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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