climate sensitivity of snow cover duration in austria

INTERNATIONAL JOURNAL OF CLIMATOLOGY

Int. J. Climatol. 20: 615–640 (2000)

CLIMATE SENSITIVITY OF SNOW COVER DURATION IN AUSTRIAMICHAEL HANTEL*, MARTIN EHRENDORFER and ANNEMARIE HASLINGER

Institut fur Meteorologie und Geophysik der Uni6ersitat Wien, Hohe Warte 38, A-1190 Vienna, Austria

Recei6ed 5 May 1998Re6ised 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 dayswithin a season, is denoted n. This seasonal relative snow cover duration is considered a function of station heightH and of the seasonal mean temperature T over Europe. When T increases, n decreases and 6ice 6ersa. The functionbecomes saturated both for high stations at low European temperature (‘always snow’, n=1) and for low stations athigh 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 assimulation 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 individualstation (local mode) as well as for all Austrian stations (global mode) with a hyperbolic tangent function. In theglobal mode, s reaches an extreme value of −0.3490.04 K−1 in winter and −0.4690.13 K−1 in spring.

The implications of these results are discussed. Included in this discussion is the fact that a rise in the Europeantemperature by 1 K may reduce the length of the snow cover period in the Austrian Alps by about 4 weeks in winterand 6 weeks in spring. However, these extreme values apply only to the height of maximum sensitivity (575 m inwinter, 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 period1961–1990

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 theimpact of climate changes upon the snow cover duration.

Snow cover duration is just one out of many climate elements upon which possible climatic changesmay have a certain impact. In more general terms, climatic changes may be reflected in different climateelements. Examples specific for the Alpine region, and relevant on time scales between about 1 and 100years, include glacier length (Oerlemans, 1994), mountain plants (Grabherr et al., 1994) or snow depthand duration (Beniston, 1997). Concerning glacier length and mountain plants, the response time is wellabove 1 year. For example, Oerlemans (1994) reports for most valley glaciers a response time of 10–50years with respect to global climate changes, while Grabherr et al. (1994) find upward moving rates fortypical 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 of1 year, is an interannual climate phenomenon. This is evidenced by the fact that the snow cover mayfluctuate from year to year, both in amount and duration. On longer time scales, there may be climateepochs 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 fur Meteorologie und Geophysik der Universitat 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

M. HANTEL ET AL.616

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 seasonalto annual pressure field over the Alpine region, a strong correlation between high (low) snow amountsand duration and low (high) values of the North Atlantic Oscillation Index. This implies that large-scaleforcing, 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, thequantitative relation is less well known, particularly in the Alps where the height dependence of the snowcover tends to blur the climate dependence of the snow cover. To quantify this, one can consider theseasonal number N of days with snow cover equal to or exceeding 5 cm (N in days per season; dps in thispaper). For example, N0=92 dps is the maximum value N can have in spring; N=0 dps would be theminimum value. It is convenient to consider, instead of N, its normalized equivalent,

n NN0

(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 theperiod of a month or a year.

In Figure 1 the mean observed n for winter (DJF, 50 stations) and spring (MAM, 66 stations) isdisplayed for climate stations of Austria as a function of station height H ; the overbar refers to the

Figure 1. Mean snow cover duration n plotted versus station height H. Each dot represents one Austrian climate station. Stationdata averaged 1961–1990. Data fitted with logistic function (ordinate, independent variable H ; abscissa, dependent variable n). (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 allseasons. For example, the mean spring snow duration is close to zero at low elevations and increases to100% at Alpine levels exceeding about 3000 m. The slope of the n(H) curves must be close to zero, bothat low levels (‘never snow’) and at high levels (‘always snow’) with an extreme value in between; the levelat which the slope becomes extreme is low in winter, high in summer and intermediate in the transitionseasons. Figure 1 reproduces, with different data, Figure 2.11 of Haiden and Hantel (1993), who studiedthe same problem in the context of possible impacts of global and European climate changes upon theregion of Austria.

However, the influence of large-scale climate changes cannot be investigated with plots of this type sinceFigure 1 represents only the climate mean. For the purpose of studying climate changes, Haiden andHantel (1992, 1993) introduced, from the observed temperature anomalies of the stations de Bilt, Berlinand Vienna, a mean European temperature TME. They related interannual shifts of N to interannual shiftsof TME and defined the climate sensitivity of snow cover duration in the form DN/DTME. They devised afirst approximation for this ratio in spring of about −10 dps K−1. This figure implies that a rise inEuropean temperature of 1 K would cause a maximum reduction of the number of spring snow days inAustria 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. Differentdatasets for n will be used (5 cm snow cover instead of 1 cm) and for T (European temperature definedby Jones instead of TME, see below). The snow cover duration shall be considered as a function of theindependent parameters station height and European temperature: n=n(H, T). Thus, the climate sensiti6-ity 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 ensembleaverage of s for the Austrian climate stations. The result will be that the approximation found by Haidenand 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 asprovided by Jones (1994), denoted T J. The current study has also experimented with the Europetemperature of Peterson and Vose (1997), denoted TP; TP and T J are quite close to each other (see nextsection).

� The natural fluctuations of n at a fixed height are presumably dependent upon: the natural fluctuationsof T ; the amount of snowfall preceding the season considered; the amount of radiation; the expositionand ground features of the station considered; and upon other possible parameters not listed oroverlooked. Of these, the dependence upon T is only considered in this statistical model. This impliesthe a priori assumption that the impact of the other parameters will be random so that the pertinenterror should be about normally distributed when the average is taken over the climate stations ofAustria. 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 meanclimate is being simulated by the observable fluctuations of present climate. This is equivalent toassuming that the parameters that specify a climate fluctuation can be gained by observing interannualnatural fluctuations of the present climate. Thus, the parameters that fit the function n(H, T) can bedetermined 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 regimeprovided its mean is located within the variance regime of the present climate. This is the prerequisitefor 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 bediscussed in Section 3, both in the local mode (for each climate station i individually) and in the globalmode (for all Austrian stations at once).


M. HANTEL ET AL.618

� 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 inSection 3. The fit procedure is considered in Section 4, and the sensitivity of the climate stations is putinto 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, forthe seasons of the cold half of the year, the main coordinates of the stations used along with the individualstation statistics of the basic dataset. The station temperatures have been added for completeness althoughthey 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 wascounted as one with snow cover when the snow height was at least 5 cm. Haiden and Hantel (1992) hadused a 1 cm dataset, but Fliri (1992a) recommends using a minimum of 2 cm. Beniston (1997) considerssnow depth thresholds from 1 cm up to 150 cm. This paper shall restrict the subsequent discussion to thethreshold of 5 cm as a relevant choice for the present purpose, because of the lower limit suggested byFliri (1992a). On the other hand, increasing the upper limit beyond 5 cm would have severely reduced theamount of available data.

Before further processing, the N values were normalized with N0, the total number of days in the actualseason considered (N0=90 or 91 for winter and 92 for spring), to obtain n, the relative snow coverduration of this season.

The geographical position of the stations is shown in Figure 2. The area above heights of about1500 m has many gaps, while the area below that level is better represented. There are few stations thatdo not have data for the whole period (e.g. Grobming at a height of 766 m a.s.1.). No attempt was madeat 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 inthe data to show low values of 6n (standard deviation of relative snow cover duration) for snow durationvalues located in the saturated parts of the curve (n=0 and n=1) and high values of 6n for snow durationvalues close to n=0.5. This is what is expected from the character of the quantity n : the relative accuracyof the observation of n (inversely proportional to the variance of n) should be at a maximum in thesaturated parts; the reason is that in these climate states (‘always snow cover’ as well as ‘never snowcover’) 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 ofaround 50% because here both observation and representativeness errors presumably reach their maxi-mum 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 Europeanclimate stations over the array 5°–25°E and 42.5°–52.5°N (see Figure 2) as provided by Jones (1994); seealso Hulme et al. (1995). The seasonal values of T J are considered as sufficiently close to what a climatemodel should predict for the large-scale temperature development representative for the seasons ofCentral Europe. In a similar manner, the data set of Peterson and Vose (1997), referred to as TP, whichis representative for the slightly larger array 5°–25°E and 40°–55°N has been used. For the definition ofa European temperature see Baur (1975) or Rocnik (1995); for the construction of a Deutschland–Temperatur, see Schonwiese and Rapp (1997).



Table I. Austrian climate stations used in the present study

Winter Spring

Number Name Elevation t( 6t n 6n m t( 6t n 6n m

5972 Groß-Enzersdorf 153 0.1 1.9 0.171 0.156 28 9.7 1.0 0.022 0.044 292600 Hohenau/March 155 −0.5 1.8 0.189 0.167 30 9.2 0.9 0.015 0.025 307704 Eisenstadt 159 0.4 1.7 0.178 0.150 28 9.9 0.9 0.021 0.038 305904 Wien-Hohe Warte 202 0.6 1.7 0.314 0.219 24 9.9 1.0 0.049 0.066 303805 Krems 207 0 1.7 0.186 0.205 24 9.3 0.9 0.027 0.046 28

905 Retz 242 −0.6 1.8 0.158 0.157 30 8.9 0.9 0.023 0.045 303202 Linz/Stadt 263 −0.4 1.6 0.287 0.216 28 9.0 1.0 0.031 0.052 277604 Wr. Neustadt 270 −0.6 1.9 0.210 0.179 25 8.9 1.0 0.037 0.057 27

16600 Furstenfeld 273 −1.2 1.7 0.406 0.255 25 8.9 0.8 0.045 0.068 275604 St. Polten 274 −0.6 1.9 0.308 0.213 30 8.7 1.1 0.051 0.065 30

19201 Bad Gleichenberg 303 −0.9 1.7 0.400 0.262 27 9.2 0.9 0.050 0.066 287011 Waidhofen/Ybbs 365 −1.2 1.9 0.472 0.250 25 7.9 1.0 0.060 0.072 28

16412 Graz-Universitat 366 −0.5 1.5 0.368 0.232 27 9.4 0.9 0.046 0.064 303110 Waizenkirchen 370 −1.7 1.9 0.418 0.261 30 7.7 0.9 0.053 0.076 304501 Ranshofen 382 −1.2 1.9 0.343 0.230 28 7.6 1.1 0.041 0.058 285010 Kremsmunster 383 −0.8 1.8 0.351 0.238 30 8.4 1.0 0.047 0.069 29

18900 Deutschlandsberg 410 −0.6 1.7 0.365 0.252 28 8.9 0.9 0.051 0.063 296620 Gmunden 424 −0.2 1.8 0.345 0.253 27 8.1 1.0 0.055 0.060 266300 Salzburg-Flughafen 434 −0.6 2.0 0.390 0.223 30 8.4 1.1 0.052 0.057 304700 Ried im Innkreis 435 −1.2 1.9 0.322 0.237 29 7.7 1.1 0.050 0.066 30

11102 Bregenz 436 0.8 1.6 0.384 0.260 30 8.6 0.9 0.073 0.073 3011115 Feldkirch 439 −0.4 1.7 0.321 0.236 30 8.4 0.9 0.038 0.049 3020211 Klagenfurt 447 −3.2 1.6 0.620 0.324 28 8.1 0.9 0.094 0.089 2918705 St. Andra-Winkling 468 −2.7 1.5 0.541 0.278 27 8.0 0.8 0.052 0.060 289610 Bad Ischl 469 −0.8 1.6 0.657 0.250 29 7.9 1.1 0.143 0.123 29

10510 Reichenau-Rax 486 −0.6 1.8 0.341 0.193 25 7.7 0.9 0.069 0.072 2813301 Bruck/Mur 489 −1.5 1.4 0.419 0.248 28 7.9 0.9 0.040 0.062 2910000 Hieflau 492 −1.6 1.3 0.766 0.207 27 7.5 1.0 0.172 0.125 309010 Kufstein 495 −1.4 1.5 0.670 0.246 30 7.7 1.0 0.147 0.121 301920 Stift Zwettl 506 −2.3 1.9 0.406 0.262 29 6.1 1.0 0.094 0.100 301601 Freistadt 548 −2.4 1.7 0.540 0.256 30 6.7 1.0 0.082 0.091 30

11803 Innsbruck-Univ. 577 −0.9 1.6 0.495 0.275 30 8.5 1.0 0.052 0.062 303410 Pabneukirchen 595 −1.8 1.7 0.535 0.237 30 7.0 1.1 0.103 0.103 30

13701 Bernstein 615 −1.3 1.7 0.421 0.241 28 7.7 0.9 0.090 0.092 289801 Aigen 640 −3.0 1.4 0.605 0.248 27 6.5 0.9 0.107 0.098 28

15001 Mayrhofen 643 −1.7 1.4 0.768 0.209 27 7.5 1.0 0.174 0.135 289900 Admont 646 −3.5 1.5 0.829 0.183 27 6.4 1.0 0.207 0.128 27

16101 Zeltweg 669 −3.8 1.8 0.563 0.284 28 6.7 0.9 0.091 0.098 2818300 Radenthein 685 −2.5 1.3 0.583 0.318 29 7.2 0.9 0.080 0.090 301400 Kollerschlag 725 −3.0 1.7 0.714 0.246 24 6.0 1.1 0.215 0.131 27

12322 Zell am See 753 −3.6 1.7 0.763 0.248 26 6.4 1.0 0.191 0.137 2910400 Murzzuschlag 755 −3.0 1.5 0.729 0.224 28 5.8 1.0 0.187 0.134 2912200 Kitzbuhel 763 −3.9 1.6 0.841 0.166 22 5.8 1.2 0.263 0.174 2412810 Grobming 766 −3.3 1.2 0.688 0.227 9 6.2 1.1 0.196 0.123 93600 Gutenbrunn 810 −3.5 1.9 0.627 0.235 21 4.8 1.1 0.162 0.115 21

15900 Oberwolz 810 −2.9 1.3 0.589 0.256 27 6.0 0.8 0.109 0.103 2714403 Landeck 818 −1.0 1.3 0.485 0.295 30 7.7 0.9 0.063 0.077 2912615 Radstadt 845 −4.4 1.1 0.898 0.151 27 5.0 1.1 0.300 0.125 2511505 Reutte 870 −2.4 1.5 0.844 0.182 29 5.1 1.0 0.321 0.160 3016015 Neumarkt 872 −3.4 1.3 0.676 0.247 24 5.8 0.8 0.122 0.107 2413110 Seckau 874 −2.5 1.4 0.734 0.225 26 6.2 1.0 0.132 0.113 287220 Mariazell 875 −2.1 1.5 0.791 0.209 30 5.1 1.3 0.282 0.169 29

15402 Rauris 945 −3.8 1.4 0.803 0.233 29 5.2 1.0 0.198 0.133 3018000 Dollach 1010 −2.6 1.3 0.695 0.245 23 5.4 0.9 0.210 0.135 2715710 Tamsweg 1012 −5.3 1.4 0.782 0.222 30 4.8 0.9 0.197 0.132 29


M. HANTEL ET AL.620

Table I. (Continued)

Winter Spring

Number Name Elevation t( 6t n 6n m t( 6t n 6n m

14630 Umhausen 1036 −2.2 1.4 0.765 0.275 29 5.9 1.0 0.218 0.143 3019710 Kornat 1037 −2.2 1.6 0.780 0.284 28 5.4 0.9 0.317 0.145 2818805 Preitenegg 1055 −2.8 1.5 0.736 0.304 29 5.2 0.9 0.229 0.133 3015100 Krimml 1062 −3.1 1.4 0.918 0.106 18 4.9 0.9 0.309 0.160 1819500 Sillian 1075 −4.5 1.5 0.775 0.299 26 4.8 1.0 0.315 0.149 2815515 Badgastein 1100 −2.9 1.5 0.869 0.182 25 4.9 1.1 0.302 0.122 2711400 Holzgau 1100 −3.4 1.3 0.920 0.144 30 4.3 1.0 0.437 0.163 3018110 Mallnitz 1185 −3.1 1.3 0.752 0.222 28 4.2 1.0 0.262 0.125 2614310 Langen/Arlberg 1218 −1.9 1.5 0.930 0.109 30 3.7 1.1 0.606 0.164 3015420 Heiligenblut 1242 −4.7 1.1 0.913 0.119 12 3.5 1.0 0.422 0.152 1011300 Schrocken 1263 −2.0 1.6 0.964 0.089 30 3.5 1.2 0.776 0.137 2814300 St. Anton/Arlb. 1280 −4.4 1.3 0.923 0.109 29 3.5 1.1 0.475 0.161 3017100 Nauders 1360 −4.0 1.3 0.776 0.238 29 3.9 1.0 0.289 0.148 2917700 St. Jakob/Def. 1400 −6.1 1.5 0.958 0.082 27 2.8 1.0 0.404 0.145 2816421 Schockl 1436 −3.7 1.6 0.859 0.218 27 2.3 1.0 0.432 0.157 2814801 Brenner 1450 −3.8 1.4 0.964 0.065 30 3.0 1.1 0.478 0.128 3020100 Kanzelhohe 1526 −3.1 1.3 0.865 0.252 29 2.7 1.1 0.516 0.162 3017001 Galtur 1583 −5.7 1.5 0.971 0.061 30 1.4 1.1 0.675 0.113 306610 Feuerkogel 1618 −3.3 1.7 0.980 0.040 29 1.5 1.2 0.836 0.118 30

15600 Obertauern 1755 −4.8 1.7 1.000 0.000 9 0.6 1.3 0.925 0.067 1112210 Hahnenkamm 1760 −4.0 1.6 0.982 0.053 10 0.8 1.1 0.789 0.127 1217300 Obergurgl 1938 −5.2 1.5 0.993 0.035 28 0.6 1.3 0.704 0.197 2812311 Schmittenhohe 1964 −4.8 1.8 0.979 0.073 23 −0.3 1.2 0.888 0.083 2415310 Mooserboden 2036 −5.6 1.6 0.994 0.028 28 −0.7 1.2 0.914 0.108 299620 Krippenstein 2050 −6.0 1.8 0.995 0.016 28 −1.3 1.3 0.975 0.042 29

20020 Villacheralpe 2140 −6.5 1.7 0.971 0.087 28 −1.8 1.1 0.890 0.112 2914810 Patscherkofel 2247 −6.6 1.5 0.969 0.059 27 −2.2 1.2 0.861 0.114 2815320 Rudolfshutte 2304 −7.8 2.1 0.992 0.025 14 −2.9 1.3 0.992 0.027 1515410 Sonnblick 3105 −12.1 1.6 1.000 0.000 30 −7.8 1.1 1.000 0.000 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 stationtemperature in °C; 6t, standard deviation of station temperature in K; n, 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) andEuropean temperatures is given in Table II. The TP data are in °C while the T J data are relative to areference temperature; this explains the large differences between T J and TP. However, both datasets canbe interpreted for the present purpose as anomalies and thus are given in absolute temperature units; thisinterpretation has evidently no impact upon the results. 6n and 6T are the observed standard deviationscorresponding 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 Austriansnow cover duration n and European temperature T

n T( /K 6n 6T/K

Winter0.37122 1.529150.28358T J 0.45989

1.436190.28424TP 1.101850.47157

Spring0.09198 0.23492 0.83902T J 0.26921

0.25555 9.10654 0.23231 0.84473TP




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Figure 3. Time series of European winter temperature between 1961 and 1990. TJ (curve with dots, full) from database of Jones(1994), TP (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 TP is presented in Figure 3. Both curves run parallel (both in winter andspring) and are highly correlated although they have been compiled independently. It appears difficult totell which one may be more appropriate for the present purpose. All comparisons below have been madewith both datasets with almost identical results. Thus, the subsequent discussion will be restricted toT 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 wasapplied: high elevation stations were excluded from further processing when their data consisted of lessthan three n values different from 1 (mostly in winter); an equivalent condition was applied for n=0. Thisexcluded eight stations in winter, two in spring.

Second, the linear correlation coefficient r between n and T was determined for each station and eachseason. 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 inconsistentwith 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, aftersome numerical experimentation, arbitrarily at �r �\0.3. Stations with �r �B0.3 were dropped (butnevertheless carried in Tables I and III). This criterion ensures that very noisy data are not used whenlocated in the transition zone between positive and negative slopes. Subjective screening indicated that theother 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 50stations in winter and 66 in spring were eventually used out of the total of 84 available stations listed inTable I. There is an insignificant impact upon the figures given in Table II: mean and standard deviationsof n are slightly different for T J and TP 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 enhancesthe negative slope dependence of n upon T in the data set. A positive slope dependence (i.e. increase inn accompanied by increase in T) appears only possible if the snow duration is governed, as mentioned in



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 additionaleffect 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 overAlpine regions in Europe. Some final justification for the data screening strategy as described above isthat 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 theclimate of today, implicitly represented by the European temperature T( , averaged for the seasons over theperiod 1961–1990. However, the snow cover duration may also be influenced by climate fluctuationsrepresented by actual values of T. In order to account for observed climate fluctuations in a model of thefunctional 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 coldclimate and shortest in a warm climate. This is supported by the relatively high linear correlation betweenn and T found for most of the Austrian climate stations. However, the functional relationship betweenthese variables cannot be linear throughout since n must approach unity at the extreme cold limit of Tadopted in this season, and approach zero at the warm limit of T. The interpolation in between may bemade with a logistic curve of the kind used in Figure 1 (Hantel, 1992)

n(T ; s0, T0)=12

{tanh[2s0(T−T0)]+1} (3)

The parameters s0, T0 are to be fitted to the data of one climate station. The derivative s (n/(T isnegative everywhere and adopts its minimum s0 for T=T0 corresponding to n=0.5. It is anticipated thatT0 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 fromdifferent fitting procedures discussed in detail in Section 4. The thick curves have been obtained with themost balanced fitting algorithm, the two other curves with more extreme fitting procedures designed toassess the limits of our sensitivity estimates. Winter (Figure 4(a)) is an example for rather small differencesbetween the s0 estimates implying relatively high credibility of the results; spring (Figure 4(b)) appears lessreliable. The value for the specific winter 1982–1983 has been marked in Figure 4(a); it will be recoveredin 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 measuresthe steepness of the snow duration–climate relationship only for this station. Yet s0, allowing for somescatter, should be sufficiently independent of the individual climate station so that it can be considered aparameter representing the climate sensitivity of a larger region. In order to test this point, a preliminaryevaluation has been run. Formula (3) has been applied to each of the Austrian climate stations of Figure1; the resulting parameters (obtained with the LE-fit) are plotted in Figure 5. The mean value of theextreme sensitivity is −0.30 K−1 in winter and −0.41 K−1 in spring. It is obvious that T0 is wellcorrelated with height, while s0 is not. The equivalent plots for the other seasons (not shown) look similarto 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


M. HANTEL ET AL.624

Figure 4. Snow cover duration for climate station Wien–Hohe Warte (dots) versus T for (a) winter and (b) spring. Parameters forcurves 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 oneAustrian climate station over the period 1961–1990. Linear correlation coefficient given in each panel, regression lines (calculatedfor 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



n(H, T ; s0, g, T00)=12!

tanh�

2s0(T−gH¿ËÀ

=t

−T00)n

+1"

(5)

The functional relationship of Equation (5) represents the global model; the parameters to be fitted ares0, 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 relation-ship, and it is what is considered to be the governing parameter of this relationship. The climatologicalsignificance 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 parametersof (3) or (5) to the data. However, the measurement error and the representativeness error of bothdatasets {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 ofthe datasets carries errors while the other is exact. Concerning the height of the station, which is the thirdindependent quantity in Equation (5), we stipulate that it is exact in all cases. Thus, in order to fit theparameters s0, T0 (local mode) or s0, g, T00 (global mode) it is sufficient in the following to consider onlythe local mode.

The corresponding relation between snow cover duration and European temperature has been ex-pressed above in formula (3) as

n(T)=12

{tanh[2s0(T−T0)]+1} (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)=1s0

E+T0 (8)

We refer to (6) as the non-linear model version and to (8) as the rectified model version. Although bothare equivalent, it is evident that different estimates for the parameters s0, T0 will result from fitting datapairs 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 withEquation (6) may be written as

n(Ti) ni (9)

We call it the non-linear model 6alue. Equivalently, the theoretical temperature value that belongs to anobserved E(ni) may, with Equations (7) and (8), be written as

T(ni) Ti (10)

We call it the rectified model 6alue. The model values ni, Ti 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).


M. HANTEL ET AL.626

4.1. The non-linear fit

Here we stipulate that the observations Ti are exact while ni carries an unknown error (assumedstochastic). Consequently, the cost function to be minimized when fitting the parameters s0, T0 to the datapairs ni, Ti can be written as

J(s0, T0)= %I

i=1

�ni−ni

s(ni)n2

(11)

The index i for the data points runs from 1 to I, ni is the measured and ni is the non-linear model valuedefined in Equation (9) via the measured Ti. s(ni) is the error that specifies the weight the correspondingdeviation 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 modeldoes not seem to be appropriate in the present case. The reason is that the possible values the measuredquantity n can adopt are limited below and above. Thus, it is assumed that in the vicinity of saturationn=0, n=1, the error is considerably smaller than at intermediate n values. For example, for a stationthat 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 astation with varying snow cover (Vienna in winter), the error of n should be a maximum. This approachis 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 thederivative of the n(T) curve. We stipulate for the non-linear model (6) that s2 is proportional to dn(T)/dT

s2[n(T)]=As0

cosh2[2s0(T−T0)](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,

s2(n)=4(1−n)n (13)

This function vanishes for n=0, n=1, and adopts its maximum s2=1 for n=0.5, as required. Theparameter combination s0, T0 for which (11) adopts the minimum is independent upon the scalingparameter A.

The theoretical extreme values ni=0, ni=1, observed or not, cannot be accepted as data since theywould yield an infinite weight for this observation and thus an infinite term in the cost function. Thepractical extreme value is ni=0.01, corresponding to an observed value of about Ni=1 dps. This yieldswith (13)

1[s(ni=0.01)]2

:25 (14)

The same result is obtained for ni=0.99. This indicates that the observations in the saturated parts of thecurve 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 mentionedabove; 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, whileTi carries stochastic errors independent of the value of Ti. In this case we run a common linear regressionfrom T to E. The parameters s0, T0 obtained with the rectified fit will evidently be different from thoseobtained with the non-linear fit.



4.3. The extended fit

Neither one of the two extreme error models just discussed can be the final answer. The reason is thatwe 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, withthe abbreviations

ni−ni

si

fi, fi= fi(s0, T0) (15)

Ti−Ti

xi

gi, gi=gi(s0, T0) (16)

( f i2+gi

2)1/2 hi, hi=hi(s0, T0) (17)

we introduce an extended error model and consider the extended cost function

%I

i=1

hi2 Je, Je=Je(s0, T0) (18)

The parameters s0, T0 for which this cost function adopts its minimum are uniquely determined and theyare 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 errormodel 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 ia ·sb, xi=x i

a ·xb (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 ia=s0[4(1−ni)ni ]1/2, x i

a=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

i=1

1(s i

a)2=12[s0

2=12

%I

i=1

1(1−ni)ni

(21)

%I

i=1

1(x i

a)2=12[xo

2=2I (22)

The second factor in products (19) is taken as the natural variability of the observed data,

sb=6n, xb=6T ; (23)

6n, 6T have been listed above in Table II. Note that 6n in Table II has been obtained by averaging the local6n 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 parameterT0 in (6) and in (15)–(18) by the two parameters g and T00, see Equation (4); the index i runs now overall years and over all climate stations. Specifically, the extended cost function (18) in the local modebecomes J e

global(s0, g, T00) with three parameters to be fitted, consistent with the function (5), in the globalmode. However, these modifications do not alter the arguments of the previous subsection.


M. HANTEL ET AL.628

4.5. Terminology for the sensiti6ity experiments

In order to distinguish the various sensitivity calculations of this study we shall use the followingcategories: local (L)/global (G) and non-linear (N)/rectified (R)/extended (E). In the local mode, theeventual results have been run with the LE-fit; for each individual station we have also run, but will notdiscuss in detail, the LN- and the LR-fit. In the global mode the final results have been obtained with theGE-fit; for the sensitivity tests also the GN- and the GR-fit have been carried out and will be discussedbelow.

5. RESULTS FOR THE LOCAL MODE

The parameters s0, T0 obtained with the LE-fit are listed in Table III. There is no obvious heightdependence of s0 in Table III (see also Figure 5), which supports the hypothesis implicit for the globalmode that assumes just one overall sensitivity which is valid for all Austria. On the other hand, T0 doesexhibit 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, correspondingto 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 andwith the LR-fit. In all cases we reproduce the results already found in Figure 4,

s0(LR)5s0(LE)5s0(LN) (24)

The s0 can be quite different for the three fits as has been demonstrated above in the example for Viennain spring (Figure 4(b)). In most cases, however, the parameters do not show the excessive spread indicatedin Figure 4(b).

Some additional snow cover duration/temperature curves for selected Austrian climate stations areshown in Figure 6 for winter and in Figure 7 for spring. They demonstrate that the concept of a logisticcurve is indeed applicable to individual station data and that it yields acceptable results, although thescatter is not small. Nevertheless, the transition from n=1 to n=0 is unique; i.e. none of the stationsincluded in the eventual evaluation has a positive (‘wrong’) slope. A second point in favor of the presentmodel is that within the period 1961–1990, the snow duration data cover most of the n values between0 and 1 in winter (less so in spring), which supports the hypothesis that the natural interannualfluctuations can be taken as a substitute for a possible climate change. The third point is that in springthe variance of both n and T is smaller than in winter (Table II); this is also reflected by the fact that thespring data points in Figure 7 are concentrated over a smaller part of the fit curve than are the winterdata points in Figure 6. This demonstrates that the model can yet be applied even in the high-curvatureparts of the logistic curve (e.g. Station Mayrhofen or Feuerkogel in Figure 7) but that the quality of thefit tends to deteriorate in these cases.

There is another problem when the fit to the logistic curve is made with data from the high-curvaturepart of the curve: The extreme sensitivity s0 would not be a particularly useful parameter for the actualclimate of this station. We may consider the slope of the curve (3)

(n(T

=s0

cosh2[2s0(T−T0)] sT (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(n) instead of T=T( in Equation (25) (seeEquation (10)); this yields sn in Table III. Now it is evident, and can be verified by the data in Table III,that

s05sn and s05sT( (26)




Tab

leII

I.Sn

owdu

rati

onse

nsit

ivit

ypa

ram

eter

sfo

und

inpr

esen

tst

udy

for

Aus

tria

ncl

imat

est

atio

nses

tim

ated

wit

hlo

cal

fit

(LE

)

Win

ter

Spri

ng

s T(

s ns 0

T0

Num

ber

s T(

s nN

ame

Ele

vati

ons 0

T0

−0

084

−0.

161

−0.

519

−1.

928

−0.

030

−0.

284

−0.

046

−1.

776

153

Gro

ß-E

nzer

sdor

f59

7226

00−

1.39

8−

0.12

6−

0.14

3–

––

–H

ohen

au/M

arch

155

−0.

232

−0.

252

−1.

555

−0.

111

−0.

148

−0.

401

−2.

266

−0.

035

−0.

034

7704

Eis

enst

adt

159

−0.

198

−0.

208

−0.

494

−1.

744

−0.

050

−0.

571

−0.

093

202

−0.

242

5904

Wie

n-H

ohe

War

te−

2.04

838

05−

0.07

2−

0.15

9−

0.38

6−

2.21

3−

0.04

2−

0.04

2K

rem

s20

7−

0.26

1−

2.34

890

5−

0.06

0−

0.13

0−

0.57

5−

1.97

1−

0.02

0−

0.05

3R

etz

242

−0.

244

−0.

159

−0.

209

−0.

436

−1.

953

−0.

047

−1.

026

−0.

054

−0.

256

263

Lin

z/St

adt

3202

−0.

104

−0.

151

−0.

601

7604

−1.

668

−0.

034

−0.

088

Wr.

Neu

stad

t27

0−

0.22

7−

1.70

8−

0.20

0−

0.20

9–

––

−0.

295

–16

600

−0.

217

273

Fur

sten

feld

−0.

768

5604

−0.

181

−0.

206

––

––

St.

Pol

ten

274

−0.

241

−0.

259

−0.

277

––

––

1920

1B

adG

leic

henb

erg

303

−0.

288

−0.

197

−0.

326

−0.

355

−0.

430

−1.

736

−0.

068

−0.

051

−0.

097

7011

−0.

356

365

Wai

dhof

en/Y

bbs

−0.

489

1641

2−

0.20

9−

0.23

4−

0.47

1−

1.70

8−

0.05

9−

0.08

3G

raz-

Uni

vers

itat

366

−0.

251

0.10

631

10−

0.33

1−

0.33

3−

0.49

5−

1.73

5−

0.05

0−

0.10

1W

aize

nkir

chen

370

−0.

342

−0.

179

−0.

206

−0.

426

−1.

982

−0.

047

−0.

725

−0.

068

382

−0.

228

4501

Ran

shof

en−

0.44

650

10−

0.24

0−

0.27

7–

––

–K

rem

smun

ster

383

−0.

304

−0.

430

1890

0−

0.22

3−

0.24

6–

––

–D

euts

chla

ndsb

erg

410

−0.

265

−0.

234

−0.

260

−0.

390

−1.

927

−0.

061

−0.

435

−0.

082

−0.

288

424

Gm

unde

n66

20−

0.23

2−

0.32

2−

0.21

0−

0.22

1–

––

–63

00Sa

lzbu

rg-F

lugh

afen

434

−0.

209

−0.

257

−0.

458

−1.

731

−0.

061

−0.

650

−0.

088

435

−0.

294

4700

Rie

dim

Innk

reis

−0.

226

1110

2−

0.27

1−

0.29

3−

0.43

8−

1.71

9−

0.06

8−

0.11

9B

rege

nz43

6−

0.30

9−

0.21

3−

0.23

6–

––

–11

115

Fel

dkir

ch43

9−

0.27

1−

0.55

4−

0.29

5−

0.35

2−

0.46

6−

1.61

9−

0.07

11.

037

−0.

160

2021

1−

0.37

444

7K

lage

nfur

t–

1870

5–

–−

0.49

8−

1.60

0−

0.06

4−

0.09

9St

.A

ndra

-Win

klin

g46

8–

1.40

696

10−

0.20

3−

0.24

9−

0.34

6−

1.38

1−

0.14

1−

0.17

0B

adIs

chl

469

−0.

276

−0.

144

−0.

154

−0.

564

−1.

495

−0.

059

−0.

859

−0.

147

486

−0.

171

1051

0R

eich

enau

-Rax

−0.

064

1330

1−

0.26

7−

0.27

6−

0.56

0−

1.60

7−

0.04

8−

0.08

8B

ruck

/Mur

489

−0.

284

2.10

710

000

−0.

127

−0.

184

−0.

484

−0.

986

−0.

190

−0.

277

Hie

flau

492

−0.

257

−0.

196

−0.

262

−0.

480

−1.

177

−0.

142

1.49

1−

0.24

2−

0.29

749

5K

ufst

ein

9010

−0.

072

1920

−0.

264

−0.

270

−0.

512

−1.

574

−0.

063

−0.

175

Stif

tZ

wet

tl50

6−

0.28

0−

0.33

5−

0.36

3−

0.48

6−

1.47

9−

0.08

30.

772

−0.

147

1601

−0.

365

548

Fre

ista

dt0.

560

1180

3−

0.33

1−

0.33

6–

––

–In

nsbr

uck-

Uni

v.57

7−

0.33

60.

714

3410

−0.

251

−0.

258

−0.

438

−1.

417

−0.

108

−0.

163

Pab

neuk

irch

en59

5−

0.25

9−

0.25

8−

0.26

4−

0.49

0−

1.59

1−

0.06

7−

0.03

2−

0.16

1−

0.27

061

5B

erns

tein

1370

1–

––

––

–98

01A

igen

640

––

––

−0.

674

−0.

907

−0.

160

–−

0.38

864

3–

1500

1M

ayrh

ofen

–99

00–

–−

0.44

5−

1.02

3−

0.18

9−

0.29

3A

dmon

t64

6–

0.76

916

101

−0.

261

−0.

270

−0.

443

−1.

682

−0.

070

−0.

147

Zel

tweg

669

−0.

274

−0.

325

−0.

391

−0.

431

−1.

499

−0.

098

0.95

7−

0.12

7−

0.40

268

5R

aden

thei

n18

300

M. HANTEL ET AL.630


Tab

leII

I.(C

onti

nued

)

Win

ter

Spri

ng

s T(

s ns 0

T0

Num

ber

s T(

s nN

ame

Ele

vati

ons 0

T0

−0.

200

−0.

265

−0.

517

−0.

930

−0.

199

−0.

325

−0.

350

1.48

872

5K

olle

rsch

lag

1400

1232

2–

––

−0.

489

−0.

879

−0.

222

−0.

303

Zel

lam

See

753

– ––

––

−0.

420

−1.

081

−0.

180

−0.

256

Mur

zzus

chla

g10

400

755

−0.

113

−0.

135

−0.

421

−0.

814

−0.

247

2.26

1−

0.32

776

3−

0.25

412

200

Kit

zbuh

el1.

873

1281

0−

0.14

9−

0.21

0−

0.26

7−

1.24

5−

0.16

7−

0.16

9G

robm

ing

766

−0.

245

1.50

836

00−

0.18

9−

0.25

2−

0.40

1−

1.24

3−

0.15

1−

0.21

8G

uten

brun

n81

0−

0.26

9–

––

––

––

–81

0O

berw

olz

1590

0−

0.39

2−

0.39

2−

0.51

814

403

−1.

357

−0.

093

−0.

124

Lan

deck

818

−0.

392

0.39

4–

–−

0.39

2−

0.63

4−

0.28

8–

−0.

330

1261

5–

845

Rad

stad

t2.

148

1150

5−

0.11

3−

0.15

9−

0.34

0−

0.60

5−

0.27

4−

0.29

7R

eutt

e87

0−

0.30

3–

–−

0.54

1−

1.20

1−

0.11

7−

0.23

216

015

Neu

mar

kt87

2–

––

–−

0.35

6−

1.59

8−

0.10

8–

−0.

163

1311

0–

874

Seck

au–

7220

––

−0.

402

−0.

678

−0.

280

−0.

325

Mar

iaze

ll87

5–

–15

402

––

−0.

386

−1.

053

−0.

192

−0.

245

Rau

ris

945

––

–−

0.33

5−

1.24

6−

0.16

4–

−0.

222

1010

–18

000

Dol

lach

–15

710

––

−0.

487

−1.

006

−0.

184

−0.

309

Tam

sweg

1012

––

1463

0–

–−

0.57

9−

0.91

4−

0.18

7−

0.39

5U

mha

usen

1036

–−

0.19

7−

0.29

1−

0.44

9−

0.73

8−

0.26

91.

471

−0.

390

−0.

425

1037

Kor

nat

1971

0−

0.46

61.

555

−0.

167

−0.

361

−0.

334

−0.

984

−0.

207

−0.

236

Pre

iten

egg

1880

510

55–

–−

0.56

4−

0.64

1−

0.30

4–

−0.

482

1062

–15

100

Kri

mm

l1.

593

1950

0−

0.17

9−

0.25

9−

0.33

3−

0.64

8−

0.26

4−

0.28

8Si

llian

1075

−0.

371

––

−0.

263

−0.

810

−0.

212

−0.

222

1551

5B

adga

stei

n11

00–

––

–−

0.26

5−

0.15

6−

0.26

0–

−0.

260

1140

0–

1100

Hol

zgau

1.97

218

110

−0.

107

−0.

296

−0.

322

−0.

973

−0.

208

−0.

250

Mal

lnit

z11

85−

0.39

72.

824

1431

0−

0.07

6−

0.06

2−

0.27

30.

643

−0.

249

−0.

260

Lan

gen/

Arl

berg

1218

−0.

240

––

−0.

209

−0.

416

−0.

200

–−

0.20

412

42–

1542

0H

eilig

enbl

ut2.

962

1130

0−

0.08

6−

0.02

2−

0.28

91.

454

−0.

164

−0.

200

Schr

ocke

n12

63−

0.15

9–

1430

0–

–−

0.28

00.

035

−0.

279

−0.

279

St.

Ant

on/A

rlb.

1280

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0.44

3−

0.78

7−

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364

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rs17

100

–17

700

––

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234

−0.

297

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226

−0.

225

St.

Jako

b/D

ef.

1400

–−

0.05

2−

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6−

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5−

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4−

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528

−0.

260

1642

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414

36Sc

hock

l–

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1–

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0.21

40.

040

−0.

214

−0.

213

Bre

nner

1450

–2.

174

2010

0−

0.01

5−

0.34

2−

0.24

90.

250

−0.

248

−0.

249

Kan

zelh

ohe

1526

−0.

734

––

−0.

219

1.11

3−

0.18

0–

−0.

192

–15

83G

altu

r17

001

––

−0.

368

1.59

2−

0.13

1−

0.20

166

10F

euer

koge

l16

18–

––

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0.46

82.

281

−0.

030

–−

0.12

817

55–

1560

0O

bert

auer

n–

1221

0–

––

––

–H

ahne

nkam

m17

60–

–17

300

––

−0.

218

1.34

5−

0.16

4−

0.18

1O

berg

urgl

1938

––

––

––

––

–19

64Sc

hmit

tenh

ohe

1231

1



Tab

leII

I.(C

onti

nued

)

Win

ter

Spri

ng

s 0T

0s T

(s n

s 0T

0s T

(s n

Nam

eN

umbe

rE

leva

tion

––

––

––

––

2036

Moo

serb

oden

1531

0–

––

––

–96

20K

ripp

enst

ein

2050

––

−0.

039

−0.

023

−0.

442

1.58

1−

0.11

13.

971

−0.

172

2140

−0.

204

2002

0V

illac

hera

lpe

–14

810

––

––

––

Pat

sche

rkof

el22

47–

––

–15

320

––

–R

udol

fshu

tte

2304

––

––

––

––

––

1541

0So

nnbl

ick

3105

Num

ber,

inte

rnal

Aus

tria

nst

atio

nco

de;

nam

e,na

me

ofst

atio

n;el

evat

ion,

heig

htof

stat

ion

a.s.

1.in

m;

s 0,

extr

eme

stat

ion

sens

itiv

ity

from

LE

-fit

inK

−1;

T0,

refe

renc

ete

mpe

ratu

refo

rth

isst

atio

nin

K;

s T(,

actu

alst

atio

nse

nsit

ivit

yin

K−

1fr

omm

ean

tem

pera

ture

;s n

,ac

tual

stat

ion

sens

itiv

ity

inK

−1

from

mea

nsn

owco

ver

dura

tion

.St

atio

nsw

ith

mis

sing

resu

lts

are

thos

eth

atfa

ilth

ecr

iter

iadi

scus

sed

inSe

ctio

n2.

3.

M. HANTEL ET AL.632

Figure 6. Snow cover duration versus European temperature for selected Austrian climate stations in winter. Full curves accordingto 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 Kanzelhohe 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. Thecorresponding extreme sensitivity s0 for these stations are in absolute value much larger than the actualsensitivities.

On the other hand, there are stations in Table III for which the three sensitivities are quite closetogether. An example is Landeck in winter when they are identical.

Table IV. Parameters for global fit

s0 g T00

(K km−1)(K−1) (K)

Winter−0.4498 1.8330 −0.7545Rectified (GR)

Extended (GE) −0.3377 2.9788 −1.34114.9576−0.1992 −2.5135Non-linear (GN)

Spring−1.15310.9662−0.9262Rectified (GR)

Extended (GE) −0.4624 1.9382 −2.5691Non-linear (GN) −0.1291 7.1951 −10.0006



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 temperaturedata has been to run the three programs GN, GR and GE. These correspond to the non-linear, therectified and the extended fit; both GN and GE runs have been carried out with the standard error modeldescribed in Section 4. The results are put together in Table IV. The various estimates within one seasondiffer more in spring than in winter; in addition, the extreme slopes are more extreme in spring than inwinter. The latter result was already apparent in the local mode; e.g. s0 (spring)5s0 (winter) for allstations below 1050 m (Table III). Note that the parameters s0 and g listed in Table IV for the GE-fitcorrespond closely to the mean values of the station sensitivities and the slopes of T0 obtained in the localmode (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 durationdays and the error scale of the European temperature. Both scales are not exactly known, so we havesubstituted the above ratio sb/xb by the observed ratio 6n/6T of the standard deviations of n and T. In thesensitivity experiment of Figure 8 we have changed this ratio in powers of ten from zero (correspondingto all errors in n, no errors in T, extreme left of abscissa) to infinity (all errors in T, no errors in n, extremeright 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 ofthe generalized fit with the extended cost function. We have carried out this test both for the local andthe global mode, for both winter and spring and for both European temperatures T J and TP; as anexample, Figure 8 is reproduced applied to the global mode, the winter case, and to the T J case.


M. HANTEL ET AL.634

Figure 8. Dependence of extreme slope of snow curve on error scale ratio between measurements of snow cover duration n andEuropean temperature T, season winter. Scale ratio has been varied through replacing sb in formula (19) by sb ·106; abscissa of plotis 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 demonstratedin Figure 4(b) for the station Vienna in spring that estimating s0 from data in the high-curvature part ofthe interpolating curve is less reliable than in the linear part. On the other hand, we maintain that theLE-fit for the local mode (Figure 4) and the GE-fit for the global mode (dot for 6=0 in Figure 8) yieldsthe best estimate for s0. Thus, it appears straightforward to take, guided by the 3-s criterion, about a sixthof the difference between s0 found with the GR- and the GN-fit as an estimate of the error of s0. Thisyields 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 thelocal climate enters t except for station height makes this parameter independent of the specific stationproperties (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 Alpineclimate stations for simulating an equivalent temperature dependence controlled by the parameter g. Theeffect is a visible increase of the observed range of the European temperature: while the standarddeviation 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 Viennain 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 theinset 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 thefollowing values for the sensitivity of the seasonal snow cover duration with respect to the Europeantemperature

Extreme sensitivity for winter: s0= (−0.3490.04) K−1 (27)

Extreme sensitivity for spring: s0= (−0.4690.13) K−1 (28)

The extreme sensitivity is adopted at a mean height level Hextr for which the argument of the hyperbolictangent in Equation (5) vanishes, yielding



Figure 9. Snow cover duration for all Austrian climate stations in (a) winter and (b) spring plotted versus European Alpinetemperature 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 toobtain Hextr as the height of extreme sensitivity valid for the seasons of the present climate. With theparameters 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.


M. HANTEL ET AL.636

Figure 10. Frequency distributions of deviation between observations and fit, valid for winter. (a) Observation=n, plotted isfrequency 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 11in form of frequency distributions of the functions fi and gi defined above, along with equivalent Gaussiancurves. The distributions are approximately normal. Thus, it appears that our standard error modelreproduces the observations with sufficient accuracy. We have also run the naive error model withconstant 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 moreadequate than the naive model. Further, with the naive model the absolute values of the extremesensitivities become somewhat smaller (s0= −0.28 K−1 for winter and −0.42 K−1 for spring); this ispractically within the estimates (27) and (28).

The concept of the European Alpine temperature has an interesting implication for the sensitivity of thecurves. By writing n(H, T)=n [t(H, T)] and taking the identity (t(H, T)/(T=1 into account we find forEquation (2)

s=dn(t)

dt(30)

Thus, the sensitivity of the snow cover duration with respect to the European Alpine temperature isindependent 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 coverduration by a maximum of 34% (see Equation (27)); this corresponds to a decrease of the absolute snow



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, andconsidering the situation for constant t, the level of extreme sensitivity Hextr (for which n=0.5) wouldincrease 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 climatetheory since the models of Budyko (1969) and Sellers (1969). In a warm climate, the slope of the n–Tcurve 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. Thisregion has been considered by climate modellers mainly from the perspective of albedo and climatesensitivity (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 smoothtransition from high to low albedo for increasing T), then the model of Ghil (1976) has just one solutionwhich is stable. Conversely, if the transition is sharp, as in the model of Held and Suarez (1974), Ghil’smodel shows three solutions, of which the intermediate is unstable. This albedo–temperature feedbackparameter as applied to global conditions is equivalent to the local snow cover–temperature sensitivity


M. HANTEL ET AL.638

parameter we have studied in the present paper. However, while the albedo–temperature feedback is adynamical parameter, relevant for global climate stability, our regional Austrian sensitivity parameter isjust a diagnostic tool for the purpose of assessing the impact of global climate changes upon the regionalclimate of an Alpine country.

We have first plotted for each individual climate station the relative snow cover duration n per seasonversus the regional temperature T valid for Europe. We have assumed that n–T pairs in consecutiveseasons at the same climate station should lie on a characteristic curve principally similar to theBudyko–Sellers curves. This is equivalent to interpreting the interannual climate noise as a substitute forlong-term climate fluctuations. Since the response time of the seasonal snow cover is supposedly muchsmaller than a year, we have stipulated that the change of n from one year to the next under the influenceof a certain increase of T would be the same no matter if the increase of T in the subsequent year wouldcontinue or would stop. This hypothesis accepted, the erratic fluctuations of T from year to year and thecorresponding 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 stationof Austria (50 in winter, 66 in spring; see Section 2.3); it is represented by two parameters, one of thembeing the extreme sensitivity s0. We have found that the values of s0 of all stations are approximatelyindependent of height (Figure 5). We have further developed the actual sensitivity valid for this station(two different estimates sn, sT( exist for the latter sensitivity; see Equation (26)). The actual sensitivity canadopt 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 issimulated in the data set of this study by the station height H ; it causes stations at low elevations torepresent a warm climate and stations at high elevations to represent a cold climate. For example, thedata points belonging to the function n(T) for Feuerkogel in spring (Figure 7) are located on the far leftof the respective curve (high elevation, representing cold climate) while for Mayrhofen they are located onthe far right (low elevation, warm climate).

Both effects caused by T and H have been combined into the parameter European Alpine temperaturet=T−gH. This quantity works as a magnifying glass—it increases the temperature variance in winterby 28% and in spring by 90%. With t we have fitted, in the global mode, a generalized logistic curve forall climate stations of Austria (Figure 9) for winter and spring. The most relevant of the three parametersthat specify the logistic curve is the sensitivity s0; it is the extreme slope of the n(T) curve and is adoptedfor t=T00. As the values of Hextr have shown the strongest effects of the decrease of n occur at heightsbetween 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 coverduration decreases by 34–46%, corresponding to a decrease of 31 dps of snow co6er duration in winter (42dps in spring) if the European mean temperature increases by 1 K. This reduction of relative snow coverduration 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 groupsof 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 haveconsidered 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), tothe 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 betweenduration and mean height of the snow cover; both quantities depend both upon temperature andprecipitation. Specifically, we have not considered the impact of precipitation upon n. For example, whenT 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 naivelyexpected 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



these blurring effects are implicitly included in our eventual sensitivities (see the discussion in Sections 1and 2.3); the fact that the deviations fi and gi are almost normally distributed (Figures 10 and 11) has beenan argument in favour of this assumption.

A further methodical limitation is our use of the hyperbolic tangent function with two parameters in thelocal and three in the global mode; other logistic functions like the error function and other parametersettings might yield slightly different results but we did not check these possibilities. The sensitivityexperiment of Figure 8 has shown that what we have called the non-linear fit underestimates the absolutevalue of s0, while the rectified fit overestimates it. This result has been consistent with Equation (24), andis a consequence of considering errors exclusively in n or exclusively in T. However, our ignorance in howto 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 forAustria as a whole; it has no immediate significance for an individual climate station. Further, we havetried to estimate the observed climate sensiti6ity, not climate change. It has yet been tempting to transformour extreme sensitivities into hypothetical extreme climate changes over the last century. For this periodthe linear temperature trend in winter over Europe is, according to Schonwiese et al. (1994), below 1 Kper 100 a, in agreement with global model experiments (e.g. Graham, 1995). This would reduce our aboveestimates 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 ofclimate temperature has been significantly larger in winter than in spring. This implies that the naturalclimate 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 fulfilledin winter than in spring. This makes the winter results more reliable than the spring results, in accord withthe 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 realaccording to Fliri, he does not find a significant linear secular trend for the snow cover duration inTyrolia. On the other hand, Cehak (1977) and Mohnl (1994) report for most stations in Austria, adecrease in snow cover days; e.g. Mohnl finds a reduction of the snow cover duration by 10–30% overthe last 100 years. Beniston (1997) concludes for the Swiss Alps that the length of the snow season andsnow 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 last20 years.

In summary, the possibility of a net maximum decrease of the seasonal snow cover duration of about4 weeks in winter and 6 weeks in spring under a hypothesized 1 K increase of European temperatureappears disturbing. Thus, we conclude that the impact of large-scale climate changes in the sense of anincrease 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 ofthe Central Institute for Meteorology and Geodynamics, Vienna; fruitful discussions with I. Auer and R.Bohm are acknowledged. The University of East Anglia and T. Peterson from the National Climatic DataCenter kindly provided the data sets of the European temperature. L. Haimberger helped with thepreparation of the figures and in many other ways. The Commission for Clean Air of the O8 sterreichischeAkademie der Wissenschaften granted funds for the project.

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M. HANTEL ET AL.640

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