would have provided a generally
satisfactory result except in the lower tongue area.
The resulting geometric fields provided an interesting test
of the capability of a newly developed HO model code to
simulate mountain glacier flow. In such a setting, where the
spatial resolution is much higher than for ice-sheet model-
ling, the robustness and numerical convergence of the ice-
dynamic model are crucial. The flow model is able to
diagnostically reproduce the observed surface velocities
closely (R
2
= 0.93). The rate factor (1.6 10
–16
Pa
–3
a
–1
) and
sliding parameter (12 10
–16
m
7
N
–3
a
–1
) obtained from
calibrating the model with observed surface velocities are
similar to values quoted in the literature. For a best model fit,
internal deformation accounts on average for 70% of the
glacier flow, and basal sliding for the remaining 30%, even
though this ratio varies locally.
The flow model reproduces the almost stagnant front and
confluence area between Vadret da Morteratsch and Vadret
Pers, which are the most vulnerable parts of the glacier
under a warming climate. This suggests a continued severe
frontal retreat under present-day climate, and an imminent
separation of the two glaciers that make up the Morteratsch
glacier complex. Further insight into the rate and timing of
this retreat would require a prognostic model in which ice
flow is coupled to a surface mass-balance model.
ACKNOWLEDGEMENTS
We are grateful to everyone from AWI and VUB who took
part in the fieldwork on Morteratsch glacier and helped with
the data collection over the past decade. We also thank
D. Maes for the help provided on the statistical aspects.
Comments and suggestions by two anonymous reviewers
helped to improve the manuscript greatly. Financial support
was provided through the German HGF-Strategiefonds
Projekt ‘SEAL’ and successive projects funded by the Belgian
Science Policy Office (BELSPO) within its Research Program
on Science for a Sustainable Development (projects MILMO,
ASTER and iCLIPS).
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