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Statistical modelling of the surface mass-balance variability of the

Morteratsch glacier, Switzerland: strong control of early melting

season meteorological conditions

HARRY ZEKOLLARI,

1,2,3

PHILIPPE HUYBRECHTS

1

1

Earth System Science & Department Geografie,Vrije Universiteit Brussel,Brussels,Belgium

2

Laboratory of Hydraulics,Hydrology and Glaciology (VAW),ETH Zürich,Zurich,Switzerland

3

Swiss Federal Institute for Forest,Snow and Landscape Research (WSL),Birmensdorf,Switzerland

Correspondence: Harry Zekollari <zharry@ethz.ch>

ABSTRACT. In this study we analyse a 15-year long time series of surface mass-balance (SMB) measure-

ments performed between 2001 and 2016 in the ablation zone of the Morteratsch glacier complex

(Engadine, Switzerland). For a better understanding of the SMB variability and its causes, multiple

linear regressions analyses are performed with temperature and precipitation series from nearby

meteorological stations. Up to 85% of the observed SMB variance can be explained by the mean

May–June–July temperature and the total precipitation from October to March. A new method is pre-

sented where the contribution of each month’s individual temperature and precipitation to the SMB

can be examined in a total sample of 2

24

(16.8 million) combinations. More than 90% of the observed

SMB can be explained with particular combinations, in which the May–June–July temperature is the most

recurrent, followed by October temperature. The role of precipitation is less pronounced, but autumn,

winter and spring precipitation are always more important than summer precipitation. Our results indi-

cate that the length of the ice ablation season is of larger importance than its intensity to explain year-to-

year variations. The widely used June–July–August temperature index may not always be the best option

to describe SMB variability through statistical correlation.

KEYWORDS: energy balance, glacier cover, glacier mass balance, glacier meteorology, glacier monitoring

1. INTRODUCTION

Mountain glaciers worldwide have retreated significantly in

the past decades as a consequence of an increase in global

temperature (Vaughan and others, 2013). This resulted in

an important contribution to global sea-level rise (Church

and others, 2013; Marzeion and others, 2017) and pro-

foundly affects water supply, hydro-electricity production,

natural hazards and tourism in mountainous regions (e.g.

Werder and others, 2010; Farinotti and others, 2012;

Vincent and others, 2012; Gilbert and others, 2015; Huss

and Hock, 2015,2018; Ragettli and others, 2016; Kääb

and others, 2018). Many studies have highlighted these

changes for glaciers in the Swiss Alps through a wide range

of approaches, based on for instance geodetic methods,

LIDAR measurements, Unmanned Aerial Vehicle (UAV)

surveys and three-dimensional glacier evolution modelling

(Jouvet and others, 2009; Gabbud and others, 2015,2016;

Zekollari and Huybrechts, 2015; Fischer and others, 2016;

Sold and others, 2016; Gindraux and others, 2017; Rossini

and others, 2018). These changes are largely driven by a

strongly negative surface mass-balance (SMB) trend (Huss

and others, 2015; Zemp and others, 2015; Vincent and

others, 2017), which has been modelled at a variety of hori-

zontal scales and through models of varying complexity for

glaciers in the European Alps (e.g. Klok and Oerlemans,

2002; Huss and others, 2008; Machguth and others, 2009;

Nemec and others, 2009; Berthier and Vincent, 2012;

Gabbi and others, 2014; Huss and Fischer, 2016; Réveillet

and others, 2017). Process-based SMB models are powerful

and very useful tools for many applications, but they rely

on parameterisations, simplifications and assumptions

that influence the relationship between the model input

(meteorological data) and output (modelled SMB). In order

to better quantify the link between meteorological data and

SMB, direct statistical methods therefore represent an attract-

ive alternative. Furthermore, such statistical methods can be

useful tools for cases where the necessary measurements

needed to setup more sophisticated SMB models (e.g.

albedo and radiation measurements) are not available.

The first studies that laid the foundation for multivariate stat-

istical regression between SMB data and meteorological

observations were performed at the end of the 1970s

(Young, 1977;Martin,1978; Tangborn, 1980). Since then,

multivariate statistical regressions have been widely applied,

under slightly varying forms, among others for glaciers in the

Rocky Mountains and Coast Mountains (Western Canada)

(Letréguilly, 1988), Svalbard (Lefauconnier and Hagen,

1990; Lefauconnier and others, 1999), Norway (e.g.

Trachsel and Nesje, 2015) and in the European Alps (e.g.

Chen and Funk, 1990;VincentandVallon,1997; Torinesi

and others, 2002). The majority of these studies highlight the

importance of summer temperature and winter precipitation,

which variables are typically used to describe the observed

SMB trends.

Here we analyse a 15-year dataset of SMB measurements

from the ablation zone of the Morteratsch glacier complex

(Switzerland) and investigate the meteorological variables

that best describe the interannual variability in SMB through

multiple linear regression analysis (MLRA). Compared with

previous studies, where the meteorological input mostly

Journal of Glaciology (2018), 64(244) 275–288 doi: 10.1017/jog.2018.18

©The Author(s) 2018. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.

org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

consists of adjacent clusters of months, we introduce a new

method in which all possible monthly combinations are con-

sidered. Our main objective is to make use of this new

method to describe the measured SMB variations through a

statistical analysis that is as simple as possible, i.e. relying

on a minimal number of predictor variables to describe as

much of the SMB variability as possible.

2. LOCATION, FIELDWORK AND DATA

2.1. Location and fieldwork

The Morteratsch glacier complex is situated on the southern

side of the European Alps (Engadine, SE Switzerland) and

consists of two glaciers, the Morteratsch glacier (Vadret da

Morteratsch) and the Pers glacier (Vadret Pers) (Fig. 1).

Until 2015, Vadret Pers was the main tributary of the

Vadret da Morteratsch (Zekollari and Huybrechts, 2015),

but now both glaciers have disconnected and act as inde-

pendent ice bodies. At present, the glacier complex covers

an area of ∼16 km

2

and has a volume of ∼1.1 km

3

(Zekollari and others, 2013).

Since 2001 we measured the SMB on the glacier complex

from an elaborate stake network emplaced in its ablation

area and around the ELA. These measurements were per-

formed at the very end of September/beginning of October,

around the time of the first snowfall events that mark the

end of the ablation season and the onset of the accumulation

season. For some years, some additional ablation occurred in

October, while for others the annual ice ablation already

stopped at the time of measurements due to an earlier

(September) snowfall event. The stake position was measured

with high precision GPS systems, which were corrected with

reference base stations until 2013 (see Zekollari and others,

2013). More recently Real Time Kinematic (RTK) GPS

systems were used.

2.2. Meteorological data

For the analysis, data from two nearby MeteoSwiss stations

are used, from a meteorological station in Samedan (1708 m

a.s.l., 46°32′N, 9°53′E) and from a station in Segl-Maria

(1804 m a.s.l., 46°26′N, 9°46E’)(seeFig. 2a). The temperature

signal is very similar for both stations (R

2

=0.99 for the period

2001–2016), but the station of Samedan shows a slightly

higher seasonal contrast with lower winter temperatures (see

Fig. 2a). This is a consequence of a stronger winter inversion

in the valley at Samedan compared with Segl-Maria.

Precipitation is higher at Segl-Maria: for the period covering

the SMB measurements, the average annual precipitation

at Segl-Maria is 932 mm a

−1

, while for Samedan it is

683 mm a

−1

(27% lower). The higher precipitation in Segl-

Maria results from the fact that the precipitation mostly

comes from the south over the Maloja pass and that the air

dries up when advancing in the Engadine valley towards

Samedan. Meteorological conditions at these stations are

good indicators for conditions on the glacier, as revealed by

measurements from an in situ meteorological station on the

Morteratsch glacier (Klok and Oerlemans, 2002,2004).

Another nearby meteorological station is also available, at

Piz Corvatsch (46°25′N, 9°49′E), but due to its high elevation

(3302 m), this station is more representative for high mountain

conditions and the free troposphere. The local valley meteoro-

logical conditions, which mostly affect the ablation area of the

glacier due to the often-present temperature inversion, are

therefore better represented by the Segl-Maria and Samedan

stations. As we are interested in the interannual variability in

meteorological parameters, which is very similar for Segl-

Maria and Samedan, we opt to use the average temperature

and precipitation of both stations for the statistical correlations

performed further below. Both stations are at a similar distance

from the glacier: 12.2 km in a direct line from the glacier snout

for the Samedan station and 13.4 km for the Segl-Maria

station.

2.3. Ablation measurements

Over the 15-year period, a total of 232 annual mass-balance

point measurements are available for the ablation area and

around the Equilibrium Line Altitude (ELA) of the

Morteratsch glacier complex (Fig. 3). These readings result

from annual visits to the glacier, which occur at the very

end of September –beginning of October, corresponding

to a floating-date system that is very close to the fixed-date

system (Cogley and others, 2011). Of the 232 readings, eight

have a positive mass balance (up to +0.6 m ice eq a

−1

)

(Fig. 3). A total of 128 readings were performed on Vadret

da Morteratsch and 104 on Vadret Pers (see Fig. 1). These

readings were obtained from 31 separate stakes (17 on

Vadret da Morteratsch, 14 on Vadret Pers), of which 12

stakes have a series of at least 10 years and eight stakes

cover the full 15-year period. The entire dataset is available

as Supplementary material. For Vadret da Morteratsch, the ele-

vation of the stakes ranges from the front (∼2030–2100 m a.s.l.

over this period) to ∼2600 m a.s.l. (just underneath the icefall,

the ‘labyrinth’). Most of this range is covered with stakes,

roughly at 100 m height intervals (see also Fig. 1). For

Vadret Pers, two SMB observations were taken at the front

(∼2450 m a.s.l.), but all other measurements are situated

between 2600 and 3050 m a.s.l. (∼the ELA). The SMB is sig-

nificantly lower on Vadret Pers compared with Vadret da

Fig. 1. Overview of the Morteratsch glacier complex and stakes for

SMB measurements in the ablation area. The eight stakes used in the

multiple linear regression analysis (MLRA) are shown in blue (Vadret

da Morteratsch) and red (Vadret Pers), the other stakes are

represented in light grey. The terminus is at ∼2100 m a.s.l., while

the highest mountain peaks are ∼4000 m. The SwissTopo Digital

Elevation Model (DEM) used to produce this figure is from 2001

(i.e. start of the field campaign). Figure created with TopoZeko

toolbox (Zekollari, 2017).

276 Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland

Morteratsch (Fig. 3), which is likely related to the orientation

and resulting daily insolation cycle for both glaciers. The abla-

tion area of the Pers glacier is oriented towards the WNW

(tongue)–NW (upper ablation area) and is more exposed to

direct insolation than the Morteratsch glacier, which is

exposed to the N and strongly shielded by the high mountain

peaks (see also Fig. 1). A simple approach in which a best

linear fit (i.e. linear regression) through all stakes is taken

clearly illustrates the higher SMB for Morteratsch in the

2000–3000 m elevation range (ablation area):

SMBMORT ¼0:01119 ð±0:00098ÞELEV

31:98 ð±2:36ÞðR2¼0:80Þ;

ð1Þ

SMBPERS ¼0:01164 ð±0:00106ÞELEV

35:31 ð±2:98ÞðR2¼0:82Þ;

ð2Þ

where SMB

MORT/PERS

is the annual measured SMB (in meter

ice equivalent), ELEV is the stake elevation (in m a.s.l.) and

the uncertainties correspond to the 95% confidence bounds.

Based on this very simple approach, for the period 2001–

2016, the average ELA for the Morteratsch glacier is expected

to occur at 2859 m, which is a slight underestimation com-

pared with remote-sensing observations (Chan and others,

2009). This is likely related to the fact that the SMB-elevation

gradient decreases towards the ELA. For the Pers glacier, the

linear correlation suggests that the ELA for this period is at

3035 m, which agrees very well with observations (see

Fig. 2. (a) Mean monthly daily temperature and monthly precipitation for the MeteoSwiss meteorological stations of Segl-Maria and Samedan.

(b) Mean seasonal mean daily temperature and seasonal precipitation averaged for the meteorological stations of Segl-Maria and Samedan.

2000 2200 2400 2600 2800 3000

Elevation (m)

-10

-8

-6

-4

-2

0

Measured SMB (m ice eq. a -1)

Morteratsch

Pers

Fig. 3. Mean annual surface mass balance against elevation for the

Pers and Morteratsch glacier for all 232-point measurements. The

coloured lines represent the best linear fit for both glaciers

individually (Eqns (1) and (2)).

277Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland

Fig. 3). The difference in ELA between both glaciers is typically

of the order of 150 m for this period. Direct SMB measure-

ments in the ablation area of the Morteratsch glacier are

∼2.1–2.5 m ice eq. a

−1

higher than these measured on the

Pers glacier at the same elevation. These are large differences

in SMB and ELA given the close proximity of both glaciers.

3. DATA HANDLING AND STATISTICAL

BACKGROUND

For the MLRA, 120 SMB measurements are considered,

which correspond to the eight stakes that cover the whole

observational period (i.e. the 15-year record). All eight

stakes are in the ablation zone and are in debris-poor

areas. Four of these stakes are located on Vadret da

Morteratsch and four on Vadret Pers (see Fig. 1). Including

SMB measurements from stakes that do not cover this

entire period would introduce bias in the anomalies due to

the gap in their data record. Since SMB data from the eight

stakes covering the full period would be needed to solve

these biases, this approach would not add information

about total SMB variance.

The SMB stakes undergo elevation changes as time evolves

due to their movement along with the glacier flow and due to

changes in local ice thickness (Zekollari and others, 2014;

Fischer and others, 2015). Over the 15-year period, the eleva-

tion change for the eight stakes ranges from −115 m (P33) to

−44 m (P21). To account for this effect, all stakes are pro-

jected back to their initial elevation (in 2001) based on the

SMB-elevation gradients found from the simple linear correl-

ation (Eqns (1) and (2)). For the Pers glacier, this corresponds

to 0.0116 m ice eq. a

−1

m

−1

, while for the Morteratsch

glacier, this is 0.0112 m ice eq. a

−1

m

−1

. Corrections based

on the annual values of the mass-balance gradient lead to

almost identical results and do not significantly alter the

results as the elevation change is relatively limited and the

SMB-elevation gradient varies only little over time.

The individual stake measurements are shown in Figures

4a, b. The highest SMB is observed for the balance years

2003–2004, 2012–2013 and 2013–14, and lowest SMB for

2002–2003 and 2014–2015. The stakes have a consistent

annual SMB signal. For these elevation change corrected

SMB values, no significant trend is observed over the 15-

year period (R

2

=0.13, p-value F-test =0.185). The std

dev. in SMB per stake over the entire period varies

between 0.6 and 0.9 m ice eq. a

−1

and is not correlated to

elevation (R

2

=0.04) (Fig. 4c). For the analysis, the eleva-

tion-corrected SMB measurements for each stake are con-

verted to perturbations with respect to the 15-year stake

mean. As shown on Figure 4d, the annual SMB perturbation

is largely similar over the whole ablation area, for both gla-

ciers taken together and for Vadret Pers and Vadret da

Morteratsch separately. No link with elevation is observed.

This is underscored by the very high correlation (R

2

=

0.85–0.91) between the SMB perturbation of individual

stakes and the mean perturbation over all stakes (Table 1).

Exceptions are M51 and M62 with a somewhat lower R

2

of between 0.62 and 0.67, where a few individual measure-

ments (M51 in 2005–06 and 2015–16 and M62 in 2008–09

and 2012–13) cause a slight deviation from the mean

perturbation. Despite this, the correlation between the

perturbation of these individual stakes and the mean

perturbation over all stakes is still very significant (p-values

of 2 × 10

−4

and 5 × 10

−4

, respectively). Furthermore, no

link between the meteorological data (temperature and pre-

cipitation) and annual SMB elevation gradient is found. The

presence of a common SMB signal, which is independent

from the location, is a prerequisite for our analyses and is

in line with the time-space decomposition as proposed by

Lliboutry (1974) and related studies (Meier and Tangborn,

1965; Kuhn, 1984; Rasmussen, 2004; Eckert and others,

2011).

The SMB perturbations over the eight stakes are averaged

on an annual basis, and the same is done for the meteoro-

logical variables, which are averaged over the two meteoro-

logical stations. To assess the glacier’s sensitivity to

temperature and precipitation changes, which are measured

in different units (°C and mm w.e., respectively), these values

are standardised through a conversion to a z-score, which

corresponds to the number of standard deviations that a

value is separated from the mean value.

The correlation between SMB perturbation and meteoro-

logical data are tested with MLRA (e.g. Legendre and

Legendre, 2012):

y¼a1x1þa2x2þ...þanxnþb:ð3Þ

Here yis the dependent variable (also referred to as

response variable; the SMB perturbation in this study), a

i

and bare the regression coefficients and x

i

are the inde-

pendent variables (also referred to as predictor variables;

the z-score of the meteorological components in this

study, i.e. temperature and precipitation). The monthly tem-

perature and precipitation series used in our analyses have

a very weak correlation (R

2

=0.17 when all months are

considered, with the highest correlation in spring (MAM):

R

2

=0.21 and lowest correlation in winter (DJF): R

2

=

0.04). They can therefore be considered as being uncorre-

lated, which is a common approach for MLRAs performed

on SMB series (e.g. Lefauconnier and others, 1999;

Trachsel and Nesje, 2015). Due to the conversion of the

meteorological data to a z-score, the regression coefficient

(a

i

) of variables represents the climatic variability of this

variable. Under this z-score approach, the standard regres-

sion coefficients for temperature and precipitation are dir-

ectly comparable and, assuming both are entirely

uncorrelated, indicate the relative importance of both for

the SMB. Furthermore, the standardisation procedure

leads to an intercept of the regression analysis (b) that is

equal to zero: i.e. under the mean 2001–2016 climato-

logical conditions (z-score of 0 for the independent vari-

ables), the mean 2001–2016 SMB is obtained (dependent

variable is 0). For each MLRA the error degrees of freedom

correspond to the difference between the total number of

years (15 in this case) and the number of independent vari-

ables in the analysis.

The outcome of each MLRA is expressed as a R

2

value and

ap-value for the F-test. The R

2

value corresponds to the frac-

tion of the variability in the response variable that the model

explains. The F-test tests for a significant linear regression

relationship between the response variable and the predictor

variables. The p-value of the F-test, also referred to as calcu-

lated probability, is the probability of obtaining a result

(a linear correlation) equal to or more extreme than what is

observed when the null hypothesis (no linear correlation)

is true. The lower the p-value, the higher the confidence

(lower significance level) at which the null hypothesis (no

linear correlation) can be rejected.

278 Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland

In our MLRA, we opt for the mean monthly tempera-

ture and not for the monthly average of the daily

maxima, as air temperature can also cause melt during

the night time (cf. Letréguilly, 1988). The difference

between results obtained with mean monthly temperature

and the monthly average of the daily maxima is however

very limited as the trends in both are very similar (R

2

=

0.97).

Table 1. Correlation (R

2

value and p-value of the F-test) between the eight selected stakes and the mean perturbation (black line on Fig. 4d)

M20 M51 M54 M62 P21 P22 P32 P33

Mean SMB R

2

=0.91

p=3.9 × 10

−8

R

2

=0.67

p=2.0 × 10

−4

R

2

=0.90

p=5.5 × 10

−8

R

2

=0.62

p=5.1 × 10

−4

R

2

=0.85

p=8.8 × 10

−7

R

2

=0.89

p=1.5 × 10

−7

R

2

=0.86

p=7.6 × 10

−7

R

2

=0.91

p=3.3 × 10

−8

Fig. 4. (a) Annual surface mass balance against elevation for different years for the eight selected stakes (before projection to initial elevation);

(b) annual surface mass balance against elevation for the eight selected stakes (before projection to initial elevation); (c) standard deviation (σ)

per stake over the 15-year period; (d) SMB perturbation for the eight selected stakes for the whole glacier complex and for both glaciers

separately. The circles and squares correspond to the individual stakes (cf. panel b).

279Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland

4. METEOROLOGICAL DATA MLRA

4.1. Variable selection

Initially, meteorological variables that best describe the

observed SMB are considered as time clusters, i.e. as continu-

ous time periods (4.2.), which is in line with most previous

studies. Subsequently, a new method is introduced, where

all possible monthly combinations (full-spectrum analysis)

are considered for describing the observed SMB signal (4.3.).

For all MLRAs, the goal is to describe most of the observed

SMB variance, by using as few predictor variables as possible.

Adding additional predictor variables increases the fraction of

the SMB variation that can be explained, but reduces the

degrees of freedom. To assess whether it is justified adding

additional predictors (i.e. to avoid overfitting), the p-value of

the F-test is considered, which should be as low as possible.

4.2. Continuous time periods

Annual periods are used at first (4.2.1.), after which a subdiv-

ision is made into half-years (4.2.2.), seasons (4.2.3.) and

finally months (4.2.4.).

4.2.1. Annual time period

As a first test, the mean annual temperature (T

ann

) and the

total annual precipitation (P

ann

) are used to explain the

observed SMB variation (i.e. MLRA with 13 error degrees of

freedom). An MLRA between these two variables and the

measured SMB reveals that ∼22% of variance in SMB can

be explained by these annual anomalies (R

2

=0.22, see

Fig. 5a and Table 2). The weak significance of this model

in describing the measured SMB variance is reflected in the

high p-value for the F-test (0.229). Despite the weak correl-

ation, the signs of the variables are logical, being negative

for T

ann

(−0.35, negative correlation between temperature

and SMB) and positive for P

ann

(+0.20, positive correlation

between precipitation and SMB).

4.2.2. Half-year time period

In the second step, the year is subdivided in a winter half-year

(WHY), consisting of fall (OND: October–November–

December) and winter (JFM: January–February–March), and a

summer half-year (SHY), which consists of spring (AMJ: April–

May–June) and summer (JAS: July–August–September). These

monthly clusters rather agree with the glaciological seasons

than with the meteorological seasons. They are chosen so

that the fall season (OND) starts just after the field measure-

ments, broadly corresponding to the beginning of the accumu-

lation season (i.e. the beginning of the water/hydrological year).

From the MLRA, the importance of the SHY temperature

(T

SHY

) is clear, as this predictor variable alone already

explains 35% of the observed SMB variance (R

2

=0.35).

Without this variable, almost none of the SMB variance

can be explained in a MLRA with two predictor variables

(<4%; p-values F-test above 0.8) (Table 2). The SHY tem-

perature and the WHY precipitation account for almost half

of the observed SMB variance (46.4%, Fig. 5b), which

rejects the null hypothesis (no linear correlation) at the 5%

level (p-value F-test 0.024), but which still leaves a very

large part of the SMB unexplained. The larger absolute

regression coefficient T

SHY

(−0.51) compared with P

WHY

(0.25) indicates a relatively higher importance of the SHY

temperature. This is also supported by the fact that the SHY

temperature combined with the (supposedly) not very rele-

vant SHY precipitation can explain almost a similar fraction

of the observed SMB variance (R

2

=0.39, see Table 2).

4.2.3. Seasonal time period

In this analysis, the predictor variables were split up in sea-

sonal components, i.e. spring (AMJ), summer (JAS), autumn

Fig. 5. Observed SMB perturbation and modelled SMB perturbation

based on multiple linear regression analysis (MLRA) using the

following two predictors: (a) annual temperature (T

ann

) and annual

precipitation (P

ann

), (b) summer half-year temperature (T

SHY

) and

winter half-year precipitation (P

WHY

), (c) average May, June, July

temperature (T

MJJ

) and winter half-year precipitation (P

WHY

). The

black line is the observed SMB (cf. black line in Fig. 4d), the grey

line is the calculated SMB signal resulting from the MLRA.

280 Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland

(OND) and winter (JFM). Combined with the possibility of

still using the variables at the half-year time period (WHY

and SHY, see previous section), this allows for 36 (6

2

) pos-

sible combinations for MLRAs with one temperature and

one precipitation predictor variable. With two predictor vari-

ables, most of the SMB variability is described by the SHY

temperature and the WHY precipitation (cf. Section 4.2.2.),

followed by combining the spring temperature and the

WHY precipitation (R

2

=0.46, p-value of F-test =0.0251).

With three predictor variables, most of the variation can be

described through the spring and summer temperature

normalised anomalies as two separate variables and the

WHY normalised precipitation anomaly as one variable:

SMB =−0.49 T

spr

−0.21 T

sum

+0.23 P

WHY

. The R

2

-value

of 0.55 is higher than the one for the model based on T

SHY

and P

WHY

(see Section 4.2.2.), but the p-value of the F-test

(0.0286) is also higher (suggesting overfitting) as the error

degrees of freedom are in this case reduced to 12 (vs 13 in

the previous analysis). Although the correlation remains rela-

tively low, which suggests that a more refined choice of time

periods is needed, the regression coefficients indicate a larger

importance for spring temperature compared with the

summer temperature.

4.2.4. Monthly time period

A more in-depth analysis, where all different combinations of

clustered (subsequent) months are allowed, suggests that the

SMB variance is best described by the average May–June–

July temperature (T

MJJ

) and the total precipitation from

October to March (i.e. the WHY precipitation, Table 2).

This MLRA is statistically very significant (p-value F-test of

1.02 × 10

−5

) and accounts for 85.3% of the variance in mea-

sured SMB (Fig. 5c), with a clear dominance of the MJJ tem-

perature (regression coefficient of −0.65) compared with the

WHY precipitation (regression coefficient of 0.24).

4.3. Discontinuous time periods (full-spectrum

analysis)

MLRAs are now performed between the SMB series and all

possible monthly combinations of temperature and precipita-

tion. This new setup is still based on two predictor variables,

i.e. one for temperature and one for precipitation, and the

error degrees of freedom are therefore still equal to 13. For

each MLRA, a different combination of temperature and pre-

cipitation months is selected and analysed, resulting in a total

of 16.8 million (=2

24

) combinations. The total number of

combinations arises from combining all the 24 different vari-

ables, 12 for the temperature months and 12 for the

precipitation months, in every possible way. Performing

such a large number of MLRAs is now possible because of

increased computational power, which was a limiting

factor in the past (e.g. Letréguilly, 1988). For each MLRA,

the temperature predictor variable is the mean temperature

of the selected temperature months, while the precipitation

variable is the precipitation sum of all selected precipitation

months. As was the case for the continuous time period

MLRAs (Section 4.2.), both predictor variables are expressed

as anomalies through a z-score conversion to allow for a

direct comparison of their individual contribution to

explain the SMB series.

The R

2

values of the 2

24

MLRAs have a right-skewed dis-

tribution (skewness of +0.759), with a peak at R

2

=0.1–0.12

(see Fig. 6a). R

2

values range between 0 and 0.9086. The

highest R

2

value (0.9086) is obtained from the mean

May–June–July–October–November temperature and the

February–May–June–December precipitation. This obtained

unique combination is however very sensitive to even

small changes in observed SMB, and it is therefore more rele-

vant to consider an ensemble of combinations. The 50 com-

binations that best describe the observed SMB variance

(Fig. 6b;Fig. 7), i.e. the ones with the highest R

2

values

(R

2

>0.86088), have a clear monthly pattern. For all 50 com-

binations, the June and July temperatures are included (100%

probability inclusion (p.i.)), while 49 combinations include

the May temperature (98% p.i.) (Fig. 6c). Subsequently, the

October temperature (40% p.i.) and November temperature

(24% p.i.) sometimes appear in the top 50 combinations,

but for all other months, the effect of the temperature on

the SMB variations is close to negligible. For precipitation

all months are well represented in the top 50 combinations,

except for July and August (Fig. 6d). The most crucial months

seem to be those before and at the onset of the ablation

season (February–March–April–May–June) and those at the

beginning of the accumulation season –early winter

(October–November–December).

When considering the top 0.1% of all combinations,

which corresponds to the 16 777 combinations with an R

2

value higher than 0.73599, the importance of the monthly

temperatures remains almost unaltered (Fig. 6e). July is the

most common month in this selection (98.8% p.i.), closely

followed by May (93.3% p.i.), June (92.4% p.i.), and subse-

quently by October (57.9% p.i.) and November (37.2% p.

i.). For the precipitation, the importance of the individual

months changes slightly when considering this larger ensem-

ble of combinations (Fig. 6f) compared with the top 50 com-

binations (Fig. 6d). The summer months still have a slightly

more limited contribution to the observed SMB, but for the

other months, no clear pattern is obtained and all months

Table 2. Multiple linear regression analysis (MLRA) between z-score standardised meteorological variables (averaged over Segl-Maria and

Samedan) and observed SMB perturbations for eight selected stakes covering the period 2001–2016

Time period Best-fit multilinear correlation R

2

p-value F-test

Annual SMB =−0.35 T

ann

+0.20 P

ann

0.22 0.229

Half-year SMB =−0.09 T

WHY

+0.002 P

WHY

0.02 0.899

SMB =−0.14 T

WHY

+0.10 P

SHY

0.04 0.803

SMB =−0.51 T

SHY

+0.25 P

WHY

0.46 0.024

SMB =−0.44 T

SHY

−0.14 P

SHY

0.39 0.052

Seasonal SMB =−0.47 T

spr

+0.14 P

SHY

0.46 0.025

SMB =−0.49 T

spr

–0.21 T

sum

+0.23 P

SHY

0.55 0.029

Individual months (continuous) SMB =−0.65 T

MayJunJul

+0.24 P

WHY

0.85 1.02 × 10

−5

281Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland

have a fairly similar occurrence in the 0.1% top combina-

tions (varying between 41.2 and 73.5% p.i.).

5. DISCUSSION

5.1. Robustness correlations and time-series length

The MLRAs performed for different time periods (annual,

half-year, seasonal and monthly) are mutually consistent,

which indicates that the correlations are robust. The full-

spectrum method, in which discontinuous periods are also

considered, supports this, and the dominant months from

the continuous time period analysis (May, June, July

temperature and WHY precipitation) are confirmed.

Furthermore, the importance of other months’temperature

(October, and to a lesser extent November) and precipitation

(spring: April, May, June) also appears.

The robustness and significance of the results is supported

by statistical evidence. For the best models, low p-values for

the F-test and high R

2

-values are obtained (R

2

>0.86 for the

best 50 combinations of the full-spectrum analysis). In con-

trast, when performing the full-spectrum analysis with

series of random SMB data and/or meteorological data, the

50 best combinations of monthly meteorological variables

are typically only able to explain between 50 and 70% of

the SMB variance, significantly less than the 86% found

here from the real data.

The length of the record, 15 years, is thus sufficient to obtain

meaningful regression coefficients. The main finding of a very

limited effect of August and September temperature in fact

already appears when a period of 10 years (e.g. 2001–2011)

is considered (not shown here). The sufficient length of the

15-year record agrees with the findings by Letréguilly and

Reynaud (1989) who state that at least 10 years are needed

to obtain qualitative regression coefficients, and Vincent and

Vallon (1997) who mention a 5–10-year period as a minimum.

5.2. Temperature dominance

The results from the MLRAs indicate that there is a strong cor-

relation between temperature and observed SMB, which is

the cornerstone for various widely used simplified SMB

models (e.g. Reeh, 1989; Hock, 2003; Braithwaite and

others, 2013). This strong SMB dependence on temperature

is also in line with the findings from a previous simple

surface energy balance (SEB) modelling study on the

Morteratsch glacier (Nemec and others, 2009). Nemec and

others (2009) parameterised the longwave radiation and the

Fig. 6. (a) Probability density of R

2

values for MLRA for all (16.8 million) possible combinations of months. The green area corresponds to R

2

range between 0.74 (lower limit 0.1% highest R

2

values) and 0.86 (lower limit 50 highest R

2

combinations), the red area is the range of the 50

combinations with the highest R

2

value. (b) Zoom on the 50 MLRAs with highest R

2

values, which are visually non-detectable in (a). (c) and (d)

Probability that a particular month’s temperature or precipitation is included in the 50 combinations with highest R

2

value. (e) and (f)

Probability that a particular month’s temperature or precipitation is included in the top 0.1% combinations (16 777 combinations) with

the highest R

2

value.

Fig. 7. SMB reconstruction for the top 50 combinations (see Fig. 6b–

d). The thick black line represents the observed SMB perturbation

(cf. Fig. 5) and the grey lines represent the 50 best combinations

(the greyer the higher the R

2

, the whiter the lower the R

2

).

282 Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland

turbulent heat fluxes as a function of temperature (cf.

Oerlemans, 2001) and were able to closely reproduce the

observed SMB.

The SEB in the ablation area of the Morteratsch glacier

complex is mainly driven by the radiation components and

the sensible heat flux, while the latent heat flux is much

smaller (Klok and Oerlemans, 2002,2004). Notice that this

is strongly related to the climatic conditions and the location

of the glacier (mid-latitude maritime glacier), and in other set-

tings (e.g. continental, polar, tropical), the importance of the

individual SMB components may strongly vary (e.g. Benn

and Evans, 2010). The fact that temperature explains a very

large fraction of the observed SMB variations partly results

from its direct influence on the sensible heat flux, which is

one of the dominant fluxes in the glacier SEB. Furthermore,

temperature correlates with the short-wave radiation compo-

nent, as a higher temperature leads to surface melt and

lowers the surface albedo, which in turn increases the net

shortwave radiation. The longwave radiation, despite being

the first source of the energy balance (Klok and Oerlemans,

2002), has a limited interannual variability and is therefore

of limited importance when considering SMB variations

over time (cf. Sicart and others, 2008).

MLRAs in which additionally radiation components are

considered (e.g. Lefauconnier and others, 1999) hardly

improve the fraction of the observed SMB that can be

explained. For instance, if the total monthly hours of sun

are added to the analysis, the fraction of the SMB variance

that can be explained is almost unaltered. The monthly

hours of sun are not significant at the 5% significance level

given the other terms (temperature and precipitation for

selected months) in the MLRA. Notice that the same is true

when a second temperature variable is introduced (i.e.

working with three predictor variables), adding additional

complexity to the MLRA, with limited to no added value

(case of overfitting, as the p-value of the F-test increases).

5.3. Limited effect of mid- to late-summer conditions

on SMB variability

In the ablation area of alpine glaciers, most of the surface

mass loss occurs during summer, which at the monthly

time period used in this study corresponds to July, August

and September. During this period, the snow cover that pro-

tects the glacier through its high albedo is removed and the

glacier ice (low albedo) is directly exposed. Due to its

lower albedo, an ice-covered surface absorbs more of the

incoming radiation compared with a snow-covered surface

and therefore experiences more melt.

According to the MLRA conducted here, the meteoro-

logical conditions in August and September have a very

limited effect on the interannual SMB variability, as models

without these components are able to explain more than

90% of the observed changes in SMB (Fig. 6). The precipita-

tion during these 2 months typically occurs as rain and also

the temperature of these months was found to have a very

limited contribution to the observed SMB variability. On

the other hand, the WHY precipitation and the meteoro-

logical conditions at the start and at the end of the ablation

season have a much larger effect on the interannual SMB

variability in the ablation area of the Morteratsch glacier

complex. The latter corresponds to the mid- to late-spring

precipitation (April, May), the late-spring to early-summer

temperature (May, June, July) and early-autumn precipitation

and temperature (October). The WHY precipitation deter-

mines the winter balance and determines the thickness of

the snow pack at the beginning of the ablation season. The

May–June–July temperatures subsequently determine the

snow ablation intensity and the precipitation type (snow/

rain). A lower mean monthly temperature increases the

chance of a precipitation event being a snowfall event,

which increases the SMB. The importance of autumn

(October and to a lesser extent November) conditions is

not surprising, as this period largely determines the transition

from the ice-covered to the snow-covered ablation area.

Often ice melt still occurs after the SMB measurements

(late September to early October), as field observations deter-

mined that the ablation area is often not entirely snow-

covered at this time.

The results from the MLRAs thus suggest that the length of

the ablation season, rather than its intensity, determine the

interannual SMB variability here. This is related to the inter-

annual melt differences between the summer seasons, which

are believed to be much smaller than the intra-annual con-

trast between snow and ice melt (contrast between the

seasons) (cf. Durand and others, 2009; Thibert and others,

2013). Results from a straightforward positive-degree-day

(PDD) calculation support the dominant effect of the ice

ablation season duration on the glacier’s ablation area

SMB. The PDD model is run at an hourly time step, and

the 2 m temperature at each stake is derived from an interpol-

ation between measured temperatures at Samedan (1708 m)

and Corvatsch (3302 m). Precipitation is upscaled from

Samedan with an elevation gradient of 1.0 mm w.e. m

−1

,

based on field measurements in the glacier’s accumulation

area (Sodemann and others, 2006; Nemec and others,

2009). A distinction is made between snowfall and rainfall

depending on the air temperature. The degree-day factors

(DDF) for snow and ice are calibrated in order to minimise

the RMSE between observed and modelled SMB. For the

eight stakes over the 15-year period, an RMSE of 0.385 m

w.e. a

−1

is obtained (Fig. 8a) with a DDF for snow/ice of

2.8/5.4 mm ice eq. per degree day for the Morteratsch

glacier, and 3.5/6.6 mm ice eq. per degree day for the Pers

glacier. These DDFs are in good agreement with the values

found for other glaciers in the European Alps (e.g. Hock

and others, 1999). Despite the PDD model’s simplicity (e.g.

shading effects from the surrounding mountains and refreez-

ing/retention of meltwater in the snowpack are not

accounted for), the observed SMB (Fig. 8a) and its interann-

ual variability (R

2

=0.93, Fig. 8b) are closely reproduced.

The PDD model indicates that the number of ice ablation

days (averaged over the eight stakes) is strongly anticorre-

lated with the SMB perturbation (Fig. 8c). The variation in

ice ablation days is ∼70% anticorrelated to the modelled

SMB variance (R

2

=−0.71) and 60% anticorrelated to the

observed SMB (R

2

=−0.60). The PDD simulations further-

more indicate that the temperature in the early- and late-abla-

tion season has the largest influence on the ice ablation

season length (Fig. 8d, shown for a temperature bias of

−1 and +1°C). The effect of changes in the July

temperature on the ice ablation season length is found to

be limited, as this is usually the warmest month of the year.

The importance of the WHY precipitation and early-

and late-ablation season meteorological conditions is also

clear when analysing the 2 years with the lowest SMB

(2002–03 and 2014–15) and the 3 years with the highest

283Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland

SMB (2003–04, 2012–13 and 2013–14) (Table 3). The

lowest SMB (2002–03) over the period can be explained

by the 2003 European heat wave (Beniston, 2004;

García-Herrera and others, 2010)(z-score of 1.41 for

2003 T

sum

), but also the high temperatures in October

2002 and May 2003 have had an important role (z-score

of 1.58 for T

OctMayJunJul

). Notice that the 2002–03 SMB

year also had a high winter precipitation (524.4 mm vs

an average of 308.8 mm, z-score of 1.89), but this is

almost entirely due to the extremely high November pre-

cipitation, which is the highest monthly precipitation for

the entire period considered in this study 16-year period

(392.2 mm, see Fig. 2). A substantial fraction of the pre-

cipitation occurred on November 14–16 when tempera-

tures in the ablation area of the glacier were varying

between −3.5 and +3.5°C (Fig. 9). It is therefore likely

that (a part of) the precipitation over this time period did

not occur as snow, which illustrates the aforementioned

role of autumn temperatures in delimiting the rain/snowfall

regime. The snow accumulation that may have occurred in

November likely melted during late autumn and early

winter, which winter had moreover a very dry period

later on (P

Dec→Mar

for this balance year is the lowest of

the record, 58.8 vs 136.3 mm on average; z-score of

−1.26). The year with the second most negative SMB

(2014–15) had similar average temperatures for the

months of May–June–July–October as 2002–2003, but

probably the SMB was slightly less negative due to the

higher December–March precipitation, which was close

to average (133.6 mm, z-score of −0.05).

Fig. 8. (a) Observed vs modelled annual SMB through a PDD approach for the eight selected stakes over the 15-year period. (b) Observed

SMB perturbation (black line, cf. Fig. 5) and PDD modelled (grey line) SMB perturbation. (c) PDD modelled SMB perturbation (grey line)

and days of ice ablation (averaged over eight selected stakes, black line). (d) Change in number of ice ablation days for a monthly

temperature increase/decrease of 1°C, as obtained from the PDD model.

Table 3. Meteorological variables averaged over the stations of Samedan and Segl-Maria for the 2001–2016 period for selected years. Values

between brackets indicate the respective z-score of the values

2001–2016

Average

2002–03

(Lowest SMB)

2014–15

(Second lowest SMB)

2003–04

(Third highest SMB)

2012–13

(Second highest SMB)

2013–14

(Highest SMB)

T

spr

(T

AprMayJun

) (°C) 6.44 7.58 (1.52) 6.68 (0.32) 5.52 (−1.23) 5.28 (−1.54) 6.02 (−0.56)

T

sum

(T

JulAugSep

) (°C) 10.69 11.72 (1.41) 11.58 (1.22) 10.60 (−0.12) 11.05 (0.50) 9.65 (−1.42)

T

MayJunJulOct

(°C) 8.34 9.51 (1.58) 9.53 (1.59) 6.88 (−1.96) 7.58 (−1.03) 7.82 (−0.70)

P

WHY

(mm) 308.8 524.4 (1.89) 436.1 (1.12) 302.9 (−0.05) 258.5 (−0.44) 467.1 (1.39)

P

Dec−Mar

(mm) 136.3 58.8 (−1.27) 133.6 (−0.05) 109.05 (−0.45) 95.1 (−0.67) 287.85 (2.48)

284 Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland

The 3 years with the highest SMB (2003–04, 2012–13 and

2013–14) are all characterised by low average May–June–

July–October temperatures. The limited importance of

August and September temperatures on the SMB variability

is also illustrated by the relatively high SMB for the years

2003–04 and 2012–13, which years respectively had an

average (z-score of −0.05) summer (July–August–September)

and an above-average warm summer (11.05 vs 10.69°C on

average, z-score of 0.50). The most positive SMB year,

2013–14, had relatively low May–June–July–October tem-

peratures, and here the very high P

WHY

(z-score of 1.39)/

P

Dec−Mar

(z-score of 2.48) played a major role.

5.4. Comparison to other studies

Most studies correlating temperature and precipitation data

to SMB variations found summer temperatures to be the

key component. Martin (1978) explained 77% (R

2

=0.77)

of the 1949–75 SMB variance of Glacier de Sarennes by

means of maximum summer temperature, precipitation

from October to May and a June precipitation component

(three predictor variables). For glaciers in the Swiss Alps,

Chen and Funk (1990) used the mean summer temperature

and the mean annual precipitation to describe the SMB var-

iations and with this they could explain between 49 and 81%

of the observed variance. For these studies on Alpine gla-

ciers, the temperature variations dominate the SMB com-

pared with the precipitation variations, which is also the

case in our analysis and confirmed in other SMB modelling

studies (e.g. Oerlemans and Reichert, 2000). In a study on

Austre Brøggerbreen (Svalbard) (Lefauconnier and Hagen,

1990), 77% of the 19-year net SMB series is explained by

the July and August PDDs, which are a measure for air tem-

perature, combined with precipitation from October to May.

Later this was updated to 66% of the SMB variance being

explained for a 28-year time series by Lefauconnier and

others (1999). In a recent study, Trachsel and Nesje (2015)

analysed the SMB of eight Scandinavian glaciers using

statistical models and found that summer temperature and

winter precipitation explained more than 70% of the vari-

ance for maritime glaciers, and between 50 and 70% for con-

tinental glaciers.

It is noteworthy that the early- and late-ablation season

conditions (e.g. temperatures and precipitation in April,

May, June and October), which are key in explaining the

SMB variations for the Morteratsch glacier complex, are

never included in the above analyses. An exception is the

study by Letréguilly (1988) where the June and July tempera-

tures (without August and September) are combined with the

October–March precipitation to explain ∼70% of the SMB

variations for Peyto glacier (Rocky Mountains, Canada). In

the same study, the late-spring conditions (T

May,June,Jul,Aug

and P

Oct→May

) are also incorporated to explain 65–75% of

the SMB variations for Sentinel glacier and Place glacier

(Coast Mountains, Canada).

Important to note as well is that the above-mentioned

studies consider the glacier-wide SMB, while our analysis

is based on stakes only from the glacier’s ablation area. In

the accumulation area, the snow-ice cover contrast is not

present, but also here a seasonal albedo contrast exists.

The winter and spring fresh snow have a high albedo, but

as the summer advances the snow cover metamorphoses

and becomes dirtier (Oerlemans and others, 2009), which

in combination with melt and rainfall events (Oerlemans

and Klok, 2004) leads to an albedo decrease. Therefore,

the differences between the ablation area SMB signal and

the glacier-wide SMB signal are expected to be limited.

The difference between our results, with a limited effect of

August and September conditions and other studies is there-

fore also believed to be partly linked to differences in meth-

odology (e.g. point measurements vs glacier-wide SMB, see

further) and the full spectrum of monthly combinations

considered here.

Also for the continuous month approach (4.2.4), i.e. the

classic approach, the correlations obtained are very high

for our 15-year record (Fig. 5c) compared with other

studies. This may again be linked to opting for a different

variable than the widely used June–July–August tempera-

tures (here May–June–July is used), but may in part also be

related to the quality and homogeneity of the data used in

this study. First of all, the data from the MeteoSwiss stations

is from a recent period and highly reliable, which is not

always the case in earlier studies, as argued for instance by

Letréguilly (1988). Especially older temperature and precipi-

tation measurements contain biases and need to be cor-

rected to homogenise the time series (e.g. Böhm and

others, 2010). Second, the systematic error on the

Morteratsch SMB is believed to be relatively limited and

similar in magnitude over time, as all measurements

occurred by the same group of scientists, using the same

methodology and with the same materials. This is often not

the case and for extended periods of time, the measurement

errors can be very large, up to 20–40 cm and even more for

older stakes (Letréguilly and Reynaud, 1989). Finally, we

opted to correlate the meteorological data directly to our

SMB measurements, instead of using a spatially averaged/

glacier-wide SMB (cf. Vincent and others, 2017). The eight

stakes used in this study cover the whole period and move

only very little, which is corrected with high-precision GPS

measurements. Other studies consider spatially averaged

SMB or (partially) modelled SMB, which adds an additional

uncertainty.

Fig. 9. Hourly precipitation at Samedan meteorological station and

hourly temperature at Samedan meteorological station (1708 m) and

Corvatsch meteorological station (3302 m). Temperature series at

2200 m (lowest elevation of SMB measurements) and 2900 m

(highest elevation of SMB measurements) are derived through

linear interpolation between the Samedan and Corvatsch series.

The shaded area roughly corresponds to the rain/snow threshold.

285Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland

6. CONCLUSION AND OUTLOOK

In this study, we analysed a dataset consisting of 15 years of

SMB measurements in the ablation area and around the

ELA of the Morteratsch glacier complex, and attempted to

describe the SMB variability through a minimal statistical

approach. The observed interannual SMB variance is very

similar for both glaciers and can largely be explained

(>90%) by combining monthly temperature and precipita-

tion data from nearby meteorological stations. The fact that

almost the entirety of the SMB variance can be explained

by a simple MLRA with one temperature and one precipita-

tion variable makes it redundant to add additional predictor

variables (e.g. radiation, additional temperature or precipita-

tion variable) to the analysis. The full-spectrum method intro-

duced in this study relies on increased computational power,

which was a limiting factor in the past, and considers the con-

tribution of every month’s temperature and precipitation,

which are combined in these two predictor variables. The

MLRA indicates that the spring-/early-summer and autumn

meteorological conditions play a crucial role, as they deter-

mine the length of the ablation season. High temperatures

and low precipitation in spring advance the transition from

a snow- to an ice-covered surface, while in autumn they

favour a prolonged ice ablation season. Furthermore, tem-

peratures during these months also influence the SMB per-

turbation by affecting the precipitation type (rain/snow) over

the ablation area of the glacier. The mid- to late-summer

(August and September) conditions are found to have only a

minor effect on the annual SMB perturbation. These findings

are supported by output from a simple PDD model, indicating

that early- and late-melting season meteorological conditions

are the main drivers for the ice ablation season duration,

which is itself strongly anticorrelated to the SMB perturbation.

The widely used June–July–August (JJA) temperature index

may therefore not always be the most appropriate variable

to describe SMB variability through statistical correlation.

Applications of the full-spectrum method to other glaciers

and other SMB series will be useful to better understand the

meteorological station representativeness, the impact of

their distance to the measurements and contrasts between

ablation/accumulation measurements and seasonal/annual

SMB measurements. Through its minimal character, relying

on as few predictor variables as possible, the method is an

interesting alternative/complementary approach to the more

classic, well-established stepwise selection procedures (e.g.

Tinsley and Brown, 2000; Whittingham and others, 2006).

In future works, the threshold delimiting the ‘best’combina-

tions (here set at the best 50 combinations and the top 0.1%

combinations) will have to be investigated and could poten-

tially be changed for other studies. In this study, the tempera-

ture dependence is very clear, but this is less the case for

precipitation. For the latter, the monthly signal changes as

the threshold changes, which is related to the fact that the pre-

cipitation contributes less to the observed SMB variance than

the temperature. Possible extension of the method presented

here could for instance separate between the SMB in the abla-

tion and accumulation area, or between winter and summer

SMB, if the necessary measurements are available.

SUPPLEMENTARY MATERIAL

The supplementary material for this article can be found at

https://doi.org/10.1017/jog.2018.18

ACKNOWLEDGEMENTS

We are grateful to everyone who helped to collect the SMB

data on the Morteratsch glacier complex over the past

15 years and thank MeteoSwiss for making all meteoro-

logical data available. Harry Zekollari performed this work

through a PhD fellowship of the Research Foundation –

Flanders (FWO-Vlaanderen) and finalised this research

during a research stay at ETH Zürich, financed by a travel

grant for a long stay abroad from the Research Foundation

Flanders (grant No. V427216N). Matthias Huss is thanked

for the fruitful discussions. We thank the Associate Chief

Editor H. Jiskoot, the Scientific Editor N. Eckert, and three

anonymous reviewers whose comments and suggestions

greatly helped to improve the manuscript.

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MS received 5 February 2017 and accepted in revised form 7 February 2018; first published online 22 March 2018

288 Zekollari and Huybrechts: Statistical modelling of the surface mass-balance variability of the Morteratsch glacier, Switzerland