INTERNATIONAL JOURNAL OF CLIMATOLOGY
Int. J. Climatol. 20: 615–640 (2000)
CLIMATE SENSITIVITY OF SNOW COVER DURATION IN AUSTRIA
MICHAEL HANTEL*, MARTIN EHRENDORFER and ANNEMARIE HASLINGER
Institut für Meteorologie und Geophysik der Uni6ersität Wien, Hohe Warte 38, A-1190 Vienna, Austria
Recei6ed 5 May 1998
Re6ised 12 July 1999
Accepted 25 July 1999
ABSTRACT
The number of days with snow cover at Austrian climate stations, normalized by the maximum possible snow days
within a season, is denoted n. This seasonal relative snow cover duration is considered a function of station height
H and of the seasonal mean temperature T over Europe. When T increases, n decreases and 6ice 6ersa. The function
becomes saturated both for high stations at low European temperature (‘always snow’, n = 1) and for low stations at
high temperature (‘never snow’, n=0). In the saturated regions, the sensitivity s (n(H, T)/(T is practically zero,
while in the transition region, s is extreme. The observed interannual fluctuations of T are considered here as
simulation of a possible climate shift. s is determined for the climate stations of Austria from its snow cover record
[1961–1990, 84 stations between 153 and 3105 m above sea level (a.s.1.)] by fitting the data of n for each individual
station (local mode) as well as for all Austrian stations (global mode) with a hyperbolic tangent function. In the
global mode, s reaches an extreme value of −0.349 0.04 K − 1 in winter and −0.469 0.13 K − 1 in spring.
The implications of these results are discussed. Included in this discussion is the fact that a rise in the European
temperature by 1 K may reduce the length of the snow cover period in the Austrian Alps by about 4 weeks in winter
and 6 weeks in spring. However, these extreme values apply only to the height of maximum sensitivity (575 m in
winter, 1373 m in spring); the actual sensitivity of individual stations located at higher or lower levels is less.
Copyright © 2000 Royal Meteorological Society.
KEY WORDS: Austria;
Alpine climate; logistic curve; snow cover duration; Europe temperature; climate sensitivity; the period
1961 – 1990
1. INTRODUCTION
The seasonal duration of snow cover at an Alpine climate station increases with station height. However,
the snow cover at one fixed station may also be influenced by climate changes. This study investigates the
impact of climate changes upon the snow cover duration.
Snow cover duration is just one out of many climate elements upon which possible climatic changes
may have a certain impact. In more general terms, climatic changes may be reflected in different climate
elements. Examples specific for the Alpine region, and relevant on time scales between about 1 and 100
years, include glacier length (Oerlemans, 1994), mountain plants (Grabherr et al., 1994) or snow depth
and duration (Beniston, 1997). Concerning glacier length and mountain plants, the response time is well
above 1 year. For example, Oerlemans (1994) reports for most valley glaciers a response time of 10–50
years with respect to global climate changes, while Grabherr et al. (1994) find upward moving rates for
typical nival plant species of the order of 1 m per decade.
On the other hand, the non-glaciated snow cover, which disappears and recovers within the seasons of
1 year, is an interannual climate phenomenon. This is evidenced by the fact that the snow cover may
fluctuate from year to year, both in amount and duration. On longer time scales, there may be climate
epochs with consistent long deep snow cover during consecutive winters, which are considered cold;
conversely, epochs with consistent short thin snow cover during consecutive winters would be considered
* Correspondence to: Institut für Meteorologie und Geophysik der Universität Wien, Hohe Warte 38, A-1190 Vienna, Austria; tel.:
+43 1 360263001; fax: + 43 1 360263020; e-mail: [email protected]
Copyright © 2000 Royal Meteorological Society
616
M. HANTEL ET AL.
warm. Beniston (1997) concludes, on the basis of snow statistics over the last 50 years in the Swiss Alps,
that temperature is the controlling factor for snow depth and duration. Further he finds, via the seasonal
to annual pressure field over the Alpine region, a strong correlation between high (low) snow amounts
and duration and low (high) values of the North Atlantic Oscillation Index. This implies that large-scale
forcing, not local or regional factors, controls the snow characteristics in the Alps.
While the qualitative anti-correlation between seasonal snow and climate warmness is evident, the
quantitative relation is less well known, particularly in the Alps where the height dependence of the snow
cover tends to blur the climate dependence of the snow cover. To quantify this, one can consider the
seasonal number N of days with snow cover equal to or exceeding 5 cm (N in days per season; dps in this
paper). For example, N0 =92 dps is the maximum value N can have in spring; N= 0 dps would be the
minimum value. It is convenient to consider, instead of N, its normalized equivalent,
n
N
N0
(1)
This percentage of seasonal number of snow-days shall be referred to as the relati6e snow co6er duration;
n can have values between 0 and 1. Definition (1) is not restricted to the season; it applies equally to the
period of a month or a year.
In Figure 1 the mean observed n̄ for winter (DJF, 50 stations) and spring (MAM, 66 stations) is
displayed for climate stations of Austria as a function of station height H; the overbar refers to the
Figure 1. Mean snow cover duration n̄ plotted versus station height H. Each dot represents one Austrian climate station. Station
data averaged 1961 – 1990. Data fitted with logistic function (ordinate, independent variable H; abscissa, dependent variable n̄). (a)
Winter, (b) spring
Copyright © 2000 Royal Meteorological Society
Int. J. Climatol. 20: 615 – 640 (2000)
AUSTRIAN SNOW COVER
617
climate average (here, a 30 year period). The relative snow cover duration increases with height in all
seasons. For example, the mean spring snow duration is close to zero at low elevations and increases to
100% at Alpine levels exceeding about 3000 m. The slope of the n̄(H) curves must be close to zero, both
at low levels (‘never snow’) and at high levels (‘always snow’) with an extreme value in between; the level
at which the slope becomes extreme is low in winter, high in summer and intermediate in the transition
seasons. Figure 1 reproduces, with different data, Figure 2.11 of Haiden and Hantel (1993), who studied
the same problem in the context of possible impacts of global and European climate changes upon the
region of Austria.
However, the influence of large-scale climate changes cannot be investigated with plots of this type since
Figure 1 represents only the climate mean. For the purpose of studying climate changes, Haiden and
Hantel (1992, 1993) introduced, from the observed temperature anomalies of the stations de Bilt, Berlin
and Vienna, a mean European temperature TME. They related interannual shifts of N to interannual shifts
of TME and defined the climate sensitivity of snow cover duration in the form DN/DTME. They devised a
first approximation for this ratio in spring of about − 10 dps K − 1. This figure implies that a rise in
European temperature of 1 K would cause a maximum reduction of the number of spring snow days in
Austria of about 10 days, to be reached at levels between 500 and 1000 m.
It is the purpose of the present study to revisit the problem considered by Haiden and Hantel. Different
datasets for n will be used (5 cm snow cover instead of 1 cm) and for T (European temperature defined
by Jones instead of TME, see below). The snow cover duration shall be considered as a function of the
independent parameters station height and European temperature: n= n(H, T). Thus, the climate sensiti6ity shall be defined as
s
(n(H, T)
(T
(2)
We want to find the extreme sensitivity s0 (which is a minimum since s is negative) of the ensemble
average of s for the Austrian climate stations. The result will be that the approximation found by Haiden
and Hantel is an underestimate.
This program includes the following steps:
“
“
“
“
“
T shall be identified with the horizontal average of the temperature of European climate stations as
provided by Jones (1994), denoted T J. The current study has also experimented with the Europe
temperature of Peterson and Vose (1997), denoted T P; T P and T J are quite close to each other (see next
section).
The natural fluctuations of n at a fixed height are presumably dependent upon: the natural fluctuations
of T; the amount of snowfall preceding the season considered; the amount of radiation; the exposition
and ground features of the station considered; and upon other possible parameters not listed or
overlooked. Of these, the dependence upon T is only considered in this statistical model. This implies
the a priori assumption that the impact of the other parameters will be random so that the pertinent
error should be about normally distributed when the average is taken over the climate stations of
Austria. This assumption will be tested a posteriori with the data available (see Section 6).
The present approach is based upon the hypothesis that an unobservable change of the future mean
climate is being simulated by the observable fluctuations of present climate. This is equivalent to
assuming that the parameters that specify a climate fluctuation can be gained by observing interannual
natural fluctuations of the present climate. Thus, the parameters that fit the function n(H, T) can be
determined by using data for a specific period, in this case for the period 1961–1990.
The above hypothesis implies that the fit can be transferred to a slightly different climate regime
provided its mean is located within the variance regime of the present climate. This is the prerequisite
for extrapolating the sensitivity s0 found for the present climate to an unknown future climate.
The fit of observed data triples (ni, Hi, Ti ) will be made as in Figure 1, with a logistic function to be
discussed in Section 3, both in the local mode (for each climate station i individually) and in the global
mode (for all Austrian stations at once).
Copyright © 2000 Royal Meteorological Society
Int. J. Climatol. 20: 615 – 640 (2000)
618
“
M. HANTEL ET AL.
Error margins will be derived for the extreme sensitivity s0.
The paper is organized as follows. The available data are discussed in Section 2, and the theory in
Section 3. The fit procedure is considered in Section 4, and the sensitivity of the climate stations is put
into a coherent picture in Sections 5 and 6. Conclusions are drawn in Section 7.
2. DATA
Two basic datasets have been used for this study, which shall be described consecutively. Table I lists, for
the seasons of the cold half of the year, the main coordinates of the stations used along with the individual
station statistics of the basic dataset. The station temperatures have been added for completeness although
they have not been entered in the subsequent analysis.
2.1. The snow day dataset
Basic Austrian data for the present study are the number N of snow days per season. A given day was
counted as one with snow cover when the snow height was at least 5 cm. Haiden and Hantel (1992) had
used a 1 cm dataset, but Fliri (1992a) recommends using a minimum of 2 cm. Beniston (1997) considers
snow depth thresholds from 1 cm up to 150 cm. This paper shall restrict the subsequent discussion to the
threshold of 5 cm as a relevant choice for the present purpose, because of the lower limit suggested by
Fliri (1992a). On the other hand, increasing the upper limit beyond 5 cm would have severely reduced the
amount of available data.
Before further processing, the N values were normalized with N0, the total number of days in the actual
season considered (N0 =90 or 91 for winter and 92 for spring), to obtain n, the relative snow cover
duration of this season.
The geographical position of the stations is shown in Figure 2. The area above heights of about
1500 m has many gaps, while the area below that level is better represented. There are few stations that
do not have data for the whole period (e.g. Gröbming at a height of 766 m a.s.1.). No attempt was made
at this stage to exclude stations from entering the subsequent analysis.
The stations in Table I have been ordered according to height above sea level. There is a tendency in
the data to show low values of 6n (standard deviation of relative snow cover duration) for snow duration
values located in the saturated parts of the curve (n̄= 0 and n̄= 1) and high values of 6n for snow duration
values close to n̄ = 0.5. This is what is expected from the character of the quantity n: the relative accuracy
of the observation of n (inversely proportional to the variance of n) should be at a maximum in the
saturated parts; the reason is that in these climate states (‘always snow cover’ as well as ‘never snow
cover’) the observation of n is practically free of errors and is also well representative for the environment.
Conversely, the relative accuracy of n should be a minimum for intermediate snow duration values of
around 50% because here both observation and representativeness errors presumably reach their maximum amplitude. This fact will be used below for the construction of the error model for the fit.
2.2. The temperature datasets
The European temperature T is identified with the horizontal average of the temperature of European
climate stations over the array 5° – 25°E and 42.5°–52.5°N (see Figure 2) as provided by Jones (1994); see
also Hulme et al. (1995). The seasonal values of T J are considered as sufficiently close to what a climate
model should predict for the large-scale temperature development representative for the seasons of
Central Europe. In a similar manner, the data set of Peterson and Vose (1997), referred to as T P, which
is representative for the slightly larger array 5°–25°E and 40°–55°N has been used. For the definition of
a European temperature see Baur (1975) or Rocnik (1995); for the construction of a Deutschland–
Temperatur, see Schönwiese and Rapp (1997).
Copyright © 2000 Royal Meteorological Society
Int. J. Climatol. 20: 615 – 640 (2000)
619
AUSTRIAN SNOW COVER
Table I. Austrian climate stations used in the present study
Winter
Number
Name
Elevation
5972
2600
7704
5904
3805
905
3202
7604
16600
5604
19201
7011
16412
3110
4501
5010
18900
6620
6300
4700
11102
11115
20211
18705
9610
10510
13301
10000
9010
1920
1601
11803
3410
13701
9801
15001
9900
16101
18300
1400
12322
10400
12200
12810
3600
15900
14403
12615
11505
16015
13110
7220
15402
18000
15710
Groß-Enzersdorf
Hohenau/March
Eisenstadt
Wien-Hohe Warte
Krems
Retz
Linz/Stadt
Wr. Neustadt
Fürstenfeld
St. Pölten
Bad Gleichenberg
Waidhofen/Ybbs
Graz-Universität
Waizenkirchen
Ranshofen
Kremsmünster
Deutschlandsberg
Gmunden
Salzburg-Flughafen
Ried im Innkreis
Bregenz
Feldkirch
Klagenfurt
St. Andrä-Winkling
Bad Ischl
Reichenau-Rax
Bruck/Mur
Hieflau
Kufstein
Stift Zwettl
Freistadt
Innsbruck-Univ.
Pabneukirchen
Bernstein
Aigen
Mayrhofen
Admont
Zeltweg
Radenthein
Kollerschlag
Zell am See
Mürzzuschlag
Kitzbühel
Gröbming
Gutenbrunn
Oberwölz
Landeck
Radstadt
Reutte
Neumarkt
Seckau
Mariazell
Rauris
Döllach
Tamsweg
153
155
159
202
207
242
263
270
273
274
303
365
366
370
382
383
410
424
434
435
436
439
447
468
469
486
489
492
495
506
548
577
595
615
640
643
646
669
685
725
753
755
763
766
810
810
818
845
870
872
874
875
945
1010
1012
Copyright © 2000 Royal Meteorological Society
Spring
t(
6t
n̄
6n
m
t(
6t
n̄
6n
m
0.1
−0.5
0.4
0.6
0
−0.6
−0.4
−0.6
−1.2
−0.6
−0.9
−1.2
−0.5
−1.7
−1.2
−0.8
−0.6
−0.2
−0.6
−1.2
0.8
−0.4
−3.2
−2.7
−0.8
−0.6
−1.5
−1.6
−1.4
−2.3
−2.4
−0.9
−1.8
−1.3
−3.0
−1.7
−3.5
−3.8
−2.5
−3.0
−3.6
−3.0
−3.9
−3.3
−3.5
−2.9
−1.0
−4.4
−2.4
−3.4
−2.5
−2.1
−3.8
−2.6
−5.3
1.9
1.8
1.7
1.7
1.7
1.8
1.6
1.9
1.7
1.9
1.7
1.9
1.5
1.9
1.9
1.8
1.7
1.8
2.0
1.9
1.6
1.7
1.6
1.5
1.6
1.8
1.4
1.3
1.5
1.9
1.7
1.6
1.7
1.7
1.4
1.4
1.5
1.8
1.3
1.7
1.7
1.5
1.6
1.2
1.9
1.3
1.3
1.1
1.5
1.3
1.4
1.5
1.4
1.3
1.4
0.171
0.189
0.178
0.314
0.186
0.158
0.287
0.210
0.406
0.308
0.400
0.472
0.368
0.418
0.343
0.351
0.365
0.345
0.390
0.322
0.384
0.321
0.620
0.541
0.657
0.341
0.419
0.766
0.670
0.406
0.540
0.495
0.535
0.421
0.605
0.768
0.829
0.563
0.583
0.714
0.763
0.729
0.841
0.688
0.627
0.589
0.485
0.898
0.844
0.676
0.734
0.791
0.803
0.695
0.782
0.156
0.167
0.150
0.219
0.205
0.157
0.216
0.179
0.255
0.213
0.262
0.250
0.232
0.261
0.230
0.238
0.252
0.253
0.223
0.237
0.260
0.236
0.324
0.278
0.250
0.193
0.248
0.207
0.246
0.262
0.256
0.275
0.237
0.241
0.248
0.209
0.183
0.284
0.318
0.246
0.248
0.224
0.166
0.227
0.235
0.256
0.295
0.151
0.182
0.247
0.225
0.209
0.233
0.245
0.222
28
30
28
24
24
30
28
25
25
30
27
25
27
30
28
30
28
27
30
29
30
30
28
27
29
25
28
27
30
29
30
30
30
28
27
27
27
28
29
24
26
28
22
9
21
27
30
27
29
24
26
30
29
23
30
9.7
9.2
9.9
9.9
9.3
8.9
9.0
8.9
8.9
8.7
9.2
7.9
9.4
7.7
7.6
8.4
8.9
8.1
8.4
7.7
8.6
8.4
8.1
8.0
7.9
7.7
7.9
7.5
7.7
6.1
6.7
8.5
7.0
7.7
6.5
7.5
6.4
6.7
7.2
6.0
6.4
5.8
5.8
6.2
4.8
6.0
7.7
5.0
5.1
5.8
6.2
5.1
5.2
5.4
4.8
1.0
0.9
0.9
1.0
0.9
0.9
1.0
1.0
0.8
1.1
0.9
1.0
0.9
0.9
1.1
1.0
0.9
1.0
1.1
1.1
0.9
0.9
0.9
0.8
1.1
0.9
0.9
1.0
1.0
1.0
1.0
1.0
1.1
0.9
0.9
1.0
1.0
0.9
0.9
1.1
1.0
1.0
1.2
1.1
1.1
0.8
0.9
1.1
1.0
0.8
1.0
1.3
1.0
0.9
0.9
0.022
0.015
0.021
0.049
0.027
0.023
0.031
0.037
0.045
0.051
0.050
0.060
0.046
0.053
0.041
0.047
0.051
0.055
0.052
0.050
0.073
0.038
0.094
0.052
0.143
0.069
0.040
0.172
0.147
0.094
0.082
0.052
0.103
0.090
0.107
0.174
0.207
0.091
0.080
0.215
0.191
0.187
0.263
0.196
0.162
0.109
0.063
0.300
0.321
0.122
0.132
0.282
0.198
0.210
0.197
0.044
0.025
0.038
0.066
0.046
0.045
0.052
0.057
0.068
0.065
0.066
0.072
0.064
0.076
0.058
0.069
0.063
0.060
0.057
0.066
0.073
0.049
0.089
0.060
0.123
0.072
0.062
0.125
0.121
0.100
0.091
0.062
0.103
0.092
0.098
0.135
0.128
0.098
0.090
0.131
0.137
0.134
0.174
0.123
0.115
0.103
0.077
0.125
0.160
0.107
0.113
0.169
0.133
0.135
0.132
29
30
30
30
28
30
27
27
27
30
28
28
30
30
28
29
29
26
30
30
30
30
29
28
29
28
29
30
30
30
30
30
30
28
28
28
27
28
30
27
29
29
24
9
21
27
29
25
30
24
28
29
30
27
29
Int. J. Climatol. 20: 615 – 640 (2000)
620
M. HANTEL ET AL.
Table I. (Continued)
Winter
Number
Name
Elevation
14630
19710
18805
15100
19500
15515
11400
18110
14310
15420
11300
14300
17100
17700
16421
14801
20100
17001
6610
15600
12210
17300
12311
15310
9620
20020
14810
15320
15410
Umhausen
Kornat
Preitenegg
Krimml
Sillian
Badgastein
Holzgau
Mallnitz
Langen/Arlberg
Heiligenblut
Schröcken
St. Anton/Arlb.
Nauders
St. Jakob/Def.
Schöckl
Brenner
Kanzelhöhe
Galtür
Feuerkogel
Obertauern
Hahnenkamm
Obergurgl
Schmittenhöhe
Mooserboden
Krippenstein
Villacheralpe
Patscherkofel
Rudolfshütte
Sonnblick
1036
1037
1055
1062
1075
1100
1100
1185
1218
1242
1263
1280
1360
1400
1436
1450
1526
1583
1618
1755
1760
1938
1964
2036
2050
2140
2247
2304
3105
Spring
t(
6t
n̄
6n
m
t(
6t
n̄
6n
m
−2.2
−2.2
−2.8
−3.1
−4.5
−2.9
−3.4
−3.1
−1.9
−4.7
−2.0
−4.4
−4.0
−6.1
−3.7
−3.8
−3.1
−5.7
−3.3
−4.8
−4.0
−5.2
−4.8
−5.6
−6.0
−6.5
−6.6
−7.8
−12.1
1.4
1.6
1.5
1.4
1.5
1.5
1.3
1.3
1.5
1.1
1.6
1.3
1.3
1.5
1.6
1.4
1.3
1.5
1.7
1.7
1.6
1.5
1.8
1.6
1.8
1.7
1.5
2.1
1.6
0.765
0.780
0.736
0.918
0.775
0.869
0.920
0.752
0.930
0.913
0.964
0.923
0.776
0.958
0.859
0.964
0.865
0.971
0.980
1.000
0.982
0.993
0.979
0.994
0.995
0.971
0.969
0.992
1.000
0.275
0.284
0.304
0.106
0.299
0.182
0.144
0.222
0.109
0.119
0.089
0.109
0.238
0.082
0.218
0.065
0.252
0.061
0.040
0.000
0.053
0.035
0.073
0.028
0.016
0.087
0.059
0.025
0.000
29
28
29
18
26
25
30
28
30
12
30
29
29
27
27
30
29
30
29
9
10
28
23
28
28
28
27
14
30
5.9
5.4
5.2
4.9
4.8
4.9
4.3
4.2
3.7
3.5
3.5
3.5
3.9
2.8
2.3
3.0
2.7
1.4
1.5
0.6
0.8
0.6
−0.3
−0.7
−1.3
−1.8
−2.2
−2.9
−7.8
1.0
0.9
0.9
0.9
1.0
1.1
1.0
1.0
1.1
1.0
1.2
1.1
1.0
1.0
1.0
1.1
1.1
1.1
1.2
1.3
1.1
1.3
1.2
1.2
1.3
1.1
1.2
1.3
1.1
0.218
0.317
0.229
0.309
0.315
0.302
0.437
0.262
0.606
0.422
0.776
0.475
0.289
0.404
0.432
0.478
0.516
0.675
0.836
0.925
0.789
0.704
0.888
0.914
0.975
0.890
0.861
0.992
1.000
0.143
0.145
0.133
0.160
0.149
0.122
0.163
0.125
0.164
0.152
0.137
0.161
0.148
0.145
0.157
0.128
0.162
0.113
0.118
0.067
0.127
0.197
0.083
0.108
0.042
0.112
0.114
0.027
0.000
30
28
30
18
28
27
30
26
30
10
28
30
29
28
28
30
30
30
30
11
12
28
24
29
29
29
28
15
30
Winter and spring refer to the northern hemisphere.
Number, internal Austrian station code; name, name of station; elevation, height of station a.s.1. in m; t( , 1961–1990 mean of station
temperature in °C; 6t, standard deviation of station temperature in K; n̄, 1961–1990 mean of relative snow cover duration at station;
6n, standard deviation of relative snow cover duration; m, number of years used for averaging.
A statistical comparison of the entire dataset of snow duration (all Austrian stations used) and
European temperatures is given in Table II. The T P data are in °C while the T J data are relative to a
reference temperature; this explains the large differences between T J and T P. However, both datasets can
be interpreted for the present purpose as anomalies and thus are given in absolute temperature units; this
interpretation has evidently no impact upon the results. 6n and 6T are the observed standard deviations
corresponding to n and T, respectively; note that they are significantly larger in winter than in spring.
Table II. Basic statistics (overbar, mean; 6, standard deviation) for data sets of Austrian
snow cover duration n and European temperature T
n̄
T( /K
6n
6T /K
Winter
TJ
TP
0.45989
0.47157
0.37122
1.10185
0.28358
0.28424
1.52915
1.43619
Spring
TJ
TP
0.26921
0.25555
0.09198
9.10654
0.23492
0.23231
0.83902
0.84473
Copyright © 2000 Royal Meteorological Society
Int. J. Climatol. 20: 615 – 640 (2000)
Figure 2. Location of Austrian climate stations considered. Inset shows political map of Central Europe with boundaries 5–25°E and 42.5–52.5°N; these correspond to area
over which Europe temperature of Jones (1994) has been averaged. Number and name of stations listed in Table I
AUSTRIAN SNOW COVER
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621
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622
M. HANTEL ET AL.
Figure 3. Time series of European winter temperature between 1961 and 1990. T J (curve with dots, full) from database of Jones
(1994), T P (curve with diamonds, dotted) from Peterson and Vose (1997). Correlation coefficient between both curves is 0.988;
correlation coefficient for spring curves (not shown) is 0.981
A comparison of T J and T P is presented in Figure 3. Both curves run parallel (both in winter and
spring) and are highly correlated although they have been compiled independently. It appears difficult to
tell which one may be more appropriate for the present purpose. All comparisons below have been made
with both datasets with almost identical results. Thus, the subsequent discussion will be restricted to
T J =T as the principal European temperature for the present study.
2.3. Data quality checks
In order to distinguish the poorer from the more representative stations, the following criterion was
applied: high elevation stations were excluded from further processing when their data consisted of less
than three n values different from 1 (mostly in winter); an equivalent condition was applied for n= 0. This
excluded eight stations in winter, two in spring.
Second, the linear correlation coefficient r between n and T was determined for each station and each
season. It should be negative since this indicates that an increase in T corresponds to a decrease in n (i.e.
negative slope of n versus T; see Section 5). This is the basic hypothesis of the model. Data inconsistent
with this hypothesis are a priori meaningless and therefore have been discarded (another 13 in winter,
three in spring).
Third, in order to enhance the data quality, the limit for the correlation coefficient has been set, after
some numerical experimentation, arbitrarily at r\ 0.3. Stations with rB 0.3 were dropped (but
nevertheless carried in Tables I and III). This criterion ensures that very noisy data are not used when
located in the transition zone between positive and negative slopes. Subjective screening indicated that the
other possible cases (noisy data with low negative slope) are virtually not existent.
After applying these criteria (the same in the local and global mode, see the next section), data from 50
stations in winter and 66 in spring were eventually used out of the total of 84 available stations listed in
Table I. There is an insignificant impact upon the figures given in Table II: mean and standard deviations
of n are slightly different for T J and T P since different stations become excluded for both temperature sets.
Justification for this approach is that it excludes a priori irrelevant and/or misleading data and enhances
the negative slope dependence of n upon T in the data set. A positive slope dependence (i.e. increase in
n accompanied by increase in T) appears only possible if the snow duration is governed, as mentioned in
Copyright © 2000 Royal Meteorological Society
Int. J. Climatol. 20: 615 – 640 (2000)
AUSTRIAN SNOW COVER
623
Section 1, by the amount of snowfall and not by the temperature after the snowfall, as is assumed here.
This would imply that the snowfall amount is higher for high temperatures (e.g. due to the additional
effect of moisture). While this may have some impact in other climates (e.g. in polar deserts; see Loewe,
1974) it is not thought that it can possibly be the governing mechanism for the snow cover duration over
Alpine regions in Europe. Some final justification for the data screening strategy as described above is
that a fair amount of data survives the procedure.
3. THEORY
Figure 1 has shown the mean snow cover duration as a function of H. This figure applies only to the
climate of today, implicitly represented by the European temperature T( , averaged for the seasons over the
period 1961–1990. However, the snow cover duration may also be influenced by climate fluctuations
represented by actual values of T. In order to account for observed climate fluctuations in a model of the
functional relationship n(H, T), we proceed as follows.
3.1. The local station mode
The snow duration at a given station (fixed through specification of H) should be longest in a cold
climate and shortest in a warm climate. This is supported by the relatively high linear correlation between
n and T found for most of the Austrian climate stations. However, the functional relationship between
these variables cannot be linear throughout since n must approach unity at the extreme cold limit of T
adopted in this season, and approach zero at the warm limit of T. The interpolation in between may be
made with a logistic curve of the kind used in Figure 1 (Hantel, 1992)
1
n(T; s0, T0)= {tanh[2s0(T − T0)] +1}
2
(3)
The parameters s0, T0 are to be fitted to the data of one climate station. The derivative s (n/(T is
negative everywhere and adopts its minimum s0 for T=T0 corresponding to n= 0.5. It is anticipated that
T0 will be a function of H (see Section 3.2).
The result of the fit for the station Vienna is shown in Figure 4. The three different curves come from
different fitting procedures discussed in detail in Section 4. The thick curves have been obtained with the
most balanced fitting algorithm, the two other curves with more extreme fitting procedures designed to
assess the limits of our sensitivity estimates. Winter (Figure 4(a)) is an example for rather small differences
between the s0 estimates implying relatively high credibility of the results; spring (Figure 4(b)) appears less
reliable. The value for the specific winter 1982 –1983 has been marked in Figure 4(a); it will be recovered
in Figure 9(a). For a detailed discussion of the snow cover climatology of Vienna see Auer et al. (1989).
Formula (3) applies only to the specific station considered. In particular, the extreme slope s0 measures
the steepness of the snow duration – climate relationship only for this station. Yet s0, allowing for some
scatter, should be sufficiently independent of the individual climate station so that it can be considered a
parameter representing the climate sensitivity of a larger region. In order to test this point, a preliminary
evaluation has been run. Formula (3) has been applied to each of the Austrian climate stations of Figure
1; the resulting parameters (obtained with the LE-fit) are plotted in Figure 5. The mean value of the
extreme sensitivity is − 0.30 K − 1 in winter and −0.41 K − 1 in spring. It is obvious that T0 is well
correlated with height, while s0 is not. The equivalent plots for the other seasons (not shown) look similar
to Figure 5: linear height dependence of T0, no conspicuous height dependence of s0.
3.2. The global Austrian mode
Guided by the result shown in Figure 5, the reference temperature in Equation (3) is eliminated through
T0(H)=gH +T00
(4)
with constant parameters g and T00. The result is
Copyright © 2000 Royal Meteorological Society
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624
M. HANTEL ET AL.
Figure 4. Snow cover duration for climate station Wien– Hohe Warte (dots) versus T for (a) winter and (b) spring. Parameters for
curves determined with nonlinear fit (LN, dashed curve), extended fit (LE, thick), and rectified fit (LR, dotted); for detailed
explanation see Section 4. Thick point refers to winter 1982 – 1983
Figure 5. Sensitivity so (left panels (a) and (c)) and reference temperature T0 (right panels (b) and (d)) as defined in Equation (3)
plotted versus station height for winter (upper panels (a) and (b)) and spring (lower panels (c) and (d)). Each dot represents one
Austrian climate station over the period 1961–1990. Linear correlation coefficient given in each panel, regression lines (calculated
for regression T0(H)) drawn for panels (b) and (d). Slopes of the regression lines 3.02 K km − 1 (winter) and 1.96 K km − 1 (spring).
Parameters s0, T0 obtained with LE-fit
Copyright © 2000 Royal Meteorological Society
Int. J. Climatol. 20: 615 – 640 (2000)
n(H, T; s0, g, T00) =
! 625
AUSTRIAN SNOW COVER
n "
1
tanh 2s0(T − gH −T00) + 1
¿ËÀ
2
=t
(5)
The functional relationship of Equation (5) represents the global model; the parameters to be fitted are
s0, g and T00. As before, the derivative s (n/(T of this function adopts its minimum value s0 for T =T0
corresponding to n =0.5. This extreme slope measures the sensitivity of the snow–temperature relationship, and it is what is considered to be the governing parameter of this relationship. The climatological
significance of s0, as well as the meaning of the parameter t, will be discussed in Sections 6 and 7.
4. ESTIMATING THE PARAMETERS FOR THE SNOW COVER/TEMPERATURE CURVES
A priori knowledge of the error characteristics of the observations is necessary for fitting the parameters
of (3) or (5) to the data. However, the measurement error and the representativeness error of both
datasets {ni } and {Ti } are unknown (i is the index of the individual year within the data period). Thus,
there is a need to specify an error model. We shall consider first the two limiting cases that only one of
the datasets carries errors while the other is exact. Concerning the height of the station, which is the third
independent quantity in Equation (5), we stipulate that it is exact in all cases. Thus, in order to fit the
parameters s0, T0 (local mode) or s0, g, T00 (global mode) it is sufficient in the following to consider only
the local mode.
The corresponding relation between snow cover duration and European temperature has been expressed above in formula (3) as
1
n(T)= {tanh[2s0(T − T0)] +1}
2
(6)
Instead of n we may likewise consider the rectified snow cover duration defined as
E 0.5 artanh(2n −1)
(7)
The relationship inverse to Equation (6) would thus be
T(E)=
1
E+ T0
s0
(8)
We refer to (6) as the non-linear model version and to (8) as the rectified model version. Although both
are equivalent, it is evident that different estimates for the parameters s0, T0 will result from fitting data
pairs ni, Ti, pertinent to formula (6), or data pairs Ti, Ei, pertinent to (8).
The theoretical snow cover duration value that belongs to an observed temperature Ti computed with
Equation (6) may be written as
n(Ti ) n i
(9)
We call it the non-linear model 6alue. Equivalently, the theoretical temperature value that belongs to an
observed E(ni ) may, with Equations (7) and (8), be written as
T(ni ) T i
(10)
We call it the rectified model 6alue. The model values n i, T i can only be evaluated once the parameters s0
and T0 have been specified; in this sense the model values are functions of these parameters: n= n(s0, T0),
T = T(s0, T0).
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626
M. HANTEL ET AL.
4.1. The non-linear fit
Here we stipulate that the observations Ti are exact while ni carries an unknown error (assumed
stochastic). Consequently, the cost function to be minimized when fitting the parameters s0, T0 to the data
pairs ni, Ti can be written as
I
J(s0, T0)= %
i=1
n
ni −n i
s(ni )
2
(11)
The index i for the data points runs from 1 to I, ni is the measured and n i is the non-linear model value
defined in Equation (9) via the measured Ti. s(ni ) is the error that specifies the weight the corresponding
deviation has in the cost function; it will in general be a function of the snow cover duration: s= s(ni ).
The simplest choice for the variance is to make it constant for all n. However, this nai6e error model
does not seem to be appropriate in the present case. The reason is that the possible values the measured
quantity n can adopt are limited below and above. Thus, it is assumed that in the vicinity of saturation
n= 0, n= 1, the error is considerably smaller than at intermediate n values. For example, for a station
that never has snow and which is located in an environment without snow (e.g. Vienna in the summer),
the observation of n must be practically free of observation and representativeness errors, while at a
station with varying snow cover (Vienna in winter), the error of n should be a maximum. This approach
is consistent with the results found above in Section 2.1.
Of the infinite set of functions that vanish for n= 0, n= 1 and are maximum in between, we choose the
derivative of the n(T) curve. We stipulate for the non-linear model (6) that s 2 is proportional to dn(T)/dT
s 2[n(T)]=
As0
cosh [2s0(T − T0)]
2
(12)
A is a constant of proportionality. By eliminating T in Equation (12) with the help of Equations (7) and
(8), we obtain by choosing As0 =1,
s 2(n)=4(1−n)n
(13)
This function vanishes for n = 0, n =1, and adopts its maximum s 2 = 1 for n= 0.5, as required. The
parameter combination s0, T0 for which (11) adopts the minimum is independent upon the scaling
parameter A.
The theoretical extreme values ni =0, ni =1, observed or not, cannot be accepted as data since they
would yield an infinite weight for this observation and thus an infinite term in the cost function. The
practical extreme value is ni =0.01, corresponding to an observed value of about Ni = 1 dps. This yields
with (13)
1
:25
[s(ni = 0.01)]2
(14)
The same result is obtained for ni =0.99. This indicates that the observations in the saturated parts of the
curve enter the fit with a weight about 25 times the weight for ni = 0.5.
The results for the non-linear fit discussed below have been obtained with our standard error model,
expressed by (13). We have also made a sensitivity experiment with the nai6e error model mentioned
above; we shall discuss it in Section 6.
4.2. The rectified fit
Here we assume that the observations ni (and hence the transformed observations Ei ) are exact, while
Ti carries stochastic errors independent of the value of Ti. In this case we run a common linear regression
from T to E. The parameters s0, T0 obtained with the rectified fit will evidently be different from those
obtained with the non-linear fit.
Copyright © 2000 Royal Meteorological Society
Int. J. Climatol. 20: 615 – 640 (2000)
627
AUSTRIAN SNOW COVER
4.3. The extended fit
Neither one of the two extreme error models just discussed can be the final answer. The reason is that
we cannot assign a priori the data errors exclusively to either one of the measurements n or T. Rather,
both will have a certain error level (si for ni, xi for Ti ), which we have to specify externally. Thus, with
the abbreviations
ni − n i
fi,
si
Ti − T i
gi,
xi
fi =fi (s0, T0)
(15)
gi =gi (s0, T0)
( f 2i + g 2i )1/2 hi,
(16)
hi =hi (s0, T0)
(17)
we introduce an extended error model and consider the extended cost function
I
% h 2i Je,
Je =Je(s0, T0)
(18)
i=1
The parameters s0, T0 for which this cost function adopts its minimum are uniquely determined and they
are optimal in the sense of the maximum likelihood principle (Taylor, 1997).
For the variances that specify the relative weights of the observations ni, Ti entering the extended error
model we stipulate that they can be written as the product of two factors. The first (superscript a)
represents the distribution error and the second (superscript b) the error scale,
si = s ai ·s b,
xi =x ai ·x b
(19)
The first factor describes the distribution of the data error over the interval of the variable considered.
For n and T this is from the considerations above
s ai =s0[4(1− ni )ni ]1/2,
x ai =x0
(20)
The constants of proportionality s0, x0 are to be chosen such that the mean weights for all ni and all Ti
are equal, i.e.
I
1
1
1
1 I
2
=
[
s
=
%
0
a 2
2
2 i = 1 (1 −ni )ni
i = 1 (s i )
%
(21)
I
1
1
= [ x 2o =2I
a 2
2
i = 1 (x i )
%
(22)
The second factor in products (19) is taken as the natural variability of the observed data,
s b =6n,
x b =6T ;
(23)
6n, 6T have been listed above in Table II. Note that 6n in Table II has been obtained by averaging the local
6n over all Austrian stations from Table I.
4.4. The global mode
Formula (11) is valid for the local mode. For the global mode one has to replace the single parameter
T0 in (6) and in (15) – (18) by the two parameters g and T00, see Equation (4); the index i runs now over
all years and over all climate stations. Specifically, the extended cost function (18) in the local mode
becomes J global
(s0, g, T00) with three parameters to be fitted, consistent with the function (5), in the global
e
mode. However, these modifications do not alter the arguments of the previous subsection.
Copyright © 2000 Royal Meteorological Society
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628
M. HANTEL ET AL.
4.5. Terminology for the sensiti6ity experiments
In order to distinguish the various sensitivity calculations of this study we shall use the following
categories: local (L)/global (G) and non-linear (N)/rectified (R)/extended (E). In the local mode, the
eventual results have been run with the LE-fit; for each individual station we have also run, but will not
discuss in detail, the LN- and the LR-fit. In the global mode the final results have been obtained with the
GE-fit; for the sensitivity tests also the GN- and the GR-fit have been carried out and will be discussed
below.
5. RESULTS FOR THE LOCAL MODE
The parameters s0, T0 obtained with the LE-fit are listed in Table III. There is no obvious height
dependence of s0 in Table III (see also Figure 5), which supports the hypothesis implicit for the global
mode that assumes just one overall sensitivity which is valid for all Austria. On the other hand, T0 does
exhibit a pronounced correlation with height with a slope of the T0 –H curve of 2–3 K km − 1 (Figure 5).
This effect is to be expected; the increase in T0 yields a ‘colder’ T− T0 at higher elevations, corresponding
to a general decrease of atmospheric temperature with height.
We did calculate but did not include into Table III the values of s0 and T0 obtained with the LN-fit and
with the LR-fit. In all cases we reproduce the results already found in Figure 4,
s0(LR)5s0(LE) 5 s0(LN)
(24)
The s0 can be quite different for the three fits as has been demonstrated above in the example for Vienna
in spring (Figure 4(b)). In most cases, however, the parameters do not show the excessive spread indicated
in Figure 4(b).
Some additional snow cover duration/temperature curves for selected Austrian climate stations are
shown in Figure 6 for winter and in Figure 7 for spring. They demonstrate that the concept of a logistic
curve is indeed applicable to individual station data and that it yields acceptable results, although the
scatter is not small. Nevertheless, the transition from n = 1 to n= 0 is unique; i.e. none of the stations
included in the eventual evaluation has a positive (‘wrong’) slope. A second point in favor of the present
model is that within the period 1961 – 1990, the snow duration data cover most of the n values between
0 and 1 in winter (less so in spring), which supports the hypothesis that the natural interannual
fluctuations can be taken as a substitute for a possible climate change. The third point is that in spring
the variance of both n and T is smaller than in winter (Table II); this is also reflected by the fact that the
spring data points in Figure 7 are concentrated over a smaller part of the fit curve than are the winter
data points in Figure 6. This demonstrates that the model can yet be applied even in the high-curvature
parts of the logistic curve (e.g. Station Mayrhofen or Feuerkogel in Figure 7) but that the quality of the
fit tends to deteriorate in these cases.
There is another problem when the fit to the logistic curve is made with data from the high-curvature
part of the curve: The extreme sensitivity s0 would not be a particularly useful parameter for the actual
climate of this station. We may consider the slope of the curve (3)
(n
s0
=
sT
(T cosh2[2s0(T − T0)]
(25)
as the actual station sensitivity. For example, by putting T= T( in Equation (25), valid for this season,
yields sT( in Table III and is, in absolute value terms, smaller than s0. Due to the non-linearity of n(T),
the actual station sensitivity will be different when using T(n̄) instead of T= T( in Equation (25) (see
Equation (10)); this yields sn̄ in Table III. Now it is evident, and can be verified by the data in Table III,
that
s0 5 sn̄
and
s0 5sT(
Copyright © 2000 Royal Meteorological Society
(26)
Int. J. Climatol. 20: 615 – 640 (2000)
Name
Groß-Enzersdorf
Hohenau/March
Eisenstadt
Wien-Hohe Warte
Krems
Retz
Linz/Stadt
Wr. Neustadt
Fürstenfeld
St. Pölten
Bad Gleichenberg
Waidhofen/Ybbs
Graz-Universität
Waizenkirchen
Ranshofen
Kremsmünster
Deutschlandsberg
Gmunden
Salzburg-Flughafen
Ried im Innkreis
Bregenz
Feldkirch
Klagenfurt
St. Andrä-Winkling
Bad Ischl
Reichenau-Rax
Bruck/Mur
Hieflau
Kufstein
Stift Zwettl
Freistadt
Innsbruck-Univ.
Pabneukirchen
Bernstein
Aigen
Mayrhofen
Admont
Zeltweg
Radenthein
Number
5972
2600
7704
5904
3805
905
3202
7604
16600
5604
19201
7011
16412
3110
4501
5010
18900
6620
6300
4700
11102
11115
20211
18705
9610
10510
13301
10000
9010
1920
1601
11803
3410
13701
9801
15001
9900
16101
18300
153
155
159
202
207
242
263
270
273
274
303
365
366
370
382
383
410
424
434
435
436
439
447
468
469
486
489
492
495
506
548
577
595
615
640
643
646
669
685
Elevation
−0.284
−0.232
−0.252
−0.242
−0.261
−0.244
−0.256
−0.227
−0.217
−0.241
−0.288
−0.356
−0.251
−0.342
−0.228
−0.304
−0.265
−0.288
−0.232
−0.294
−0.309
−0.271
−0.374
–
−0.276
−0.171
−0.284
−0.257
−0.297
−0.280
−0.365
−0.336
−0.259
−0.270
–
–
–
−0.274
−0.402
s0
Winter
−1.776
−1.398
−1.555
−0.571
−2.048
−2.348
−1.026
−1.708
−0.295
−0.768
−0.197
−0.051
−0.489
0.106
−0.725
−0.446
−0.430
−0.435
−0.322
−0.650
−0.226
−0.554
1.037
–
1.406
−0.859
−0.064
2.107
1.491
−0.072
0.772
0.560
0.714
−0.032
–
–
–
0.769
0.957
T0
−0 084
−0.126
−0.111
−0.198
−0.072
−0.060
−0.159
−0.104
−0.200
−0.181
−0.259
−0.326
−0.209
−0.331
−0.179
−0.240
−0.223
−0.234
−0.210
−0.209
−0.271
−0.213
−0.295
–
−0.203
−0.144
−0.267
−0.127
−0.196
−0.264
−0.335
−0.331
−0.251
−0.258
–
–
–
−0.261
−0.325
sT(
−0.161
−0.143
−0.148
−0.208
−0.159
−0.130
−0.209
−0.151
−0.209
−0.206
−0.277
−0.355
−0.234
−0.333
−0.206
−0.277
−0.246
−0.260
−0.221
−0.257
−0.293
−0.236
−0.352
–
−0.249
−0.154
−0.276
−0.184
−0.262
−0.270
−0.363
−0.336
−0.258
−0.264
–
–
–
−0.270
−0.391
sn̄
−0.519
–
−0.401
−0.494
−0.386
−0.575
−0.436
−0.601
–
–
–
−0.430
−0.471
−0.495
−0.426
–
–
−0.390
–
−0.458
−0.438
–
−0.466
−0.498
−0.346
−0.564
−0.560
−0.484
−0.480
−0.512
−0.486
–
−0.438
−0.490
–
−0.674
−0.445
−0.443
−0.431
s0
Spring
−1.928
–
−2.266
−1.744
−2.213
−1.971
−1.953
−1.668
–
–
–
−1.736
−1.708
−1.735
−1.982
–
–
−1.927
–
−1.731
−1.719
–
−1.619
−1.600
−1.381
−1.495
−1.607
−0.986
−1.177
−1.574
−1.479
–
−1.417
−1.591
–
−0.907
−1.023
−1.682
−1.499
T0
−0.030
–
−0.035
−0.050
−0.042
−0.020
−0.047
−0.034
–
–
–
−0.068
−0.059
−0.050
−0.047
–
–
−0.061
–
−0.061
−0.068
–
−0.071
−0.064
−0.141
−0.059
−0.048
−0.190
−0.142
−0.063
−0.083
–
−0.108
−0.067
–
−0.160
−0.189
−0.070
−0.098
sT(
−0.046
–
−0.034
−0.093
−0.042
−0.053
−0.054
−0.088
–
–
–
−0.097
−0.083
−0.101
−0.068
–
–
−0.082
–
−0.088
−0.119
–
−0.160
−0.099
−0.170
−0.147
−0.088
−0.277
−0.242
−0.175
−0.147
–
−0.163
−0.161
–
−0.388
−0.293
−0.147
−0.127
sn̄
Table III. Snow duration sensitivity parameters found in present study for Austrian climate stations estimated with local fit (LE)
AUSTRIAN SNOW COVER
Copyright © 2000 Royal Meteorological Society
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Name
Kollerschlag
Zell am See
Mürzzuschlag
Kitzbühel
Gröbming
Gutenbrunn
Oberwölz
Landeck
Radstadt
Reutte
Neumarkt
Seckau
Mariazell
Rauris
Döllach
Tamsweg
Umhausen
Kornat
Preitenegg
Krimml
Sillian
Badgastein
Holzgau
Mallnitz
Langen/Arlberg
Heiligenblut
Schröcken
St. Anton/Arlb.
Nauders
St. Jakob/Def.
Schöckl
Brenner
Kanzelhöhe
Galtür
Feuerkogel
Obertauern
Hahnenkamm
Obergurgl
Schmittenhöhe
Number
1400
12322
10400
12200
12810
3600
15900
14403
12615
11505
16015
13110
7220
15402
18000
15710
14630
19710
18805
15100
19500
15515
11400
18110
14310
15420
11300
14300
17100
17700
16421
14801
20100
17001
6610
15600
12210
17300
12311
Table III. (Continued)
725
753
755
763
766
810
810
818
845
870
872
874
875
945
1010
1012
1036
1037
1055
1062
1075
1100
1100
1185
1218
1242
1263
1280
1360
1400
1436
1450
1526
1583
1618
1755
1760
1938
1964
Elevation
−0.325
–
–
−0.254
−0.245
−0.269
–
−0.392
–
−0.303
–
–
–
–
–
–
–
−0.425
−0.466
–
−0.371
–
–
−0.397
−0.240
–
−0.159
–
–
–
−0.384
–
−0.734
–
–
–
–
–
–
s0
Winter
1.488
–
–
2.261
1.873
1.508
–
0.394
–
2.148
–
–
–
–
–
–
–
1.471
1.555
–
1.593
–
–
1.972
2.824
–
2.962
–
–
–
2.528
–
2.174
–
–
–
–
–
–
T0
−0.200
–
–
−0.113
−0.149
−0.189
–
−0.392
–
−0.113
–
–
–
–
–
–
–
−0.197
−0.167
–
−0.179
–
–
−0.107
−0.076
–
−0.086
–
–
–
−0.052
–
−0.015
–
–
–
–
–
–
sT(
−0.265
–
–
−0.135
−0.210
−0.252
–
−0.392
–
−0.159
–
–
–
–
–
–
–
−0.291
−0.361
–
−0.259
–
–
−0.296
−0.062
–
−0.022
–
–
–
−0.186
–
−0.342
–
–
–
–
–
–
sn̄
−0.517
−0.489
−0.420
−0.421
−0.267
−0.401
–
−0.518
−0.392
−0.340
−0.541
−0.356
−0.402
−0.386
−0.335
−0.487
−0.579
−0.449
−0.334
−0.564
−0.333
−0.263
−0.265
−0.322
−0.273
−0.209
−0.289
−0.280
−0.443
−0.234
−0.265
−0.214
−0.249
−0.219
−0.368
−0.468
–
−0.218
–
s0
Spring
−0.930
−0.879
−1.081
−0.814
−1.245
−1.243
–
−1.357
−0.634
−0.605
−1.201
−1.598
−0.678
−1.053
−1.246
−1.006
−0.914
−0.738
−0.984
−0.641
−0.648
−0.810
−0.156
−0.973
0.643
−0.416
1.454
0.035
−0.787
−0.297
−0.234
0.040
0.250
1.113
1.592
2.281
–
1.345
–
T0
−0.199
−0.222
−0.180
−0.247
−0.167
−0.151
–
−0.093
−0.288
−0.274
−0.117
−0.108
−0.280
−0.192
−0.164
−0.184
−0.187
−0.269
−0.207
−0.304
−0.264
−0.212
−0.260
−0.208
−0.249
−0.200
−0.164
−0.279
−0.255
−0.226
−0.257
−0.214
−0.248
−0.180
−0.131
−0.030
–
−0.164
–
sT(
−0.350
−0.303
−0.256
−0.327
−0.169
−0.218
–
−0.124
−0.330
−0.297
−0.232
−0.163
−0.325
−0.245
−0.222
−0.309
−0.395
−0.390
−0.236
−0.482
−0.288
−0.222
−0.260
−0.250
−0.260
−0.204
−0.200
−0.279
−0.364
−0.225
−0.260
−0.213
−0.249
−0.192
−0.201
−0.128
–
−0.181
–
sn̄
630
M. HANTEL ET AL.
Copyright © 2000 Royal Meteorological Society
Int. J. Climatol. 20: 615 – 640 (2000)
Copyright © 2000 Royal Meteorological Society
Mooserboden
Krippenstein
Villacheralpe
Patscherkofel
Rudolfshütte
Sonnblick
15310
9620
20020
14810
15320
15410
2036
2050
2140
2247
2304
3105
Elevation
–
–
−0.204
–
–
–
s0
Winter
–
–
3.971
–
–
–
T0
–
–
−0.039
–
–
–
sT(
–
–
−0.023
–
–
–
sn̄
–
–
−0.442
–
–
–
s0
Spring
–
–
1.581
–
–
–
T0
–
–
−0.111
–
–
–
sT(
–
–
−0.172
–
–
–
sn̄
Number, internal Austrian station code; name, name of station; elevation, height of station a.s.1. in m; s0, extreme station sensitivity from LE-fit in
K−1; T0, reference temperature for this station in K; sT( , actual station sensitivity in K−1 from mean temperature; sn̄, actual station sensitivity in K−1
from mean snow cover duration.
Stations with missing results are those that fail the criteria discussed in Section 2.3.
Name
Number
Table III. (Continued)
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M. HANTEL ET AL.
Figure 6. Snow cover duration versus European temperature for selected Austrian climate stations in winter. Full curves according
to model (3) with parameters s0, T0 from LE-fit (these are entered in Table III). Measured values n = 0, n = 1 have been drawn in
figure but did not enter curve fit
For example, the low-level station Retz has a small actual temperature sensitivity sT( both in winter
( − 0.060 K − 1) and in spring (− 0.020 K − 1); likewise, the high-level station Kanzelhöhe has a value of
− 0.015 K − 1 in winter and Obertauern (also high-level) has a value of − 0.030 K − 1 in spring. The
corresponding extreme sensitivity s0 for these stations are in absolute value much larger than the actual
sensitivities.
On the other hand, there are stations in Table III for which the three sensitivities are quite close
together. An example is Landeck in winter when they are identical.
Table IV. Parameters for global fit
s0
(K−1)
g
(K km−1)
Winter
Rectified (GR)
Extended (GE)
Non-linear (GN)
−0.4498
−0.3377
−0.1992
1.8330
2.9788
4.9576
−0.7545
−1.3411
−2.5135
Spring
Rectified (GR)
Extended (GE)
Non-linear (GN)
−0.9262
−0.4624
−0.1291
0.9662
1.9382
7.1951
−1.1531
−2.5691
−10.0006
Copyright © 2000 Royal Meteorological Society
T00
(K)
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AUSTRIAN SNOW COVER
633
Figure 7. Like Figure 6 but for spring
6. RESULTS FOR THE GLOBAL MODE
The first step in fitting the theoretical model to the observed snow duration and European temperature
data has been to run the three programs GN, GR and GE. These correspond to the non-linear, the
rectified and the extended fit; both GN and GE runs have been carried out with the standard error model
described in Section 4. The results are put together in Table IV. The various estimates within one season
differ more in spring than in winter; in addition, the extreme slopes are more extreme in spring than in
winter. The latter result was already apparent in the local mode; e.g. s0 (spring)5s0 (winter) for all
stations below 1050 m (Table III). Note that the parameters s0 and g listed in Table IV for the GE-fit
correspond closely to the mean values of the station sensitivities and the slopes of T0 obtained in the local
mode (Figure 5). This agreement highlights the consistency of the local and the global fitting approach.
The second step was to run the GE-fit for a variable ratio between the error scale of the snow duration
days and the error scale of the European temperature. Both scales are not exactly known, so we have
substituted the above ratio s b/x b by the observed ratio 6n /6T of the standard deviations of n and T. In the
sensitivity experiment of Figure 8 we have changed this ratio in powers of ten from zero (corresponding
to all errors in n, no errors in T, extreme left of abscissa) to infinity (all errors in T, no errors in n, extreme
right of abscissa). The horizontal lines (non-linear and rectified limit) correspond to the special cases GN,
GR listed in Table IV; they have been programmed independent from the GE-fit.
The first purpose of Figure 8 is to serve as an effective test for the correct computer programming of
the generalized fit with the extended cost function. We have carried out this test both for the local and
the global mode, for both winter and spring and for both European temperatures T J and T P; as an
example, Figure 8 is reproduced applied to the global mode, the winter case, and to the T J case.
Copyright © 2000 Royal Meteorological Society
Int. J. Climatol. 20: 615 – 640 (2000)
634
M. HANTEL ET AL.
Figure 8. Dependence of extreme slope of snow curve on error scale ratio between measurements of snow cover duration n and
European temperature T, season winter. Scale ratio has been varied through replacing s b in formula (19) by s b ·106; abscissa of plot
is 6. Dots: s0 estimated with cost function for GE-fit. Upper horizontal line: s0 estimated with GN-fit; lower horizontal line: s0
estimated with GR-fit
The second purpose of Figure 8 is to obtain error estimates for the sensitivity. It has been demonstrated
in Figure 4(b) for the station Vienna in spring that estimating s0 from data in the high-curvature part of
the interpolating curve is less reliable than in the linear part. On the other hand, we maintain that the
LE-fit for the local mode (Figure 4) and the GE-fit for the global mode (dot for 6= 0 in Figure 8) yields
the best estimate for s0. Thus, it appears straightforward to take, guided by the 3-s criterion, about a sixth
of the difference between s0 found with the GR- and the GN-fit as an estimate of the error of s0. This
yields error estimates of 0.0418 K − 1 for winter and 0.1328K − 1 for spring.
The main result of this study is drawn in Figure 9. It shows the snow cover duration as function of t
(inset of Figure 9) according to Equation (5). This transformation allows us to plot the function n(H, T)
in a two-dimensional diagram together with all observed data. The fact that no information about the
local climate enters t except for station height makes this parameter independent of the specific station
properties (such as, e.g. north – south exposition, location in valley or on mountain top, and the like).
Thus, t can be considered a European Alpine temperature that uses the height dependence of the Alpine
climate stations for simulating an equivalent temperature dependence controlled by the parameter g. The
effect is a visible increase of the observed range of the European temperature: while the standard
deviation of Ti in winter (spring) is about 1.5 K (0.8 K, see Table II), the standard deviation of ti in winter
(spring) is about 1.7 K (1.1 K).
We may illustrate the use of the nomograms in Figure 9 by the following example. The point for Vienna
in winter 1982–1983 in Figure 4(a) with the coordinates n= 0.25, T= 1.8 K (marked with a thick dot)
yields with the station height of Vienna (H= 202 m, see Table I) and g= 3.0 K km − 1 (Table IV) in the
inset of Figure 9(a) the value t = 1.2 K. This gives the coordinates n= 0.25, t= 1.2 K for the main plot;
the corresponding point in Figure 9(a) is located on the far right.
With the error estimate for s0 found above we arrive at the basis of Table IV and Figure 9 at the
following values for the sensitivity of the seasonal snow cover duration with respect to the European
temperature
Extreme sensitivity for winter: s0 =( −0.34 90.04) K − 1
(27)
Extreme sensitivity for spring: s0 =( − 0.469 0.13) K − 1
(28)
The extreme sensitivity is adopted at a mean height level Hextr for which the argument of the hyperbolic
tangent in Equation (5) vanishes, yielding
Copyright © 2000 Royal Meteorological Society
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635
AUSTRIAN SNOW COVER
Figure 9. Snow cover duration for all Austrian climate stations in (a) winter and (b) spring plotted versus European Alpine
temperature defined as t T− gH. For nomogram of t see inset (labels for isolines of t identical to T at H = 0); for parameters of
theoretical curves from GE-fit see Table IV. Point for Vienna in 1983 marked in Figure 4(a) is repeated in (a)
Hextr T( −T00
g
(29)
For T( we choose the seasonally dependent mean European temperature T( J from Table II, in order to
obtain Hextr as the height of extreme sensitivity valid for the seasons of the present climate. With the
parameters g, T00 from Table IV, we find
Hextr(winter)= 575 m;
Hextr(spring) = 1373 m
Stations located below or above these levels have reduced actual sensitivity.
Copyright © 2000 Royal Meteorological Society
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636
M. HANTEL ET AL.
Figure 10. Frequency distributions of deviation between observations and fit, valid for winter. (a) Observation =n, plotted is
frequency distribution of fi, defined in Equation (15); (b) observation = T, plotted is frequency distribution of gi, defined in Equation
(16)
The deviation of the measured data from the fitted curves in Figure 9 is drawn in Figures 10 and 11
in form of frequency distributions of the functions fi and gi defined above, along with equivalent Gaussian
curves. The distributions are approximately normal. Thus, it appears that our standard error model
reproduces the observations with sufficient accuracy. We have also run the naive error model with
constant si and constant xi (see Section 4). It yields slightly larger residuals (10–20% bigger cost function)
than the standard error model. This supports our hypothesis that the standard error model is more
adequate than the naive model. Further, with the naive model the absolute values of the extreme
sensitivities become somewhat smaller (s0 = −0.28 K − 1 for winter and − 0.42 K − 1 for spring); this is
practically within the estimates (27) and (28).
The concept of the European Alpine temperature has an interesting implication for the sensitivity of the
curves. By writing n(H, T) =n[t(H, T)] and taking the identity (t(H, T)/(T = 1 into account we find for
Equation (2)
s=
dn(t)
dt
(30)
Thus, the sensitivity of the snow cover duration with respect to the European Alpine temperature is
independent of height; it is the same for all Alpine climate stations in Austria.
A hypothesized increase of DT = 1 K in winter would for fixed H reduce the relative snow cover
duration by a maximum of 34% (see Equation (27)); this corresponds to a decrease of the absolute snow
Copyright © 2000 Royal Meteorological Society
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AUSTRIAN SNOW COVER
637
Figure 11. Like Figure 10, but for spring
cover duration by a maximum of about 31 dps. Looked at from a different but equivalent perspective, and
considering the situation for constant t, the level of extreme sensitivity Hextr (for which n= 0.5) would
increase by DH = DT/g = 336 m; the equivalent figure for spring is DH = 516 m.
7. CONCLUSIONS
The snow cover as a function of global temperature has been considered a relevant parameter in climate
theory since the models of Budyko (1969) and Sellers (1969). In a warm climate, the slope of the n–T
curve is close to zero and it takes a large decrease in temperature before any snow can occur. Similarly,
in a cold climate, the slope of the curve is also near zero because the snow cover duration is a maximum;
again it requires a large increase in temperature before the snow cover duration begins to diminish.
Relevant for the snow cover – temperature sensitivity is the intermediate temperature interval. This
region has been considered by climate modellers mainly from the perspective of albedo and climate
sensitivity (e.g. Held and Suarez, 1974 or Ghil, 1976; for reviews see North et al., 1981 or Hantel, 1989).
One parameter of relevance here is the albedo – temperature feedback. If it is weak (i.e. if there is a smooth
transition from high to low albedo for increasing T), then the model of Ghil (1976) has just one solution
which is stable. Conversely, if the transition is sharp, as in the model of Held and Suarez (1974), Ghil’s
model shows three solutions, of which the intermediate is unstable. This albedo–temperature feedback
parameter as applied to global conditions is equivalent to the local snow cover–temperature sensitivity
Copyright © 2000 Royal Meteorological Society
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638
M. HANTEL ET AL.
parameter we have studied in the present paper. However, while the albedo–temperature feedback is a
dynamical parameter, relevant for global climate stability, our regional Austrian sensitivity parameter is
just a diagnostic tool for the purpose of assessing the impact of global climate changes upon the regional
climate of an Alpine country.
We have first plotted for each individual climate station the relative snow cover duration n per season
versus the regional temperature T valid for Europe. We have assumed that n –T pairs in consecutive
seasons at the same climate station should lie on a characteristic curve principally similar to the
Budyko–Sellers curves. This is equivalent to interpreting the interannual climate noise as a substitute for
long-term climate fluctuations. Since the response time of the seasonal snow cover is supposedly much
smaller than a year, we have stipulated that the change of n from one year to the next under the influence
of a certain increase of T would be the same no matter if the increase of T in the subsequent year would
continue or would stop. This hypothesis accepted, the erratic fluctuations of T from year to year and the
corresponding fluctuations of n could be interpreted as if they had happened in a consecutive manner.
Using this concept we have fitted, in the local mode, a logistic curve for each individual climate station
of Austria (50 in winter, 66 in spring; see Section 2.3); it is represented by two parameters, one of them
being the extreme sensitivity s0. We have found that the values of s0 of all stations are approximately
independent of height (Figure 5). We have further developed the actual sensitivity valid for this station
(two different estimates sn̄, sT( exist for the latter sensitivity; see Equation (26)). The actual sensitivity can
adopt all values between about 0 and −0.39 K − 1 in winter (−0.48 K − 1 in spring).
The impact of T upon the snow cover becomes augmented in the Alps through the vertical coordinate:
within the same year, with a fixed climate temperature, the warm climate/cold climate mechanism is
simulated in the data set of this study by the station height H; it causes stations at low elevations to
represent a warm climate and stations at high elevations to represent a cold climate. For example, the
data points belonging to the function n(T) for Feuerkogel in spring (Figure 7) are located on the far left
of the respective curve (high elevation, representing cold climate) while for Mayrhofen they are located on
the far right (low elevation, warm climate).
Both effects caused by T and H have been combined into the parameter European Alpine temperature
t= T− gH. This quantity works as a magnifying glass—it increases the temperature variance in winter
by 28% and in spring by 90%. With t we have fitted, in the global mode, a generalized logistic curve for
all climate stations of Austria (Figure 9) for winter and spring. The most relevant of the three parameters
that specify the logistic curve is the sensitivity s0; it is the extreme slope of the n(T) curve and is adopted
for t = T00. As the values of Hextr have shown the strongest effects of the decrease of n occur at heights
between 500 and 1500 m depending on the season; this supports the results of Haiden and Hantel (1992,
1993) and Beniston (1997).
The prominent result of this study is that at the extreme sensitivity heights the seasonal snow cover
duration decreases by 34 – 46%, corresponding to a decrease of 31 dps of snow co6er duration in winter (42
dps in spring) if the European mean temperature increases by 1 K. This reduction of relative snow cover
duration would have serious implications for nature and for the economy.
However, this result should not be overinterpreted for a couple of reasons. We shall discuss two groups
of reasons: limitations of the method, and limits of interpretation. Concerning limitations of the method,
we may stress again that we have restricted the investigation to 5 cm snow data. We could have
considered the multi-dimensional quantity number of days with depth of snow]X cm (X =1, 5, 10, 15, 20,
30, 50, 100), but have decided to leave this for later studies. We may refer here to Auer et al. (1989), to
the comprehensive investigation of the snow conditions in Tyrolia over the last 100 years of Fliri (1992b),
or to the study of Beniston (1997) for the Swiss Alps. Fliri as well as Beniston distinguish between
duration and mean height of the snow cover; both quantities depend both upon temperature and
precipitation. Specifically, we have not considered the impact of precipitation upon n. For example, when
T increases in winter and spring, snow fall may also increase at the sensitive elevations so that the n(T)
sensitivity becomes damped. On the other hand, fresh snow may have less impact upon n than naively
expected because the integrating effect of weather after the snowfall (temperature, precipitation, radiation)
may have a compensating influence on the snow cover. Nevertheless, we have tried to argue that most of
Copyright © 2000 Royal Meteorological Society
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AUSTRIAN SNOW COVER
639
these blurring effects are implicitly included in our eventual sensitivities (see the discussion in Sections 1
and 2.3); the fact that the deviations fi and gi are almost normally distributed (Figures 10 and 11) has been
an argument in favour of this assumption.
A further methodical limitation is our use of the hyperbolic tangent function with two parameters in the
local and three in the global mode; other logistic functions like the error function and other parameter
settings might yield slightly different results but we did not check these possibilities. The sensitivity
experiment of Figure 8 has shown that what we have called the non-linear fit underestimates the absolute
value of s0, while the rectified fit overestimates it. This result has been consistent with Equation (24), and
is a consequence of considering errors exclusively in n or exclusively in T. However, our ignorance in how
to do the fit exactly is a further source of error in the final results.
Concerning limits in interpretation we have argued that s0 in the global mode is an extreme estimate for
Austria as a whole; it has no immediate significance for an individual climate station. Further, we have
tried to estimate the observed climate sensiti6ity, not climate change. It has yet been tempting to transform
our extreme sensitivities into hypothetical extreme climate changes over the last century. For this period
the linear temperature trend in winter over Europe is, according to Schönwiese et al. (1994), below 1 K
per 100 a, in agreement with global model experiments (e.g. Graham, 1995). This would reduce our above
estimates of maximum secular decrease of the seasonal snow days accordingly.
A further remark concerns our finding that the standard deviation of both snow cover duration and of
climate temperature has been significantly larger in winter than in spring. This implies that the natural
climate fluctuations are larger in winter than in spring. Consequently, the basic assumption of this study,
i.e. to model a hypothetical climate change by the observed interannual fluctuations, will be better fulfilled
in winter than in spring. This makes the winter results more reliable than the spring results, in accord with
the error estimates of Equations (27) and (28).
Snow cover has been a stable climate element during the 20th century in western Austria (Fliri,
1992a,b). Despite the increase of temperature during winter over the last century, which appears real
according to Fliri, he does not find a significant linear secular trend for the snow cover duration in
Tyrolia. On the other hand, Cehak (1977) and Mohnl (1994) report for most stations in Austria, a
decrease in snow cover days; e.g. Mohnl finds a reduction of the snow cover duration by 10–30% over
the last 100 years. Beniston (1997) concludes for the Swiss Alps that the length of the snow season and
snow amount have substantially decreased since the mid-1980s. Also, Figure 3.23(a) in Nicholls et al.
(1996) shows a 10% decrease of northern hemisphere spring snow cover with high confidence over the last
20 years.
In summary, the possibility of a net maximum decrease of the seasonal snow cover duration of about
4 weeks in winter and 6 weeks in spring under a hypothesized 1 K increase of European temperature
appears disturbing. Thus, we conclude that the impact of large-scale climate changes in the sense of an
increase of European temperature upon the regional Alpine climate may be significant.
ACKNOWLEDGEMENTS
The snow cover data and the station temperatures were made accessible to us by the Climate Section of
the Central Institute for Meteorology and Geodynamics, Vienna; fruitful discussions with I. Auer and R.
Böhm are acknowledged. The University of East Anglia and T. Peterson from the National Climatic Data
Center kindly provided the data sets of the European temperature. L. Haimberger helped with the
preparation of the figures and in many other ways. The Commission for Clean Air of the O8 sterreichische
Akademie der Wissenschaften granted funds for the project.
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Copyright © 2000 Royal Meteorological Society
Int. J. Climatol. 20: 615 – 640 (2000)