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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).

REFERENCES

Bauder A, Funk M and Huss M (2007) Ice-volume changes of

selected glaciers in the Swiss Alps since the end of the

19th century. Ann. Glaciol., 46, 145–149 (doi: 10.3189/

172756407782871701)

Blatter H (1995) Velocity and stress fields in grounded glaciers: a

simple algorithm for including deviatoric stress gradients.

J. Glaciol.,41(138), 333–344

Bueler E and Brown J (2009) Shallow shelf approximation as a

‘sliding law’ in a thermomechanically coupled ice sheet model.

J. Geophys. Res.,114(F3), F03008 (doi: 10.1029/2008JF001179)

Cuffey KM and Paterson WSB (2010) The physics of glaciers, 4th

edn. Butterworth-Heinemann, Oxford

Eisen O, Nixdorf U, Keck L and Wagenbach D (2003) Alpine ice

cores and ground penetrating radar: combined investigations for

glaciological and climatic interpretations of a cold Alpine ice

body. Tellus B,55(5), 1007–1017 (doi: 10.1034/j.1600-0889.

2003.00080.x)

Elsasser H and Bu

¨rki R (2002) Climate change as a threat to tourism in

the Alps. Climate Res.,20(3), 253–257 (doi: 10.3354/cr020253)

Farinotti D, Huss M, Bauder A, Funk M and Truffer M (2009) A

method to estimate ice volume and ice-thickness distribution of

alpine glaciers. J. Glaciol.,55(191), 422–430 (doi: 10.3189/

002214309788816759)

Fu

¨rst JJ, Rybak O, Goelzer H, De Smedt B, de Groen P and

Huybrechts P (2011) Improved convergence and stability proper-

ties in a three-dimensional higher-order ice sheet model. Geosci.

Model Dev.,4(4), 1133–1149 (doi: 10.5194/gmd-4-1133-2011)

Glaciological Reports (1881–2011) The Swiss Glaciers. Yearbooks

of the Cryospheric Commission of the Swiss Academy of

Sciences (SCNAT), 1–128. Published since 1964 by the Labora-

tory of Hydraulics, Hydrology and Glaciology (VAW) of ETH,

Zu

¨rich. http://glaciology.ethz.ch/swiss-glaciers

Glen JW (1955) The creep of polycrystalline ice. Proc. R. Soc.

London, Ser. A,228(1175), 519–538 (doi: 10.1098/rspa.1955.

0066)

Gudmundsson GH (1999) A three-dimensional numerical model of

the confluence area of Unteraargletscher, Bernese Alps,

Switzerland. J. Glaciol.,45(150), 219–230 (doi: 10.3189/

002214399793377086)

Hantz D and Lliboutry L (1983) Waterways, ice permeability at

depth, and water pressures at Glacier d’Argentie`re, French Alps.

J. Glaciol.,29(102), 227–239

Hindmarsh RCA (2004) A numerical comparison of approximations

to the Stokes equations used in ice sheet and glacier modeling.

J. Geophys. Res.,109(F1), F01012 (doi: 10.1029/2003JF000065)

Hock R, Iken A and Wangler A (1999) Tracer experiments and

borehole observations in the overdeepening of Aletschgletscher,

Switzerland. Ann. Glaciol., 28, 253–260 (doi: 10.3189/

172756499781821742)

Huss M, Usselmann S, Farinotti D and Bauder A (2010) Glacier

mass balance in the south-eastern Swiss Alps since 1900 and

perspectives for the future. Erdkunde,64(2), 119–140 (doi:

10.3112/erdkunde.2010.02.02)

Huybrechts P, de Nooze P and Decleir H (1989) Numerical

modelling of Glacier d’Argentie`re and its historic front vari-

ations. In Oerlemans J ed. Glacier fluctuations and climatic

change. Kluwer Academic Publishers, Dordrecht, 373–389

Iken A and Truffer M (1997) The relationship between subglacial

water pressure and velocity of Findelengletscher, Switzerland,

during its advance and retreat. J. Glaciol.,43(144), 328–338

Zekollari and others: Calibration of higher-order 3-D ice-flow model350

Jacob T, Wahr J, Pfeffer WT and Swenson S (2012) Recent

contributions of glaciers and ice caps to sea level rise. Nature,

482(7386), 514–518 (doi: 10.1038/nature10847)

Jouvet G, Huss M, Blatter H, Picasso M and Rappaz J (2009)

Numerical simulation of Rhonegletscher from 1874 to 2100.

J. Comput. Phys.,228(17), 6426–6439 (doi: 10.1016/j.jcp.2009.

05.033)

Jouvet G, Huss M, Funk M and Blatter H (2011) Modelling the

retreat of Grosser Aletschgletscher, Switzerland, in a changing

climate. J. Glaciol.,57(206), 1033–1045 (doi: 10.3189/

002214311798843359)

Kaser G, Cogley JG, Dyurgerov MB, Meier MF and Ohmura A

(2006) Mass balance of glaciers and ice caps: consensus

estimates for 1961–2004. Geophys. Res. Lett.,33(19), L19501

(doi: 10.1029/2006GL027511)

Le Meur E and Vincent C (2003) A two-dimensional shallow ice-

flow model of Glacier de Saint-Sorlin, France. J. Glaciol.,

49(167), 527–538 (doi: 10.3189/172756503781830421)

Le Meur E, Gagliardini O, Zwinger T and Ruokolainen J (2004)

Glacier flow modelling: a comparison of the shallow ice

approximation and the full-Stokes equation. C. R. Phys.,5(7),

709–722

Lemke P and 10 others (2007) Observations: changes in snow, ice

and frozen ground. In Solomon S and 7 others eds. Climate

change 2007: the physical science basis. Contribution of Working

Group I to the Fourth Assessment Report of the Intergovernmental

Panel on Climate Change. Cambridge University Press, Cam-

bridge, 339–383

Li H, Li Z, Zhang M and Li W (2011) An improved method based on

shallow ice approximation to calculate ice thickness along

flow-line and volume of mountain glaciers. J. Earth Sci.,22(4),

441–338 (doi: 10.1007/s12583-011-0198-1)

Li H, Ng F, Li Z, Qin D and Cheng G (2012) An extended ‘perfect-

plasticity’ method for estimating ice thickness along the flow

line of mountain glaciers. J. Geophys. Res.,117(F1), F01020

(doi: 10.1029/2011JF002104)

Moran ML, Greenfield RJ, Arcone SA and Delaney AJ (2000)

Delineation of a complexly dipping temperate glacier bed using

short-pulse radar arrays. J. Glaciol.,46(153), 274–286 (doi:

10.3189/172756500781832882)

Narod BB and Clarke GKC (1994) Miniature high-power impulse

transmitter for radio-echo sounding. J. Glaciol.,40(134),

190–194

Nemec J, Huybrechts P, Rybak O and Oerlemans J (2009)

Reconstruction of the annual balance of Vadret da Morteratsch,

Switzerland, since 1865. Ann. Glaciol.,50(50), 126–134 (doi:

10.3189/172756409787769609)

Nye JF (1957) The distribution of stress and velocity in glaciers

and ice sheets. Proc. R. Soc. London, Ser. A,239(1216),

113–133

Oerlemans J (1986) An attempt to simulate historic front variations

of Nigardsbreen, Norway. Theor. Appl. Climatol.,37(3),

126–135 (doi: 10.1007/BF00867846)

Paterson WSB (1994) The physics of glaciers, 3rd edn. Elsevier,

Oxford

Pattyn F (2003) A new three-dimensional higher-order thermo-

mechanical ice-sheet model: basic sensitivity, ice stream

development, and ice flow across subglacial lakes. J. Geophys.

Res.,108(B8), 2382 (doi: 10.1029/2002JB002329)

Paul F, Ka

¨a

¨b A, Maisch M, Kellenberger T and Haeberli W (2004)

Rapid disintegration of Alpine glaciers observed with satellite

data. Geophys. Res. Lett.,31(21), L21402 (doi: 10.1029/

2004GL020816)

Radic

´V and Hock R (2011) Regionally differentiated

contribution of mountain glaciers and ice caps to future

sea-level rise. Nature Geosci.,4(2), 91–94 (doi: 10.1038/

ngeo1052)

Van der Veen CJ and Whillans IM (1989) Force budget: I. Theory

and numerical methods. J. Glaciol.,35(119), 53–60 (doi:

10.3189/002214389793701581)

Weertman J (1964) The theory of glacier sliding. J. Glaciol.,5(39),

287–303

Zwinger T and Moore JC (2009) Diagnostic and prognostic

simulations with a full Stokes model accounting for super-

imposed ice of Midtre Love

´nbreen, Svalbard. Cryosphere,3(2),

217–229 (doi: 10.5194/tc-3-217-2009)

Zwinger T, Greve R, Gagliardini O, Shiraiwa T and Lyly M (2007) A

full Stokes-flow thermo-mechanical model for firn and ice

applied to the Gorshkov crater glacier, Kamchatka. Ann.

Glaciol., 45, 29–37 (doi: 10.3189/172756407782282543)

Zekollari and others: Calibration of higher-order 3-D ice-flow model 351